自动控制原理（双语教学）：04.LMI Control Toolbox

For Use with MATLAB Pascal Gahinet Arkadi Nemirovski Alan J. Laub Mahmoud Chilali ? User’s Guide Version 1 LMI Control Toolbox How to Contact The MathWorks: www.mathworks.com Web comp.soft-sys.matlab Newsgroup support@mathworks.com Technical support suggest@mathworks.com Product enhancement suggestions bugs@mathworks.com Bug reports doc@mathworks.com Documentation error reports service@mathworks.com Order status, license renewals, passcodes info@mathworks.com Sales, pricing, and general information 508-647-7000 Phone 508-647-7001 Fax The MathWorks, Inc. Mail 3 Apple Hill Drive Natick, MA 01760-2098 For contact information about worldwide offices, see the MathWorks Web site. LMI Control Toolbox User’s Guide ? COPYRIGHT 1995 by The MathWorks, Inc. The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. 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Printing History: May 1995 First printing New for Version 1 i Contents Preface About the Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1 Introduction Linear Matrix Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2 Toolbox Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3 LMIs and LMI Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4 The Three Generic LMI Problems . . . . . . . . . . . . . . . . . . . . . . . 1-5 Further Mathematical Background . . . . . . . . . . . . . . . . . . . . . 1-9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-10 2 Uncertain Dynamical Systems Linear Time-Invariant Systems . . . . . . . . . . . . . . . . . . . . . . . . 2-3 SYSTEM Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3 Time and Frequency Response Plots . . . . . . . . . . . . . . . . . . . 2-6 Interconnections of Linear Systems . . . . . . . . . . . . . . . . . . . . 2-9 ii Contents Model Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-12 Uncertain State-Space Models . . . . . . . . . . . . . . . . . . . . . . . . . 2-14 Polytopic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-14 Affine Parameter-Dependent Models . . . . . . . . . . . . . . . . . . . . 2-15 Quantification of Parameter Uncertainty . . . . . . . . . . . . . . . . 2-17 Simulation of Parameter-Dependent Systems . . . . . . . . . . . . . 2-19 From Affine to Polytopic Models . . . . . . . . . . . . . . . . . . . . . . . . 2-20 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-21 Linear-Fractional Models of Uncertainty . . . . . . . . . . . . . . . 2-23 How to Derive Such Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-23 Specification of the Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . 2-26 From Affine to Linear-Fractional Models . . . . . . . . . . . . . . . . . 2-32 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-35 3 Robustness Analysis Quadratic Lyapunov Functions . . . . . . . . . . . . . . . . . . . . . . . . 3-3 LMI Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-4 Quadratic Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-6 Maximizing the Quadratic Stability Region . . . . . . . . . . . . . . . . 3-8 Decay Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9 Quadratic H∞ Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-10 Parameter-Dependent Lyapunov Functions . . . . . . . . . . . . 3-12 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-14 μ Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-17 Structured Singular Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-17 Robust Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-19 Robust Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-21 The Popov Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-24 Real Parameter Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-25 iii Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-28 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-32 4 State-Feedback Synthesis Multi-Objective State-Feedback . . . . . . . . . . . . . . . . . . . . . . . . 4-3 Pole Placement in LMI Regions . . . . . . . . . . . . . . . . . . . . . . . . 4-5 LMI Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-7 Extension to the Multi-Model Case . . . . . . . . . . . . . . . . . . . . . . . 4-9 The Function msfsyn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-11 Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-18 5 Synthesis of H∞ Controllers H∞ Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3 Riccati- and LMI-Based Approaches . . . . . . . . . . . . . . . . . . . . . . 5-7 H∞ Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-10 Validation of the Closed-Loop System . . . . . . . . . . . . . . . . . . . 5-13 Multi-Objective H∞ Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . 5-15 LMI Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-16 The Function hinfmix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-20 Loop-Shaping Design with hinfmix . . . . . . . . . . . . . . . . . . . . . . 5-20 iv Contents References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-22 6 Loop Shaping The Loop-Shaping Methodology . . . . . . . . . . . . . . . . . . . . . . . . 6-2 The Loop-Shaping Methodology . . . . . . . . . . . . . . . . . . . . . . . . 6-3 Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5 Specification of the Shaping Filters . . . . . . . . . . . . . . . . . . . . 6-10 Nonproper Filters and sderiv . . . . . . . . . . . . . . . . . . . . . . . . . . 6-12 Specification of the Control Structure . . . . . . . . . . . . . . . . . 6-14 Controller Synthesis and Validation . . . . . . . . . . . . . . . . . . . 6-16 Practical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-18 Loop Shaping with Regional Pole Placement . . . . . . . . . . . 6-19 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-24 7 Robust Gain-Scheduled Controllers Gain-Scheduled Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-3 Synthesis of Gain-Scheduled H? Controllers . . . . . . . . . . . . . 7-7 Simulation of Gain-Scheduled Control Systems . . . . . . . . . . 7-9 Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-10 v References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-15 8 The LMI Lab Background and Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . 8-3 Overview of the LMI Lab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-6 Specifying a System of LMIs . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-8 A Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-9 setlmis and getlmis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-11 lmivar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-11 lmiterm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-13 The LMI Editor lmiedit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-16 How It All Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-18 Retrieving Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-21 lmiinfo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-21 lminbr and matnbr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-21 LMI Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-22 From Decision to Matrix Variables and Vice Versa . . . . . . 8-28 Validating Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-29 Modifying a System of LMIs . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-30 dellmi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-30 dellmi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-30 setmvar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-31 Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-33 Structured Matrix Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-33 Complex-Valued LMIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-35 Specifying c T x Objectives for mincx . . . . . . . . . . . . . . . . . . . . . 8-38 Feasibility Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-39 vi Contents Well-Posedness Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-40 Semi-Definite B(x) in gevp Problems . . . . . . . . . . . . . . . . . . . . 8-41 Efficiency and Complexity Issues . . . . . . . . . . . . . . . . . . . . . . . 8-41 Solving M + PTXQ + QTXTP < 0 . . . . . . . . . . . . . . . . . . . . . . . . 8-42 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-44 9 Command Reference List of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-3 H∞ Control and Loop Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . 9-6 LMI Lab: Specifying and Solving LMIs . . . . . . . . . . . . . . . . . . 9-7 LMI Lab: Additional Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . 9-8 Preface Preface viii About the Authors Dr. Pascal Gahinet is a reserach fellow at INRIA Rocquencourt, France. His research interests include robust control theory, linear matrix inequalities, numerical linear algebra, and numerical software for control. Prof. Arkadi Nemirovski is with the Faculty of Industrial Engineering and Management at Technion, Haifa, Israel. His research interests include convex optimization, complexity theory, and non-parametric statistics. Prof. Alan J. Laub is with the Electrical and Computer Engineering Department of the University of California at Santa Barbara, USA. His research interests are in numerical analysis, mathematical software, scientific computation, computer-aided control system design, and linear and large-scale control and filtering theory. Mahmoud Chilali is completing his Ph.D. at INRIA Rocquencourt, France. His thesis is on the theory and applications of linear matrix inequalities in control. Acknowledgments ix Acknowledgments The authors wish to express their gratitude to all colleagues who directly or indirectly contributed to the making of the LMI Control Toolbox. Special thanks to Pierre Apkarian, Gregory Becker, Hiroyuki Kajiwara, and Anca Ignat for their help and contribution. Many thanks also to those who tested and helped refine the software, including Bobby Bodenheimer, Markus Brandstetter, Eric Feron, K.C. Goh, Anders Helmersson, Ted Iwasaki, Jianbo Lu, Roy Lurie, Jason Ly, John Morris, Ravi Prasanth, Michael Safonov, Carsten Scherer, Andy Sparks, Mario Rotea, Matthew Lamont Tyler, Jim Tung, and John Wen. Apologies, finally, to those we may have omitted. The work of Pascal Gahinet was supported in part by INRIA. Preface x 1 Introduction Linear Matrix Inequalities . . . . . . . . . . . . . 1-2 Toolbox Features . . . . . . . . . . . . . . . . . . 1-3 LMIs and LMI Problems . . . . . . . . . . . . . . . 1-4 The Three Generic LMI Problems . . . . . . . . . . . . 1-5 Further Mathematical Background . . . . . . . . . 1-9 References . . . . . . . . . . . . . . . . . . . . . 1-10 1 Introduction 1-2 Linear Matrix Inequalities Linear Matrix Inequalities (LMIs) and LMI techniques have emerged as powerful design tools in areas ranging from control engineering to system identification and structural design. Three factors make LMI techniques appealing: ?A variety of design specifications and constraints can be expressed as LMIs. Once formulated in terms of LMIs, a problem can be solved exactly by efficient convex optimization algorithms (the “LMI solvers”). While most problems with multiple constraints or objectives lack analytical solutions in terms of matrix equations, they often remain tractable in the LMI framework. This makes LMI-based design a valuable alternative to classical “analytical” methods. See [9] for a good introduction to LMI concepts. The LMI Control Toolbox is designed as an easy and progressive gateway to the new and fast-growing field of LMIs: For users mainly interested in applying LMI techniques to control design, the LMI Control Toolbox features a variety of high-level tools for the analysis and design of multivariable feedback loops (see Chapters 2 through 7). For users who occasionally need to solve LMI problems, the “LMI Editor” and the tutorial introduction to LMI concepts and LMI solvers provide for quick and easy problem solving. For more experienced LMI users, the “LMI Lab” (Chapter 8) offers a rich, flexible, and fully programmable environment to develop customized LMI-based tools. The LMI Control Toolbox implements state-of-the-art interior-point LMI solvers. While these solvers are significantly faster than classical convex optimization algorithms, it should be kept in mind that the complexity of LMI computations remains higher than that of solving, say, a Riccati equation. For instance, problems with a thousand design variables typically take over an hour on today's workstations. However, research on LMI optimization is still very active and substantial speed-ups can be expected in the future. Thanks to its efficient “structured” representation of LMIs, the LMI Control Toolbox is geared to making the most out of such improvements. Toolbox Features 1-3 Toolbox Features The LMI Control Toolbox serves two purposes: Provide state-of-the-art tools for the LMI-based analysis and design of robust control systems Offer a flexible and user-friendly environment to specify and solve general LMI problems (the LMI Lab) The control design tools can be used without a priori knowledge about LMIs or LMI solvers. These are dedicated tools covering the following applications of LMI techniques: Specification and manipulation of uncertain dynamical systems (linear-time invariant, polytopic, parameter-dependent, etc.) Robustness analysis. Various measures of robust stability and performance are implemented, including quadratic stability, techniques based on parameter-dependent Lyapunov functions, analysis, and Popov analysis. Multi-model/multi-objective state-feedback design Synthesis of output-feedback H ∞ controllers via Riccati- and LMI-based techniques, including mixed H 2 /H ∞ synthesis with regional pole placement constraints Loop-shaping design Synthesis of robust gain-scheduled controllers for time-varying parameter-dependent systems For users interested in developing their own applications, the LMI Lab provides a general-purpose and fully programmable environment to specify and solve virtually any LMI problem. Note that the scope of this facility is by no means restricted to control-oriented applications. 1 Introduction 1-4 LMIs and LMI Problems A linear matrix inequality (LMI) is any constraint of the form (1-1) where x = (x 1 , . . . , x N ) is a vector of unknown scalars (the decision or optimization variables) A 0 , . . . , A N are given symmetric matrices < 0 stands for “negative definite,” i.e., the largest eigenvalue of A(x) is negative Note that the constraints A(x) > 0 and A(x) < B(x) are special cases of (1-1) since they can be rewritten as –A(x) < 0 and A(x) – B(x) < 0, respectively. The LMI (1-1) is a convex constraint on x since A(y) < 0 and A(z) < 0 imply that . As a result, Its solution set, called the feasible set, is a convex subset of R N Finding a solution x to (1-1), if any, is a convex optimization problem. Convexity has an important consequence: even though (1-1) has no analytical solution in general, it can be solved numerically with guarantees of finding a solution when one exists. Note that a system of LMI constraints can be regarded as a single LMI since where diag (A 1 (x), . . . , A K (x)) denotes the block-diagonal matrix with A 1 (x), . . . , A K (x) on its diagonal. Hence multiple LMI constraints can be imposed on the vector of decision variables x without destroying convexity. Ax() := A 0 x 1 A 1 … x N A N 0<+++ A yz+ 2 ----------- ?? ?? 0< A 1 x() 0< . . . A K x() 0< is equivalent to ? ? ? ? ? ? ? ? ? Ax() := diag A( 1 x(),… A K x()) 0<, LMIs and LMI Problems 1-5 In most control applications, LMIs do not naturally arise in the canonical form (1-1), but rather in the form L(X 1 , . . . , X n ) < R(X 1 , . . . , X n ) where L( . ) and R( . ) are affine functions of some structured matrix variables X 1 , . . . , X n . A simple example is the Lyapunov inequality (1-2) where the unknown X is a symmetric matrix. Defining x 1 , . . . , x N as the independent scalar entries of X, this LMI could be rewritten in the form (1-1). Yet it is more convenient and efficient to describe it in its natural form (1-2), which is the approach taken in the LMI Lab. The Three Generic LMI Problems Finding a solution x to the LMI system (1-3) is called the feasibility problem. Minimizing a convex objective under LMI constraints is also a convex problem. In particular, the linear objective minimization problem (1-4) plays an important role in LMI-based design. Finally, the generalized eigenvalue minimization problem (1-5) is quasi-convex and can be solved by similar techniques. It owes its name to the fact that is related to the largest generalized eigenvalue of the pencil (A(x),B(x)). Many control problems and design specifications have LMI formulations [9]. This is especially true for Lyapunov-based analysis and design, but also for A T XXA0<+ Ax() 0< Minimize c T x subject to Ax() 0< Minimize λ subject to Ax() λBx()< Bx() 0> Cx() 0< ? ? ? ? ? 1 Introduction 1-6 optimal LQG control, H ∞ control, covariance control, etc. Further applications of LMIs arise in estimation, identification, optimal design, structural design [6, 7], matrix scaling problems, and so on. The main strength of LMI formu- lations is the ability to combine various design constraints or objectives in a numerically tractable manner. A nonexhaustive list of problems addressed by LMI techniques includes the following: Robust stability of systems with LTI uncertainty (μ-analysis) [24, 21, 27] Robust stability in the face of sector-bounded nonlinearities (Popov criterion) [22, 28, 13, 16] Quadratic stability of differential inclusions [15, 8] Lyapunov stability of parameter-dependent systems [12] Input/state/output properties of LTI systems (invariant ellipsoids, decay rate, etc.) [9] Multi-model/multi-objective state feedback design [4, 17, 3, 9, 10] Robust pole placement Optimal LQG control [9] Robust H ∞ control [11, 14] Multi-objective H ∞ synthesis [17, 23, 10, 18] Design of robust gain-scheduled controllers [5, 2] Control of stochastic systems [9] Weighted interpolation problems [9] To hint at the principles underlying LMI design, let’s review the LMI formulations of a few typical design objectives. Stability. the stability of the dynamical system is equivalent to the feasibility of Find P = P T such that A T P + P A < 0, P > I. x · Ax= LMIs and LMI Problems 1-7 This can be generalized to linear differential inclusions (LDI) where A(t) varies in the convex envelope of a set of LTI models: A sufficient condition for the asymptotic stability of this LDI is the feasibility of Find P = P T such that RMS gain . the random-mean-squares (RMS) gain of a stable LTI system is the largest input/output gain over all bounded inputs u(t). This gain is the global minimum of the following linear objective minimization problem [1, 26, 25]. Minimize γ over X = X T and γ such that LQG performance . for a stable LTI system x · At()x= At() Co A 1 ,…,A n {}∈ a i A i : i 1= n ∑ a i 0≥ a i i 1= N ∑ 1=, ?? ?? ?? ?? = A i T PPA i 0 PI.>,<+ x · Ax Bu+= yCxDu+= ? ? ? A T XXA+ XB C T B T X γ– ID T CDγ– I?? ?? ?? 0< X 0> G x · Ax Bw+= yCx= ? ? ? 1 Introduction 1-8 where w is a white noise disturbance with unit covariance, the LQG or H 2 performance ||G|| 2 is defined by It can be shown that Hence is the global minimum of the LMI problem Minimize Trace (Q) over the symmetric matrices P,Q such that Again this is a linear objective minimization problem since the objective Trace (Q) is linear in the decision variables (free entries of P,Q). G 2 2 : = E T ∞→ lim 1 T --- y T 0 T ∫ t()yt()dt ?? ?? ?? 1 2π ------ G H ∞– ∞ ∫ jω()Gjω()dω= G 2 2 inf Trace CPC T (){ : AP PA T BB T 0 }<++= G 2 2 AP PA T BB T 0<++ QCP PC T P ?? ?? ?? 0> Further Mathematical Background 1-9 Further Mathematical Background Efficient interior-point algorithms are now available to solve the three generic LMI problems (1-3)–(1-5) defined in “The Three Generic LMI Problems” on page 1-5. These algorithms have a polynomial-time complexity. That is, the number N(ε) of flops needed to compute an ε-accurate solution is bounded by N(ε) e M N 3 log(V/ε) where M is the total row size of the LMI system, N is the total number of scalar decision variables, and V is a data-dependent scaling factor. The LMI Control Toolbox implements the Projective Algorithm of Nesterov and Nemirovski [20, 19]. In addition to its polynomial-time complexity, this algorithm does not require an initial feasible point for the linear objective minimization problem (1-4) or the generalized eigenvalue minimization problem (1-5). Some LMI problems are formulated in terms of inequalities rather than strict inequalities. For instance, a variant of (1-4) is Minimize c T x subject to A(x) e 0. While this distinction is immaterial in general, it matters when A(x) can be made negative semi-definite but not negative definite. A simple example is (1-6) Such problems cannot be handled directly by interior-point methods which require strict feasibility of the LMI constraints. A well-posed reformulation of (1-6) would be Minimize c T x subject to x ? 0. Keeping this subtlety in mind, we always use strict inequalities in this manual. Minimize c T x subject to xx xx?? ?? ?? 0.≥ 1 Introduction 1-10 References [1] Anderson, B.D.O, and S. Vongpanitlerd, Network Analysis, Prentice-Hall, Englewood Cliffs, 1973. [2] Apkarian, P., P. Gahinet, and G. Becker, “Self-Scheduled H ∞ Control of Linear Parameter-Varying Systems,” Proc. Amer. Contr. Conf., 1994, pp. 856-860. [3] Bambang, R., E. Shimemura, and K. Uchida, “Mixed H 2 /H ∞ Control with Pole Placement,” State-Feedback Case,“ Proc. Amer. Contr. Conf., 1993, pp. 2777-2779. [4] Barmish, B.R., “Stabilization of Uncertain Systems via Linear Control,“ IEEE Trans. Aut. Contr., AC–28 (1983), pp. 848-850. [5] Becker, G., Packard, P., “Robust Performance of Linear-Parametrically Varying Systems Using Parametrically-Dependent Linear Feedback,” Systems and Control Letters, 23 (1994), pp. 205-215. [6] Bendsoe, M.P., A. Ben-Tal, and J. Zowe, “Optimization Methods for Truss Geometry and Topology Design,” to appear in Structural Optimization. [7] Ben-Tal, A., and A. Nemirovski, “Potential Reduction Polynomial-Time Method for Truss Topology Design,” to appear in SIAM J. Contr. Opt. [8] Boyd, S., and Q. Yang, “Structured and Simultaneous Lyapunov Functions for System Stability Problems,” Int. J. Contr., 49 (1989), pp. 2215-2240. [9] Boyd, S., L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, SIAM books, Philadelphia, 1994. [10] Chilali, M., and P. Gahinet, “H ∞ Design with Pole Placement Constraints: an LMI Approach,” to appear in IEEE Trans. Aut. Contr. Also in Proc. Conf. Dec. Contr., 1994, pp. 553-558. [11] Gahinet, P., and P. Apkarian, “A Linear Matrix Inequality Approach to H ∞ Control,” Int. J. Robust and Nonlinear Contr., 4 (1994), pp. 421-448. [12] Gahinet, P., P. Apkarian, and M. Chilali, “Affine Parameter-Dependent Lyapunov Functions for Real Parametric Uncertainty,” Proc. Conf. Dec. Contr., 1994, pp. 2026-2031. References 1-11 [13] Haddad, W.M. and D.S. Berstein,“Parameter-Dependent Lyapunov Functions, Constant Real Parameter Uncertainty, and the Popov Criterion in Robust Analysis and Synthesis: Part 1 and 2,” Proc. Conf. Dec. Contr., 1991, pp. 2274-2279 and 2632-2633. [14] Iwasaki, T., and R.E. Skelton, “All Controllers for the General H ∞ Control Problem: LMI Existence Conditions and State-Space Formulas,” Automatica, 30 (1994), pp. 1307-1317. [15] Horisberger, H.P., and P.R. Belanger, “Regulators for Linear Time-Varying Plants with Uncertain Parameters,” IEEE Trans. Aut. Contr., AC–21 (1976), pp. 705-708. [16] How, J.P., and S.R. Hall, “Connection between the Popov Stability Criterion and Bounds for Real Parameter Uncertainty,” Proc. Amer. Contr. Conf., 1993, pp. 1084-1089. [17] Khargonekar, P.P., and M.A. Rotea,“Mixed H 2 /H ∞ Control: a Convex Optimization Approach,” IEEE Trans. Aut. Contr., 39 (1991), pp. 824-837. [18] Masubuchi, I., A. Ohara, and N. Suda, “LMI-Based Controller Synthesis: A Unified Formulation and Solution,” submitted to Int. J. Robust and Nonlinear Contr., 1994. [19] Nemirovski, A., and P. Gahinet, “The Projective Method for Solving Linear Matrix Inequalities,” Proc. Amer. Contr. Conf., 1994, pp. 840-844. [20] Nesterov, Yu, and A. Nemirovski, Interior Point Polynomial Methods in Convex Programming: Theory and Applications, SIAM Books, Philadelphia, 1994. [21] Packard, A., and J.C. Doyle, “The Complex Structured Singular Value,” Automatica, 29 (1994), pp. 71-109. [22] Popov, V.M., “Absolute Stability of Nonlinear Systems of Automatic Control,” Automation and Remote Control, 22 (1962), pp. 857-875. [23] Scherer, C., “Mixed H 2 H ∞ Control,” to appear in Trends in Control: A European Perspective, volume of the special contributions to the ECC 1995. [24] Stein, G. and J.C. Doyle, “Beyond Singular Values and Loop Shapes,” J. Guidance, 14 (1991), pp. 5-16. [25] Vidyasagar, M., Nonlinear System Analysis, Prentice-Hall, Englewood Cliffs, 1992. 1 Introduction 1-12 [26] Willems, J.C., “Least-Squares Stationary Optimal Control and the Algebraic Riccati Equation,” IEEE Trans. Aut. Contr., AC–16 (1971), pp. 621-634. [27] Young, P. M., M. P. Newlin, and J. C. Doyle, “Let's Get Real,” in Robust Control Theory, Springer Verlag, 1994, pp. 143-174. [28] Zames, G., “On the Input-Output Stability of Time-Varying Nonlinear Feedback Systems, Part I and II,” IEEE Trans. Aut. Contr., AC–11 (1966), pp. 228-238 and 465-476. 2 Uncertain Dynamical Systems Linear Time-Invariant Systems . . . . . . . . . . . 2-3 SYSTEM Matrix . . . . . . . . . . . . . . . . . . . 2-3 Time and Frequency Response Plots . . . . . . . . . 2-6 Interconnections of Linear Systems . . . . . . . . . 2-9 Model Uncertainty . . . . . . . . . . . . . . . . .2-12 Uncertain State-Space Models . . . . . . . . . . . .2-14 Polytopic Models . . . . . . . . . . . . . . . . . . .2-14 Affine Parameter-Dependent Models . . . . . . . . . . .2-15 Quantification of Parameter Uncertainty . . . . . . . . .2-17 Simulation of Parameter-Dependent Systems . . . . . . .2-19 From Affine to Polytopic Models . . . . . . . . . . . . .2-20 Example . . . . . . . . . . . . . . . . . . . . . .2-21 Linear-Fractional Models of Uncertainty . . . . . . .2-23 How to Derive Such Models . . . . . . . . . . . . . . .2-23 Specification of the Uncertainty . . . . . . . . . . . . .2-26 From Affine to Linear-Fractional Models . . . . . . . . .2-32 References . . . . . . . . . . . . . . . . . . . . .2-35 2 Uncertain Dynamical Systems 2-2 The LMI Control Toolbox offers a variety of tools to facilitate the description and manipulation of uncertain dynamical systems. These include functions to: Manipulate the state-space realization of linear time-invariant (LTI) systems as a single SYSTEM matrix Form interconnections of linear systems Specify linear systems with uncertain state-space matrices (polytopic differential inclusions, systems with uncertain or time-varying physical parameters, etc.) Describe linear-fractional models of uncertainty The next sections provide a tutorial introduction to uncertain dynamical systems and an overview of these facilities. Linear Time-Invariant Systems 2-3 Linear Time-Invariant Systems The LMI Control Toolbox provides streamlined tools to manipulate state-space representations of linear time-invariant (LTI) systems. These tools handle general LTI models of the form where A, B, C, D, E are real matrices and E is invertible, as well as their discrete-time counterpart Recall that the vectors x(t), u(t), y(t) denote the state, input, and output trajectories. Similarly, x k , u k , y k denote the values of the state, input, and output vectors at the sample time k. The “descriptor” formulation ( ) proves useful when specifying parameter-dependent systems (see “Affine Parameter-Dependent Models” on page 2-15) and also avoids inverting E when this inversion is poorly conditioned. Moreover, many dynamical systems are naturally written in descriptor form. For instance, the second-order system . admits the state-space representation where . SYSTEM Matrix For convenience, the state-space realization of LTI systems is stored as a single MATLAB matrix called a SYSTEM matrix. Specifically, a continuous- or E dx dt ------ Ax t() Bu t()+= yt() Cx t() Du t()+= Ex k 1+ Ax k Bu k += y k Cx k Du k += EI≠ mx ·· fx · kx++ uyx=,= 10 0 m?? ?? ?? dX dt -------- 01 k– f–?? ?? ?? X 0 1?? ?? ?? uy10,()X=,+= Xt() := xt() x · t()?? ?? ?? 2 Uncertain Dynamical Systems 2-4 discrete-time LTI system with state-space matrices A, B, C, D, E is represented by the structured matrix where . The upper right entry n corresponds to the number of states (i.e., A ∈ R n×n ) while the entry Inf is used to differentiate SYSTEM matrices from regular matrices. This data structure is similar to that used in the μ Analysis and Synthesis Toolbox. The functions ltisys and ltiss create SYSTEM matrices and extract state- space data from them. For instance, sys = ltisys( 1,1,1,0) specifies the LTI system To retrieve the values of A, B, C, D from the SYSTEM matrix sys, type [a,b,c,d] = ltiss(sys) For single-input/single-output (SISO) systems, the function ltitf returns the numerator/denominator representation of the transfer function Conversely, the command sys = ltisys('tf',n,d) returns a state-space realization of the SISO transfer function n(s)/d(s) in SYSTEM format. Here n and d are the vector representation of n(s) and d(s) (type help poly for details). The number of states, inputs, and outputs of a linear system are retrieved from the SYSTEM matrix with sinfo: … AjEI–()+ B n 0 CD 0 0 –Inf j 1–= x · x– uyx=,+= Gs() DCsEA–() 1– B ns() ds() -----------=+= Linear Time-Invariant Systems 2-5 sinfo(sys) System with 1 state(s), 1 input(s), and 1 output(s) Similarly, the poles of the system are given by spol: spol(sys) ans = 1 The function ssub selects particular inputs and outputs of a system and returns the corresponding subsystem. For instance, if G has two inputs and three outputs, the subsystem mapping the first input to the second and third outputs is given by ssub(g,1,2:3) The function sinv computes the inverse H(s) = G(s) –1 of a system G(s) with square invertible D matrix: h = sinv(g) Finally, the state-space realization of an LTI system can be “balanced” with sbalanc. This function seeks a diagonal scaling similarity that reduces the norms of A, B, C. 2 Uncertain Dynamical Systems 2-6 Time and Frequency Response Plots Time and frequency responses of linear systems are plotted directly from the SYSTEM matrix with the function splot. For the sake of illustration, consider the second-order system with m = 2, f = 0.01, and k = 0.5. This system is specified in descriptor form by sys = ltisys([0 1, 0.5 0.01],[0,1],[1 0],0,[1 0,0 2]) the last input argument being the E matrix. To plot its Bode diagram, type splot(sys,'bo') The second argument is the string consisting of the first two letters of “bode.” This command produces the plot of Figure 2-1. A third argument can be used to adjust the frequency range. For instance, splot(sys,'bo',logspace( 1,1,50)) displays the Bode plot of Figure 2-2 instead. Figure 2-1: splot (sys,'bo') mx ·· fx · kx++ uyx=,= 10 ?1 10 0 ?50 0 50 Frequency (rad/sec) Gain dB 10 ?1 10 0 ?90 ?180 0 Frequency (rad/sec) Phase deg Time and Frequency Response Plots 2-7 Figure 2-2: splot (sys,'bo',logspace (–1,1,50)) To draw the singular value plot of this second-order system, type splot(sys,'sv'). For a system with transfer function G(s) = D + C(sE – A) –1 B, the singular value plot consists of the curves (ω, σ i (G(jω))) where σ i denotes the i-th singular value of a matrix M in the order σ 1 (M) ? σ 2 (M) ? . . . ? σ p (M). Similarly, the step and impulse responses of this system are plotted by splot(sys,'st') and splot(sys,'im'), respectively. See the “Command Reference” chapter for a complete list of available diagrams. The function splot is also applicable to discrete-time systems. For such systems, the sampling period T must be specified to plot frequency responses. For instance, the Bode plot of the system 10 ?1 10 0 10 1 ?50 0 50 Frequency (rad/sec) Gain dB 10 ?1 10 0 10 1 ?90 ?180 0 Frequency (rad/sec) Phase deg 2 Uncertain Dynamical Systems 2-8 with sampling period t = 0.01 s is drawn by splot(ltisys(0.1,0.2, 1),0.01,'bo') x k 1+ 0.1x k 0.2u k += y k x– k = ? ? ? Interconnections of Linear Systems 2-9 Interconnections of Linear Systems Tools are provided to form series, parallel, and simple feedback inter- connections of LTI systems. More complex interconnections can be built either incrementally with these basic facilities or directly with sconnect (see“Polytopic Models” on page 2-14 for details). The interconnection functions work on the SYSTEM matrix representation of dynamical systems and their names start with an s. Note that most of these functions are also applicable to polytopic or parameter- dependent models (see “Polytopic Models” on page 2-14). Series and parallel interconnections are performed by sadd and smult. In terms of transfer functions, sadd(g1,g2) returns the system with transfer function G 1 (s) + G 2 (s) while smult(g1,g2) returns the system with transfer function G 2 (s)G 1 (s). Both functions take up to ten input arguments, at most one of which can be a parameter-dependent system. Similarly, the command sdiag(g1,g2) appends (concatenates) the systems G 1 and G 2 and returns the system with transfer function This corresponds to combining the input/output relations y 1 = G 1 (s)u 1 , y 2 = G 2 (s)u 2 into the single relation Gs() G 1 s() 0 0 G 2 s() ?? ?? ?? = y 1 y 2?? ?? ?? ?? Gs() u 1 u 2?? ?? ?? ?? = 2 Uncertain Dynamical Systems 2-10 Finally, the functions sloop and slft form basic feedback interconnections. The function sloop computes the closed-loop mapping between r and y in the loop of Figure 2-3. The result is a state-space realization of the transfer function (I – εG 1 G 2 ) –1 G 1 where ε = ±1. Figure 2-3: sloop This function is useful to specify simple feedback loops. For instance, the closed-loop system clsys corresponding to the two-degree-of-freedom tracking loop is derived by setting G 1 (s) = G(s)K(s) and G 2 (s) = 1: clsys = smult(c, sloop(smult(k,g),1)) The function slft forms the more general feedback interconnection of Figure 2-4 and returns the closed-loop mapping from to . To form this interconnection when u ∈ R 2 and y ∈ R 3 , the command is slft(P1,P2,2,3) r y+ ?ε G 1 (s) G 2 (s) r y+ ? K GC ω 1 ω 2 ?? ?? z 1 z 2 ?? ?? Interconnections of Linear Systems 2-11 The last two arguments dimension u and y. This function is useful to compute linear-fractional interconnections such as those arising in H ∞ theory and its extensions (see “How to Derive Such Models” on page 2-23). Figure 2-4: slft w 1 z 1 u y w 2 z 2 P 1 (s) P 2 (s) 2 Uncertain Dynamical Systems 2-12 Model Uncertainty The notion of uncertain dynamical system is central to robust control theory. For control design purposes, the possibly complex behavior of dynamical systems must be approximated by models of relatively low complexity. The gap between such models and the true physical system is called the model uncertainty. Another cause of uncertainty is the imperfect knowledge of some components of the system, or the alteration of their behavior due to changes in operating conditions, aging, etc. Finally, uncertainty also stems from physical parameters whose value is only approximately known or varies in time. Note that model uncertainty should be distinguished from exogenous actions such as disturbances or measurement noise. The LMI Control Toolbox focuses on the class of dynamical systems that can be approximated by linear models up to some possibly nonlinear and/or time-varying model uncertainty. When deriving the nominal model and estimating the uncertainty, two fundamental principles must be remembered: Uncertainty should be small where high performance is desired (tradeoff between performance and robustness). In other words, the linear model should be sufficiently accurate in the control bandwidth. The more information you have about the uncertainty (phase, structure, time invariance, etc.), the higher the achievable performance will be. There are two major classes of uncertainty: Dynamical uncertainty, which consists of dynamical components neglected in the linear model as well as of variations in the dynamical behavior during operation. For instance, high-frequency flexible modes, nonlinearities for large inputs, slow time variations, etc. Parameter uncertainty, which stems from imperfect knowledge of the physical parameter values, or from variations of these parameters during operation. Examples of physical parameters include stiffness and damping coefficients in mechanical systems, aerodynamical coefficients in flying devices, capacitors and inductors in electric circuits, etc. Other important characteristics of uncertainty include whether it is linear or nonlinear, and whether it is time invariant or time varying. Model uncertainty is generally a combination of dynamical and parametric uncertainty, and may arise at several different points in the control loop. For instance, there may be Model Uncertainty 2-13 dynamical uncertainty on the system actuators, and parametric uncertainty on some sensor coefficients. Two representations of model uncertainty are used in the LMI Control Toolbox: Uncertain state-space models. This representation is relevant for systems described by dynamical equations with uncertain and/or time-varying coefficients. Linear-fractional representation of uncertainty. Here the uncertain system is described as an interconnection of known LTI systems with uncertain components called “uncertainty blocks.” Each uncertainty block ? i ( . ) represents a family of systems of which only a few characteristics are known. For instance, the only available information about ? i may be that it is a time-invariant nonlinearity with gain less than 0.01. Determinant factors in the choice of representation include the available model (state-space equations, frequency-domain model, etc.) and the analysis or synthesis tool to be used. 2 Uncertain Dynamical Systems 2-14 Uncertain State-Space Models Physical models of a system often lead to a state-space description of its dynamical behavior. The resulting state-space equations typically involve physical parameters whose value is only approximately known, as well as approximations of complex and possibly nonlinear phenomena. In other words, the system is described by an uncertain state-space model E = Ax + Bu, y = Cx + Du where the state-space matrices A, B, C, D, E depend on uncertain and/or time-varying parameters or vary in some bounded sets of the space of matrices. Of particular relevance to this toolbox are the classes of polytopic or parameter-dependent models discussed next. We collectively refer to such models as P-systems, “P–” standing for “polytopic or parameter-dependent.” P-systems are specified with the function psys and manipulated like ordinary LTI systems except for a few specific restrictions. Polytopic Models We call polytopic system a linear time-varying system E(t) = A(t) x + B(t) u y = C(t) x + D(t) u whose SYSTEM matrix varies within a fixed polytope of matrices, i.e., where S 1 , . . . , S k are given vertex systems: (2-1) In other words, S(t) is a convex combination of the SYSTEM matrices S 1 , . . . , S k . The nonnegative numbers α 1 , . . . , α k are called the polytopic coordinates of S. x · x · St() At() jE t()+ Bt() Ct() Dt() = St() Co S 1 … S k ,,{}∈ := α i S i i 1= k ∑ : α i 0 α i i 1= k ∑ 1=,≥ ?? ?? ?? ?? S 1 A 1 jE 1 + B 1 C 1 D 1 ,…S k A k jE k + B k C k D k == Uncertain State-Space Models 2-15 Such models are also called polytopic linear differential inclusions in the literature [3] and arise in many practical situations, including: Multimodel representation of a system, each model being derived around particular operating conditions Nonlinear systems of the form = A(x) x + B(x) u, y = C(x) x + D(x) u State-space models depending affinely on time-varying parameters (see “From Affine to Polytopic Models” on page 2-20) A simple example is the system = (sin x) x whose state matrix A = sin x ranges in the polytope A ? Co{–1, 1} = [–1, 1]. Polytopic systems are specified by the list of their vertex systems, i.e., by the SYSTEM matrices S 1 , . . . , S k in (2-1). For instance, a polytopic model taking values in the convex envelope of the three LTI systems s1, s2, s3 is declared by polsys = psys([s1 s2 s3]) Affine Parameter-Dependent Models The equations of physics often involve uncertain or time-varying coefficients. When the system is linear, this naturally gives rise to parameter-dependent models (PDS) of the form E(p) = A(p) x + B(p) u y = C(p) x + D(p) u where A( . ), . . . , E( . ) are known functions of some parameter vector p = (p 1 , . . . , p n ). Such models commonly arise from the equations of motion, aerodynamics, circuits, etc. The LMI Control Toolbox offers various tools to analyze the stability and performance of parameter-dependent systems with an affine dependence on the parameter vector p = (p 1 , . . . , p n ). That is, PDSs where A(p) = A 0 + p 1 A 1 + . . .+ p n A n , B(p) = B 0 + p 1 B 1 + . . .+ p n B n , x · x · x · 2 Uncertain Dynamical Systems 2-16 and so on. Affine parameter-dependent models are well-suited for Lyapunov- based analysis and synthesis and are easily converted to linear-fractional uncertainty models for small-gain-based design (see the nonlinear spring example in “Sector-Bounded Uncertainty” on page 2-30). With the notation the affine dependence on p is written more compactly in SYSTEM matrix terms as S(p) = S 0 + p 1 S 1 + . . . + p n S n . The system “coefficients” S 0 , . . . , S n fully characterize the dependence on the uncertain parameters p 1 , . . . , p n . Note that S 0 , . . . , S n need not represent meaningful dynamical systems. Only their combination S(p) is a relevant description of the problem. Affine parameter-dependent systems are specified with psys by providing A description of the parameter vector p in terms of bounds on the parameter values and rates of variation The list of SYSTEM matrix coefficients S 0 , . . . , S n For instance, the system S(p) = S 0 + p 1 S 1 + p 2 S 2 is defined by s0 = ltisys(a0,b0,c0,d0,e0) s1 = ltisys(a1,b1,c1,d1,e1) s2 = ltisys(a2,b2,c2,d2,e2) affsys = psys(pv, [s0 s1 s2]) where pv is the parameter description returned by pvec (see next subsection for details). The output affsys is a structured matrix storing all relevant data. Sp() Ap() jE p()+ Bp() Cp() Dp()?? ?? ?? S i A i jE i + B i C i D i?? ?? ?? ,=,= Uncertain State-Space Models 2-17 Important: By default, ltisys sets the E matrix to the identity. Omitting the arguments e0, e1, e2 altogether results in setting E(p) = I + (p 1 + p 2 )I. To specify an affine PDS with a parameter-independent E matrix (e.g., E(p) = I), you must explicitly set E 1 = E 2 = 0 by typing s0 = ltisys(a0,b0,c0,d0) s1 = ltisys(a1,b1,c1,d1,0) s2 = ltisys(a2,b2,c2,d2,0) Quantification of Parameter Uncertainty Parameter uncertainty is quantified by the range of parameter values and possibly the rates of parameter variation. This is done with the function pvec. The characteristics of parameter vectors defined with pvec are retrieved with pvinfo. The parameter uncertainty range can be described as a box in the parameter space. This corresponds to cases where each uncertain or time-varying parameter p i ranges between two empirically determined extremal values and : (2-1) If p = (p 1 , . . . , p n ) is the vector of all uncertain parameters, (2-2) delimits a hyperrectangle of the parameter space R n called the parameter box. Consider the example of an electrical circuit with uncertain resistor ρ and capacitor c ranging in ρ ∈ [600, 1000] c ∈ [1, 5] The corresponding parameter vector p = (ρ, c) takes values in the box drawn in Figure 2-5. This uncertainty range is specified by the commands: range = [600 1000,1 5] p = pvec('box',range) The i-th row of the matrix range lists the lower and upper bounds on p i . p i p i p i p i p i ,[].∈ 2 Uncertain Dynamical Systems 2-18 Similarly, bounds on the rate of variation of p i (t) are specified by adding a third argument rate to the calling list of pvec. For instance, the constraints are incorporated by rate = [0.1 1, 0.001 0.001] p = pvec('box',range,rate) All parameters are assumed to be time-invariant when rate is omitted. Slowly varying parameters can be specified in this manner. In general, robustness against fast parameter variations is more stringent than robustness against constant but uncertain parameters. Figure 2-5: Parameter box Alternatively, uncertain parameter vectors can be specified as ranging in a polytopic of the parameter space R n like the one drawn in Figure 2-6 for n = 2. This polytope is characterized by the three vertices: Π 1 = (1, 4), Π 2 = (3, 8), Π 3 = (10, 1). An uncertain parameter vector p with this range of values is defined by pi1=[1,4], pi2=[3,8], pi3=[10,1] p = pvec('pol',[pi1,pi2,pi3]) p · i 0.1 ρ · t() 1 c · t() 0.001≤,≤≤ c p 600 1000 1 5 range of parameter values Uncertain State-Space Models 2-19 The string 'pol' indicates that the parameter range is defined as a polytope. Figure 2-6: Polytopic parameter range Simulation of Parameter-Dependent Systems The function pdsimul simulates the time response of affine parameter- dependent systems along given parameter trajectories. The parameter vector p(t) is required to range in a box (type 'box' of pvec). The parameter trajectory is defined by a function p = fun(t) returning the value of p at time t. For instance, pdsimul(pds,'traj') plots the step response of a single-input parameter-dependent system pds along the parameter trajectory defined in traj.m. To obtain other time responses, specify an input signal as in pdsimul(pds,'traj',1,'sin') This command plots the response to a sine wave between t = 0 and t = 1 s. Multi-input multi-output (MIMO) systems are simulated similarly by specifying an appropriate input function. p 2 p 1 3 10 1 8 4 1 Π 2 Π 3 Π 1 range of values of p 2 Uncertain Dynamical Systems 2-20 From Affine to Polytopic Models Affine parameter-dependent models are readily converted to polytopic ones. Suppose for instance that each parameter p i ranges in some interval [ ]. The parameter vector p = (p 1 , . . . , p n ) then takes values in a parameter box with 2 n corners Π 1 , Π 2 , . . . . If the function S(p) is affine in p, it maps this parameter box to some polytope of SYSTEM matrices. More precisely, this polytope is the convex envelope of the images S(Π 1 ), S(Π 2 ), . . . of the parameter box corners Π 1 , Π 2 , . . . as illustrated by the figure below. Figure 2-7: Mapping the parameter box to a polytope of systems Given an affine parameter-dependent system, the function aff2pol performs this mapping and returns an equivalent polytopic model. The syntax is polsys = aff2pol(affsys) where affsys is the affine model. The resulting polytopic model polsys consists of the instances of affsys at the vertices of the parameter range. Sp() Ap() jE p()+ Bp() Cp() Dp()?? ?? ?? = p i p i , p 1 p 2 Π 4 Π 3 Π 1 Π 2 parameter box polytope of matrices S(p) Uncertain State-Space Models 2-21 Example We conclude with an example illustrating the manipulation of polytopic and parameter-dependent models in the LMI Control Toolbox. Example 2.1. Consider a simple electrical circuit with equation where the inductance L, the resistor R, and the capacitor C are uncertain parameters ranging in L ∈ [10, 20], R ∈ [1, 2], C ∈ [100, 150]. A state-space representation of its undriven response is E(L, R, C) = A(L, R, C)x where and This affine system is specified with psys as follows: a0 = [0 1;0 0]; e0 = [1 0;0 0]; s0 = ltisys(a0,e0) aL = zeros(2); eL = [0 0;0 1]; sL = ltisys(aL,eL) aR = [0 0; 1 0]; sR = ltisys(aR,0) aC = [0 0;0 1]; sC = ltisys(aC,0) pv=pvec('box',[10 20;1 2;100 150]) pds = psys(pv,[s0 sL sR sC]) The first SYSTEM matrix s0 contains the state-space data for L = R = C = 0 while sL, sR, sC define the coefficient matrices of L, R, C. The range of parameter values is specified by the pvec command. The results can be checked with psinfo and pvinfo: L d 2 i dt -------- R di dt ----- Ci+ V=+ x · x T i di dt ------, ?? ?? = ALRC,,() 01 R– C–?? ?? ?? 01 00?? ?? ?? L 0 R 00 1– 0?? ?? ?? C 00 01–?? ?? ?? ++×+== ELRC,,() 10 0 L?? ?? ?? 10 00?? ?? ?? L 00 01?? ?? ?? R 0 C 0×+× .++== 2 Uncertain Dynamical Systems 2-22 psinfo(pds) Affine parameter-dependent system with 3 parameters (4 systems) Each system has 2 state(s), 0 input(s), and 0 output(s) pvinfo(pv) Vector of 3 parameters ranging in a box The system can also be evaluated for given values of L, R, C: sys = psinfo(pds,'eval',[15 1.2 150]) [a,b,c,d,e] = ltiss(sys) The matrices a and e now contain the values of A(L, R, C) and E(L, R, C) for L = 15, R = 1.2, and C = 150. Finally, the polytopic counterpart of this affine model is given by pols = aff2pol(pds) psinfo(pols) Polytopic model with 8 vertex systems Each system has 2 state(s), 0 input(s), and 0 output(s) Linear-Fractional Models of Uncertainty 2-23 Linear-Fractional Models of Uncertainty For systems with both dynamical and parametric uncertainty, a more general representation of uncertainty is the linear-fractional model of Figure 2-8 [1, 2]. In this generic model: The LTI system P(s) gathers all known LTI components (controller, nominal models of the system, sensors, and actuators, . . .) The input vector u includes all external actions on the system (disturbance, noise, reference signal, . . .) and the vector y consists of all output signals generated by the system ? is a structured description of the uncertainty. Specifically, ? = diag(? 1 , . . . , ? r ) where each uncertainty block ? i accounts for one particular source of uncertainty (neglected dynamics, nonlinearity, uncertain parameter, etc.). The diagonal structure of ? reflects how each uncertainty component ? i enters the loop and affects the overall behavior of the true system. Figure 2-8: Linear-fractional uncertainty How to Derive Such Models Before proceeding with the specification of such uncertainty models, we illustrate how natural uncertainty descriptions can be recast into the standard linear-fractional model of Figure 2-8. u y P(s) ? w q 2 Uncertain Dynamical Systems 2-24 Example 2.2 . Consider the feedback loop where G(s) is an uncertain LTI system approximated within 5% accuracy by the second-order nominal model In other words, G(s) = G 0 (s)(I + ?(s)) where the multiplicative dynamical uncertainty ?(s) is any stable LTI system with RMS gain less than 0.05 k(t) is a fluctuating gain modeled by k(t) = k 0 + δ(t) where k 0 is the nominal value and |δ(t)| < 0.1 k 0 . To derive a linear-fractional representation of this uncertain feedback system, first separate the uncertainty from the nominal models by rewriting the loop as r y+ ? G(s) k(t) G 0 s() 1 s 2 0.01s 1++ ------------------------------------= r y+ ? δ(.) k 0 + + G 0 + + ? Linear-Fractional Models of Uncertainty 2-25 Then pull out all uncertain components and lump them together into a single block as follows: If and denote the input and output vectors of the uncertainty block, the plant P(s) is simply the transfer function from to in the diagram. r + k 0 y + + + G 0 + δ 0 0 ??? ?? ?? ? e q 1 q 2 P 2 p 1 p 1 p 2?? ?? ?? q 1 q 2?? ?? ?? q 1 q 2 r ?? ?? ?? ?? p 1 p 2 y ?? ?? ?? ?? r y P(s) δ 0 0 ??? ?? ?? q 1 q 2 ?? ?? p 1 p 2 ?? ?? 2 Uncertain Dynamical Systems 2-26 Hence the SYSTEM matrix of P(s) can be computed with sconnect or by specifying the following interconnection: Specification of the Uncertainty In linear-fractional uncertainty models, each uncertainty block ? i is a dynamical system characterized by Its dynamical nature: linear time-invariant or time-varying, nonlinear memoryless, arbitrary nonlinear Its dimensions and structure (full block or scalar block ? i = δ i × I). Scalar blocks are used to represent uncertain parameters. Whether αii is real or complex in the case of scalar uncertainty ? i = δ i by I Quantitative information such as norm bounds or sector bounds The properties of each individual uncertainty block ? i are specified with the function ublock. These individual descriptions are then appended with udiag to form a complete description of the structured uncertainty ? = diag(? 1 , . . . , ? r ) Proper quantification of the uncertainty is important since this determines achievable levels of robust stability and performance. The basics about quantification issues are reviewed next. r + k 0 y + + + G 0 + ? p 2 q 2 q 1 p 1 e Linear-Fractional Models of Uncertainty 2-27 Norm-Bounded Uncertainty Norm bounds specify the amount of uncertainty in terms of RMS gain. Recall that the RMS gain or L 2 gain of a BIBO-stable system is defined as the worst-case ratio of the input and output energies: where For additive uncertainty G = G 0 + ?, the bound on ||?|| ∞ is an estimate of the worst-case gap G – G 0 between the true system G and its linear model G 0 (s). For multiplicative uncertainty G = G 0 (I + ?), this bound quantifies the relative gap between G and its model G 0 . Norm bounds are also useful to quantify parametric uncertainty. For instance, an uncertain parameter p can be represented as p = p 0 (1 + δ), |δ| < δ max where p 0 is the nominal value and δ max bounds the relative deviation from p 0 . If p ∈ [ ], then p 0 and δ max are given by When ? is LTI (i.e., when the physical system G itself is LTI), the uncertainty can be quantified more accurately by using frequency-dependent bounds of the form (2-2) ? ∞ sup ?w L 2 w L 2 ------------------= wL 2 ∈ w 0≠ w L 2 2 w T t()wt() t.d 0 ∞ ∫ = pp, p 0 pp+ 2 ------------- δ max pp– 2p 0 ------------=,= W 1– s()?s() ∞ 1< 2 Uncertain Dynamical Systems 2-28 where W(s) is some SISO shaping filter. Such bounds specify different amounts of uncertainty across the frequency range and the uncertainty level at each frequency is adjusted through theshaping filter shaping filter W(s). For instance, choosing results in the frequency-weighted uncertainty bound fquan2.ps drawn in Figure 2-9. Figure 2-9: Frequency-dependent bound on |?(jw)| These various norm bounds are easily specified with ublock. For instance, a 2-by- 3 LTI uncertainty block ?(s) with RMS gain smaller than 10 is declared by delta = ublock([2 3],10) while a scalar nonlinearity Φ( . ) with unit RMS gain is defined by Ws() 2s 1+ s 100+ ------------------= ? jω()βω()< := 4ω 2 1+ ω 2 10 4 + ---------------------- 10 ?1 10 0 10 1 10 2 10 3 10 ?2 10 ?1 10 0 10 1 Frequency w Bound beta(w) Linear-Fractional Models of Uncertainty 2-29 phi = ublock(1,1,'nl') In these commands, the first argument dimensions the block, the second argument quantifies the uncertainty by a norm bound or sector bounds, and the optional third argument is a string specifying the nature and structure of the uncertainty block. The default properties are full, complex-valued, and linear time invariant (LTI). Scalar blocks and real parameter uncertainty are denoted by the letters s and r, respectively. For example, the command delta = ublock(3,0.5,'sr') declares a 3 ?βψ?3 block ? = δ ?βψ?I with repeated real parameter δ bounded by |δ| e 0.5. Finally, setting the second argument of ublock to a SISO system W(s) specifies a frequency-dependent norm bound on the LTI uncertainty block ?(s). Thus, delta = ublock(2, ltisys('tf',[100 0],[1 100])) declares an LTI uncertainty ?(s) ∈ C 2×2 satisfying at all frequencies ω. After specifying each block ? i with ublock, call udiag to derive a complete description of the uncertainty structure ? = diag(? 1 , . . . , ? n ). For instance, the commands delta1 = ublock([3 1],0.1) delta2 = ublock(2,2,'s') delta3 = ublock([1 2],100,'nl') delta = udiag(delta1,delta2,delta3) specify the uncertainty structure ? = diag(? 1 , ? 2 , ? 3 ) where ? 1 (s) ∈ C 3×1 and ||? 1 || ∞ < 0.1 σ max ? jω()()Wjω()<<, Ws() 100s s 100+ ------------------= 2 Uncertain Dynamical Systems 2-30 ? 2 = δ by I 2 with δ ∈ C and |δ| < 2 ? 3 ( . ) is a nonlinear operator with two inputs and one output and satisfies ||? 3 || ∞ < 100. Finally, uinfo displays the characteristics of uncertainty structures declared with ublock and udiag. uinfo(delta) Sector-Bounded Uncertainty The behavior of uncertain dynamical components can also be quantified in terms of sector bounds on their time-domain response [5, 4]. A (possibly nonlinear) BIBO-stable system φ( . ) is said to be in the sector {a, b} if the mapping φ : u ∈ L 2 α y∈L 2 satisfies a quadratic constraint of the form As a special case, passive systems satisfy which corresponds to the sector {0, +∞}. Note that sector bounds can always be converted to norm bounds since φ is in the sector {a, b} if and only if If φ is passive, a bilinear transformation maps φ to the operator (I – φ)(I + φ) –1 of norm less than one. block dims type real/cplx full/scal bounds 1 3x1 LTI c f norm <= 0.1 2 2x2 LTI c s norm <= 2 3 1x2 NL r f norm <= 100 yt() au t()– T 0 ∞ ∫ yt() bu t()– dt 0for all uL 2 ∈≤ yt() T 0 ∞ ∫ ut()dt 0≥ φ ab+ 2 ------------– ∞ ab– 2 ---------------< Linear-Fractional Models of Uncertainty 2-31 The system φ( . ) is called memoryless if y(t) only depends on the input value u(t) at time t, i.e., y(t) = φ(u(t)). For memoryless systems, the sector condition reduces to the “instantaneous” property (y(t) – au(t)) T (y(t) – bu(t)) e 0 at all time t. In the scalar case, this means that the response y = φ(u) lies in the sector delimited by the two lines y = au and y = bu as illustrated below. A simple example is the nonlinear spring y = k(u) with The response of this spring lies in the sector {1, 2} as is seen from Figure 2-10. This scalar memoryless nonlinearity is specified with ublock by k = ublock(1,[1 2],'nlm') y u y=au y=bu y=φ (υ) ku() ? ? ? ? ? = u 1– if u 1–< 2u if u 1≤ u 1+ if u 1> 2 Uncertain Dynamical Systems 2-32 Robustness in the face of sector-bounded uncertainty is analyzed by the circle and Popov criteria. From Affine to Linear-Fractional Models Since some uncertain systems are more naturally specified as affine parameter-dependent systems, the function aff2lft is provided to derive a linear-fractional representation of their parametric uncertainty. All uncertain parameters are assumed real and are repeated in the ? structure when necessary. For instance, the system (2-3) Figure 2-10: Nonlinear spring y = k(u) with constant p 1 ∈ [2, 10] and time-varying p 2 ∈ [–1, 3] is specified by s0 = ltisys([0 1,0 0]) s1 = ltisys([1 0, 1 1],0) % p1 dx dt ------- p 1 1 p– 1 p 1 2p 2 + ?? ?? ?? x= -1 1 2 k(u) Linear-Fractional Models of Uncertainty 2-33 s2 = ltisys([0 0,0 2],0) % p2 pv = pvec('box',[2 10, 1 3],[0 0,-Inf Inf]) pds = psys(pv,[s0 s1 s2]) To derive the linear-fractional representation of the uncertainty on p 1 , p 2 , type [P,delta] = aff2lft(pds) On output: The nominal plant P corresponds to the average values of p 1 and p 2 . Specifically, closing the interconnection of Figure 2-8 with ? ≡ 0 yields the system (2-4) for p 1 = 6 and p 2 = 1, as confirmed by cls = slft(zeros(3),P) a = ltiss(cls) a = 6 1 6 8 The uncertainty structure delta consists of real scalar blocks as confirmed by uinfo(delta) Note that the parameter p 1 is repeated and that the norm bound corresponds to the maximal deviation from the average value of the parameter. block dims type real/cplx full/sc al bounds 1 2-by-2 LTI r s norm <= 4 2 1-by-1 LTV r s norm <= 2 2 Uncertain Dynamical Systems 2-34 The corresponding block diagram appears in Figure 2-11. Figure 2-11: Linear-fractional representation of the system (2-4) u y P(s) δp 1 I× 0 0 δp 2?? ?? ?? References 2-35 References [1] Doyle, J.C. and G. Stein, “Multivariable Feedback Design: Concepts for a Classical/Modern Synthesis,” IEEE Trans. Aut. Contr., AC-26 (1981), pp. 4-16. [2] Doyle, J.C., “Analysis of Feedback Systems with Structured Uncertainties,” IEE Proc., vol. 129, pt. D (1982), pp. 242–250. [3] Boyd, S., L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, SIAM books, Philadelphia, 1994. [4] Vidyasagar, M., Nonlinear System Analysis, Prentice-Hall, Englewood Cliffs, 1992. [5] Zames, G., “On the Input-Output Stability of Time-Varying Nonlinear Feedback Systems, Part I and II,” IEEE Trans. Aut. Contr., AC–11 (1966), pp. 228-238 and 465-476. 2 Uncertain Dynamical Systems 2-36 3 Robustness Analysis Quadratic Lyapunov Functions . . . . . . . . . . . 3-3 LMI Formulation . . . . . . . . . . . . . . . . . . . 3-4 Quadratic Stability . . . . . . . . . . . . . . . . . . 3-6 Maximizing the Quadratic Stability Region . . . . . . . . 3-8 Decay Rate . . . . . . . . . . . . . . . . . . . . . 3-9 Quadratic H∞ Performance . . . . . . . . . . . . . . .3-10 Parameter-Dependent Lyapunov Functions . . . . . .3-12 Stability Analysis . . . . . . . . . . . . . . . . . . .3-14 μ Analysis . . . . . . . . . . . . . . . . . . . . .3-17 Structured Singular Value . . . . . . . . . . . . . . .3-17 Robust Stability Analysis . . . . . . . . . . . . . . .3-19 Robust Performance . . . . . . . . . . . . . . . . . .3-21 The Popov Criterion . . . . . . . . . . . . . . . .3-24 Real Parameter Uncertainty . . . . . . . . . . . . . .3-25 Example . . . . . . . . . . . . . . . . . . . . . .3-28 References . . . . . . . . . . . . . . . . . . . . .3-32 3 Robustness Analysis 3-2 Control systems are often designed for a simplified model of the physical plant that does not take into account all sources of uncertainty. A-posteriori robustness analysis is then necessary to validate the design and obtain guarantees of stability and performance in the face of plant uncertainty. The LMI Control Toolbox offers a variety of tools to assess robust stability and robust performance. These tools cover most available Lyapunov-based and frequency-domain analysis techniques. Quadratic stability/performance Tests involving parameter-dependent Lyapunov functions Mixed-μ analysis The Popov criterion Each test is geared to a particular class of uncertainty, for example, real parameter uncertainty for parameter-dependent Lyapunov functions, or memoryless nonlinearities for the Popov criterion. Hence users should select the most appropriate tool for their problem. Since all of these tests are based on sufficient conditions, they are only productive when they succeed in establishing robust stability or performance. Quadratic Lyapunov Functions 3-3 Quadratic Lyapunov Functions The notions of quadratic stability and quadratic performance are useful to analyze linear time-varying systems E(t)(t) = A(t)x(t), x(0) = x 0 . Given such systems, a sufficient condition for asymptotic stability is the existence of a positive-definite quadratic Lyapunov function V(x) =x T Px such that along all state trajectories. In terms of Q := P –1 , this is equivalent to (3-1) at all times t. Assessing quadratic stability is not tractable in general since (3-1) places an infinite number of constraints on Q. However, (3-1) can be reduced to a finite set of LMI contraints in the following cases: 1 A(t) and E(t) are fixed affine functions of some time-varying parameters p 1 (t), . . . , p n (t). A(t) = A 0 + p 1 (t)A 1 + . . . + p n (t)A n E(t) = E 0 + p 1 (t)E 1 + . . . + p n (t)E n . This is referred to as an affine parameter-dependent model. 2 A(t) + jE(t) ranges in a fixed polytope of matrices, that is, A(t) = α 1 (t)A 1 + . . . + α n (t)A n E(t) = α 1 (t)E 1 + . . . + α n (t)E n with α i (t) ? 0 and . This is referred to as a polytopic model. The first case corresponds to systems whose state-space equations depend affinely on time-varying physical parameters, and the second case to x · dV x t()() dt ----------------------- 0< At()QE t() T Et()QA t() T + 0< α i t() i 1= n ∑ 1= 3 Robustness Analysis 3-4 time-varying systems modeled by an envelope of LTI systems (see “Uncertain State-Space Models” on page 2-14 for details). Note that quadratic Lyapunov functions guarantee stability for arbitrarily fast time variations, which is their main source of conservatism. Quadratic Lyapunov functions are also useful to assess robust RMS performance. Specifically, a time-varying system (3-2) with zero initial state satisfies ||y|| L 2 < γ||u|| L 2 for all bounded input u(t) if there exists a positive-definite Lyapunov function V(x) =x T Px such that In terms of Q := P –1 , this is equivalent to the family of “Bounded Real Lemma” inequalities The smallest γ for which such a Lyapunov function exists is called the quadratic H ∞ performance. This is an upper bound on the worst-case RMS gain of the system (3-2). LMI Formulation Sufficient LMI conditions for quadratic stability are as follows [8, 1, 2]: Affine models: consider the parameter-dependent model ? ? ?Et()x · At()xBt()u+= yCt()xDt()u+= dV x() dt ---------------- y T y γ 2 u T u 0<–+ At()QE t() T Et()QA t() T + Bt()Et()QC t() T Bt() T γI– Dt() T Ct()QE t() T Dt() γI–?? ?? ?? 0< Quadratic Lyapunov Functions 3-5 (3-3) where p i (t) ∈ [ ], and let denote the set of corners of the corresponding parameter box. The dynamical system (3-3) is quadratically stable if there exist symmetric matrices Q and such that (3-4) (3-5) (3-6) (3-7) Polytopic models: the polytopic system E(t)= A(t)x, A(t) + jE(t) ∈ Co{A 1 + jE 1 , . . . , A n + jE n }, is quadratically stable if there exist a symmetric matrix Q and scalars t ij = t ji such that (3-8) (3-9) Ep()x · Ap()x= , Ap() A 0 p 1 A 1 … p n A n +++= , Et() E 0 p 1 E 1 … p n E n +++= p i p i , V ω 1 …ω n ,,() : ω i p i p i ,{}∈{}= M i {} i 1= n A ω()QE ω() T E ω()QA ω() T ω i 2 i ∑ M i 0 for all ω V∈<++ A i PE i T E i PA i T M i 0 for i 1,…,n=≥++ M i 0 ≥ QI> x · A i QE j T E j QA i T A j QE i T E i QA j T 2t ij I for ij, 1,…,n{}∈<+++ QI> 3 Robustness Analysis 3-6 (3-10) Note that these LMI conditions are in fact necessary and sufficient for quadratic stability whenever In the affine case, no parameter p i enters both A(t) and E(t), that is, A i = 0 or E i = 0 for all i. The conditions (3-5)–(3-6) can then be deleted and it suffices to check (3-1) at the corners ω of the parameter box. In the polytopic case, either A(t) or E(t) is constant. It then suffices to solve (3-8)–(3-9) for t ij = 0. Similar sufficient conditions are available for robust RMS performance. For polytopic systems, for instance, LMI conditions guaranteeing the RMS performance γ are as follows. A symmetric matrix Q and scalars t ij = t ji for i,j ∈ {1, . . . , n} exist such that Quadratic Stability The function quadstab tests the quadratic stability of polytopic or affine parameter-dependent systems. The corresponding LMI feasibility problem is solved with feasp. t 11 … t 1n t 1n … t nn 0< . . . …… A i QE j T E j QA i T A j QE i T E j QA i +++ HH B i T B j T + 2γI– H C i QE i T C i QE j T + D i D j + 2γI– ?? ?? ?? 2t ij 1×< t 11 … t 1n t 1n … t nn 0< Q 0> … … Quadratic Lyapunov Functions 3-7 Example 3.1. Consider the time-varying system = A(t)x where A(t) ∈ Co{A 1 , A 2 , A 3 } with This polytopic system is specified by s1 = ltisys(a1) s2 = ltisys(a2) s3 = ltisys(a3) polsys = psys([s1 s2 s3]) To test its quadratic stability, type [tmin,P] = quadstab(polsys) Solver for LMI feasibility problems L(x) < R(x) This solver minimizes t subject to L(x) < R(x) + t*I The best value of t should be negative for feasibility Iteration : Best value of t so far 1 0.040521 2 0.032876 Result: best value of t: 0.032876 f-radius saturation: 0.000% of R = 1.00e+08 This system is quadratically stable tmin = 3.2876e 02 Collectively denoting the LMI system (3-8)–(3-10) by A(x) < 0, quadstab assesses its feasibility by minimizing τ subject to A(x) < τI and returns the global minimum tmin of this problem. Hence the system is quadratically stable if and only if tmin < 0. In this case, the second output P is the Lyapunov matrix proving stability. x · A 1 01 2– 0.2–?? ?? ?? = , A 2 01 2.2– 0.3–?? ?? ?? = , A 3 01 1.9– 0.1–?? ?? ?? = 3 Robustness Analysis 3-8 Maximizing the Quadratic Stability Region This option of quadstab is only available for affine parameter-dependent systems where the E matrix is constant and each parameter p i (t) ranges in an interval p i (t) ∈ [] Denoting the center and radius of each interval by and , the maximal quadratic stability region is defined as the largest portion of the parameter box where quadratic stability can be established. In other words, the largest dilatation factor such that the system is quadratically stable whenever p i (t) ∈ [μi – θδ i , i + θδ i ] Computing this largest factor is a generalized eigenvalue minimization problem (see gevp). This optimization is performed by quadstab when the first entry of the options vector is set to 1. Example 3.2. Consider the second-order system = A(k, f)x where This system is entered by s0 = ltisys([0 1;0 0]) s1 = ltisys([0 0; 1 0],0) % k s2 = ltisys([0 0;0 1],0) % f pv = pvec('box',[10 12;0.5 0.7]) % parameter box affsys = psys(pv,[s0 s1 s2]) To compute the largest dilatation of the parameter box where quadratic stability holds, type [marg,P] = quadstab(affsys,[1 0 0]) The result is p i p i , u i 1 2 --- p(= i p i )+ δ i 1 2 --- p(= i p i – ) x · Akf,() 01 k– f–?? ?? ?? kt() 10 12,[]ft() 0.5 0.7,[]∈,∈,= Quadratic Lyapunov Functions 3-9 marg = 1.4908e+00 which means 149% of the specified box, that is, k(t) ∈ [9.51, 12.49], f(t) ∈ [0.451, 0.749] Note that marg ? 1 implies quadratic stability in the prescribed parameter box. Decay Rate For the time-varying system E(t) = A(t)x, the quadratic decay rate α* is the smallest α such that A(t)QE(t) T + E(t)QA(t) T < αE(t)QE(t) T holds at all times for some fixed Q > 0. The system is quadratically stable if and only if α* < 0, in which case α* is an upper bound on the rate of return to equilibrium. Specifically, x T (t)Q –1 x(t) < e α*t (x(0) T Q –1 x(0)). Note that is also the largest β such that the shifted system E(t) = (A(t) + βE(t))x is quadratically stable. For affine or polytopic models, the decay rate computation is attacked as a generalized eigenvalue minimization problem (see gevp). This task is performed by the function decay. Example 3.3. For the time-varying system considered in Example 3.1, the decay rate is computed by [drate,P] = decay(polsys) This command returns drate = 5.6016e 02 x · α* 2 -------– x · 3 Robustness Analysis 3-10 P = 6.0707e 01 1.9400e 02 1.9400e 02 3.1098e 01 Quadratic H ∞ Performance The function quadperf computes the quadratic H ∞ performance of an affine parameter-dependent system E(p) = A(p)x + B(p)u y = C(p)x + D(p)u or of a polytopic system E(t) = A(t)x + B(t)u y = C(t)x + D(t)u where Recall that the quadratic H ∞ performance γ qp is an upper bound on the worst-case input/output RMS gain of the system. Specifically ||y|| L 2 < γ qp ||u|| L 2 holds for all bounded input u when the initial state is zero. The function quadperf can also be used to test whether the worst-case RMS gain does not exceed a given value γ > 0, or to maximize the portion of the parameter box where this level γ is not exceeded. Example 3.4. Consider the second-order system with time-varying mass and damping This affine parameter-dependent model is specified as x · x · St() At() jE t()+ Bt() Ct() Dt()?? ?? ?? Co S 1 ,…,S n {}∈= 10 0 mt()?? ?? ?? x · x ·· ?? ?? ?? 01 2– ft()–?? ?? ?? x x · ?? ?? ?? = 0 1?? ?? ?? u+ yx mt() 0.8 1.2,[]ft() 0.4 0.6,[]∈,∈,= Quadratic Lyapunov Functions 3-11 s0 = ltisys([0 1; 2 0],[0;1],[1 0],0,[1 0;0 0]) s1 = ltisys(zeros(2),[0;0],[0 0],0,[0 0;0 1]) % m s2 = ltisys([0 0;0 1],[0;0],[0 0],0,0) % f pv = pvec('box',[0.8 1.2 ; 0.4 0.6]) affsys = psys(pv,[s0 s1 s2]) and its quadratic H ∞ performance from u to y is computed by qperf = quadperf(affsys) qperf = 6.083e+00 This value is larger than the nominal LTI performance for m = 1 and f = 0.5 as confirmed by nomsys = psinfo(affsys,'eval',[1 0.5]) norminf(nomsys) 1.4311e+00 3 Robustness Analysis 3-12 Parameter-Dependent Lyapunov Functions The robustness test discussed next is applicable to affine parameter-dependent systems or time-invariant uncertain systems described by a polytope of models. To prove the robust stability of such systems, we seek a quadratic Lyapunov function that depends on the uncertain parameters or the polytopic coordinates in the case of a polytopic model. The resulting tests are less conservative than quadratic stability when the parameters are constant or slowly varying [6]. Moreover, available bounds on the rates of parameter variation can be explicitly taken into account. Affine models: for an affine parameter-dependent system E(p) = A(p)x with parameter vector p = (p 1 , . . . , p n ) ∈ R n , we seek positive definite Lyapunov functions of the form V(x, p) = x T Q(p) –1 x where Q(p) = Q 0 + p 1 Q 1 + . . . + p n Q n For such Lyapunov functions, the stability condition is equivalent to (3-11) Given interval bounds on each p i and its time derivative , the vectors p and range in n-dimensional “boxes.” If V and T list the corners of these boxes, (3-11) holds for all parameter trajectories if the following LMI problem is feasible [6]. Find symmetric matrices Q 0 , Q 1 , . . . , Q n , and such that x · dV x p,() dt ---------------------- 0< Ep()Qp()A T p() Ap()Qp()Ep() T Ep() dQ dt -------- Ep() T 0.<–+ p i p i p i ,[] ν i ν i ,[],∈,∈ dp i dt --------- dp dt ------- M i {} i 1= n Parameter-Dependent Lyapunov Functions 3-13 For all (ω, τ) ∈ V × T For ω ∈ V and i = 1, . . . , n Q(ω) > I for all ω ∈ V M i ? 0 Note that these conditions reduce to quadratic stability when the rates of variation are allowed to range in (–∞, +∞). In such cases indeed, Q 1 , . . . , Q n must be set to zero for feasibility. Polytopic models: A similar extension of the quadratic stability test is available for time-invariant polytopic systems E = Ax where one of the matrices A, E is constant and the other uncertain. Assuming that E is constant and A ranges in the polytope A ∈ {α 1 A 1 + . . .+ α n A n : α i ? 0, α 1 + . . .+ α n = 1}, we seek a Lyapunov function of the form V(x, α) = x T Q(α) –1 x where Q(α) = α 1 Q 1 + . . . + α n Q n Using such Lyapunov functions, sufficient conditions for stability over the entire polytope are as follows. E ω()Q ω()A ω() T A ω()Q ω()E ω() T E ω()Q τ() Q 0 – E ω() T – ω i 2 M i 0< i ∑ ++ A i T Q i E ω() E ω() T Q i A i A i T Q ω()E i E i T Q ω()A i ++++ A ω() T Q i E i E i T Q i A ω() E i T Q τ() Q 0 –– M i 0≥ dp i dt --------- x · 3 Robustness Analysis 3-14 There exist symmetric matrices Q 1 , . . . , Q n , and scalars t ij = t ji such that Stability Analysis Given an affine or polytopic system, the function pdlstab seeks a parameter- dependent Lyapunov function establishing robust stability over a given parameter range or polytope of models. The syntax is similar to that of quadstab. For an affine parameter-dependent system with two uncertain parameters p 1 and p 2 , for instance, pdlstab is invoked by [tmin,Q0,Q1,Q2] = pdlstab(ps) Here ps is the system description and includes available information on the range of values and rates of variation of each parameter (see psys and pvec for details). If no rates of variation are specified, the parameters are regarded as time-invariant. The function pdlstab determines whether the sufficient LMI conditions listed above are feasible. Feasibility is established when tmin < 0. In such cases, a parameter-dependent Lyapunov function proving stability is V(x, p) =x T Q(p) –1 x with Q(p) = Q 0 + p 1 Q 1 + p 2 Q 2 Remark In the case of time-invariant uncertain parameters, pdlstab can be combined with a branch-and-bound scheme to reduce conservatism and enlarge the computed stability region. Specifically, if pdlstab fails to prove stability over the entire parameter range, you can split the range into smaller regions and try to establish stability on each subregion independently. Example 3.5. Consider the second-order system A i Q j E T EQ j A i T A j Q i E T EQ i A j T 2t ij <++ + Q j I> t 11 … t 1n t 1n … t nn 0< … … . . . x · x ·· ?? ?? ?? 01 kt()– ft()–?? ?? ?? x x · ?? ?? ?? = Parameter-Dependent Lyapunov Functions 3-15 where the stiffness k(t) and damping f(t) range in k(t) ∈ [5, 10], f(t) ∈ [0.01, 0.1] and their rates of variation are bounded by (3-12) This system and parameter data are specified by s0 = ltisys([0 1;0 0]) s1 = ltisys([0 0; 1 0],0) s2 = ltisys([0 0;0 1],0) pv = pvec('box',[5 10 ; 0.01 0.1],[ 0.01 0.01 ; 1 1]) ps = psys(pv,[s0 s1 s2]) The second argument of pvec defines the range of parameter values while the third argument specifies the bounds on their rate of variation. This system is not quadratically stable over the specified parameter box as confirmed by tmin = quadstab(ps) tmin = 8.0118e 04 Nevertheless, it turns out to be robustly stable when the parameter variations do not exceed the maximum rates (3-12), as shown by [tmin,Q0,Q1,Q2] = pdlstab(ps) Solver for LMI feasibility problems L(x) < R(x) This solver minimizes t subject to L(x) < R(x) + t*I The best value of t should be negative for feasibility Iteration : Best value of t so far 1 0.084914 2 0.011826 3 0.011826 4 1.878414e 03 dk dt ------- 0.01 df dt ------ 1<,< 3 Robustness Analysis 3-16 5 1.878414e 03 6 4.846478e 04 7 1.528537e 04 8 7.832830e 04 Result: best value of t: 7.832830e 04 f-radius saturation: 0.003% of R = 1.00e+07 This system is stable for the specified parameter trajectories This makes pdlstab a useful tool to analyze systems with constant or slowly-varying parameters. μ Analysis 3-17 μ Analysis μ analysis investigates the robust stability or performance of systems with linear time-invariant linear-fractional uncertainty. It is also applicable to time-invariant parameter-dependent systems by first deriving an equivalent linear-fractional representation with aff2lft (see “From Affine to Linear-Fractional Models” on page 2-32 for details). Nonlinear and/or time-varying uncertainty is addressed by the Popov criterion discussed in the next section. Structured Singular Value μ analysis is based on the concept of structured singular value (SSV or μ) [3, 12, 13, 10]. Consider the algebraic loop of Figure 3-1 where M ∈ C n×m and ? = diag(? 1 , . . ., ? n ) is a structured perturbation characterized by The dimensions of each block ? i Whether ? i is a complex or real-valued matrix Whether ? i is a full matrix or a scalar matrix of the form ? i = δ i × I We denote by ? the set of perturbations with this particular structure. The algebraic loop of Figure 3-1 is well posed if and only if I – M? is invertible. Measuring the smallest amount of perturbation ? ∈ ? needed to destroy well-posedness is the purpose of the SSV. Figure 3-1: Static μ problem ? M qw 3 Robustness Analysis 3-18 Formally, the SSV of M with respect to the perturbation structure ? is defined as [10] with the convention μ ? (M) = 0 if I – M? is invertible for all ? ∈ ?. From this definition, I – M? remains invertible as long as ? ∈ ? satisfies σ max (?) < 1/μ ? (M) That is, as long as the size of ? does not exceed K ? := 1/μ ? (M). The critical size K is called thewell-posedness margin well-posedness margin. Note that μ ? (M) = σ max (M) when ? = C m×n (unstructured case). Thus μ ? (M) extends the notion of largest singular value to the case of structured perturbations. Computing μ ? (M) is an NP-hard problem in general. However, a conservative estimate of the margin K ? (upper bound μ) can be computed by solving an LMI problem [4]. Assuming for simplicity that the blocks ? i are square, the variables in this LMI problem are two scaling matrices D = diag(D 1 , . . . , D n ), G = diag(G 1 , . . . , G n ) with the same block-diagonal structure as ? and where D i = d i × I with d i > 0 if ? i is a full block, and otherwise. G i = 0 if ? i is complex-valued, G i = g i × I with g i ∈ R if ? i is a real-valued full block, and if ? i = δ i × I with δ i real. Denoting by D and G the sets of D, G scalings with such properties, an upper bound on μ ? (M) is given by where α* is the global minimum of the following generalized eigenvalue minimization problem [4]: Minimize α over D ∈ D and G ∈ G subject to M H DM + j(GM – M H G) < αD, D > I The μ upper bound ν ? (M) and the optimal D, G scalings are computed by the function mubnd. Its syntax is [mu,D,G] = mubnd(M,delta) μ M() := 1 min σ max ?(): ?∈ ? and IM– ? is singular{} ---------------------------------------------------------------------------------------------------------------------------------------- D i D i H = 0> G i G i H = ν ? max 0 α*,()= μ Analysis 3-19 where delta specifies the perturbation structure ? (use ublock and udiag to define delta, see “Norm-Bounded Uncertainty” on page 2-27 and “Sector-Bounded Uncertainty” for details). When each block ? i is bounded by 1, the output mu is equal to ν ? (M). If different bounds β i are used for each ? i , the interpretation of mu is as follows: The interconnection of Figure 3-1 is well posed for all ? ∈ ? satisfying For instance, 1/mu = 0.9 means that well-posedness is guaranteed for perturbation sizes that do not exceed 90% of the prescribed bounds. Robust Stability Analysis The structured singular value μ is useful to assess the robust stability of time-invariant linear-fractional interconnections (see Figure 3-2). Here P(s) is a given LTI system and ?(s) = diag(? 1 (s), . . . , ? n (s)) is a norm- or sector- bounded linear time-invariant uncertainty with some prescribed structure. Figure 3-2: Robust stability analysis This interconnection is stable for all structured ?(s) satisfying ||?|| ∞ < 1 if and only if [3, 12, 10, 16] The reciprocal K m = 1/μ m represents the robust stability margin, that is, the largest amount of uncertainty ? that can be tolerated without losing stability. σ max ? i () β i mu ------< ?(s) P(s) w q u y μ m := sup μ ? Pjω()()1.< ω 3 Robustness Analysis 3-20 As mentioned earlier, we can only compute a conservative estimate of K m based on the upper bound ν ? of μ ? . We refer to as the guaranteed stability margin. A grid-based estimation of is performed by mustab: [margin,peakf] = mustab(P,delta,freqs) On input, delta describes the uncertainty ? (see ublock and udiag for details) and freqs is an optional vector of frequencies where to evaluate ν ? (P(jω)). On output, margin is the computed value of and peakf is the frequency near which ν ? (P(jω) peaks, i.e., where the margin is the smallest. The uncertainty quantification may involve different bounds for each block ? i . In this case, margin represents the relative stability margin as a percentage of the specified bounds. More precisely, margin = θ means that stability is guaran- teed as long as each block ? i satisfies For norm-bounded blocks σ max (? i (jω)) < θ β i (ω) where β i (ω) is the (possibly frequency-dependent) norm bound specified with ublock For sector-bounded blocks where a < b are the sector bounds specified with ublock. In the special case b = +∞, this condition reads for θ e 1 and for θ > 1 For instance, margin = 0.7 for a single-block uncertainty ?(s) bounded by 3 means that stability is guaranteed for ||?|| ∞ < 0.7-by-3 = 2.1. The syntax K ? m K ? m K ? m K ? m ? i is in the sector ab+ 2 ------------- θ ba– 2 ------------– , ab+ 2 ------------ θ ba– 2 ------------+ ?? ?? ?? ? i is in the sector a 10– 10+ -------------+ ,a 10+ 10– -------------+ ?? ?? ?? ? i is outside the sector a 10– 10+ -------------+ ,a 10+ 10– -------------+ ?? ?? ?? outside μ Analysis 3-21 [margin,peakf,fs,ds,gs] = mustab(P,delta) also returns the D, G scaling matrices at the tested frequencies fs. To retrieve the values of D and G at the frequency fs(i), type Di = getdg(ds,i), Gi = getdg(gs,i) Note that Di and Gi are not necessarily the optimal scalings at fs(i). Indeed, the LMI optimization is often stopped before completion to speed up compu- tations. Caution: is computed by frequency sweep, i.e., by evaluating ν ? (P(jω)) over a finite grid of frequencies. The coarser the gridding, the higher the risk of underestimating the peak value of ν ? (P(jω)). Worse, this function may be discontinuous at its peaks when the uncertainty contains real blocks. In such cases, any “blind” frequency sweep is likely to miss the peak value and yield an overoptimistic stability margin. The function mustab has built-in tests to detect such discontinuities and minimize the risks of missing the critical peaks. Robust Performance In robust control, it is customary to formulate the design specifications as abstract disturbance rejection objectives. The performance of a control system is then measured in terms of the closed-loop RMS gain from disturbances to outputs (see “H Control” on page 5-3 for details). The presence of uncertainty typically deteriorates performance. For an LTI system with linear-fractional uncer- tainty (Figure 3-3), the robust performance γ rob is defined as the worst-case RMS gain from u to y in the face of the uncertainty ?(s). K ? m 3 Robustness Analysis 3-22 This robust performance is generally worse (larger) than the nominal performance, i.e., the RMS gain from u to y for (s) ≡ 0. Figure 3-3: Robust performance problem Assessing whether the performance γ can be robustly sustained, that is, whether γ rob < γ, is equivalent to a robust stability problem. Indeed, ||y|| L 2 < γ||u|| L 2 holds for all ?(s) in the prescribed uncertainty set if and only if the interconnection of Figure 3-4 is stable for all ?(s) and for ? perf (s) satisfying ||? perf (s)|| ∞ < γ –1 Given a norm- or sector-bounded linear time-invariant uncertainty ?(s), the function muperf assesses the robust performance γ rob via this equivalent robust stability problem. Figure 3-4: Equivalent robust stability problem Two problems can be solved with muperf: Compute the robust performance γ rob for the specified uncertainty ?(s). This is done by ?(s) P(s) uy P(s) ? perf s()0 0 ? s() ?? ?? ?? μ Analysis 3-23 grob = muperf(P,delta) The value grob is finite if and only if the interconnection of Figure 3-3 is robustly stable. Assess the robustness of a given performance γ = g > 0 in the face of the uncertainty ?(s). The command margin = muperf(P,delta,g) computes a fraction margin of the specified uncertainty bounds for which the RMS gain from u to y is guaranteed not to exceed γ. Note that g should be larger than the nominal RMS gain for ? = 0. The performance g is robust if and only if margin ? 1. The function muperf uses the following characterization of γ rob . Assuming ||?|| ∞ < 1 and denoting by D and G the related scaling sets, where γ rob (ω) is the global minimum of the LMI problem Minimize γ over D ∈ D, G ∈ G such that D > 0 and γ rob = max γ rob ω() ω Pjω() H D 0 0 I?? ?? ?? Pjω()j G 0 00?? ?? ?? Pjω()Pjω() H G 0 00?? ?? ?? – ?? ?? ??D 0 0 γ 2 I ?? ?? ?? <+ 3 Robustness Analysis 3-24 The Popov Criterion The Popov criterion gives a sufficient condition for the robust stability of the interconnection of Figure 3-5 where G(s) is a given LTI system and φ = diag (φ 1 , . . . , φ n ) is a sector-bounded BIBO-stable uncertainty satisfying φ(0) = 0. The operator φ can be nonlinear and/or time-varying which makes the Popov criterion more general than the μ stability test. For purely linear time- invariant uncertainty, however, the μ test is generally less conservative. Figure 3-5: Popov criterion A state-space representation of this interconnection is or equivalently in block-partitioned form: Suppose that φ j ( . ) satisfies the sector bound (3-13) (this includes norm bounds as the special case β j = –α j ). To establish robust stability, the Popov criterion seeks a Lyapunov function of the form G(s) w q u y φ · () ? ? ? ? ? x · Ax Bw+= qCxDw+= w φ q()= ? ? ? ? ? x · Ax Σj B j w j += q j C j xD j w ∈ R r j ()+= w φ q j ()= T? 0 w j α j q j –() T 0 T ∫ w j β j q j –()dt 0≤,> The Popov Criterion 3-25 (3-14) where σ j > 0 and ν j are scalars and ν j = 0 unless D j = 0 and φ j ( . ) is a scalar memoryless nonlinearity (i.e., w j (t) depends only on q j (t); in such case, the sector constraint (3-13) reduces to . Letting Kα := diag(α j I r j ), K β := diag(β j I r j ) S := diag(σ j I r j ), N := diag(ν j I r j ) and assuming nominal stability, the conditions V(x, t) > 0 and dV/dt < 0 are equivalent to the following LMI feasibility problem (with the notation H(M) := M + M T ): Find P = P T and structured matrices S, N such that S > 0 and (3-15) When φ( . ) is nonlinear time-varying, (3-15) reduces to the scaled small-gain criterion. The Popov stability test is performed by the command [t,P,S,N] = popov(G,phi) where G is the SYSTEM matrix representation of G(s) and phi is the uncertainty description specified with ublock and udiag. The function popov returns the output t of the LMI solver feasp. Hence the interconnection is robust stable if t < 0. In this case, P, S, and N are solutions of the LMI problem (3-15). Real Parameter Uncertainty A sharper version of the Popov criterion can be used to analyze systems with uncertain real parameters. Assume that D j = 0 and w j = φ j (q j ) := δ j q j where δ j is an uncertain real parameter ranging in [α j , β j ]. Then a more general form for the corresponding term in the Lyapunov function (3-14) is Vxt,() 1 2 ---x T Px ν j j ∑ φ j z()zd 0 q j ∫ σ j j ∑ w j α j q j –() T w j β j q j –()τd 0 t ∫ –+= α j q j 2 q j φ j q j ()β j q j 2 )≤≤ H I 0?? ?? ?? PAB,() 0 I?? ?? ?? NCACB,() C T K α D T K α 1–?? ?? ?? –+ ? ? ? ? ? S C T K β D T K β 1–?? ?? ?? T ? ? ? ? ? 0< 3 Robustness Analysis 3-26 (3-16) where S j and N j are r j × r j matrices subject to , and N j = 0 when the parameter δ j is time varying. As a result, the variables S and N in the Popov condition (3-15) assume a block-diagonal structure where σ j I rj and ν j I rj are replaced by the full blocks S j and N j , respectively. When all φ j ( . ) are real time-invariant parameters, the Lyapunov function (3-14) becomes The first term can be interpreted as a parameter-dependent quadratic function of the state. This is analogous to the notion of parameter-dependent Lyapunov function discussed in “Parameter-Dependent Lyapunov Functions” on page 3-12, even though the Popov condition (3-15) is not equivalent to the robust stability conditions discussed in that section.. Finally, it is possible to obtain an even sharper test by applying the Popov criterion to the equivalent interconnection (3-17) The Lyapunov function now becomes with S j , N j subject to . The resulting test is discussed in [5, 6] and often performs as well as the real stability test while eliminating the hazards of a frequency sweep (see “Caution” on page 3-21). To use this refined test, invoke popov as follows. φ z() T N j zd 0 q j ∫ w j α j q j –() T S j w j β j q j –()τd 0 t ∫ =– δ j x T C T N j Cx δ j α j –()δ j β j –()q j T 0 t ∫ S j S j T +()q j τd– S j S j T ON, N j T =>+ Vxt,() 1 2 ---x T P δ j j ∑ C T N j C+ ?? ?? ?? x δ j α j –()δ j β j –() j ∑ – q j T 0 t ∫ S j S j T +()q j τd= ? ? ? ? ? x · = Ax Σj B j C j w ? j +() q ? j = x w ? j δ j q ? j = Vxt,() 1 2 ---x T P δ j j ∑ N j + ?? ?? ?? x δ j α j –()δ j β j –() j ∑ – q j T 0 t ∫ S j S j T +()q j τd= S j S j T O and N j N j T =>+ The Popov Criterion 3-27 [t,P,S,N] = popov(G,phi,1) The third input 1 signals popov to perform the loop transformation (3-17) before solving the feasibility problem (3-15). 3 Robustness Analysis 3-28 Example The use of these various robustness tests is illustrated on the benchmark problem proposed in [14]. The physical system is sketched in Figure 3-6. Figure 3-6: Benchmark problem The goal is to design an output-feedback law u = K(s)x 2 that adequately rejects the disturbances w 1 , w 2 and guarantees stability for values of the stiffness parameter k ranging in [0.5, 2.0]. For m 1 = m 2 = 1, a state-space description of this system is (3-18) A solution of this problem proposed in [15] is the fourth-order controller u = C K (sI – A K ) –1 B K x 2 where u w 1 w 2 k x 1 x 2 m 1 m 2 x · 1 x · 2 x ·· 1 x ·· 2 0010 0001 k– k 00 kk– 00 x 1 x 2 x · 1 x · 2 0 0 1 0 uw 1 +() 0 0 0 1 w 2 ++= A K 00.7195– 10 02.9732– 01 2.5133– 4.8548 1.7287– 0.9616– 1.0063 5.4097– 0.0081– 0.0304 B K 0.720 2.973 3.37– 4.419 =,= C K 1.506– 0.494 1.738– 0.932– = Example 3-29 The concern here is to assess, for this particular controller, the closed-loop stability margin with respect to the uncertain real parameter k. Since the plant equations (3-18) depend affinely on k, we can define the uncertain physical system as an affine parameter-dependent system G with psys. A0=[0 0 1 0;0 0 0 1;0 0 0 0;0 0 0 0]; B0=[0;0;1;0]; C0=[0 1 0 0]; D0=0; S0=ltisys(A0,B0,C0,D0); % system for k=0 A1=[0 0 0 0;0 0 0 0; 1 1 0 0;1 1 0 0]; B1=zeros(4,1); C1=zeros(1,4); D1=0; S1=ltisys(A1,B1,C1,D1,0); % k component pv=pvec('box',[0.5 2]) % range of values of k G=psys(pv,[S0,S1]) After entering the controller data as a SYSTEM matrix K, close the loop with slft. Cl = slft(G,K) The result Cl is a parameter-dependent description of the closed-loop system. To estimate the robust stability margin with respect to k, we can now apply to Cl the various tests described in this chapter: Quadratic stability: to determine the portion of the interval [0.5, 2] where quadratic stability holds, type marg = quadstab(Cl,[1 0 0]) marg = 4.1919e 01 The value marg = 0.419 means 41% of [0.5, 2] (with respect to the center 1.25). That is, quadratic stability in the interval [0.943, 1.557]. Since quadratic stability assumes arbitrarily fast time variations of the parameter k, we can expect this answer to be conservative when k is time invariant. Parameter-dependent Lyapunov functions: when k does not vary in time, a less conservative estimate is provided by pdlstab. To test stability for k ∈ [0.5, 2], type 3 Robustness Analysis 3-30 t = pdlstab(Cl) t = 2.1721e 01 Since t < 0, the closed loop is robustly stable for this range of values of k. Assume now that k slowly varies with a rate of variation bounded by 0.1. To test if the closed loop remains stable in the face of such slow variations, redefine the parameter vector and update the description of the closed-loop system by pv1 = pvec('box',[0.5 2],[ 0.1 0.1]) G1 = psys(pv1,[S0,S1]) Cl1 = slft(G1,K) Then call pdlstab as earlier. t = pdlstab(Cl1) t = 2.0089e 02 Since t is again negative, this level of time variations does not destroy robust stability. μ analysis: to perform μ analysis, first convert the affine parameter-dependent model Cl to an equivalent linear-fractional uncertainty model. [P0,deltak] = aff2lft(Cl) uinfo(deltak) Here P0 is the closed loop system for the nominal value k 0 = 1.25 of k and the uncertainty on k is represented as a real scalar block deltak. Note that k must be assumed time invariant in the μ framework. To get the relative parameter margin, type block dims type real/cplx full/scal bounds 1 1x1 LTI r s norm <= 0.75 Example 3-31 [pmarg,peakf] = mustab(P0,deltak) pmarg = 1.0670e+00 peakf = 6.3959e 01 The value pmarg = 1.064 means stability as long as the deviation |δk| from k 0 = 1.25 does not exceed 0.75 × 1.067 ≈ 0.80. That is, as long as k remains in the interval [0.45, 2.05]. This estimate is sharp since the closed loop becomes unstable for δk = –0.81, that is, for k = k 0 – 0.81 = 0.44: spol(slft(P0, 0.81)) ans = 1.0160e+00 + 1.9208e+00i 1.0160e+00 1.9208e+00i 1.0512e+00 + 9.7820e 01i 1.0512e+00 9.7820e 01i 6.2324e 03 + 6.3884e 01i 6.2324e 03 6.3884e 01i 2.7484e 01 + 1.9756e 01i 2.7484e 01 1.9756e 01i Popov criterion: finally, we can apply the Popov criterion to the linear-fractional model returned by aff2lft [t,P,D,N] = popov(P0,deltak) t = 1.3767e 02 This test is also successful since t < 0. Note that the Popov criterion also proves stability for k = k 0 + δk( . ) where δk( . ) is any memoryless nonlinearity with gain less than 0.75. 3 Robustness Analysis 3-32 References [1] Barmish, B.R., “Stabilization of Uncertain Systems via Linear Control,” IEEE Trans. Aut. Contr., AC–28 (1983), pp. 848–850. [2] Boyd, S., and Q. Yang, “Structured and Simultaneous Lyapunov Functions for System Stability Problems,” Int. J. Contr., 49 (1989), pp. 2215–2240. [3] Doyle, J.C., “Analysis of Feedback Systems with Structured Uncertainties,” IEE Proc., vol. 129, pt. D (1982), pp. 242–250. [4] Fan, M.K.H., A.L. Tits, and J.C. Doyle,“Robustness in the Presence of Mixed Parametric Uncertainty and Unmodeled Dynamics,” IEEE Trans. Aut. Contr., 36 (1991), pp. 25–38. [5] Feron, E, P. Apkarian, and P. Gahinet, “S-Procedure for the Analysis of Control Systems with Parametric Uncertainties via Parameter-Dependent Lyapunov Functions,” Third SIAM Conf. on Contr. and its Applic., St. Louis, Missouri, 1995. [6] Gahinet, P., P. Apkarian, and M. Chilali, “Affine Parameter-Dependent Lyapunov Functions for Real Parametric Uncertainty,” Proc. Conf. Dec. Contr., 1994, pp. 2026–2031. [7] Haddad, W.M. and D.S. Berstein,“Parameter-Dependent Lyapunov Functions, Constant Real Parameter Uncertainty, and the Popov Criterion in Robust Analysis and Synthesis: Part 1 and 2,” Proc. Conf. Dec. Contr., 1991, pp. 2274–2279 and 2632–2633. [8] Horisberger, H.P., and P.R. Belanger, “Regulators for Linear Time-Varying Plants with Uncertain Parameters,” IEEE Trans. Aut. Contr., AC–21 (1976), pp. 705–708. [9] How, J.P., and S.R. Hall, “Connection between the Popov Stability Criterion and Bounds for Real Parameter Uncertainty,” Proc. Amer. Contr. Conf., 1993, pp. 1084–1089. [10] Packard, A., and J.C. Doyle, “The Complex Structured Singular Value,” Automatica, 29 (1994), pp. 71–109. [11] Popov, V.M., “Absolute Stability of Nonlinear Systems of Automatic Control,” Automation and Remote Control, 22 (1962), pp. 857–875. [12] Safonov, M.G., “L1 Optimal Sensitivity vs. Stability Margin,” Proc. Conf. Dec. Contr., 1983. References 3-33 [13] Stein, G. and J.C. Doyle, “Beyond Singular Values and Loop Shapes,” J. Guidance, 14 (1991), pp. 5–16. [14] Wie, B., and D.S. Berstein, “Benchmark Problem for Robust Control Design,” J. Guidance and Contr., 15 (1992), pp. 1057–1059. [15] Wie, B., Q. Liu, and K.-W. Byun, “Robust H ∞ Control Synthesis Method and Its Application to Benchmark Problems,” J. Guidance and Contr., 15 (1992), pp 1140–1148. [16] Young, P. M., M. P. Newlin, and J. C. Doyle, “Let's Get Real,” in Robust Control Theory, Springer Verlag, 1994, pp. 143–174. [17] Zames, G., “On the Input-Output Stability of Time-Varying Nonlinear Feedback Systems, Part I and II,” IEEE Trans. Aut. Contr., AC–11 (1966), pp. 228–238 and 465–476. 3 Robustness Analysis 3-34 4 State-Feedback Synthesis Multi-Objective State-Feedback . . . . . . . . . . . 4-3 Pole Placement in LMI Regions . . . . . . . . . . . 4-5 LMI Formulation . . . . . . . . . . . . . . . . . . 4-7 Extension to the Multi-Model Case . . . . . . . . . . . 4-9 The Function msfsyn . . . . . . . . . . . . . . . .4-11 Design Example . . . . . . . . . . . . . . . . . .4-13 References . . . . . . . . . . . . . . . . . . . . .4-18 4 State-Feedback Synthesis 4-2 In many control problems, the design specifications are a mix of performance and robustness objectives expressed both in the time and frequency domains. These various objectives are rarely encompassed by a single synthesis criterion. While some tracking and robustness are best captured by an H ∞ criterion, noise insensitivity is more naturally expressed in LQG terms, and transient behaviors are more easily tuned in terms of l 1 norm or closed-loop damping. The LMI framework is particularly well suited to multi-objective state-feedback synthesis. As an illustration, the LMI Control Toolbox offers tools for state-feedback design with a combination of the following objectives: H ∞ performance (for tracking, disturbance rejection, or robustness aspects) H 2 performance (for LQG aspects) Robust pole placement specifications (to ensure fast and well-damped transient responses, reasonable feedback gain, etc.) These tools apply to multi-model problems, i.e., when the objectives are to be robustly achieved over a polytopic set of plant models. Multi-Objective State-Feedback 4-3 Multi-Objective State-Feedback The function msfsyn performs multi-model H 2 /H ∞ state-feedback synthesis with pole placement constraints. For simplicity, we describe the underlying problem in the case of a single LTI model. The control structure is depicted by Figure 4-1. The plant P(s) is a given LTI system and we assume full measurement of its state vector x. Figure 4-1: State-feedback control Denoting by T ∞ (s) and T 2 (s) the closed-loop transfer functions from w to z ∞ and z 2 , respectively, our goal is to design a state-feedback law u = Kx that Maintains the RMS gain (H ∞ norm) of T ∞ below some prescribed value γ 0 > 0 Maintains the H 2 norm of T 2 (LQG cost) below some prescribed value ν 0 > 0 Minimizes an H 2 /H ∞ trade-off criterion of the form Places the closed-loop poles in a prescribed region D of the open left-half plane. This abstract formulation encompasses many practical situations. For instance, consider a regulation problem with disturbance d and white measurement noise n, and let e denote the regulation error. Setting z 2 P(s) K w u x z ∞ α T ∞ ∞ 2 β T 2 2 2 + w d n?? ?? ?? , z ∞ e, z 2 x u?? ?? ?? ,=== 4 State-Feedback Synthesis 4-4 the mixed H 2 /H ∞ criterion takes into account both the disturbance rejection aspects (RMS gain from d to e) and the LQG aspects (H 2 norm from n to z 2 ). In addition, the closed-loop poles can be forced into some sector of the stable half-plane to obtain well-damped transient responses. Pole Placement in LMI Regions 4-5 Pole Placement in LMI Regions The concept of LMI region [3] is useful to formulate pole placement objectives in LMI terms. LMI regions are convex subsets D of the complex plane characterized by where M and L = L T are fixed real matrices. The matrix-valued function is called the characteristic function of the region D. The class of LMI regions is fairly general since its closure is the set of convex regions symmetric with respect to the real axis. More practically, LMI regions include relevant regions such as sectors, disks, conics, strips, etc., as well as any intersection of the above. Another strength of LMI regions is the availability of a “Lyapunov theorem” for such regions. Specifically, if denote the entries of the matrices L and M, a matrix A has all its eigenvalues in D if and only if there exists a positive definite matrix P such that [3] with the notation Note that this condition is an LMI in P and that the classical Lyapunov theorem corresponds to the special case Next we list a few examples of useful LMI regions along with their characteristic function f D : D z C∈ : LMzM T z 0<++{}= f D z(): LMzM T z++= λ ij {} 1 i≤ ,jm≤ and μ ij {} 1 i≤ ,jm≤ λ ij P μ ij AP μ ji PA T ++[] 1 i≤ ,jm≤ 0< S ij [] 1 i≤ ,jm≤ := S 11 … S i1 S m1 … S mm ?? ?? ?? … … . . . f D z() zz+= 4 State-Feedback Synthesis 4-6 Disk with center at (–q, 0) and radius r: Conic sector centered at the origin and with inner angle θ: The damping ratio of poles lying in this sector is at least cos . Vertical strip h 1 < x < h 2 : r –q f D z() r– zq+ zq+ r–?? ?? ?? = θ f D z() θ 2 --- zz+()sin cos– θ 2 --- zz–() cos θ 2 --- zz–() θ 2 --- zz–()sin ?? ?? ?? = θ 2 --- h 1 h 2 f D z() 2h 1 zz+()– 0 0 zz+()2h 2 – ?? ?? ?? = LMI Formulation 4-7 LMI Formulation Given a state-space realization of the plant P, the closed-loop system is given in state-space form by Taken separately, our three design objectives have the following LMI formulation: H ∞ performance: the closed-loop RMS gain from w to z ∞ does not exceed γ if and only if there exists a symmetric matrix X ∞ such that [5] H 2 performance: the closed-loop H 2 norm of T 2 does not exceed ν if there exist two symmetric matrices X 2 and Q such that ? ? ? ? ? x · = Ax B 1 wB 2 u++ z ∞ = C 1 xD 11 wD 12 u++ z 2 = C 2 xD 22 u+ ? ? ? ? ? x · = AB 2 K+()xB 1 w+ z ∞ = C 1 D 12 K+ xD 11 w+ z 2 = C 2 D 22 K+()x AB 2 K+()X ∞ X ∞ AB 2 K+() T + B 1 X ∞ C 1 D 12 K+() T B 1 T I– D 11 T C 1 D 12 K+()X ∞ D 11 γ 2 I– ?? ?? ?? 0< X ∞ 0> 4 State-Feedback Synthesis 4-8 (see “LQG performance” on page 1-7 for details) Pole placement: the closed-loop poles lie in the LMI region where if and only if there exists a symmetric matrix X pol satisfying These three sets of conditions add up to a nonconvex optimization problem with variables Q, K, X ∞ , X 2 and X pol . For tractability in the LMI framework, we seek a single Lyapunov matrix (4-1) that enforces all three objectives. With the change of variable Y := KX, this leads to the following suboptimal LMI formulation of our multi-objective state-feedback synthesis problem [4, 3, 2]: AB 2 K+()X 2 X 2 AB 2 K+() T + B 1 B 1 T I– ?? ?? ?? 0< QC 2 D 22 K+()X 2 X 2 C 2 D 22 K+() T X 2 ?? 0> Trace Q() ν 2 < D z C∈ : LMzM T z 0<++{}= LL T λ ij {}== 1 i≤ ,jm≤ M μ ij {}= 1 i≤ ,jm≤ λ ij X pol μ ij AB 2 K+()X pol μ ij X pol μ ji X pol AB 2 K+() T +++[] 1 i≤ ,jm≤ 0< X pol 0> X := X ∞ X 2 X pol == LMI Formulation 4-9 Minimize α γ 2 + β Trace(Q) over Y, X, Q, and γ 2 satisfying (4-2) (4-3) (4-4) (4-5) (4-6) Denoting the optimal solution by (X*, Y*, Q*, γ* ), the corresponding state-feedback gain is given by K* = Y*(X* ) –1 and this gain guarantees the worst-case performances: Note that K* does not yield the globally optimal trade-off in general due to the conservatism of Assumption (4-1). Extension to the Multi-Model Case The LMI approach outlined above extends to uncertain linear time-varying plants AX XA T B+ 2 Y+ Y T B 2 T + B 1 XC 1 T Y T D 12 T + B 1 T I– D 11 T C 1 XD 12 Y+ D 11 γ 2 I– ?? ?? ?? 0< QC 2 XD 22 Y+ XC 2 T Y T D 22 T + X ?? ?? ?? 0> λ ij μ ij AX B 2 Y+()X pol μ ji XA T Y T B 2 T +()++[] 1 i≤ ,jm≤ 0< Trace Q() ν 0 2 < γ 2 γ 0 2 < T ∞ ∞ γ* T 2 2 Trace Q*()≤,≤ 4 State-Feedback Synthesis 4-10 with state-space matrices varying in a polytope: Such polytopic models are useful to represent plants with uncertain and/or time-varying parameters (see “Polytopic Models” on page 2-14 and the design example below). Seeking a single quadratic Lyapunov function that enforces the design objectives for all plants in the polytope leads to the following multi-model counterpart of the LMI conditions (4-2)–(4-6): Minimize α γ 2 + β Trace(Q) over Y, X, Q, and γ 2 subject to P: ? ? ? ? ?x · = At()xB 1 t()wB 2 t()u++ z ∞ = C 1 t()xD 11 t()wD 12 t()u z 2 = C 2 t()xD 22 t()u+ At() B 1 t() B 2 t() C 1 t()D 11 t()D 12 t() C 2 t() 0 D 22 t() ?? ?? ?? Co A k B 1k B 2k C 1k D 11k D 12k C 2k 0 D 22k?? ?? ?? : k 1 … · ,,= K ?? ?? ?? ?? ∈ A k XXA k T B+ 2k Y+ Y T B 2k T + B 1k XC 1k T Y T D 12k T + B 1k T I– D 11k T C 1k XD 12k Y+ D 11k γ 2 I– ?? ?? ?? 0< QC 2k XD 22k Y+ XC 2k T Y T D 22 T k+ X ?? ?? ?? 0> γ 2 γ 0 2 < Trace Q() ν 0 2 < λ ij μ ij A k XB 2k Y+()μ ji XA k T Y T B 2k T +()++[] 1 i≤ ,jm≤ 0< The Function msfsyn 4-11 The Function msfsyn The function msfsyn implements the LMI approach to multi-model H 2 /H ∞ synthesis outlined above. The pole placement objectives are expressed in terms of LMI regions characterized by the two matrices L and M. LMI regions are specified interactively with the function lmireg. Denoting the closed-loop transfer functions from w to z ∞ and z 2 by T ∞ and T 2 , msfsyn computes a suboptimal solution to the mixed problem: Minimize subject to ||T ∞ || ∞ < γ 0 ||T 2 || 2 < ν0 The closed-loop poles lie in D. The syntax is [gopt,h2opt,K,Pcl] = msfsyn(P,r,obj,region) where P is the plant SYSTEM matrix in the single-model case, or the polytopic/parameter-dependent description of the plant in the multi-model case (see psys for details). The dimensions of the D 22 matrix are specified by r obj = [γ 0 , ν 0 , α, β] is a four-entry vector specifying the H 2 /H ∞ criterion region specifies the LMI region to be used for pole placement, the default being the open left-half plane. Use lmireg to generate the matrix region, or set it to [L, M] if the characteristic matrices L and M are readily available. On output, gopt and h2opt are the guaranteed H ∞ and H 2 performances, K is the state-feedback gain, and Pcl is the closed-loop system in SYSTEM matrix or polytopic model format. D z C∈ : LMzMz0<++{}= α T ∞ 0 2 β T 2 2 2 + 4 State-Feedback Synthesis 4-12 Several mixed or unmixed designs can be performed with msfsyn. The various possibilities are summarized in the table below. obj Corresponding Design [0 0 0 0] pole placement only [0 0 1 0] H ∞ -optimal design [0 0 0 1] H 2 -optimal design [g 0 0 1] minimize ||T 2 || 2 subject to ||T ∞ || ∞ < g [0 h 1 0] minimize ||T ∞ || ∞ subject to ||T 2 || 2 < h [0 0 a b] minimize a T ∞ ∞ 2 b T 2 2 2 + Design Example 4-13 Design Example This example is adapted from [1] and covered by the demo sateldem. The system is a satellite consisting of two rigid bodies (main body and instrumen- tation module) joined by a flexible link (the “boom”). The boom is modeled as a spring with torque constant k and viscous damping f and finite-element analysis gives the following uncertainty ranges for k and f : 0.09 e k 0.4 0.0038 e f 0.04: The dynamical equations are where θ 1 and θ 2 are the yaw angles for the main body and the sensor module, T is the control torque, and w is a torque disturbance on the main body. Figure 4-1: Satellite The control purpose is to minimize the influence of the disturbance w on the angular position θ 2 . This goal is expressed through the following objectives: J 1 θ ·· 1 f θ · 1 θ · 2 –()k θ 1 θ 2 –()Tw+=++ J 2 θ ·· 2 f θ · 2 θ · 1 –()k θ 2 θ 1 –()0=++?? ?? ?? θ 1 θ 2 main body sensor module 4 State-Feedback Synthesis 4-14 Obtain a good trade-off between the RMS gain from w to θ 2 and the H 2 norm of the transfer function from w to (LQG cost of control) Place the closed-loop poles in the region shown in Figure 4-3 to guarantee some minimum decay rate and closed-loop damping Figure 4-2: Pole placement region Achieve these objectives for all possible values of the varying parameters k and f. Since these parameters enter the plant state matrix in an affine manner, we can model the parameter uncertainty by a polytopic system with four vertices corresponding to the four combinations of extremal parameter values (see “From Affine to Polytopic Models” on page 2-20). θ 1 θ 2 T ?? ?? ?? ?? ?? –0.1 Design Example 4-15 To solve this design problem with the LMI Control Toolbox, first specify the plant as a parameter-dependent system with affine dependence on k and f. A state-space description is readily derived from the dynamical equations as: This parameter-dependent model is entered by the commands a0 = [zeros(2) eye(2); zeros(2,4)] ak = [zeros(2,4) ; [-1 1;1 -1] zeros(2)] af = [zeros(2,4) ; zeros(2) [-1 1;1 -1]] e0 = diag([1 1 J1 J2]) b = [0 0;0 0;1 1;0 0] % b = [b1 b2] c = [0 1 0 0;1 0 0 0;0 1 0 0;0 0 0 0] % c = [c1;c2] d = [0 0;0 0;0 0;0 1] % range of parameter values pv = pvec('box',[0.09 0.4 ; 0.0038 0.04]) % parameter-dependent plant P = psys(pv,[ ltisys(a0,b,c,d,e0) , ... ltisys(ak,0*b,0*c,0*d,0) , ... ltisys(af,0*b,0*c,0*d,0) ]) Next, specify the LMI region for pole placement as the intersection of the half-plane x < –0.1 and of the sector centered at the origin and with inner angle 3π/4. This is done interactively with the function lmireg: 10 0 0 01 0 0 00J 1 0 00 0 J 2 θ · 1 θ · 2 θ ·· 1 θ ·· 2 0010 0001 k– kf– f kk– ff– θ 1 θ 2 θ · 1 θ · 2 0 0 0 1 w( T )++ += z ∞ θ 2 , z 2 1000 0100 0000 θ 1 θ 2 θ · 1 θ · 2 0 0 1 T+== 4 State-Feedback Synthesis 4-16 region = lmireg To assess the trade-off between the H ∞ and H 2 performances, first compute the optimal quadratic H ∞ performance subject to the pole placement constraint by gopt = msfsyn(P,[1 1],[0 0 1 0],region) This yields gopt ≈ 0. For a prescribed H ∞ performance g > 0, the best H 2 performance h2opt is computed by [gopt,h2opt,K,Pcl] = msfsyn(P,[1 1],[g 0 0 1],region) Here obj = [g 0 0 1] asks to optimize the H 2 performance subject to ||T ∞ || ∞ < g and the pole placement constraint. Repeating this operation for the values g ∈ {0.01, 0.1, 0.2, 0.5} yields the Pareto-like trade-off curve shown in Figure 4-4. By inspection of this curve, the state-feedback gain K obtained for g = 0.1 yields the best compromise between the H ∞ and H 2 objectives. For this choice of K, Figure 4-5 superimposes the impulse responses from w to θ 2 for the four combinations of extremal values of k and f. Finally, the closed-loop poles for these four extremal combinations are displayed in Figure 4-6. Note that they are robustly placed in the prescribed LMI region. Figure 4-3: Trade-off between the H ∞ and H 2 performances 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 2.5 3 3.5 4 4.5 5 5.5 6 6.5 H?infinity performance H2 performance Design Example 4-17 Figure 4-4: Impulse responses for the extremal values of k and f Figure 4-5: Corresponding closed-loop poles 0 1 2 3 4 5 6 7 8 9 10 ?2 0 2 4 6 8 10 12 x 10 ?3 Time (secs) Amplitude Impulse responses w ?> x2 for the robust design ?14 ?12 ?10 ?8 ?6 ?4 ?2 0 ?10 ?8 ?6 ?4 ?2 0 2 4 6 8 10 Closed?loop poles for the four extremal sets of parameter values 4 State-Feedback Synthesis 4-18 References [1] Biernacki, R.M., H. Hwang, and S.P. Battacharyya, “Robust Stability with Structured Real Parameter Perturbations,” IEEE Trans. Aut. Contr., AC–32 (1987), pp. 495–506. [2] Boyd, S., L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, SIAM books, 1994. [3] Chilali, M., and P. Gahinet, “H ∞ Design with Pole Placement Constraints: an LMI Approach,” to appear in IEEE Trans. Aut. Contr. Also in Proc. Conf. Dec. Contr., 1994, pp. 553–558. [4] Khargonekar, P.P., and M.A. Rotea,“Mixed H 2 /H ∞ Control: A Convex Optimization Approach,” IEEE Trans. Aut. Contr., 39 (1991), pp. 824-837. [5] Scherer, C., “H ∞ Optimization without Assumptions on Finite or Infinite Zeros,” SIAM J. Contr. Opt., 30 (1992), pp. 143–166. 5 Synthesis of H ∞ Controllers H∞ Control . . . . . . . . . . . . . . . . . . . . . 5-3 Riccati- and LMI-Based Approaches . . . . . . . . . . . 5-7 H∞ Synthesis . . . . . . . . . . . . . . . . . . . . 5-10 Validation of the Closed-Loop System . . . . . . . . . . 5-13 Multi-Objective H∞ Synthesis . . . . . . . . . . . . 5-15 LMI Formulation . . . . . . . . . . . . . . . . . . . 5-16 The Function hinfmix . . . . . . . . . . . . . . . . . 5-20 Loop-Shaping Design with hinfmix . . . . . . . . . . . 5-20 References . . . . . . . . . . . . . . . . . . . . . 5-22 5-21 5 Synthesis of H∞ Controllers 5-2 Disturbance rejection, input/output decoupling, and general robust stability or performance objectives can all be formulated as gain attenuation problems in the face of dynamical or parametric uncertainty. H ∞ control is the cornerstone of design techniques for this class of problems. The LMI Control Toolbox addresses three variants of output-feedback H ∞ synthesis: Standard H ∞ control for continuous-time systems. Both the Riccati-based and LMI-based solutions of this problem are implemented Multi-objective H ∞ design (mixed H 2 /H ∞ synthesis with closed-loop pole placement in LMI regions) Discrete-time H ∞ control via either Riccati- or LMI-based approaches Additional facilities for loop-shaping design are described in Chapter 6, “Loop Shaping.” H? Control 5-3 H ∞ Control The H ∞ norm of a stable transfer function G(s) is its largest input/output RMS gain, i.e., where L 2 is the space of signals with finite energy and y(t) is the output of the system G for a given input u(t). This norm also corresponds to the peak gain of the frequency response G(jω), that is, In its abstract “standard” formulation, the H ∞ control problem is one of disturbance rejection. Specifically, it consists of minimizing the closed-loop RMS gain from w to z in the control loop of Figure 5-1. This can be interpreted as minimizing the effect of the worst-case disturbance w on the output z. Figure 5-1: H ∞ control Partitioning the plant P(s) as G ∞ sup y L 2 u L 2 -------------= uL 2 ∈ u 0≠ G ∞ sup ω σ max Gjω()()= K(s) P(s) u y w z Zs() Ys()?? ?? ??P 11 s()P 12 s() P 21 s()P 22 s() ?? ?? ?? Ws() Us()?? ?? ?? ,= 5 Synthesis of H∞ Controllers 5-4 the closed-loop transfer function from w to z is given by the linear-fractional expression (5-1) Hence the optimal H ∞ control seeks to minimize ||F(P, K)|| ∞ over all stabilizing LTI controllers K(s). Alternatively, we can specify some maximum value γ for the closed-loop RMS gain and ask the following question: Does there exist a stabilizing controller K(s) that ensures ||F(P, K)|| ∞ < γ ? This is known as the suboptimal H ∞ control problem, and γ is called the prescribed H ∞ performance. A number of control problems can be recast in this standard formulation. Example 5.1. Consider the following disturbance rejection problem relative to the tracking loop of Figure 5-2: For disturbances d with spectral density concentrated between 0 and 1 rad/s, maintain the closed-loop RMS gain from d to the output y below 1%. Since the RMS gain is the largest gain over all square-integrable inputs, external signals with uneven spectral density must be specified by means of shaping filters. For instance, a disturbance d with all its energy in the Figure 5-2: Tracking loop with external disturbance frequency band [0,1] is described as where is an arbitrary square-integrable signal and W lp (s) is a low-pass filter with cutoff frequency at 1 rad/s. From the closed-loop equation F PK,() := P 11 P 12 KI P 22 K–() 1– P 21 + r y+ ? K G d eu + ds() W lp s()d ? s()= d ? H Control 5-5 y = Sd + Tr with S := (I + GK) –1 and T := GKS, the transfer function from the equalized disturbance to y is SW lp . Hence our disturbance rejection objective is equivalent to finding a stabilizing controller K(s) such that ||S W lp || ∞ < 1. This RMS gain constraint is readily turned into a standard H ∞ problem by observing that, in conformance with (5-1), Example 5.2. Decoupling constraints are handled in a similar fashion. For instance, consider the loop of Figure 5-3 and the problem of decoupling the action of r 1 on y 1 from that of r 2 on y 2 in the control bandwidth ω e 10 rad/s. The closed-loop transfer function from to is T = GK(I + GK) –1 . Decoupling is achieved if T(jω) ≈ I in the bandwidth ω e 10 rad/s, or equivalently if S(jω) = I – T(jω) has a small gain in this bandwidth. This can be expressed as d ? SW lp F PK,() with Ps := W lp s() G– s() W lp s() G– s() ?? ?? ?? = r 1 r 2?? ?? ?? ?? y 1 y 2?? ?? ?? ?? 10 s 10+ --------------- Ss() ∞ ε,< 5 Synthesis of H∞ Controllers 5-6 Figure 5-3: Decoupling problem which is again of the standard form Example 5.3. H ∞ optimization is also useful for the design of robust controllers in the face of unstructured uncertainty. Consider the uncertain system where the uncertainty ?( . ) is an arbitrary BIBO-stable system satisfying the norm bound ||?|| ∞ < 1. From the small gain theorem, the controller u = K(s)y robustly stabilizes this uncertain system if the nominal closed-loop transfer function F (P, K) from w to z satisfies ||F (P, K)|| ∞ < 1. Finally, a useful application of H ∞ synthesis is the loop shaping design procedure discussed in the next chapter. r 1 y 1 + ? K(s) G(s) u y 2 ? + r 2 F PK,() ∞ ε< with Ps() := 10 s 10+ --------------- I 2 10 s 10+ ---------------– Gs() I 2 Gs()– ?? ?? ?? . P(s) w z uy ? · () H Control 5-7 Riccati- and LMI-Based Approaches Since the continuous- and discrete-time cases are essentially similar, we review only continuous-time H ∞ synthesis. Both the Riccati-based and LMI-based approaches are implemented in the LMI Control Toolbox. While the LMI-based approach is computationally more involved for large problems, it has the merit of eliminating the regularity restrictions attached to the Riccati-based solution. Since both approaches are state-space methods, the plant P(s) is given in state-space form by The Riccati-based approach is applicable to plants P satisfying (A1) D 12 and D 21 have full rank, (A2) P 12 (s) and P 21 (s) have no zeros on the jω-axis. Given γ > 0, it gives necessary and sufficient conditions for the existence of internally stabilizing controllers K(s) such that ||F (P, K)|| ∞ < γ Specifically, the H ∞ performance γ is achievable if and only if [2]: (i) γ > max x · = Ax B 1 wB 2 u++ z = C 1 xD 11 wD 12 u++ y = C 2 xD 21 wD+ 22 u+ σ max ID 12 D 12 + –()D 11 ), σ max D 11 I( D 21 + D 21 –()) 5 Synthesis of H∞ Controllers 5-8 (ii) the Riccati equations associated with the Hamiltonian pencils have stabilizing solutions X ∞ and Y ∞ , respectively (iii) X ∞ and Y ∞ further satisfy X ∞ ? 0, Y ∞ ? 0, ρ(X ∞ Y ∞ ) < γ 2 Using this characterization, the optimal H ∞ performance γ opt can be computed by bisection (the so-called γ-iterations). An H ∞ controller with performance γ ≥ γ opt is then given by explicit formulas involving X ∞ , Y ∞ , and the plant state-space matrices. Singular H ∞ problems (those violating (A1)–(A2)) require regularization by small perturbation. A notable exception is the direct computation of the optimal H ∞ performance when D 12 or D 21 is singular [5]. In comparison, the LMI approach is applicable to any plant and does not involve γ-iterations. Rather, the H ∞ performance is directly optimized by solving the following LMI problem [4, 6]: H λε A 00γ 1– B 1 B 2 0 A T – C 1 T – 00 C 1 0 I– γ 1– D 11 D 12 0 γ 1– B 1 T γ 1– D 11 T I– 0 0 B 2 T D 12 T 00 ?? ?? ?? ?? λ I 0 00?? ?? ?? –=– J λε A T 00γ 1– C 1 T C 2 T 0 A– B 1 – 00 B 1 T 0 I– γ 1– D 11 T D 21 T 0 γ 1– C 1 γ 1– D 11 I– 0 0 C 2 D 21 00 ?? ?? ?? ?? λ I 0 00?? ?? ?? –=– H Control 5-9 Minimize γ over R = R T and S = S T such that (5-2) (5-3) (5-4) where N 12 and N 21 denote bases of the null spaces of and (C 2 , D 21 ), respectively. This problem falls within the scope of the LMI solver mincx. Note that the LMI constraints (5-2)–(5-3) amount to the inequality counterpart of the H ∞ Riccati equations, R –1 and S –1 being solutions of these inequalities. Again, explicit formulas are available to derive H ∞ controllers from any solution (R, S, γ) of (5-2)–(5-4) [3]. N 12 0 0 I ?? ?? ?? T AR RA T + RC 1 T B 1 C 1 R γI– D 11 B 1 T D 11 T γI– ?? ?? ?? N 12 0 0 I ?? ?? ?? 0< N 21 0 0 I ?? ?? ?? T A T SSA+ SB 1 C 1 T B 1 T S γI– D 11 T C 1 D 11 γI– ?? ?? ?? ?? N 21 0 0 I ?? ?? ?? 0< RI IS?? ?? ?? 0≥ B 2 T D 12 T ,() 5 Synthesis of H∞ Controllers 5-10 H ∞ Synthesis The LMI Control Toolbox supports continuous- and discrete-time H ∞ synthesis using either Riccati- or LMI-based approaches. Transparency and numerical reliability were primary concerns in the development of the related tools. Original features of the Riccati-based functions include: The use of pencil-based Riccati solvers for highly accurate computation of Riccati solutions (see care, ricpen, and their discrete-time counterparts) Direct computation of the optimal H ∞ performance when D 12 and D 21 are rank-deficient (without using regularization) [5] Automatic regularization of singular problems for the controller computation. In addition, reduced-order controllers are computed whenever the matrices I – γ –2 X ∞ Y ∞ or I – RS are nearly rank-deficient. The functions for standard H ∞ synthesis are listed in Table 5-1. To illustrate the use of hinfric and hinflmi, consider the simple first-order plant P(s) with state-space equations This plant is specified as a SYSTEM matrix by a=0; b1=1; b2=2; c1=1; d11=0; d12=0; c2= 1; d21=1; d22=0; P=ltisys(a,[b1 b2],[c1;c2],[d11 d12;d21 d22]) Note that the plant P(s) is “singular” since D 12 = 0. Table 5-1: Functions for H ∞ synthesis H ∞ synthesis Riccati-based LMI-based Continuous-time hinfric hinflmi Discrete-time dhinfric dhinflmi ? ? ? ? ? x · w 2u+= zx= yx– w+= H Synthesis 5-11 To determine the optimal H ∞ performance over stabilizing control laws u = K(s)y, type gopt = hinfric(P,[1 1]) where the second argument [1 1] specifies the numbers of measurements and controls (lengths of the vectors y and u). The γ-iterations performed by hinfric are displayed on the screen as follows: Gamma-Iteration: Best closed-loop gain (GAMMA_OPT): 1.008789 The optimal H ∞ performance for P(s) is therefore approximately 1. For each tested value of γ, hinfric gives a feasible/infeasible diagnosis as well as the cause for infeasibility when applicable. Alternatively, you can use the LMI-based function hinflmi with the same syntax: gopt = hinflmi(P,[1 1]) This function optimizes the H ∞ performance using mincx: Gamma Diagnosis 10.0000 : feasible 1.0000 : infeasible (Y is not positive semi-definite) 5.5000 : feasible 3.2500 : feasible 2.1250 : feasible 1.5625 : feasible 1.2812 : feasible 1.1406 : feasible 1.0703 : feasible 1.0352 : feasible 1.0176 : feasible 1.0088 : feasible 5 Synthesis of H∞ Controllers 5-12 Minimization of gamma: Solver for linear objective minimization under LMI constraints Iterations : Best objective value so far Result:feasible solution of required accuracy best objective value: 1.002517 guaranteed relative accuracy: 9.11e 03 f-radius saturation: 0.018% of R = 1.00e+08 The value displayed in the second column is the current best estimate of γ opt . The optimal value 1.0025 approximately matches the value found by hinfric. When a second output argument is provided, these two functions also return an optimal H ∞ controller K(s): [gopt,K] = hinfric(P,[1 1]) 1 2.196542 *** new lower bound: 0.571370 2 1.381964 3 1.381964 4 1.094665 5 1.052469 6 1.008499 7 1.008499 8 1.008499 *** new lower bound: 0.656673 9 1.002517 *** new lower bound: 0.878172 10 1.002517 *** new lower bound: 0.982223 11 1.002517 *** new lower bound: 0.993382 H Synthesis 5-13 The output K is the SYSTEM matrix of K(s). To compute a suboptimal H ∞ controller with guaranteed performance γ < 10, type [g,K] = hinfric(P,[1 1],10,10) In this case only the value γ = 10 is tested. In the LMI approach, the same problem is solved by the command [g,K] = hinflmi(P,[1 1],10) Finally, the syntax [g,K,x1,x2,y1,y2] = hinfric(P,[1,1],10,10) also returns the stabilizing solutions X ∞ = x2/x1 and Y ∞ = y2/y1 of the H ∞ Riccati equations for γ = 10. Similarly, [g,K,x1,x2,y1,y2] = hinflmi(P,[1,1],10) returns solutions X = x2/x1 and Y = y2/y1 of the H ∞ Riccati inequalities. Equivalently, R = x1 and S = y1 are solutions of the characteristic LMIs (5-2)-(5-4) since x2 and y2 are always set to g × I. Further options and control parameters are discussed in the “Command Reference” chapter. Finally, the counterpart of these functions for discrete-time H ∞ synthesis are called dhinfric, dhinflmi, and dnorminf and follow the exact same syntax. Validation of the Closed-Loop System Given the H ∞ controller K computed by hinfric or hinflmi, the closed-loop mapping represented in Figure 5-4 is formed with the function slft: Figure 5-4: Closed-loop system P(s) u y w z K(s) 5 Synthesis of H∞ Controllers 5-14 Closed-loop stability is then checked by inspecting the closed-loop poles with spol spol(clsys) while the closed-loop RMS gain from w to z is computed by norminf(clsys) You can also plot the time- and frequency-domain responses of the closed- loop clsys with splot. Finally, hinfpar extracts the state-space matrices A, B 1 , . . . from the plant SYSTEM matrix P: [a,b1,b2,c1,c2,d11,d12,d21,d22] = hinfpar(P,r) Set the second argument r to [p2 m2] if D 22 ∈ R p 2 ×m 2 . Multi-Objective H Synthesis 5-15 Multi-Objective H ∞ Synthesis In many real-world applications, standard H ∞ synthesis cannot adequately capture all design specifications. For instance, noise attenuation or regulation against random disturbances are more naturally expressed in LQG terms. Similarly, pure H ∞ synthesis only enforces closed-loop stability and does not allow for direct placement of the closed-loop poles in more specific regions of the left-half plane. Since the pole location is related to the time response and transient behavior of the feedback system, it is often desirable to impose additional damping and clustering constraints on the closed-loop dynamics. This makes multi-objective synthesis highly desirable in practice, and LMI theory offers powerful tools to attack such problems. Mixed H 2 /H ∞ synthesis with regional pole placement is one example of multi-objective design addressed by the LMI Control Toolbox. The control problem is sketched in Figure 5-5. The output channel z ∞ is associated with the H ∞ performance while the channel z 2 is associated with the LQG aspects (H 2 performance). Denoting by T ∞ (s) and T 2 (s) the closed-loop transfer functions from w to z ∞ and z 2 , respectively, we consider the following multi-objective synthesis problem: Figure 5-5: Multi-objective H ∞ synthesis Design an output-feedback controller u = K(s)y that Maintains the H ∞ norm of T ∞ (s) (RMS gain) below some prescribed value γ 0 > 0 Maintains the H 2 norm of T 2 (s) (LQG cost) below some prescribed value ν 0 > 0 P(s) u y w z 2 K(s) z ∞ 5 Synthesis of H∞ Controllers 5-16 Minimizes a trade-off criterion of the form with α ? 0 and β ≥ 0 Places the closed-loop poles in some prescribed LMI region D Recall that LMI regions are general convex subregions of the open left-half plane (see “Pole Placement in LMI Regions” on page 4-5 for details). LMI Formulation Let and be state-space realizations of the plant P(s) and controller K(s), respectively, and let be the corresponding closed-loop state-space equations. Our three design objectives can be expressed as follows: α T ∞ ∞ 2 β T 2 2 2 + x · = Ax B 1 wB 2 u++ z ∞ = C ∞ xD ∞1 wD ∞2 u++ z 2 = C 2 xD 21 wD 22 u++ y = C y xD y1 w+ ?? ?? ?? ?? ?? ζ · A Kζ B Ky += u · C Kζ D Ky += ?? ?? ?? ? ? ? ? ? x · cl = A cl x cl B cl w+ z ∞ = C cl1 x cl D cl1 w+ z 2 = C cl2 x cl D cl2 w+ Multi-Objective H Synthesis 5-17 H ∞ performance: the closed-loop RMS gain from w to z ∞ does not exceed γ if and only if there exists a symmetric matrix X ∞ such that H 2 performance: the H 2 norm of the closed-loop transfer function from w to z 2 does not exceed ν if and only if D cl2 = 0 and there exist two symmetric matrices χ 2 and Q such that Pole placement: the closed-loop poles lie in the LMI region with L = L T = and M = if and only if there exists a symmetric matrix χ pol satisfying A cl χ ∞ χ ∞ A cl T + B cl X ∞ C cl1 T B cl T I– D cl1 T C cl1 χ ∞ D cl1 γ– 2 I ?? ?? ?? 0< χ ∞ 0> A cl χ 2 χ 2 A cl T + B cl B cl T I– ?? ?? ?? 0< QC cl2 χ 2 χ 2 C cl2 T χ 2 ?? ?? ?? 0> Trace Q() ν 2 < D z C∈ : LMzM T z 0<++{}= λ ij {} 1 i≤ ,jm≤ μ ij [] 1 i≤ ,jm≤ 5 Synthesis of H∞ Controllers 5-18 For tractability in the LMI framework, we must seek a single Lyapunov matrix χ := χ ∞ = χ 2 = χ pol that enforces all three sets of constraints. Factorizing χ as and introducing the change of controller variables [3]: the inequality constraints on χ are readily turned into LMI constraints in the variables R, S, Q, A K , B K , C K , and D K [8, 1]. This leads to the following suboptimal LMI formulation of our multi-objective synthesis problem: λ ij χ pol μ ij A cl χ pol μ ji X pol A cl T ++[] 1 i≤ ,jm≤ 0< χ pol 0> χχ 1 χ 2 1– , χ 1 := RI M T 0 ?? ?? ?? , χ 2 := 0 S IN T ?? ?? ?? = ? ? ? ? ? B K := NB K SB 2 D K + C K := C K M T D K C y R+ A K := NA K M T NB K C y RSB 2 C K M T SA B 2 D K C y +()R,++ + Multi-Objective H Synthesis 5-19 Minimize α γ 2 + β Trace(Q) over R, S, Q, A K , B K , C K , D K , and γ 2 satisfying: Given optimal solutions γ*, Q* of this LMI problem, the closed-loop H ∞ and H 2 performances are bounded by AR RA T B 2 C K C K T B 2 T ++ + A K T AB+ 2 D K C y + B 1 B 2 D K D y1 + H H A T SSA+ B K C y C y T B K T ++ SB 1 B K D y1 + H HHI– H C ∞ RD ∞2 C K + C ∞ D ∞2 D K C y + D ∞1 D ∞2 D K D y1 + γ– 2 I ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 0< μ ji RA T C K T B 2 T + A K T AB+ 2 D K C y () T A T SC y T B K T + ?? ?? ?? 1 i≤ ,jm≤ 0< QC 2 RD 22 C K + C 2 D 22 D K C y + H RI H IS ?? ?? ?? 0> λ ij RI IS?? ?? ?? μ ij AR B 2 C K + AB+ 2 D K C y A K SA B K C y + ?? ?? ?? ++ γ 2 γ 0 2 < TraceQ ν 0 2 < D 21 D 22 D K D y1 + 0= T ∞ ∞ γ* T 2 2 Trace Q*().≤,≤ 5 Synthesis of H∞ Controllers 5-20 The Function hinfmix The function hinfmix implements the LMI approach to mixed H 2 /H ∞ synthesis with regional pole placement described above. Its syntax is [gopt,h2opt,K,R,S] = hinfmix(P,r,obj,region) where P is the SYSTEM matrix of the LTI plant P(s). Note that z 2 or z ∞ can be empty, or even both when performing pure pole placement r is a three-entry vector listing the lengths of z 2 , y, and u obj = [γ 0 , ν 0 , α, β] is a four-entry vector specifying the H 2 /H ∞ constraints and criterion region specifies the LMI region for pole placement, the default being the open left-half plane. Use lmireg to interactively generate the matrix region. The outputs gopt and h2opt are the guaranteed H ∞ and H 2 performances, K is the controller SYSTEM matrix, and R,S are optimal values of the variables R, S (see “LMI Formulation” on page 5-16). You can perform the following mixed and unmixed designs by setting obj appropriately: Loop-Shaping Design with hinfmix In the loop-shaping context (see next chapter), the closed-loop poles also include the poles of the shaping filters. Since these poles are not affected by the obj Corresponding Design [0 0 0 0] pole placement only [0 0 1 0] H ∞ -optimal design [0 0 0 1] H 2 -optimal design [g 0 0 1] minimize ||T 2 || 2 subject to ||T ∞ || ∞ < g [0 h 1 0] minimize ||T ∞ || ∞ subject to ||T 2 || 2 < h [0 0 a b] minimize [g h a b] most general problem a T ∞ 0 2 b T 2 2 2 + Multi-Objective H Synthesis 5-21 controller, they impose hard limitations on achievable closed-loop clustering objectives. In particular, shaping filters with modes close to the imaginary axis will doom any attempt to push the closed-loop modes into a well-damped region. When the purpose of such filters is to enforce integral control action, you can eliminate this difficulty by moving the integrator from the shaping filter to the control loop itself. This amounts to reparametrizing the controller as K(s) = (s) and performing the synthesis for . An example can be found at the end of Chapter 6. 1 s ---K ? K ? 5 Synthesis of H∞ Controllers 5-22 References [1] Chilali, M., and P. Gahinet, “H ∞ Design with Pole Placement Constraints: an LMI Approach,” to appear in IEEE Trans. Aut. Contr., 1995. [2] Doyle, J.C., Glover, K., Khargonekar, P., and Francis, B., “State-Space Solutions to Standard H 2 and H ∞ Control Problems,” IEEE Trans. Aut. Contr., AC–34 (1989), pp. 831–847. [3] Gahinet, P., “Explicit Controller Formulas for LMI-based H ∞ Synthesis,” submitted to Automatica. Also in Proc. Amer. Contr. Conf., 1994, pp. 2396– 2400. [4] Gahinet, P., and P. Apkarian, “A Linear Matrix Inequality Approach to H ∞ Control,” Int. J. Robust and Nonlinear Contr., 4 (1994), pp. 421–448. [5] Gahinet, P., and A.J. Laub, “Reliable Computation of γ opt in Singular H ∞ Control,” to appear in SIAM J. Contr. Opt., 1995. Also in Proc. Conf. Dec. Contr., Lake Buena Vista, Fl., 1994, pp. 1527–1532. [6] Iwasaki, T., and R.E. Skelton, “All Controllers for the General H ∞ Control Problem: LMI Existence Conditions and State-Space Formulas,” Automatica, 30 (1994), pp. 1307–1317. [7] Scherer, C., “H ∞ Optimization without Assumptions on Finite or Infinite Zeros,” SIAM J. Contr. Opt., 30 (1992), pp. 143–166. [8] Scherer, C., “Mixed H 2 H ∞ Control,” to appear in Trends in Control: A European Perspective, volume of the special contributions to the ECC 1995. 6 Loop Shaping The Loop-Shaping Methodology . . . . . . . . . . . 6-2 The Loop-Shaping Methodology . . . . . . . . . . . 6-3 Design Example . . . . . . . . . . . . . . . . . . 6-5 Specification of the Shaping Filters . . . . . . . . . 6-10 Nonproper Filters and sderiv . . . . . . . . . . . . . . 6-12 Specification of the Control Structure . . . . . . . . 6-14 Controller Synthesis and Validation . . . . . . . . . 6-16 Practical Considerations . . . . . . . . . . . . . . 6-18 Loop Shaping with Regional Pole Placement . . . . . 6-19 References . . . . . . . . . . . . . . . . . . . . . 6-24 6 Loop Shaping 6-2 The Loop-Shaping Methodology Loop shaping is a popular technique for combining specifications such as tracking performance, bandwidth, disturbance rejection, roll-off, robustness to model uncertainty, gain limitation, etc. In this frequency domain method, the design specifications are reflected as gain constraints on the various closed-loop transfer functions. Loop-shaping design involves four steps: 1 Express the design specifications as constraints on the gain response of the closed-loop transfer functions. This defines ideal gain profiles called loop shapes. 2 Specify the desired loop shapes graphically with the GUI magshape. 3 Specify the control structure, i.e., how the feedback loop is organized and which input/output transfer functions are relevant to the loop-shaping objectives. 4 Perform a standard H ∞ synthesis on the resulting control structure to compute an adequate controller. This chapter contains a tutorial introduction to the loop-shaping methodology as well as the available tools for loop-shaping design. The Loop-Shaping Methodology 6-3 The Loop-Shaping Methodology Loop shaping is a design procedure to formulate frequency-domain specifications as H ∞ constraints and standard H ∞ synthesis problems. In loop-shaping design, the performance and robustness specifications are first expressed in terms of loop shapes, i.e., of desired gain responses for the various closed-loop transfer functions. These shaping objectives are then turned into uniform H ∞ bounds by means of shaping filters, and standard H ∞ synthesis algorithms are applied to compute an adequate controller. To get a feeling for the loop-shaping methodology, consider the MIMO tracking loop of Figure 6-1 and suppose that we want to design a controller with integral behavior. This requirement is naturally expressed in terms of the open-loop response GK(s), for instance by a gain constraint of the form (6-1) Figure 6-1: Simple tracking loop Here the main closed-loop transfer functions are: The sensitivity function S = (I + GK) –1 (the transfer function from r to e) The complementary sensitivity function T = GK(I + GK) –1 (the transfer function from r to y). To express the open-loop specification (6-1) in terms of these closed-loop transfer functions, observe that σ min (GK(jω)) ≈ 1/σ max (S(jω)) whenever the open-loop gain is large, i.e., σ min (GK(jω)) >> 1. Thus, (6-1) is equivalent to requiring that (6-2) σ min GK jω()()1 ω? for ω 1?> + ? K G r r 1 r 2 ?? ??= y y 1 y 2 ?? ??= e u σ max Sjω()() < ω for ω 1? 6 Loop Shaping 6-4 To turn this frequency-dependent gain constraint into a normalized H ∞ bound, a useful tool is the notion of shaping filter. Shaping filters are rational filters whose magnitude plot reflects the desired loop shape. Here for instance, the shaping filter W(s) = I 2 can be used to normalize the constraint (6-2) to (6-3) Indeed, (6-3) is equivalent to (1 + GK(jω)) for all ω, which in turn constrains GK to σ min (GK(jω)) > for ω << 1 Likewise, roll-off requirements on GK(jω) are enforced by H ∞ bounds of the form ||W(s)T(s)|| ∞ < 1 where W(s) is some appropriate high-pass filter. This reformulation is rooted in the approximation σ max (GK(jω)) ≈ σ max (T(jω)) valid whenever the open-loop gain is small, i.e., σ max (GK(jω)) << 1. Such constraints are also associated with robust stability in the face of plant uncertainty of the form G(s) = (I +?(s))G 0 (s) where ? satisfies the frequency-dependent bound σ max (?(jω)) < |W(jω)| Finally, shaping filters are useful to incorporate information about the spectral density of exogenous signals such as disturbances or measurement noise (see “Example 5.1” on page 5-4). 1 s --- Ws()Ss() ∞ 1< 1 ω --- σ min < 1 ω --- 1 1 ω ---≈– Design Example 6-5 Design Example Loop shaping design with the LMI Control Toolbox involves four steps: 1 Express the design specifications in terms of loop shapes and shaping filters. 2 Specify the shaping filters by their magnitude profile. This is done interactively with the graphical user interface magshape. 3 Specify the control loop structure with the functions sconnect and smult, or alternatively with Simulink. 4 Solve the resulting H ∞ problem with one of the H ∞ synthesis functions. A simple tracking problem is used to illustrate this design procedure. Type radardem to run the corresponding demo. Example 6.1. Figure 6-2 shows a simplified mechanical model of a radar antenna. The stiffness K accounts for flexibilities in the coupling between the motor and the antenna. The corresponding nominal transfer function from the motor torque u = T to the angular velocity y = of the antenna isθ · a G 0 s() 30000 s 0.02+()s 2 0.99s 30030++() -------------------------------------------------------------------------------- · .= 6 Loop Shaping 6-6 Our goal is to control through the torque T. The control structure is the tracking loop of Figure 6-3 where the dynamical uncertainty ?(s) accounts for neglected high-frequency dynamics and flexibilities. Figure 6-2: Second-order model of a radar antenna The design specifications are as follows: (i)integral control action, open-loop gain of at least 30 dB at 1 rad/s, and bandwidth of at least 10 rad/s, (ii)99% rejection of output disturbances d with spectral density in [0, 1] rad/s, (iii)robustness against the neglected high-frequency dynamics represented by the multiplicative model uncertainty ?(s). A bound on |?(jω)| determined from experimental data is given in Figure 6-4. Figure 6-3: Velocity loop for the radar antenna θ · a θ a θ m K J m J a antenna motor T r + ? G 0 (s) K d + + G(s) ? y θ · = Design Example 6-7 Figure 6-4: Empirical bound on the uncertainty ?(s) We first convert these specifications into RMS gain constraints by means of shaping filters. While the performance requirement (i) amounts to a lower bound on the open-loop gain, the disturbance rejection target (ii) imposes |S(jω)| < 0.01 for ω < 1 rad/s. Both requirements can be combined into a single constraint (6-4) where S = 1 = (1 + G 0 K) and W 1 (s) is a low-pass filter whose gain lies above the shaded region in Figure 6-5. From the small gain theorem [2], robust stability in the face of the uncertainty ?(s) is equivalent to (6-5) 10 1 10 2 10 3 10 4 ?30 ?20 ?10 0 10 20 30 40 Frequency (rad/sec) Magnitude (dB) Frequency?dependent bound on Delta W 1 s()Ss() ∞ 1< W 2 s()Ts() ∞ 1< 6 Loop Shaping 6-8 where T = G 0 K = (1 + G 0 K) and W 2 is a high pass filter whose gain follows the profile of Figure 6-4. For tractability in the H ∞ framework, (6-4)–(6-5) are replaced by the single RMS gain constraint (6-6) Figure 6-5: Magnitude profile for W 1 (s) The transfer function maps r to the signals , in the control loop of Figure 6-6. Hence our original design problem is equivalent to the following H ∞ synthesis problem: Find a stabilizing controller K(s) that makes the closed-loop RMS gain from r to less than one. W 1 S W 2 T ∞ 1.< dB rad/s 1 10 0 40 30 integral action disturbance rejection bandwidth W 1 S W 2 T ?? ?? ?? e ? y ? e ? y ? ?? ?? ?? Design Example 6-9 Figure 6-6: Control structure To perform this H ∞ synthesis, we need to 1 Compute two shaping filters W 1 (s) and W 2 (s) whose magnitude responses match the profiles of Figures 6-5 and 6-4. 2 Specify the control structure of Figure 6-6 and derive the corresponding plant. 3 Call one of the H ∞ -optimization functions to compute an adequate controller K(s) for this plant. The next three sections discuss these steps. r y + ? K G 0 e W 2 W 1 u e e ? y ? 6 Loop Shaping 6-10 Specification of the Shaping Filters The graphical user interface magshape allows you to specify shaping filters directly by their magnitude profile. Typing magshape brings up the worksheet shown in Figure 6-7. To construct the two shaping filters W 1 (s) and W 2 (s) involved in our problem: 1 Type w1,w2 after the prompt Filter names. This tells magshape to write the filter SYSTEM matrices in MATLAB variables called w1 and w2. 2 Indicate which filter you are about to specify by clicking on the corresponding button on the right-hand side of the window. 3 Click on the add point button: you can now specify the desired magnitude profile with the mouse. This profile is sketched with a few characteristic points marking the asymptotes and the slope changes. Clicking on the delete point or move point buttons allows you to delete or move particular points. 4 Once the desired profile is sketched, specify the filter order and click the fit data button to interpolate the data points by a rational filter. The gain of the resulting filter appears in solid on the plot, and its realization is written in the MATLAB variable of the same name. To make adjustments, you can move some of the points or change the filter order, and then redo the fitting. 5 If w1 is used as a filter name, magshape checks if a SYSTEM matrix of the same name already exists in the MATLAB environment. When this is the case, its magnitude plot is drawn when the w1 button is clicked. This is useful to load existing filters in the magshape window for modification or comparison. Figure 6-7 is a snapshot of the magshape window after specifying the profiles of W 1 and W 2 and fitting the data points (marked by o in the plot). The solid lines show the rational fits obtained with an order 3 for W 1 and an order 1 for W 2 . Specification of the Shaping Filters 6-11 The transfer functions of these filters (as given by ltitf(w1) and ltitf(w2)) are approximately Note that horizontal asymptotes have been added in both filters. Such asymptotes prevent humps in S and T near the crossover frequency, thus reducing the overshoot in the step response. The function magshape always returns stable and proper filters. If you do not use magshape to generate your filters, make sure that they are stable with poles sufficiently far from the imaginary axis. To avoid difficulties in the subsequent H ∞ optimization, their poles should satisfy W 1 s() 0.64s 3 10.45s 2 149.28s 25.87+++ s 3 1.20s 2 0.83s 8.31 10 4– ×+++ ----------------------------------------------------------------------------------------------= W 2 s() 100 3.41s 217.75+ s 3.54 10 4 ×+ ---------------------------------------= Re s() 10 3– –≤ 6 Loop Shaping 6-12 Figure 6-7: The magshape interface Nonproper Filters and sderiv To impose a given roll-off rate in the open-loop response, it is often desirable to use nonproper shaping filters, i.e., filters with a derivative action in the high-frequency range. Because such filters cannot be represented in SYSTEM matrix form, magshape approximates them by high-pass filters. A drawback of this approximation is the introduction of fast parasitic modes in the filter and augmented plant realizations, which in turn may cause numerical difficulties. Alternatively, you can use sderiv to include nonproper shaping filters in the loop-shaping criterion. This function appends a SISO proportional-derivator Specification of the Shaping Filters 6-13 component ns + d to selected inputs and/or outputs of a given LTI system. For instance, the interconnection is specified by Pd = sderiv(P,[ 1 2],[0.01 1]) In the calling list, [1 2] lists the input and output channels to be filtered by ns + d (here 1 for “first input” and 2 for “second output”) and the vector [0.01 1] lists the values of n and d. An error is generated if the resulting system Pd is not proper. To specify more complex nonproper filters, 1 Specify the proper “low-frequency” part of the filters with magshape. 2 Augment the plant with these low-pass filters. 3 Add the derivative action of the nonproper filters by applying sderiv to the augmented plant. This procedure is illustrated in the design example presented in “Loop Shaping with Regional Pole Placement” on page 6-19. 0.01s+1 0.01s+1 P(s) 6 Loop Shaping 6-14 Specification of the Control Structure To perform the H ∞ synthesis, the control structure of Figure 6-6 must be put in the standard linear-fractional form of Figure 6-8. In this equivalent representation, P(s) is the nominal plant determined by G 0 (s) and the control structure, and P aug (s) is the augmented plant obtained by appending the shaping filters W 1 (s) and W 2 (s). While the H ∞ synthesis is performed on P aug (s), the physical closed-loop transfer functions from r to e and y are fully specified by P(s) and K(s) (recall that the shaping filters are not part of the physical control loop). Figure 6-8: Equivalent standard H ∞ problem The function sconnect computes the SYSTEM matrices of P(s) or P aug (s) given G 0 , W 1 , W 2 , and the qualitative description of the control structure (Figure 6-6). With this function, general control structures are described by listing the input and output signals and by specifying the input of each dynamical system. In our problem, the plant P(s) corresponding to the control loop P(s) K(s) W 1 W 2 e y r u y P aug (s) e ? y ? r y + ? K G 0 e e Specification of the Control Structure 6-15 is specified by g0 = ltisys('tf',3e4,conv([1.02],[1.99 3.03e4])) P = sconnect('r(1)','e=r-G0 ; G0','K:e','G0:K',g0) The ltisys command returns a state-space realization of G 0 (s). In the sconnect command, the input arguments are strings or SYSTEM matrices specifying the control structure as follows: The first argument 'r(1)' lists and dimensions the input signals. Here the only input is the scalar reference signal r. The second argument 'e=r-G0 ; G0' lists the two output signals separated by a semicolon. Outputs are defined as combinations of the input signals and the outputs of the dynamical systems. For instance, the first output e is specified as r minus the output y of G 0 . For systems with several outputs, the syntax G0([1,3:4]) would select the first, third, and fourth outputs. The third argument 'K:e' names the controller and specifies its inputs. A two-degrees-of-freedom controller with inputs e and r would be specified by 'K: e;r'. The remaining arguments come in pairs and specify, for each known LTI system in the loop, its input list and its SYSTEM matrix. Here 'G0:K' means that the input of G 0 is the output of K, and g0 is the SYSTEM matrix of G 0 (s). You can give arbitrary names to G 0 and K in the string definitions, provided that you use the same names throughout. Similarly, the augmented plant P aug (s) corresponding to the loop of Figure 6-6 would be specified by Paug = sconnect('r(1)','W1;W2','K:e = r-G0','G0:K',g0,... 'W1:e',w1,'W2:G0',w2) where w1 and w2 are the shaping filter SYSTEM matrices returned by magshape. However, the same result is obtained more directly by appending the shaping filters to P(s) according to Figure 6-8: Paug = smult(P,sdiag(w1,w2,1)) Finally, note that sconnect is also useful to compute the SYSTEM matrix of ordinary system interconnections. In such cases, the third argument should be set to [] to signal the absence of a controller. 6 Loop Shaping 6-16 Controller Synthesis and Validation After forming the augmented plant P aug (s) with sconnect, it suffices to call hinfric or hinflmi to test whether the specifications are feasible and compute an adequate controller K(s). The loop-shaping design is feasible if and only if the H ∞ performance γ = 1 is achievable for P aug . With our choice of shaping filters, we obtain [gopt,k] = hinfric(Paug,[1 1],1,1); Gamma-Iteration: Gamma Diagnosis 1.0000 : feasible Best closed-loop gain (GAMMA_OPT): 1.000000 ** regularizing D12 or D21 to compute K(s): .. Hence the specifications are met by the controller k returned by hinfric. The last two lines indicate that the H ∞ problem associated with P aug is singular and has been regularized. To validate the controller k, you can form the open-loop response GK by typing gk = slft(k,g) or the closed-loop transfer function from r to e and y by typing Pcl = slft(P,k) The Bode diagram of GK and the nominal step response are then plotted by splot(gk,'bo') splot(ssub(Pcl,1,2),'st') Controller Synthesis and Validation 6-17 In the second command, ssub(Pcl,1,2) selects the transfer function from r to y as a subsystem of Pcl. The step response with the controller computed above appears in Figure 6-9. Figure 6-9: Step response of the closed-loop system 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.2 0.4 0.6 0.8 1 1.2 Time (secs) Amplitude 6 Loop Shaping 6-18 Practical Considerations When the model G 0 (s) of the system has poles on the imaginary axis, loop- shaping design often leads to “singular” H ∞ problems with jω-axis zeros in the (1,2) or (2,1) blocks of P aug . While immaterial in the LMI approach, jω-axis zeros bar the direct use of standard Riccati-based synthesis algorithms [1]. The function hinfric automatically regularizes such problems by perturbing the A matrix of P aug to A + εI for some positive ε. The message ** jw-axis zeros of P12(s) regularized or ** jw-axis zeros of P21(s) regularized is then displayed to inform users of the presence of a singularity. In doing so, however, hinfric shifts the shaping filter poles as part of the poles of P aug . Since the shaping filters must remain stable for closed-loop stabilizability, the amount of regularization ε is thereby limited. When some shaping filters have modes close to the jω-axis, this is likely to result in poor closed-loop damping because the central controller computed by hinfric tends, in such problems, to place closed-loop poles within ε of the jω-axis. If this difficulty arises, a simple remedy consists of perturbing the poles of the plant P(s) prior to appending the shaping filters. Provided that all poles of G 0 (s) are controllable and observable, this typically tolerates much larger ε, which in turn results in better closed-loop damping. In our example, this would amount to forming P aug as follows: P = sconnect('r(1)','e=r-G0 ; G0','K:e','G0:K',g0) % pre-regularization [a,b,c,d] = ltiss(P) a = a + reg * eye(size(a)) Preg = ltisys(a,b,c,d) Paug = smult(Preg, sdiag(w1,w2,1)) Note that the regularization level reg should always be chosen positive. Finally, a careful validation of the resulting controller on the true plant P is necessary since the H ∞ synthesis is performed on a perturbed system. Loop Shaping with Regional Pole Placement 6-19 Loop Shaping with Regional Pole Placement A weakness of the design outlined above is the plant inversion performed by the controller K(s). Specifically, the resonant poles of G(s) are cancelled by zeros of K(s) as seen when comparing the magnitude plots of G and K (see Figure 6-10). Since such cancellations leave resonant modes in the closed-loop system, oscillatory responses may result when these modes are excited, e.g., by a disturbance at the plant input (see Figure 6-11). There are several ways of circumventing this difficulty. One possible approach is to impose some minimum closed-loop damping via an additional regional pole placement constraint. In addition to avoiding cancellations, this is helpful to tune the transient responses and prevent fast controller dynamics. Loop-shaping design with pole placement is addressed by the function hinfmix. As mentioned in the previous chapter, using hinfmix requires extra care because of possible interferences between the root clustering objective and the shaping filter poles. Indeed, recall that the “closed-loop” dynamics of the loop-shaping criterion always comprise the shaping filter modes. Figure 6-10: Pole-zero cancellation between G and K 10 ?1 10 0 10 1 10 2 10 3 ?100 ?50 0 50 Frequency (rad/sec) Singular Values dB G(s) K(s) 6 Loop Shaping 6-20 Figure 6-11: Response y for an impulse disturbance at the plant input For “Example 6.1” on page 6-5, we chose as pole placement region the disk centered at (–1000, 0) and with radius 1000. By inspection of the shaping filters used in the previous design, this region clashes with the pseudo-integrator in W 1 (s) and the fast pole at s = –3.54 × 10 4 in W 2 (s). To alleviate this difficulty, 1 Remove the pseudo-integrator from W 1 (s) and move it to the main loop as shown in Figure 6-12. This defines an equivalent loop-shaping problem where the controller is reparametrized as K(s) = (s)/s and the integrator is now assignable via output feedback. 0 1 2 3 4 5 6 7 8 9 10 ?1.5 ?1 ?0.5 0 0.5 1 1.5 Time (secs) Amplitude K ? Loop Shaping with Regional Pole Placement 6-21 2 Specify W 2 (s) as a nonproper filter with derivative action. This is done with sderiv. Figure 6-12: Modified control structure A satisfactory design was obtained for the following values of W 1 and W 2 : A summary of the commands needed to specify the control structure and solve the constrained H ∞ problem is listed below. Run the demo radardem to perform this design interactively. % control structure, sint = integrator sint = ltisys('tf',1,[1 0]) [P,r] = sconnect('r','Int;G','Kt:Int','G:Kt',G0,'Int:e=r-G', sint) % add w1 w1=ltisys('tf',[1 16 200],[1 1.2 0.8]) Paug = smult(P,sdiag(w1,1,1)) % add w2 Paug = sderiv(Paug,2,[1/200 0.9]) % interactively specify the region for pole placement region = lmireg % perform the H-infinity synthesis with pole placement r = [0 1 1] r y+ ? G 0 e e 1/s W 1 W 2 e ? y ? K ? W 1 s() s 2 16s 200++ s 2 1.2s 0.8++ --------------------------------------, W 2 s() 0.9 s 200 ----------+== 6 Loop Shaping 6-22 [gopt,xx,Kt] = hinfmix(Paug,r,[0 0 1 0],region) The full controller is then given by K = smult(sint,Kt) The resulting open-loop response and the time response to a step r and an impulse disturbance at the plant input appear in Figure 6-13 and 6-14. Note that the new controller no longer inverts the plant and that input disturbances are now rejected with satisfactory transient behavior. Figure 6-13: Open-loop response GK(s) 10 ?1 10 0 10 1 10 2 10 3 ?50 0 50 100 Frequency (rad/sec) Gain dB 10 ?1 10 0 10 1 10 2 10 3 ?360 ?720 0 Frequency (rad/sec) Phase deg Loop Shaping with Regional Pole Placement 6-23 Figure 6-14: Transient behaviors with the new controller 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 ?0.5 0 0.5 1 1.5 Time (secs) Amplitude Response to a step r 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 ?0.5 0 0.5 1 1.5 2 Time (secs) Amplitude Response to an impulse disturbance on u 6 Loop Shaping 6-24 References [1] Doyle, J.C., Glover, K., Khargonekar, P., and Francis, B., “State-Space Solutions to Standard H 2 and H ∞ Control Problems,” IEEE Trans. Aut. Contr., AC–34 (1989), pp. 831–847. [2] Zames, G., “On the Input-Output Stability of Time-Varying Nonlinear Feedback Systems, Part I and II,” IEEE Trans. Aut. Contr., AC–11 (1966), pp. 228–238 and 465–476. 7 Robust Gain-Scheduled Controllers Gain-Scheduled Control . . . . . . . . . . . . . . . 7-3 Synthesis of Gain-Scheduled H Controllers . . . . . 7-7 Simulation of Gain-Scheduled Control Systems . . . . 7-9 Design Example . . . . . . . . . . . . . . . . . . 7-10 References . . . . . . . . . . . . . . . . . . . . . . 7 Robust Gain-Scheduled Controllers 7-2 Gain scheduling is a widely used technique for controlling certain classes of nonlinear or linear time-varying systems. Rather than seeking a single robust LTI controller for the entire operating range, gain scheduling consists in designing an LTI controller for each operating point and in switching controller when the operating conditions change. This section presents systematic tools to design gain-scheduled H ∞ controllers for linear parameter-dependent systems. These tools are applicable to time-varying and/or nonlinear systems whose linearized dynamics are reasonably well approximated by affine parameter-dependent models. Gain-Scheduled Control 7-3 Gain-Scheduled Control The synthesis technique discussed below is applicable to affine parameter-dependent plants with equations where p(t) = (p 1 (t), . . ., p n (t)), is a time-varying vector of physical parameters (velocity, angle of attack, stiffness,. . .) and A( . ), B 1 ( . ), C 1 ( . ), D 11 ( . ) are affine functions of p(t). This is a simple model of systems whose dynamical equations depend on physical coefficients that vary during operation. When these coefficients undergo large variations, it is often impossible to achieve high performance over the entire operating range with a single robust LTI controller. Provided that the parameter values are measured in real time, it is then desirable to use controllers that incorporate such measurements to adjust to the current operating conditions [4]. Such controllers are said to be scheduled by the parameter measurements. This control strategy typically achieves higher performance in the face of large variations in operating conditions. Note that p(t) may include part of the state vector x itself provided that the corresponding states are accessible to measurement. If the parameter vector p(t) takes values in a box of R n with corners , the plant SYSTEM matrix P .,p() ? ? ? ? ? x · = Ap()xB 1 p()wB 2 u++ z = C 1 p()xD 11 p()wD 12 u y = C 2 xD 21 wD 22 u++ pi pi t() pi≤≤ Π i {} i 1= N N 2 n =() Sp() := Ap() B 1 p() B 2 C 1 p()D 11 p()D 12 C 2 D 21 D 22?? ?? ?? 7 Robust Gain-Scheduled Controllers 7-4 ranges in a matrix polytope with vertices S(Π i ). Specifically, given any convex decomposition of p over the corners of the parameter box, the SYSTEM matrix S(p) is given by S(p) = α 1 S(Π 1 ) + . . . + α N S(Π N ) This suggests seeking parameter-dependent controllers with equations and having the following vertex property: Given the convex decomposition of the current parameter value p(t), the values of A K (p), B K (p), . . . are derived from the values A K (Π i ), B K (Π i ), . . . at the corners of the parameter box by In other words, the controller state-space matrices at the operating point p(t) are obtained by convex interpolation of the LTI vertex controllers This yields a smooth scheduling of the controller matrices by the parameter measurements p(t). pt() α 1 Π 1 …α N Π N , α i 0, α i i 1= N ∑ 1=≥++= K .,p() ζ A K = p()ζ B K + p()y uC K = p()ζ D K + p()y ? ? ? p t() α i Π i i 1= N ∑ = A K p()B K p() C K p()D K p() ?? ?? ?? α i i 1= N ∑ A K Π i ()B K Π i () C K Π i ()D K Π i () ?? ?? ?? = K i := A K Π i ()B K Π i () C K Π i ()D K Π i () ?? ?? ?? Gain-Scheduled Control 7-5 For this class of controllers, consider the following H ∞ -like synthesis problem relative to the interconnection of Figure 7-1: Figure 7-1: Gain-scheduled H ∞ problem Design a gain-scheduled controller K(., p) satisfying the vertex property and such that The closed-loop system is stable for all admissible parameter trajectories p(t) The worst-case closed-loop RMS gain from w to z does not exceed some level γ > 0. Using the notion of quadratic H ∞ performance to enforce the RMS gain constraint, and observing that the closed-loop system is a polytopic system, this synthesis problem can be reduced to the following LMI problem [3, 2] Find two symmetric matrices R and S such that (7-1) u y P(. ,p) K(., p) w z p(t) N 12 0 0 I ?? ?? ?? T A i RRA i T + RC 1i T B 1i C 1i R γI– D 11i B 1 T D 11i T γI– ?? ?? ?? N 12 0 0 I ?? ?? ?? 0, i 1 … N,,=< 7 Robust Gain-Scheduled Controllers 7-6 (7-2) (7-3) where and N 12 and N 21 are bases of the null spaces of ( ) and (C 2 , D 21 ), respectively. Remark: Requiring the plant matrices B 2 , C 2 , D 12 , D 21 to be independent of p(t) may be restrictive in practice. When B 2 and D 12 depend on p(t), a simple trick to enforce these requirements consists of filtering u through a low-pass filter with sufficiently large bandwidth. Appending this filter to the plant rejects the parameter dependence of B 2 and D 12 into the A matrix of the augmented plant. Similarly, parameter dependences in C 2 and D 21 can be eliminated by filtering the output (see [1] for details). N 21 0 0 I ?? ?? ?? T A i T SSA+ SB 1i C 1i T B 1i T S γI– D 11i T C 1i D 11i γI– ?? ?? ?? N 21 0 0 I ?? ?? ?? 0, i 1 … N,,=< RI IS?? ?? ?? 0≥ A i B 1i C 1i D 11i?? ?? ?? := A Π i ()B 1 Π i () C 1 Π i ()D 11 Π i () ?? ?? ?? B 2 T D 12 T ,() Synthesis of Gain-Scheduled H Controllers 7-7 Synthesis of Gain-Scheduled H ∞ Controllers The synthesis of gain-scheduled H ∞ controllers is performed by the function hinfgs. Since such controllers are characterized by their vertex values, both the plant and the controller are treated as polytopic models by hinfgs. The calling sequence is [gopt,pdK] = hinfgs(pdP,r) Here pdP is an affine or polytopic model of the plant (see psys) and the two- entry vector r specifies the dimensions of the D 22 matrix (set r=[p2 m2] if y ∈ R p 2 and u ∈ R m 2 ). On output, gopt is the best achievable RMS gain and pdK is the polytopic system consisting of the vertex controllers. Given any value p of the parameter vector p(t), the corresponding state-space parameters of the gain-scheduled controller are returned in SYSTEM matrix format by c = polydec(pv,p) Kp = psinfo(pdK,'eval',c) The function polydec computes the convex decomposition and returns c = (α 1 , . . ., α N ). The controller state-space data at time t is then computed by psinfo. Note that polydec ensures the continuity of the polytopic coordinates α i as a function of p, thereby guaranteeing the continuity of the controller state-space matrices along parameter trajectories. K i = A K Π i ()B K Π i () C K Π i ()D K Π i () ?? ?? ?? K i = A K Π i ()B K Π i () C K Π i ()D K Π i () ?? ?? ?? Kp() = A K p()B K p() C K p()D K p() ?? ?? ?? p t() α i Π i i 1= N ∑ = Kp() α i K i i 1= N ∑ = 7 Robust Gain-Scheduled Controllers 7-8 Given the polytopic controller representation pdK, a polytopic model of the closed-loop system is formed by pcl = slft(pdP,pdK) and evaluated for a given parameter value p of p(t) by pclt = psinfo(pcl,'eval',polydec(pv,p)) Simulation of Gain-Scheduled Control Systems 7-9 Simulation of Gain-Scheduled Control Systems Being time-varying systems, gain-scheduled control systems cannot be simulated with splot. The function pdsimul is provided to simulate the time response of such systems along a given parameter trajectory. With pdsimul, you must first define the parameter trajectory and the input signal as two functions of time with syntax p=trajfun(t) and u=inputfun(t). You can then plot the corresponding time response by pdsimul(pcl,'trajfun',tf,'inputfun') where pcl is the polytopic representation of the closed-loop system and tf is the desired duration of the simulation. If 'inputfun' is omitted, the step response is plotted by default. See the next paragraph for an example. 7 Robust Gain-Scheduled Controllers 7-10 Design Example This example is drawn from [2] and deals with the design of a gain-scheduled autopilot for the pitch axis of a missile. Type misldem to run the corresponding demo. The missile dynamics are highly dependent on the angle of attack α, the speed V, and the altitude H. These three parameters undergo large variations during operation. A simple model of the linearized dynamics of the missile is the parameter-dependent model (7-4) where a zv is the normalized vertical acceleration, q the pitch rate, δm the fin deflection, and Z α , M α are aerodynamical coefficients depending on α, V, and H. These two coefficients are “measured” in real time. As V, H, and α vary in V ∈ [0.5, 4] Mach, H ∈ [0, 18000] m, α ∈ [0, 40] degrees during operation, the coefficients Z α and M α range in Z α ∈ [0.5, 4], M α ∈ [0, 106]. Our goal is to control the vertical acceleration a zv over this operating range. The stringent performance specifications (settling time < 0.5 s) and the bandwidth limitation imposed by unmodeled high frequency dynamics make gain scheduling desirable in this context. The control structure is sketched in Figure 7-2. To enforce the performance and robustness requirements, we can use the loop-shaping criterion (7-5) G α · q · ?? ?? ?? Zα– 1 Mα– 0?? ?? ?? α q?? ?? ?? 0 1?? ?? ?? δ m += a zv q ?? ?? ?? 1– 0 01?? ?? ?? α q?? ?? ?? = ? ? ? ? ? ? ? ? ? W 1 S W 2 KS ∞ 1< Design Example 7-11 Note that S = (I + GK) –1 is now a time-varying system and that the H ∞ norm must be understood in terms of input/output RMS gain. Adequate shaping filters are derived with magshape as Figure 7-2: Missile autopilot The gain-scheduled controller synthesis parallels the standard loop-shaping procedure described in Chapter 6. First enter the parameter-dependent model (7-4) of the missile pitch axis by: % parameter range: Zmin=.5; Zmax=4; Mmin=0; Mmax=106; pv = pvec('box',[Zmin Zmax ; Mmin Mmax]) % affine model: s0 = ltisys([0 1;0 0],[0;1],[-1 0;0 1],[0;0]) s1 = ltisys([-1 0;0 0],[0;0],zeros(2),[0;0],0) % Z_al s2 = ltisys([0 0;-1 0],[0;0],zeros(2),[0;0],0) % M_al pdG = psys(pv,[s0 s1 s2]) For loop-shaping purposes, we must form the augmented plant associated with the criterion (7-5). This is done with sconnect and smult as follows: W 1 s() 2.01 s 0.201+ -----------------------= W 2 s() 9.678s 3 0.029s 2 + s 3 1.206 10 4 s 2 1.136 10 7 s 1.066 10 10 ×+×+×+ --------------------------------------------------------------------------------------------------------------------------------= r a zv + ? K G autopilot missile e u q 7 Robust Gain-Scheduled Controllers 7-12 % form the plant interconnection [pdP,r] = sconnect('r','e=r-GP1;K','K:e;G(2)','G:K',pdG); % append the shaping filters Paug = smult(pdP,sdiag(w1,w2,eye(2))) Note that pdP and Paug are polytopic models at this stage. We are now ready to perform the gain-scheduled controller synthesis with hinfgs: [gopt,pdK] = hinfgs(Paug,r) gopt = 0.205 Our specifications are achievable since the best performance γ opt = 0.205 is smaller than 1, and the polytopic description of a controller with such performance is returned in pdK. To validate the design, form the closed-loop system with pCL = slft(pdP,pdK) You can now simulate the step response of the gain-scheduled system along particular parameter trajectories. For the sake of illustration, consider the spiral trajectory Z α (t) = 2.25 + 1.70 e –4t cos(100 t) M α (t) = 50 + 49 e –4t sin(100 t) shown in Figure 7-3. After defining this trajectory is a function spiralt.m: function p = spiral(t) p = [2.25 + 1.70*exp(-4*t).*cos(100*t) ; ... 50 + 49*exp(-4*t).*sin(100*t)]; you can plot the closed-loop step response along p(t) by [t,x,y]=pdsimul(pCL,'spiralt',0.5) plot(t,1-y(:,1)) Design Example 7-13 The function pdsimul returns the output trajectories y(t) = (e(t), u(t)) T and the response a zv (t) is deduced by a zv (t) = 1 – e(t) The second command plots a zv (t) versus time as shown in Figure 7-4. Figure 7-3: Parameter trajectory 0.5 1 1.5 2 2.5 3 3.5 4 0 10 20 30 40 50 60 70 80 90 100 Z_alpha M_alpha Parameter trajectory (duration : 1 s) 7 Robust Gain-Scheduled Controllers 7-14 Figure 7-4: Corresponding step response 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.2 0.4 0.6 0.8 1 1.2 azv step response of the gain?scheduled autopilot References 7-15 References [1] Apkarian, P., and P. Gahinet,“A Convex Characterization of Gain-Scheduled H ∞ Controllers,” to appear in IEEE Trans. Aut. Contr., 1995. [2] Apkarian, P., P. Gahinet, and G. Becker, “Self-Scheduled H ∞ Control of Linear Parameter-Varying Systems,” to appear in Automatica, 1995. [3] Becker, G., Packard, P., “Robust Performance of Linear Parametrically Varying Systems Using Parametrically Dependent Linear Feedback,” Systems and Control Letters, 23 (1994), pp. 205–215. [4] Packard, A., “Gain Scheduling via Linear Fractional Transformations,” Syst. Contr. Letters, 22 (1994), pp. 79–92. 7 Robust Gain-Scheduled Controllers 7-16 8 The LMI Lab Background and Terminology . . . . . . . . . . . . 8-3 Overview of the LMI Lab . . . . . . . . . . . . . . 8-6 Specifying a System of LMIs . . . . . . . . . . . . . 8-8 Retrieving Information . . . . . . . . . . . . . . 8-21 LMI Solvers . . . . . . . . . . . . . . . . . . . 8-22 From Decision to Matrix Variables and Vice Versa . . 8-28 Validating Results . . . . . . . . . . . . . . . . 8-29 Modifying a System of LMIs . . . . . . . . . . . . 8-30 Advanced Topics . . . . . . . . . . . . . . . . . 8-33 References . . . . . . . . . . . . . . . . . . . . 8-44 8 The LMI Lab 8-2 The LMI Lab is a high-performance package for solving general LMI problems. It blends simple tools for the specification and manipulation of LMIs with powerful LMI solvers for three generic LMI problems. Thanks to a structure-oriented representation of LMIs, the various LMI constraints can be described in their natural block-matrix form. Similarly, the optimization variables are specified directly as matrix variables with some given structure. Once an LMI problem is specified, it can be solved numerically by calling the appropriate LMI solver. The three solvers feasp, mincx, and gevp constitute the computational engine of the LMI Control Toolbox. Their high performance is achieved through C-MEX implementation and by taking advantage of the particular structure of each LMI. The LMI Lab offers tools to Specify LMI systems either symbolically with the LMI Editor or incrementally with the lmivar and lmiterm commands Retrieve information about existing systems of LMIs Modify existing systems of LMIs Solve the three generic LMI problems (feasibility problem, linear objective minimization, and generalized eigenvalue minimization) Validate results This chapter gives a tutorial introduction to the LMI Lab as well as more advanced tips for making the most out of its potential. The tutorial material is also covered by the demo lmidem. Background and Terminology 8-3 Background and Terminology Any linear matrix inequality can be expressed in the canonical form L(x) = L 0 + x 1 L 1 + . . . + x N L N < 0 where L 0 , L 1 , . . . , L N are given symmetric matrices x = (x 1 , . . . , x N ) T ∈ R N is the vector of scalar variables to be determined. We refer to x 1 , . . . , x N as the decision variables. The names “design variables” and “optimization variables” are also found in the literature. Even though this canonical expression is generic, LMIs rarely arise in this form in control applications. Consider for instance the Lyapunov inequality (8-1) where and the variable is a symmetric matrix. Here the decision variables are the free entries x 1 , x 2 , x 3 of X and the canonical form of this LMI reads (8-2) Clearly this expression is less intuitive and transparent than (8-1). Moreover, the number of matrices involved in (8-2) grows roughly as n 2 /2 if n is the size of the A matrix. Hence, the canonical form is very inefficient from a storage viewpoint since it requires storing o(n 2 /2) matrices of size n when the single n-by-n matrix A would be sufficient. Finally, working with the canonical form is also detrimental to the efficiency of the LMI solvers. For these various reasons, the LMI Lab uses a structured representation of LMIs. For instance, the expression A T X + XA in the Lyapunov inequality (8-1) is explicitly described as a function of the matrix variable X, and only the A matrix is stored. A T XXA0<+ A 1– 2 02–?? ?? ?? = X x 1 x 2 x 2 x 3?? ?? ?? = x 1 2– 2 20?? ?? ?? x 2 03– 3– 4?? ?? ?? x 3 00 04–?? ?? ?? 0<++ 8 The LMI Lab 8-4 In general, LMIs assume a block matrix form where each block is an affine combination of the matrix variables. As a fairly typical illustration, consider the following LMI drawn from H ∞ theory (8-3) where A, B, C, D, N are given matrices and the problem variables are X = X T ∈ R n×n and γ ∈ R. We use the following terminology to describe such LMIs: N is called the outer factor, and the block matrix is called the inner factor. The outer factor needs not be square and is often absent. X and γ are the matrix variables of the problem. Note that scalars are considered as 1-by-1 matrices. The inner factor L(X, γ) is a symmetric block matrix, its block structure being characterized by the sizes of its diagonal blocks. By symmetry, L(X, γ) is entirely specified by the blocks on or above the diagonal. Each block of L(X, γ) is an affine expression in the matrix variables X and γ. This expression can be broken down into a sum of elementaryterm terms. For instance, the block (1,1) contains two elementary terms: A T X and XA. Terms are either constant or variable. Constant terms are fixed matrices like B and D above. Variable terms involve one of the matrix variables, like XA, XC T , and –γI above. The LMI (8-3) is specified by the list of terms in each block, as is any LMI regardless of its complexity. N T A T XXA+ XC T B CX γI– D B T D T γI– ?? ?? ?? N 0< LXγ,() A T XXA+ XC T B CX γI– D B T D T γI– ?? ?? ?? = Background and Terminology 8-5 As for the matrix variables X and γ, they are characterized by their dimensions and structure. Common structures include rectangular unstructured, symmetric, skew-symmetric, and scalar. More sophisticated structures are sometimes encountered in control problems. For instance, the matrix variable X could be constrained to the block-diagonal structure Another possibility is the symmetric Toeplitz structure Summing up, structured LMI problems are specified by declaring the matrix variables and describing the term content of each LMI. This term-oriented description is systematic and accurately reflects the specific structure of the LMI constraints. There is no built-in limitation on the number of LMIs that can be specified or on the number of blocks and terms in any given LMI. LMI systems of arbitrary complexity can therefore, be defined in the LMI Lab. X x 1 00 0 x 2 x 3 0 x 3 x 4?? ?? ?? ?? = X x 1 x 2 x 3 x 2 x 1 x 2 x 3 x 2 x 1?? ?? ?? ?? = 8 The LMI Lab 8-6 Overview of the LMI Lab The LMI Lab offers tools to specify, manipulate, and numerically solve LMIs. Its main purpose is to Allow for straightforward description of LMIs in their natural block-matrix form Provide easy access to the LMI solvers (optimization codes) Facilitate result validation and problem modification The structure-oriented description of a given LMI system is stored as a single vector called the internal representation and generically denoted by LMISYS in the sequel. This vector encodes the structure and dimensions of the LMIs and matrix variables, a description of all LMI terms, and the related numerical data. It must be stressed that users need not attempt to read or understand the content of LMISYS since all manipulations involving this internal representation can be performed in a transparent manner with LMI-Lab tools. The LMI Lab supports the following functionalities: Specification of a System of LMIs LMI systems can be either specified as symbolic matrix expressions with the interactive graphical user interface lmiedit, or assembled incrementally with the two commands lmivar and lmiterm. The first option is more intuitive and transparent while the second option is more powerful and flexible. Information Retrieval The interactive function lmiinfo answers qualitative queries about LMI systems created with lmiedit or lmivar and lmiterm. You can also use lmiedit to visualize the LMI system produced by a particular sequence of lmivar/lmiterm commands. Solvers for LMI Optimization Problems General-purpose LMI solvers are provided for the three generic LMI problems defined on page 1-5. These solvers can handle very general LMI systems and matrix variable structures. They return a feasible or optimal vector of decision variables x*. The corresponding values of the matrix variables are given by the function dec2mat. X 1 * … X K * ,, Overview of the LMI Lab 8-7 Result Validation The solution x* produced by the LMI solvers is easily validated with the functions evallmi and showlmi. This allows a fast check and/or analysis of the results. With evallmi, all variable terms in the LMI system are evaluated for the value x* of the decision variables. The left- and right-hand sides of each LMI then become constant matrices that can be displayed with showlmi. Modification of a System of LMIs An existing system of LMIs can be modified in two ways: An LMI can be removed from the system with dellmi. A matrix variable X can be deleted using delmvar. It can also be instantiated, that is, set to some given matrix value. This operation is performed by setmvar and allows, for example, to fix some variables and solve the LMI problem with respect to the remaining ones. 8 The LMI Lab 8-8 Specifying a System of LMIs The LMI Lab can handle any system of LMIs of the form N T L(X 1 , . . . , X K ) N < MT R(X 1 , . . . , X K ) M where X 1 , . . . , X K are matrix variables with some prescribed structure The left and right outer factors N and M are given matrices with identical dimensions The left and right inner factors L( . ) and R( . ) are symmetric block matrices with identical block structures, each block being an affine combination of X 1 , . . . , X K and their transposes. Important: Throughout this chapter, “left-hand side” refers to what is on the “smaller” side of the inequality, and “right-hand side” to what is on the “larger” side. Accordingly, X is called the right-hand side and 0 the left-hand side of the LMI 0 < X even when this LMI is written as X > 0. The specification of an LMI system involves two steps: 1 Declare the dimensions and structure of each matrix variable X 1 , . . . , X K . 2 Describe the term content of each LMI. This process creates the so-called internal representation of the LMI system. This computer description of the problem is used by the LMI solvers and in all subsequent manipulations of the LMI system. It is stored as a single vector which we consistently denote by LMISYS in the sequel (for the sake of clarity). There are two ways of generating the internal description of a given LMI system: (1) by a sequence of lmivar/lmiterm commands that build it incrementally, or (2) via the LMI Editor lmiedit where LMIs can be specified directly as symbolic matrix expressions. Though somewhat less flexible and powerful than the command-based description, the LMI Editor is more straightforward to use, hence particularly well-suited for beginners. Thanks to its coding and decoding capabilities, it also constitutes a good tutorial introduction to lmivar and lmiterm. Accordingly, beginners may elect to skip Specifying a System of LMIs 8-9 the subsections on lmivar and lmiterm and to concentrate on the GUI-based specification of LMIs with lmiedit. A Simple Example The following tutorial example is used to illustrate the specification of LMI systems with the LMI Lab tools. Run the demo lmidem to see a complete treatment of this example. Example 8.1. Consider a stable transfer function (8-4) with four inputs, four outputs, and six states, and consider the set of input/output scaling matrices D with block-diagonal structure (8-5) The following problem arises in the robust stability analysis of systems with time-varying uncertainty [4 ]: Find, if any, a scaling D of structure (8-5) such that the largest gain across frequency of D G(s) D –1 is less than one. Gs() CsI A–() 1– B= D d 1 000 0 d 1 00 00d 2 d 3 00d 4 d 5 ?? ?? ?? ?? = 8 The LMI Lab 8-10 This problem has a simple LMI formulation: there exists an adequate scaling D if the following feasibility problem has solutions: Find two symmetric matrices X ∈ R 6×6 and S = D T D ∈ R 4×4 such that (8-6) (8-7) (8-8) The LMI system (8-6)–(8-8) can be described with the LMI Editor as outlined below. Alternatively, its internal description can be generated with lmivar and lmiterm commands as follows: setlmis([]) X=lmivar(1,[6 1]) S=lmivar(1,[2 0;2 1]) % 1st LMI lmiterm([1 1 1 X],1,A,'s') lmiterm([1 1 1 S],C',C) lmiterm([1 1 2 X],1,B) lmiterm([1 2 2 S], 1,1) % 2nd LMI lmiterm([ 2 1 1 X],1,1) % 3rd LMI lmiterm([ 3 1 1 S],1,1) lmiterm([3 1 1 0],1) LMISYS = getlmis Here the lmivar commands define the two matrix variables X and S while the lmiterm commands describe the various terms in each LMI. Upon completion, A T XXAC T SC++ XB B T XS–?? ?? ?? < 0 X 0> S 1> Specifying a System of LMIs 8-11 getlmis returns the internal representation LMISYS of this LMI system. The next three subsections give more details on the syntax and usage of these various commands. More information on how the internal representation is updated by lmivar/lmiterm can also be found in “How It All Works” on page 8-18. setlmis and getlmis The description of an LMI system should begin with setlmis and end with getlmis. The function setlmis initializes the LMI system description. When specifying a new system, type setlmis([]) To add on to an existing LMI system with internal representation LMIS0, type setlmis(LMIS0) When the LMI system is completely specified, type LMISYS = getlmis This returns the internal representation LMISYS of this LMI system. This MATLAB description of the problem can be forwarded to other LMI-Lab functions for subsequent processing. The command getlmis must be used only once and after declaring all matrix variables and LMI terms. lmivar The matrix variables are declared one at a time with lmivar and are characterized by their structure. To facilitate the specification of this structure, the LMI Lab offers two predefined structure types along with the means to describe more general structures: Type 1: Symmetric block diagonal structure. This corresponds to matrix variables of the form X D 1 0 … 0 0 D 2 0 0 … 0 D r ?? ?? ?? ?? = … … . . . . . . . . . 8 The LMI Lab 8-12 where each diagonal block D j is square and is either zero, a full symmetric matrix, or a scalar matrix D j = d × I, d ∈ R This type encompasses ordinary symmetric matrices (single block) and scalar variables (one block of size one). Type 2: Rectangular structure. This corresponds to arbitrary rectangular matrices without any particular structure. Type 3: General structures. This third type is used to describe more sophisticated structures and/or correlations between the matrix variables. The principle is as follows: each entry of X is specified independently as either 0, x n , or –x n where x n denotes the n-th decision variable in the problem. For details on how to use Type 3, see “Structured Matrix Variables” on page 8-33 below as well as the lmivar entry of the “Command Reference” chapter. In “Example 8.1” on page 8-9, the matrix variables X and S are of Type 1. Indeed, both are symmetric and S inherits the block-diagonal structure (8-5) of D. Specifically, S is of the form After initializing the description with the command setlmis([]), these two matrix variables are declared by lmivar(1,[6 1]) % X lmivar(1,[2 0;2 1]) % S In both commands, the first input specifies the structure type and the second input contains additional information about the structure of the variable: For a matrix variable X of Type 1, this second input is a matrix with two columns and as many rows as diagonal blocks in X. The first column lists the S s 1 000 0 s 1 00 00s 2 s 3 00s 3 s 4 ?? ?? ?? ?? = Specifying a System of LMIs 8-13 sizes of the diagonal blocks and the second column specifies their nature with the following convention: 1 → full symmetric block 0 → scalar block –1 → zero block In the second command, for instance, [2 0;2 1] means that S has two diagonal blocks, the first one being a 2-by-2 scalar block and the second one a 2?βψ?2 full block. For matrix variables of Type 2, the second input of lmivar is a two-entry vector listing the row and column dimensions of the variable. For instance, a 3-by-5 rectangular matrix variable would be defined by lmivar(2,[3 5]) For convenience, lmivar also returns a “tag” that identifies the matrix variable for subsequent reference. For instance, X and S in “Example 8.1” could be defined by X = lmivar(1,[6 1]) S = lmivar(1,[2 0;2 1]) The identifiers X and S are integers corresponding to the ranking of X and S in the list of matrix variables (in the order of declaration). Here their values would be X=1 and S=2. Note that these identifiers still point to X and S after deletion or instantiation of some of the matrix variables. Finally, lmivar can also return the total number of decision variables allocated so far as well as the entry-wise dependence of the matrix variable on these decision variables (see the lmivar entry of the “Command Reference” chapter for more details). lmiterm After declaring the matrix variables with lmivar, we are left with specifying the term content of each LMI. Recall that LMI terms fall into three categories: The constant terms, i.e., fixed matrices like I in the left-hand side of the LMI S > I The variable terms, i.e., terms involving a matrix variable. For instance, A T X and C T SC in (8-6). Variable terms are of the form PXQ where X is a variable and P, Q are given matrices called the left and right coefficients, respectively. 8 The LMI Lab 8-14 The outer factors The following rule should be kept in mind when describing the term content of an LMI: Important: Specify only the terms in the blocks on or above the diagonal. The inner factors being symmetric, this is sufficient to specify the entire LMI. Specifying all blocks results in the duplication of off-diagonal terms, hence in the creation of a different LMI. Alternatively, you can describe the blocks on or below the diagonal. LMI terms are specified one at a time with lmiterm. For instance, the LMI is described by lmiterm([1 1 1 1],1,A,'s') lmiterm([1 1 1 2],C',C) lmiterm([1 1 2 1],1,B) lmiterm([1 2 2 2], 1,1) These commands successively declare the terms A T X + XA, C T SC, XB, and –S. In each command, the first argument is a four-entry vector listing the term characteristics as follows: The first entry indicates to which LMI the term belongs. The value m means “left-hand side of the m-th LMI,” and m means “right-hand side of the m-th LMI” The second and third entries identify the block to which the term belongs. For instance, the vector [1 1 2 1] indicates that the term is attached to the (1, 2) block The last entry indicates which matrix variable is involved in the term. This entry is 0 for constant terms, k for terms involving the k-th matrix variable A T XXAC T SC++ XB B T XS–?? ?? ?? < 0 Specifying a System of LMIs 8-15 X k , and k for terms involving (here X and S are first and second variables in the order of declaration). Finally, the second and third arguments of lmiterm contain the numerical data (values of the constant term, outer factor, or matrix coefficients P and Q for variable terms PXQ or PX T Q). These arguments must refer to existing MATLAB variables and be real-valued. See“Complex-Valued LMIs” on page 8-35 for the specification of LMIs with complex-valued coefficients. Some shorthand is provided to simplify term specification. First, blocks are zero by default. Second, in diagonal blocks the extra argument 's' allows you to specify the conjugated expression AXB + B T X T A T with a single lmiterm command. For instance, the first command specifies A T X + XA as the “symmetrization” of XA. Finally, scalar values are allowed as shorthand for scalar matrices, i.e., matrices of the form αI with α scalar. Thus, a constant term of the form αI can be specified as the “scalar” α. This also applies to the coefficients P and Q of variable terms. The dimensions of scalar matrices are inferred from the context and set to 1 by default. For instance, the third LMI S > I in “Example 8.3” on page 8-33 is described by Recall that by convention S is considered as the right-hand side of the inequality, which justifies the –3 in the first command. Finally, to improve readability it is often convenient to attach an identifier (tag) to each LMI and matrix variable. The variable identifiers are returned by lmivar and the LMI identifiers are set by the function newlmi. These identifiers can be used in lmiterm commands to refer to a given LMI or matrix variable. For the LMI system of “Example 8.1”, this would look like: setlmis([]) X = lmivar(1,[6 1]) S = lmivar(1,[2 0;2 1]) BRL = newlmi lmiterm([BRL 1 1 X],1,A,'s') lmiterm([BRL 1 1 S],C',C) lmiterm([BRL 1 2 X],1,B) lmiterm([BRL 2 2 S], 1,1) lmiterm([–3 1 1 2],1,1) % 1*S*1 = S lmiterm([3 1 1 0],1) % 1*I = I X k T 8 The LMI Lab 8-16 Xpos = newlmi lmiterm([-Xpos 1 1 X],1,1) Slmi = newlmi lmiterm([-Slmi 1 1 S],1,1) lmiterm([Slmi 1 1 0],1) LMISYS = getlmis Here the identifiers X and S point to the variables X and S while the tags BRL, Xpos, and Slmi point to the first, second, and third LMI, respectively. Note that Xpos refers to the right-hand side of the second LMI. Similarly, X would indicate transposition of the variable X. The LMI Editor lmiedit The LMI Editor lmiedit is a graphical user interface (GUI) to specify LMI systems in a straightforward symbolic manner. Typing lmiedit calls up a window with several editable text areas and various pushbuttons. To specify your LMI system, 1 Declare each matrix variable (name and structure) in the upper half of the worksheet. The structure is characterized by its type (S for symmetric block diagonal, R for unstructured, and G for other structures) and by an additional “structure” matrix. This matrix contains specific information about the structure and corresponds to the second argument of lmivar (see “lmivar” on page 8-11 for details). Please use one line per matrix variable in the text editing areas. Specifying a System of LMIs 8-17 2 Specify the LMIs as MATLAB expressions in the lower half of the worksheet. For instance, the LMI is entered by typing [a'*x+x*a x*b; b'*x 1] < 0 if x is the name given to the matrix variable X in the upper half of the worksheet. The left- and right-hand sides of the LMIs should be valid MATLAB expressions. Once the LMI system is fully specified, the following tasks can be performed by pressing the corresponding button: Visualize the sequence of lmivar/lmiterm commands needed to describe this LMI system (view commands buttons). Conversely, the LMI system defined by a particular sequence of lmivar/lmiterm commands can be displayed as a MATLAB expression by clicking on the describe... buttons. Beginners can use this facility as a tutorial introduction to the lmivar and lmiterm commands. Save the symbolic description of the LMI system as a MATLAB string (save button). This description can be reloaded later on by pressing the load button. Read a sequence of lmivar/lmiterm commands from a file (read button). You can then click on “describe the matrix variables” or “describe the LMIs...” to visualize the symbolic expression of the LMI system specified by these commands. The file should describe a single LMI system but may otherwise contain any sequence of MATLAB commands. This feature is useful for code validation and debugging. Write in a file the sequence of lmivar/lmiterm commands needed to describe a particular LMI system (write button). This is helpful to develop code and prototype MATLAB functions based on the LMI Lab. A T XXA+ XB B T XI–?? ?? ?? < 0 8 The LMI Lab 8-18 Generate the internal representation of the LMI system by pressing create. The result is written in a MATLAB variable named after the LMI system (if the “name of the LMI system” is set to mylmi, the internal representation is written in the MATLAB variable mylmi). Note that all LMI-related data should be defined in the MATLAB workspace at this stage. The internal representation can be passed directly to the LMI solvers or any other LMI Lab function. As with lmiterm, you can use various shortcuts when entering LMI expressions at the keyboard. For instance, zero blocks can be entered simply as 0 and need not be dimensioned. Similarly, the identity matrix can be entered as 1 without dimensioning. Finally, upper diagonal LMI blocks need not be fully specified. Rather, you can just type (*) in place of each such block. Though fairly general, lmiedit is not as flexible as lmiterm and the following limitations should be kept in mind: Parentheses cannot be used around matrix variables. For instance, the expression (a*x+b)'*c + c'*(a*x+b) is invalid when x is a variable name. By contrast, (a+b)'*x + x'*(a+b) is perfectly valid. Loops and if statements are ignored. When turning lmiterm commands into a symbolic description of the LMI system, an error is issued if the first argument of lmiterm cannot be evaluated. Use the LMI and variable identifiers supplied by newlmi and lmivar to avoid such difficulties. Figure 8-1 shows how to specify the feasibility problem of “Example 8.1” on page 8-9 with lmiedit. How It All Works Users familiar with MATLAB may wonder how lmivar and lmiterm physically update the internal representation LMISYS since LMISYS is not an argument to these functions. In fact, all updating is performed through global variables for maximum speed. These global variables are initialized by setlmis, cleared by Specifying a System of LMIs 8-19 getlmis, and not visible in the workspace. Even though this artifact is transparent from the user's viewpoint, be sure to Invoke getlmis only once and after completely specifying the LMI system Refrain from using the command clear global before the LMI system description is ended with getlmis 8 The LMI Lab 8-20 Figure 8-1: Graphical user interface lmiedit Retrieving Information 8-21 Retrieving Information Recall that the full description of an LMI system is stored as a single vector called the internal representation. The user should not attempt to read or retrieve information directly from this vector. Three functions called lmiinfo, lminbr, and matnbr are provided to extract and display all relevant information in a user-readable format. lmiinfo lmiinfo is an interactive facility to retrieve qualitative information about LMI systems. This includes the number of LMIs, the number of matrix variables and their structure, the term content of each LMI block, etc. To invoke lmiinfo, enter lmiinfo(LMISYS) where LMISYS is the internal representation of the LMI system produced by getlmis. lminbr and matnbr These two functions return the number of LMIs and the number of matrix variables in the system. To get the number of matrix variables, for instance, enter matnbr(LMISYS) 8 The LMI Lab 8-22 LMI Solvers LMI solvers are provided for the following three generic optimization problems (here x denotes the vector of decision variables, i.e., of the free entries of the matrix variables X 1 , . . . , X K ): Feasibility problem Find x ∈ R N (or equivalently matrices X 1 , . . . , X K with prescribed structure) that satisfies the LMI system A(x) < B(x) The corresponding solver is called feasp. Minimization of a linear objective under LMI constraints Minimize c T x over x ∈ R N subject to A(x) < B(x) The corresponding solver is called mincx. Generalized eigenvalue minimization problem Minimize λ over x ∈ R N subject to C(x) < D(x) 0 < B(x) A(x) < λB(x). The corresponding solver is called gevp. Note that A(x) < B(x) above is a shorthand notation for general structured LMI systems with decision variables x = (x 1 , . . . , x N ). The three LMI solvers feasp, mincx, and gevp take as input the internal representation LMISYS of an LMI system and return a feasible or optimizing value x* of the decision variables. The corresponding values of the matrix variables X 1 , . . . , X K are derived from x* with the function dec2mat. These solvers are C-MEX implementations of the polynomial-time Projective Algorithm Projective Algorithm of Nesterov and Nemirovski [3, 2]. For generalized eigenvalue minimization problems, it is necessary to distinguish between the standard LMI constraints C(x) < D(x) and the linear-fractional LMIs A(x) < λB(x) LMI Solvers 8-23 attached to the minimization of the generalized eigenvalue λ. When using gevp, you should follow these three rules to ensure proper specification of the problem: Specify the LMIs involving λ as A(x) < B(x) (without the λ) Specify them last in the LMI system. gevp systematically assumes that the last L LMIs are linear-fractional if L is the number of LMIs involving λ Add the constraint 0 < B(x) or any other constraint that enforces it. This positivity constraint is required for well-posedness of the problem and is not automatically added by gevp (see the “Command Reference” chapter for details). An initial guess xinit for x can be supplied to mincx or gevp. Use mat2dec to derive xinit from given values of the matrix variables X 1 , . . . , X K . Finally, various options are available to control the optimization process and the solver behavior. These options are described in detail in the “Command Reference” chapter. The following example illustrates the use of the mincx solver. Example 8.2. Consider the optimization problem Minimize Trace(X) subject to (8-9) with data It can be shown that the minimizer X* is simply the stabilizing solution of the algebraic Riccati equation A T X + XA + XBB T X + Q = 0 This solution can be computed directly with the Riccati solver care and compared to the minimizer returned by mincx. A T X XA XBB T XQ++ +< A 1– 2– 1 321 12– 1– ?? ?? ?? ; B 1 0 1 ?? ?? ?? ?? ?? ; Q 11– 0 1– 3– 12– 012– 36– ?? ?? ?? === 8 The LMI Lab 8-24 From an LMI optimization standpoint, problem (8-9) is equivalent to the following linear objective minimization problem: (8-10) Since Trace(X) is a linear function of the entries of X, this problem falls within the scope of the mincx solver and can be numerically solved as follows: 1 Define the LMI constraint (8-9) by the sequence of commands setlmis([]) X = lmivar(1,[3 1]) % variable X, full symmetric lmiterm([1 1 1 X],1,a,'s') lmiterm([1 1 1 0],q) lmiterm([1 2 2 0], 1) lmiterm([1 2 1 X],b',1) LMIs = getlmis 2 Write the objective Trace(X) as c T x where x is the vector of free entries of X. Since c should select the diagonal entries of X, it is obtained as the decision vector corresponding to X = I, that is, c = mat2dec(LMIs,eye(3)) Note that the function defcx provides a more systematic way of specifying such objectives (see “Specifying c T x Objectives for mincx” on page 8-38 for details). 3 Call mincx to compute the minimizer xopt and the global minimum copt = c'*xopt of the objective: options = [1e 5,0,0,0,0] Minimize Trace(X) subject to A T XXAQ++XB B T XI–?? ?? ?? < 0 LMI Solvers 8-25 [copt,xopt] = mincx(LMIs,c,options) Here 1e 5 specifies the desired relative accuracy on copt. The following trace of the iterative optimization performed by mincx appears on the screen: Solver for linear objective minimization under LMI constraints Iterations : Best objective value so far 1 2 8.511476 3 13.063640 *** new lower bound: 34.023978 4 15.768450 *** new lower bound: 25.005604 5 17.123012 *** new lower bound: 21.306781 6 17.882558 *** new lower bound: 19.819471 7 18-339853 *** new lower bound: 19.189417 8 18.552558 *** new lower bound: 18.919668 9 18.646811 *** new lower bound: 18.803708 10 18.687324 *** new lower bound: 18.753903 11 18.705715 *** new lower bound: 18.732574 12 18.712175 8 The LMI Lab 8-26 Result: feasible solution of required accuracy best objective value: 18.716695 guaranteed relative accuracy: 9.50e 06 f-radius saturation: 0.000% of R = 1.00e+09 The iteration number and the best value of c T x at the current iteration appear in the left and right columns, respectively. Note that no value is displayed at the first iteration, which means that a feasible x satisfying the constraint (8-10) was found only at the second iteration. Lower bounds on the global minimum of c T x are sometimes detected as the optimization progresses. These lower bounds are reported by the message *** new lower bound: xxx Upon termination, mincx reports that the global minimum for the objective Trace(X) is –18.716695 with relative accuracy of at least 9.5-by-10 –6 . This is the value copt returned by mincx. 4 mincx also returns the optimizing vector of decision variables xopt. The corresponding optimal value of the matrix variable X is given by *** new lower bound: 18.723491 13 18.714880 *** new lower bound: 18.719624 14 18.716094 *** new lower bound: 18.717986 15 18.716509 *** new lower bound: 18.717297 16 18.716695 *** new lower bound: 18.716873 LMI Solvers 8-27 Xopt = dec2mat(LMIs,xopt,X) which returns This result can be compared with the stabilizing Riccati solution computed by care: Xst = care(a,b,q, 1) norm(Xopt-Xst) ans = 6.5390e 05 X opt 6.3542– 5.8895– 2.2046 5.8895– 6.2855– 2.2201 2.2046 2.2201 6.0771– ?? ?? ?? = 8 The LMI Lab 8-28 From Decision to Matrix Variables and Vice Versa While LMIs are specified in terms of their matrix variables X 1 , . . . , X K , the LMI solvers optimize the vector x of free scalar entries of these matrices, called the decision variables. The two functions mat2dec and dec2mat perform the conversion between these two descriptions of the problem variables. Consider an LMI system with three matrix variables X 1 , X 2 , X 3 . Given particular values X1, X2, X3 of these variables, the corresponding value xdec of the vector of decision variables is returned by mat2dec: xdec = mat2dec(LMISYS,X1,X2,X3) An error is issued if the number of arguments following LMISYS differs from the number of matrix variables in the problem (see matnbr). Conversely, given a value xdec of the vector of decision variables, the corresponding value of the k-th matrix is given by dec2mat. For instance, the value X2 of the second matrix variable is extracted from xdec by X2 = dec2mat(LMISYS,xdec,2) The last argument indicates that the second matrix variable is requested. It could be set to the matrix variable identifier returned by lmivar. The total numbers of matrix variables and decision variables are returned by matnbr and decnbr, respectively. In addition, the function decinfo provides precise information about the mapping between decision variables and matrix variable entries (see the “Command Reference” chapter). Validating Results 8-29 Validating Results The LMI Lab offers two functions to analyze and validate the results of an LMI optimization. The function evallmi evaluates all variable terms in an LMI system for a given value of the vector of decision variables, for instance, the feasible or optimal vector returned by the LMI solvers. Once this evaluation is performed, the left- and right-hand sides of a particular LMI are returned by showlmi. In the LMI problem considered in “Example 8.2” on page 8-23, you can verify that the minimizer xopt returned by mincx satisfies the LMI constraint (8-10) as follows: evlmi = evallmi(LMIs,xopt) [lhs,rhs] = showlmi(evlmi,1) The first command evaluates the system for the value xopt of the decision variables, and the second command returns the left- and right-hand sides of the first (and only) LMI. The negative definiteness of this LMI is checked by eig(lhs-rhs) ans = 2.0387e 04 3.9333e 05 1.8917e 07 4.6680e+01 8 The LMI Lab 8-30 Modifying a System of LMIs Once specified, a system of LMIs can be modified in several ways with the functions dellmi, delmvar, and setmvar. dellmi The first possibility is to remove an entire LMI from the system with dellmi. For instance, suppose that the LMI system of “Example 8.1” on page 8-9 is described in LMISYS and that we want to remove the positivity constraint on X. This is done by NEWSYS = dellmi(LMISYS,2) where the second argument specifies deletion of the second LMI. The resulting system of two LMIs is returned in NEWSYS. The LMI identifiers (initial ranking of the LMI in the LMI system) are not altered by deletions. As a result, the last LMI S > I remains known as the third LMI even though it now ranks second in the modified system. To avoid confusion, it is safer to refer to LMIs via the identifiers returned by newlmi. If BRL, Xpos, and Slmi are the identifiers attached to the three LMIs (8-6)–(8-8), Slmi keeps pointing to S > I even after deleting the second LMI by NEWSYS = dellmi(LMISYS,Xpos) dellmi Another way of modifying an LMI system is to delete a matrix variable, that is, to remove all variable terms involving this matrix variable. This operation is performed by delmvar. For instance, consider the LMI A T X + XA + BW + W T B T + I < 0 with variables X = X T ∈ R 4×4 and W ∈ R 2×4 . This LMI is defined by setlmis([]) X = lmivar(1,[4 1]) % X W = lmivar(2,[2 4]) % W Modifying a System of LMIs 8-31 lmiterm([1 1 1 X],1,A,'s') lmiterm([1 1 1 W],B,1,'s') lmiterm([1 1 1 0],1) LMISYS = getlmis To delete the variable W, type the command NEWSYS = delmvar(LMISYS,W) The resulting NEWSYS now describes the Lyapunov inequality A T X + XA + I < 0 Note that delmvar automatically removes all LMIs that depended only on the deleted matrix variable. The matrix variable identifiers are not affected by deletions and continue to point to the same matrix variable. For subsequent manipulations, it is therefore advisable to refer to the remaining variables through their identifier. Finally, note that deleting a matrix variable is equivalent to setting it to the zero matrix of the same dimensions with setmvar. setmvar The function setmvar is used to set a matrix variable to some given value. As a result, this variable is removed from the problem and all terms involving it become constant terms. This is useful, for instance, to fixsetmvar some variables and optimize with respect to the remaining ones. Consider again “Example 8.1” on page 8-9 and suppose we want to know if the peak gain of G itself is less than one, that is, if ||G|| ∞ < 1 This amounts to setting the scaling matrix D (or equivalently, S = D T D) to a multiple of the identity matrix. Keeping in mind the constraint S > I, a legitimate choice is S = 2?βψ?I. To set S to this value, enter NEWSYS = setmvar(LMISYS,S,2) 8 The LMI Lab 8-32 The second argument is the variable identifier S, and the third argument is the value to which S should be set. Here the value 2 is shorthand for 2?βψ?I. The resulting system NEWSYS reads Note that the last LMI is now free of variable and trivially satisfied. It could, therefore, be deleted by NEWSYS = dellmi(NEWSYS,3) or NEWSYS = dellmi(NEWSYS,Slmi) if Slmi is the identifier returned by newlmi. A T XXA2C T C++ XB B T X 2I–?? ?? ?? < 0 X 0> 2II> Advanced Topics 8-33 Advanced Topics This last section gives a few hints for making the most out of the LMI Lab. It is directed toward users who are comfortable with the basics described above. Structured Matrix Variables Fairly complex matrix variable structures and interdependencies can be specified with lmivar. Recall that the symmetric block-diagonal or rectangular structures are covered by Types 1 and 2 of lmivar provided that the matrix variables are independent. To describe more complex structures or correlations between variables, you must use Type 3 and specify each entry of the matrix variables directly in terms of the free scalar variables of the problem (the so-called decision variables). With Type 3, each entry is specified as either 0 or ±x n where x n is the n-th decision variable. The following examples illustrate how to specify nontrivial matrix variable structures with lmivar. We first consider the case of uncorrelated matrix variables. Example 8.3. Suppose that the problem variables include a 3-by-3 symmetric matrix X and a 3-by-3 symmetric Toeplitz matrix The variable Y has three independent entries, hence involves three decision variables. Since Y is independent of X, these decision variables should be labeled n + 1, n + 2, n + 3 where n is the number of decision variables involved in X. To retrieve this number, define the variable X (Type 1) by setlmis([]) [X,n] = lmivar(1,[3 1]) The second output argument n gives the total number of decision variables used so far (here n = 6). Given this number, Y can be defined by Y = lmivar(3,n+[1 2 3;2 1 2;3 2 1]) Y y 1 y 2 y 3 y 2 y 1 y 2 y 3 y 2 y 1?? ?? ?? ?? = 8 The LMI Lab 8-34 or equivalently by Y = lmivar(3,toeplitz(n+[1 2 3])) where toeplitz is a standard MATLAB function. For verification purposes, we can visualize the decision variable distributions in X and Y with decinfo: lmis = getlmis decinfo(lmis,X) ans = 124 235 456 decinfo(lmis,Y) ans = 789 878 987 The next example is a problem with interdependent matrix variables. Example 8.4. Consider three matrix variables X, Y, Z with structure where x, y, z, t are independent scalar variables. To specify such a triple, first define the two independent variables X and Y (both of Type 1) as follows: setlmis([]) [X,n,sX] = lmivar(1,[1 0;1 0]) [Y,n,sY] = lmivar(1,[1 0;1 0]) The third output of lmivar gives the entry-wise dependence of X and Y on the decision variables (x 1 , x 2 , x 3 , x 4 ) := (x, y, z, t): sX = 1 0 0 2 X x 0 0 y?? ?? ?? , Y z 0 0 t?? ?? ?? , Z 0 x– t– 0?? ?? ?? === Advanced Topics 8-35 sY = 3 0 0 4 Using Type 3 of lmivar, you can now specify the structure of Z in terms of the decision variables x 1 = x and x 4 = t: [Z,n,sZ] = lmivar(3,[0 sX(1,1);–sY(2,2) 0]) Since sX(1,1) refers to x 1 while sY(2,2) refers to x 4 , this defines the variable as confirmed by checking its entry-wise dependence on the decision variables: sZ = 0 1 4 0 Complex-Valued LMIs The LMI solvers are written for real-valued matrices and cannot directly handle LMI problems involving complex-valued matrices. However, complex-valued LMIs can be turned into real-valued LMIs by observing that a complex Hermitian matrix L(x) satisfies L(x) < 0 if and only if This suggests the following systematic procedure for turning complex LMIs into real ones: Decompose every complex matrix variable X as X = X 1 + jX 2 where X 1 and X 2 are real Z 0 x 1 – x 4 – 0 ?? ?? ?? 0 x– t– 0?? ?? ?? == Re Lx()()Im Lx()() Im– Lx()()Re Lx()()?? ?? ?? 0< 8 The LMI Lab 8-36 Decompose every complex matrix coefficient A as A = A 1 + jA 2 where A 1 and A 2 are real Carry out all complex matrix products. This yields affine expressions in X 1 , X 2 for the real and imaginary parts of each LMI, and an equivalent real-valued LMI is readily derived from the above observation. For LMIs without outer factor, a streamlined version of this procedure consists of replacing any occurrence of the matrix variable X = X 1 + jX 2 by and any fixed matrix A = A 1 + jA 2 , including real ones,by For instance, the real counterpart of the LMI system (8-11) reads (given the decompositions M = M 1 + jM 2 and X = X 1 + jX 2 with M j , X j real): Note that X = X H in turn requires that and . Consequently, X 1 and X 2 should be declared as symmetric and skew- symmetric matrix variables, respectively. Assuming, for instance, that M ∈ C 5 × 5 , the LMI system (8-11) would be specified as follows: X 1 X 2 X– 2 X 1 ?? ?? ?? A 1 A 2 A– 2 A 1?? ?? ?? M H XM X, XX H I>=< M 1 M 2 M– 2 M 1?? ?? ?? T X 1 X 2 X– 2 X 1?? ?? ?? M 1 M 2 M– 2 M 1?? ?? ?? X 1 X 2 X– 2 X 1?? ?? ?? < X 1 X 2 X– 2 X 1?? ?? ?? I< X 1 X H = X 2 X 2 T + 20= Advanced Topics 8-37 M1=real(M), M2=imag(M) bigM=[M1 M2;-M2 M1] setlmis([]) % declare bigX=[X1 X2;-X2 X1] with X1=X1' and X2+X2'=0: [X1,n1,sX1] = lmivar(1,[5 1]) [X2,n2,sX2] = lmivar(3,skewdec(5,n1)) bigX = lmivar(3,[sX1 sX2;-sX2 sX1]) % describe the real counterpart of (1.12): lmiterm([1 1 1 0],1) lmiterm([ 1 1 1 bigX],1,1) lmiterm([2 1 1 bigX],bigM',bigM) lmiterm([ 2 1 1 bigX],1,1) lmis = getlmis Note the three-step declaration of the structured matrix variable bigX = 1 Specify X 1 as a (real) symmetric matrix variable and save its structure description sX1 as well as the number n1 of decision variables used in X 1 . 2 Specify X 2 as a skew-symmetric matrix variable using Type 3 of lmivar and the utility skewdec. The command skewdec(5,n1) creates a 5-by–5 skew-symmetric structure depending on the decision variables n1 + 1, n1 + 2,... 3 Define the structure of bigX in terms of the structures sX1 and sX2 of X 1 and X 2 . See the previous subsection for more details on such structure manipulations. X 1 X 2 X– 2 X 1?? ?? ?? : 8 The LMI Lab 8-38 Specifying c T x Objectives for mincx The LMI solver mincx minimizes linear objectives of the form c T x where x is the vector of decision variables. In most control problems, however, such objectives are expressed in terms of the matrix variables rather than of x. Examples include Trace(X) where X is a symmetric matrix variable, or u T Xu where u is a given vector. The function defcx facilitates the derivation of the c vector when the objective is an affine function of the matrix variables. For the sake of illustration, consider the linear objective Trace(X) + Px 0 where X and P are two symmetric variables and x 0 is a given vector. If lmisys is the internal representation of the LMI system and if x 0 , X, P have been declared by x0 = [1;1] setlmis([]) X = lmivar(1,[3 0]) P = lmivar(1,[2 1]) : : lmisys = getlmis the c vector such that c T x = Trace(X) + Px 0 can be computed as follows: n = decnbr(lmisys) c = zeros(n,1) for j=1:n, [Xj,Pj] = defcx(lmisys,j,X,P) c(j) = trace(Xj) + x0'*Pj*x0 end The first command returns the number of decision variables in the problem and the second command dimensions c accordingly. Then the for loop performs the following operations: 1 Evaluate the matrix variables X and P when all entries of the decision vector x are set to zero except x j := 1. This operation is performed by the function defcx. Apart from lmisys and j, the inputs of defcx are the identifiers X and x 0 T x 0 T Advanced Topics 8-39 P of the variables involved in the objective, and the outputs Xj and Pj are the corresponding values. 2 Evaluate the objective expression for X := Xj and P := Pj. This yields the j-th entry of c by definition. In our example the result is c = 3 1 2 1 Other objectives are handled similarly by editing the following generic skeleton: n = decnbr( LMI system ) c = zeros(n,1) for j=1:n, [ matrix values ] = defcx( LMI system,j, matrix identifiers) c(j) = objective(matrix values) end Feasibility Radius When solving LMI problems with feasp, mincx, or gevp, it is possible to constrain the solution x to lie in the ball x T x < R 2 where R > 0 is called the feasibility radius. This specifies a maximum (Euclidean norm) magnitude for x and avoids getting solutions of very large norm. This may also speed up computations and improve numerical stability. Finally, the feasibility radius bound regularizes problems with redundant variable sets. In rough terms, the set of scalar variables is redundant when an equivalent problem could be formulated with a smaller number of variables. The feasibility radius R is set by the third entry of the options vector of the LMI solvers. Its default value is R = 109 . Setting R to a negative value means “no rigid bound,” in which case the feasibility radius is increased during the 8 The LMI Lab 8-40 optimization if necessary. This “flexible bound” mode may yield solutions of large norms. Well-Posedness Issues The LMI solvers used in the LMI Lab are based on interior-point optimization techniques. To compute feasible solutions, such techniques require that the system of LMI constraints be strictly feasible, that is, the feasible set has a nonempty interior. As a result, these solvers may encounter difficulty when the LMI constraints are feasible but not strictly feasible, that is, when the LMI L(x) e 0 has solutions while L(x) < 0 has no solution. For feasibility problems, this difficulty is automatically circumvented by feasp, which reformulates the problem (8-12) as (8-13) In this modified problem, the LMI constraint is always strictly feasible in x, t and the original LMI (8-12) is feasible if and only if the global minimum t min of (8-13) satisfies t min e 0 For feasible but not strictly feasible problems, however, the computational effort is typically higher as feasp strives to approach the global optimum t min = 0 to a high accuracy. For the LMI problems addressed by mincx and gevp, nonstrict feasibility generally causes the solvers to fail and to return an “infeasibility” diagnosis. Although there is no universal remedy for this difficulty, it is sometimes possible to eliminate underlying algebraic constraints to obtain a strictly feasible problem with fewer variables. Find x such that Lx() 0≤ Minimize t subject to Lx t I.×< Advanced Topics 8-41 Another issue has to do with homogeneous feasibility problems such as A T P + P A < 0, P > 0 While this problem is technically well-posed, the LMI optimization is likely to produce solutions close to zero (the trivial solution of the nonstrict problem). To compute a nontrivial Lyapunov matrix and easily differentiate between feasibility and infeasibility, replace the constraint P > 0-by-P > αI with α > 0. Note that this does not alter the problem due to its homogeneous nature. Semi-Definite B(x) in gevp Problems Consider the generalized eigenvalue minimization problem (8-14) Technically, the positivity of B(x) for some x ∈ R n is required for the well-posedness of the problem and the applicability of polynomial-time interior-point methods. Hence problems where cannot be directly solved with gevp. A simple remedy consists of replacing the constraints A(x) < B(x), B(x) > 0 by where Y is an additional symmetric variable of proper dimensions. The resulting problem is equivalent to (8-14) and can be solved directly with gevp. Efficiency and Complexity Issues As explained in the beginning of the chapter, the term-oriented description of LMIs used in the LMI Lab typically leads to higher efficiency than the canonical representation Minimize λ subject to Ax() λBx(), < Bx() 0, > Cx() 0.< Bx() B 1 x()0 00 ?? ?? ?? , with B 1 x() 0> strictly feasible= Ax() Y 0 00?? ?? ?? , < Y λ< B 1 x() B 1 x() 0>, 8 The LMI Lab 8-42 (8-15) This is no longer true, however, when the number of variable terms is nearly equal to or greater than the number N of decision variables in the problem. If your LMI problem has few free scalar variables but many terms in each LMI, it is therefore preferable to rewrite it as (8-15) and to specify it in this form. Each scalar variable x j is then declared independently and the LMI terms are of the form x j A j . If M denotes the total row size of the LMI system and N the total number of scalar decision variables, the flop count per iteration for the feasp and mincx solvers is proportional to N 3 when the least-squares problem is solved via. Cholesly factorization of the Hessian matrix (default) [2 ] M-by-N 2 when numerical instabilities warrant the use of QR factorization instead While the theory guarantees a worst-case iteration count proportional to M, the number of iterations actually performed grows slowly with M in most problems. Finally, while feasp and mincx are comparable in complexity, gevp typically demands more computational effort. Make sure that your LMI problem cannot be solved with mincx before using gevp. Solving M + P T XQ + Q T X T P < 0 In many output-feedback synthesis problems, the design can be performed in two steps: 1 Compute a closed-loop Lyapunov function via LMI optimization. 2 Given this Lyapunov function, derive the controller state-space matrices by solving an LMI of the form (8-16) where M, P, Q are given matrices and X is an unstructured m-by-n matrix variable. A 0 x 1 A 1 … x N A N +++ 0.< MP T XQ Q T X T P++ 0< Advanced Topics 8-43 It turns out that a particular solution X c of (8-16) can be computed via simple linear algebra manipulations [1 ]. Typically, X c corresponds to the center of the ellipsoid of matrices defined by (8-16). The function basiclmi returns the “explicit” solution X c : Xc = basiclmi(M,P,Q) Since this central solution sometimes has large norm, basiclmi also offers the option of computing an approximate least-norm solution of (8-16). This is done by X = basiclmi(M,P,Q,'Xmin') and involves LMI optimization to minimize ||X||. 8 The LMI Lab 8-44 References [1] Gahinet, P., and P. Apkarian, “A Linear Matrix Inequality Approach to H ∞ Control,” Int. J. Robust and Nonlinear Contr., 4 (1994), pp. 421–448. [2] Nemirovski, A., and P. Gahinet, “The Projective Method for Solving Linear Matrix Inequalities,” Proc. Amer. Contr. Conf., 1994, pp. 840–844. [3] Nesterov, Yu, and A. Nemirovski, Interior Point Polynomial Methods in Convex Programming: Theory and Applications, SIAM Books, Philadelphia, 1994. [4] Shamma, J.S., “Robustness Analysis for Time-Varying Systems,” Proc. Conf. Dec. Contr., 1992, pp. 3163–3168. 9 Command Reference 9 Command Reference 9-2 This chapter gives a detailed description of all LMI Control Toolbox functions. Functions are grouped by application in tables at the beginning of this chapter. In addition, information on each function is aavailable through the online Help facility. List of Functions 9-3 List of Functions Linear Algebra basiclmi Solve M + P T XQ + Q T X T P < 0 care Continuous-time Riccati solver dare Discrete-time Riccati solver dricpen Generalized Riccati solver (discrete-time) ricpen Generalized Riccati solver (continuous-time) Linear Time-Invariant Systems ltisys Specify a SYSTEM matrix ltiss Extract state-space data ltitf Compute the transfer function (SISO) sbalanc Balance a state-space realization via diagonal similarity sinfo Return information about a SYSTEM matrix sinv Invert an LTI system splot Plot the time and frequency responses of LTI systems spol Return the poles of an LTI system sresp Frequency response of an LTI system ssub Extract a subsystem from a system Interconnection of Systems sadd Form the parallel interconnection of systems sconnect Describe general control loops 9 Command Reference 9-4 sdiag Append (concatenate) several systems slft Form linear-fractional interconnections sloop Form elementary feedback interconnections smult Form the series interconnection of systems Polytopic and Parameter-Dependent Systems (P-Systems) aff2pol Polytopic representation of an affine parameter-dependent system pdsimul Simulation of parameter-dependent systems along a parameter trajectory psinfo Retrieve information about a P-system psys Specify a P-system pvec Specify a vector of uncertain or time-varying parameters pvinfo Interpret the output of pvec Dynamical Uncertainty ublock Specify an uncertainty block udiag Specify block-diagonal uncertainty from individual uncertainty blocks uinfo Display the characteristics of uncertainty structures aff2lft Linear-fractional representation of an affine parameter-dependent system Interconnection of Systems List of Functions 9-5 Robustness Analysis decay Quadratic decay rate dnorminf RMS gain of a discrete-time LTI system muperf Robust performance of an uncertain LTI system mustab Robust stability of an uncertain LTI system norminf H ∞ norm (RMS gain) of an LTI system norm2 H 2 performance of an LTI system quadperf Quadratic performance quadstab Quadratic stability pdlstab Robust stability analysis via parameter-dependent Lyapunov functions popov Popov criterion 9 Command Reference 9-6 H ∞ Control and Loop Shaping Continuous Time hinflmi LMI-based H ∞ synthesis hinfmix Mixed H 2 /H ∞ synthesis with regional pole placement hinfric Riccati-based H ∞ synthesis hinfpar Extract state-space data from the plant SYSTEM matrix lmireg Specify LMI regions for pole placement magshape Graphical specification of the shaping filters msfsyn Multi-model/objective state-feedback synthesis sconnect Specify general control structures Discrete Time dhinflmi LMI-based H ∞ synthesis dhinfric Riccati-based H ∞ synthesis hinfpar Retrieve the plant state-space data Gain Scheduling hinfgs Synthesis of robust gain-scheduled controllers pdsimul Simulation along parameter trajectories LMI Lab: Specifying and Solving LMIs 9-7 LMI Lab: Specifying and Solving LMIs Specification of LMIs lmiedit GUI for LMI specification setlmis Initialize the LMI description lmivar Define a new matrix variable lmiterm Specify the term content of an LMI newlmi Attach an identifying tag to new LMIs getlmis Get the internal description of the LMI system LMI Solvers feasp Feasibility of a system of LMIs gevp Generalized eigenvalue minimization under LMI constraints mincx Minimization of a linear objective under LMI constraints dec2mat Convert the output of the solvers into values of the matrix variables defcx Help define c T x objectives for mincx Evaluation of LMIs/Validation of Results evallmi Evaluate all variable terms for given values of the decision variables showlmi Return the left- and right-hand sides of an evaluated LMI setmvar Set a matrix variable to a given value 9 Command Reference 9-8 LMI Lab: Additional Facilities Information Retrieval decinfo Visualize the mapping between matrix variable entries and decision variables decnbr Get the number of decision variables, that is, the number of free scalar variables in a problem lmiinfo Inquire about an existing system of LMIs lminbr Get the number of LMIs in a problem matnbr Get the number of matrix variables in a problem Manipulations of LMI Variables decinfo Express each matrix variable entry in terms of the decision variables dec2mat Return the matrix variable values given a vector of decision variables mat2dec Return the vector of decision variables given the matrix variable values Modification of LMI Systems dellmi Delete an LMI from the system delmvar Remove a matrix variable from the problem setmvar Instantiate a matrix variable aff2lft 9-9 9aff2lft Purpose Compute the linear-fractional representation of an affine parameter- dependent system Syntax [sys,delta] = aff2lft(affsys) Description The input argument affsys describes an affine parameter-dependent system of the form E(p) = A(p)x + B(p)u y = C(p)x + D(p)u where p = (p 1 , . . ., p n ) is a vector of uncertain or time-varying real parameters taking values in a box: Such systems are specified with psys. Given this affine parameter-dependent model, the function aff2lft returns the equivalent linear-fractional representation where P(s) is an LTI system called the nominal plant The block-diagonal operator ? = diag(δ 1 I r 1 , . . ., δ n I r n ) accounts for the uncertainty on the real parameters p j . Each scalar δ j is real, possibly repeated (r j > 1), and bounded by x · p j pj p j ≤≤ ? P(s) uy δ j p j p j – 2 ---------------≤ aff2lft 9-10 The SYSTEM matrix of P and the description of ? are returned in sys and delta. Closing the loop with ? ≡ 0 yields the LTI system E(p av ) = A(p av )x + B(p av )u y = C(p av )x + D(p av )u where p av = is the vector of average parameter values. Example See example in “Sector-Bounded Uncertainty” on page 2-30. See Also psys, pvec, ublock, udiag x · 1 2 --- pp+() aff2pol 9-11 9aff2pol Purpose Convert affine parameter-dependent models to polytopic ones Syntax polsys = aff2pol(affsys) Description aff2pol derives a polytopic representation polsys of the affine parameter- dependent system (9-1) (9-2) where p = (p 1 , . . ., p n ) is a vector of uncertain or time-varying real parameters taking values in a box or a polytope. The description affsys of this system should be specified with psys. The vertex systems of polsys are the instances of (9-1)–(9-2) at the vertices p ex of the parameter range, i.e., the SYSTEM matrices for all corners p ex of the parameter box or all vertices p ex of the polytope of parameter values. Example “Example 2.1” on page 2-21. See Also psys, pvec, aff2lft Ep()x · Ap()xBp()u+= yCp()xDp()u+= Ap ex ()jE p ex ()+ Bp ex () Cp ex () Dp ex () ?? ?? ?? basiclmi 9-12 9basiclmi Purpose Compute a solution X of M + P T XQ + Q T X T P < 0 Syntax X = basiclmi(M,P,Q,option) Description basiclmi computes a solution X of the LMI M + P T XQ + Q T X T P < 0 where M is symmetric and P,Q are matrices of compatible dimensions. If W P and W Q are orthonormal bases of the null space of P and Q, this LMI is solvable if and only if W P < 0 and W Q < 0 The output is [] when these conditions are not satisfied. The solution X is derived by standard linear algebra manipulations. When option is set to 'Xmin', basiclmi computes the least-norm solution by solving the LMI problem Minimize τ subject to M + P T XQ + Q T X T P < 0 See Also feasp, mincx W P T MW Q T M τIX X T τI ?? ?? ?? 0≥ care 9-13 9care Purpose Solve continuous-time Riccati equations (CARE) Syntax X = care(A,B,Q,R,S,E) [X,L,G,RR] = care(A,B,Q,R,S,E) Description This function computes the stabilizing solution X of the Riccati equation A T XE + E T XA – (E T XB + S)R –1 (B T XE + S T ) + Q = 0 where E is an invertible matrix. The solution is computed by unitary deflation of the associated Hamiltonian pencil For solvability, the pencil H – λL must have no finite generalized eigenvalue on the imaginary axis. If the input arguments S and E are omitted, the matrices S and E are set to the default values S = 0 and E = I. The function care also returns the vector L of closed-loop eigenvalues, the gain matrix G given by G = –R –1 (B T XE + S T ), and the Frobenius norm RR of the relative residual matrix. Reference Laub, A. J., “A Schur Method for Solving Algebraic Riccati Equations,” IEEE Trans. Aut. Contr., AC–24 (1979), pp. 913–921. Arnold, W.F., and A.J. Laub, “Generalized Eigenproblem Algorithms and Software for Algebraic Riccati Equations,” Proc. IEEE, 72 (1984), pp. 1746– 1754. See Also ricpen, cares, dare H λL– A0B Q– A– T S– S T B T R λ E00 0E T 0 000 –= dare 9-14 9dare Purpose Solve discrete-time Riccati equations (DARE) Syntax X = dare(A,B,Q,R,S,E) [X,L,G,RR] = dare(A,B,Q,R,S,E) Description dare computes the stabilizing solution X of the algebraic Riccati equation (9-3) where E is an invertible matrix. The solution is computed by unitary deflation of the associated symplectic pencil For solvability, the pencil H – λL must have no finite generalized eigenvalue on the unit circle. If the input arguments S and E are omitted, the matrices S and E are set to the default values S = 0 and E = I. The function dare also returns the vector L of closed-loop eigenvalues, the gain matrix G given by G = –(B T XB + R) –1 (B T XA + S T ), and the Frobenius norm RR of the relative residual matrix. Remark The function dricpen is a variant of dare where the inputs are the two matrices H and L defined above. The syntax is X = dricpen(H,L) or [X1,X2] = dricpen(H,L) This second syntax returns two matrices X 1 , X 2 such that has orthonormal columns, A T XA E T XE– A T XB S+()B T XB R+() 1– B T XA S T +()– Q+0= H λL– A0B Q– E– T S– S T 0R λ E00 0A T 0 0B– T 0 –= X 1 X 2 dare 9-15 is the stabilizing solution of (9-3). Reference Pappas, T., A.J. Laub, and N.R. Sandell, “On the Numerical Solution of the Discrete-Time Algebraic Riccati Equation,” IEEE Trans. Aut. Contr., AC–25 (1980), pp. 631–641. Arnold, W.F., and A.J. Laub, “Generalized Eigenproblem Algorithms and Software for Algebraic Riccati Equations,” Proc. IEEE, 72 (1984), pp. 1746– 1754. See Also dricpen, dares, care XX 2 X 1 1– = decay 9-16 9decay Purpose Quadratic decay rate of polytopic or affine P-systems Syntax [drate,P] = decay(ps,options) Description For affine parameter-dependent systems E(p) = A(p)x, p(t) = (p 1 (t), . . ., p n (t)) or polytopic systems E(t) = A(t)x, (A, E) ∈ Co{(A 1 , E 1 ), . . ., (A n , E n )}, decay returns the quadratic decay rate drate, i.e., the smallest α ∈ R such that A T QE + EQA T < αQ holds for some Lyapunov matrix Q > 0 and all possible values of (A, E). Two control parameters can be reset via options(1) and options(2): If options(1)=0 (default), decay runs in fast mode, using the least expensive sufficient conditions. Set options(1)=1 to use the least conservative conditions. options(2) is a bound on the condition number of the Lyapunov matrix P. The default is 10 9 . Example See “Example 3.3” on page 3-9. See Also quadstab, pdlstab, mustab, psys x · x · decinfo 9-17 9decinfo Purpose Describe how the entries of a matrix variable X relate to the decision variables Syntax decinfo(lmisys) decX = decinfo(lmisys,X) Description The function decinfo expresses the entries of a matrix variable X in terms of the decision variables x 1 , . . ., x N . Recall that the decision variables are the free scalar variables of the problem, or equivalently, the free entries of all matrix variables described in lmisys. Each entry of X is either a hard zero, some decision variable x n , or its opposite –x n . If X is the identifier of X supplied by lmivar, the command decX = decinfo(lmisys,X) returns an integer matrix decX of the same dimensions as X whose (i, j) entry is 0 if X(i, j) is a hard zero n if X(i, j) = x n (the n-th decision variable) –n if X(i, j) = –x n decX clarifies the structure of X as well as its entry-wise dependence on x 1 , . . ., x N . This is useful to specify matrix variables with atypical structures (see lmivar). decinfo can also be used in interactive mode by invoking it with a single argument. It then prompts the user for a matrix variable and displays in return the decision variable content of this variable. Example 1 Consider an LMI with two matrix variables X and Y with structure: X = x I 3 with x scalar Y rectangular of size 2-by-1 If these variables are defined by setlmis([]) X = lmivar(1,[3 0]) Y = lmivar(2,[2 1]) : : lmis = getlmis decinfo 9-18 the decision variables in X and Y are given by dX = decinfo(lmis,X) dX = 1 0 0 0 1 0 0 0 1 dY = decinfo(lmis,Y) dY = 2 3 This indicates a total of three decision variables x 1 , x 2 , x 3 that are related to the entries of X and Y by Note that the number of decision variables corresponds to the number of free entries in X and Y when taking structure into account. Example 2 Suppose that the matrix variable X is symmetric block diagonal with one 2-by-2 full block and one 2-by-2 scalar block, and is declared by setlmis([]) X = lmivar(1,[2 1;2 0]) : lmis = getlmis The decision variable distribution in X can be visualized interactively as follows: decinfo(lmis) There are 4 decision variables labeled x1 to x4 in this problem. X x 1 00 0x 1 0 00x 1?? ?? ?? ?? , Y x 2 x 3?? ?? ?? ?? == decinfo 9-19 Matrix variable Xk of interest (enter k between 1 and 1, or 0 to quit): ?> 1 The decision variables involved in X1 are among {-x1,...,x4}. Their entry-wise distribution in X1 is as follows (0,j>0,-j<0 stand for 0,xj,-xj, respectively): X1 : 1 2 0 0 2 3 0 0 0 0 4 0 0 0 0 4 ********* Matrix variable Xk of interest (enter k between 1 and 1, or 0 to quit): ?> 0 See Also lmivar, mat2dec, dec2mat, decnbr, decinfo decnbr 9-20 9decnbr Purpose Give the total number of decision variables in a system of LMIs Syntax ndec = decnbr(lmisys) Description The function decnbr returns the number ndec of decision variables (free scalar variables) in the LMI problem described in lmisys. In other words, ndec is the length of the vector of decision variables. Example For an LMI system lmis with two matrix variables X and Y such that X is symmetric block diagonal with one 2-by-2 full block, and one 2-by-2 scalar block Y is 2-by-3 rectangular, the number of decision variables is ndec = decnbr(LMIs) ndec = 10 This is exactly the number of free entries in X and Y when taking structure into account (see decinfo for more details). See Also dec2mat, decinfo, mat2dec dec2mat 9-21 9dec2mat Purpose Given values of the decision variables, derive the corresponding values of the matrix variables Syntax valX = dec2mat(lmisys,decvars,X) Description Given a value decvars of the vector of decision variables, dec2mat computes the corresponding value valX of the matrix variable with identifier X. This identifier is returned by lmivar when declaring the matrix variable. Recall that the decision variables are all free scalar variables in the LMI problem and correspond to the free entries of the matrix variables X 1 , . . ., X K . Since LMI solvers return a feasible or optimal value of the vector of decision variables, dec2mat is useful to derive the corresponding feasible or optimal values of the matrix variables. Example See the description of feasp. See Also mat2dec, decnbr, decinfo defcx 9-22 9defcx Purpose Help specify c T x objectives for the mincx solver Syntax [V1,...,Vk] = defcx(lmisys,n,X1,...,Xk) Description defcx is useful to derive the c vector needed by mincx when the objective is expressed in terms of the matrix variables. Given the identifiers X1,...,Xk of the matrix variables involved in this objective, defcx returns the values V1,...,Vk of these variables when the n-th decision variable is set to one and all others to zero. See the example in “Specifying cTx Objectives for mincx” on page 8-38 for more details on how to use this function to derive the c vector. See Also mincx, decinfo dellmi 9-23 9dellmi Purpose Remove an LMI from a given system of LMIs Syntax newsys = dellmi(lmisys,n) Description dellmi deletes the n-th LMI from the system of LMIs described in lmisys. The updated system is returned in newsys. The ranking n is relative to the order in which the LMIs were declared and corresponds to the identifier returned by newlmi. Since this ranking is not modified by deletions, it is safer to refer to the remaining LMIs by their identifiers. Finally, matrix variables that only appeared in the deleted LMI are removed from the problem. Example Suppose that the three LMIs X 1 + X 1 A 1 + Q 1 < 0 X 2 + X 2 A 2 + Q 2 < 0 X 3 + X 3 A 3 + Q 3 < 0 have been declared in this order, labeled LMI1, LMI2, LMI3 with newlmi, and stored in lmisys. To delete the second LMI, type lmis = dellmi(lmisys,LMI2) lmis now describes the system of LMIs (9-4) (9-5) and the second variable X 2 has been removed from the problem since it no longer appears in the system (9-4)–(9-5). To further delete (9-5), type lmis = dellmi(lmis,LMI3) or equivalently lmis = dellmi(lmis,3) A 1 T A 2 T A 3 T A 1 T X 1 X 1 A 1 Q 1 ++0< A 3 T X 3 X 3 A 3 Q 3 ++0< dellmi 9-24 Note that (9-5) has retained its original ranking after the first deletion. See Also newlmi, lmiedit, lmiinfo delmvar 9-25 9delmvar Purpose Delete one of the matrix variables of an LMI problem Syntax newsys = delmvar(lmisys,X) Description delmvar removes the matrix variable X with identifier X from the list of variables defined in lmisys. The identifier X should be the second argument returned by lmivar when declaring X. All terms involving X are automatically removed from the list of LMI terms. The description of the resulting system of LMIs is returned in newsys. Example Consider the LMI involving two variables X and Y with identifiers X and Y. To delete the variable X, type lmisys = delmvar(lmisys,X) Now lmisys describes the LMI with only one variable Y. Note that Y is still identified by the label Y. See Also lmivar, setmvar, lmiinfo 0 A T YB T YA Q++CX D+ X T C T D T + XX T +()–?? ?? ?? < 0 A T YB B T YA Q++D D T 0?? ?? ?? < dhinflmi 9-26 9dhinflmi Purpose LMI-based H ∞ synthesis for discrete-time systems Syntax gopt = dhinflmi(P,r) [gopt,K] = dhinflmi(P,r) [gopt,K,x1,x2,y1,y2] = dhinflmi(P,r,g,tol,options) Description dhinflmi is the counterpart of hinflmi for discrete-time plants. Its syntax and usage mirror those of hinflmi. The H ∞ performance γ is achievable if and only if the following system of LMIs has symmetric solutions R and S: where N 12 and N 21 denote bases of the null spaces of and (C 2 , D 21 ), respectively. The function dhinflmi returns solutions R = x1 and S = y1 of this LMI system for γ = gopt (the matrices x2 and y2 are set to gopt-by-I). Reference Gahinet, P. and P. Apkarian, “A Linear Matrix Inequality Approach to H ∞ Control,” Int. J. Robust and Nonlinear Contr., 4 (1994), pp. 421–448. See Also dhinfric, hinflmi N 12 0 0I ?? ?? ?? T ARA T R– ARC 1 T B 1 C 1 RA T γI– C 1 RC 1 T + D 11 B 1 T D 11 T γI– ?? ?? ?? N 12 0 0I ?? ?? ?? 0< N 21 0 0I ?? ?? ?? T A T SA S– A T SB 1 C 1 T B 1 T SA γI– B 1 T SB 1 + D 11 T C 1 D 11 γI– ?? ?? ?? N 21 0 0I ?? ?? ?? 0< RI IS?? ?? ?? 0≥ B 2 T D 12 T ,() dhinfric 9-27 9dhinfric Purpose Riccati-based H ∞ synthesis for discrete-time systems Syntax gopt = dhinfric(P,r) [gopt,K] = dhinfric(P,r) [gopt,K,x1,x2,y1,y2] = dhinfric(P,r,gmin,gmax,tol,options) Description dhinfric is the counterpart of hinfric for discrete-time plants P(s) with equations: x k+1 = Ax k + B 1 w k + B 2 u k z k = C 1 x k + D 11 w k + D 12 u k y k = C 2 x k + D 21 w k + D 22 u k . See dnorminf for a definition of the RMS gain (H ∞ performance) of discrete-time systems. The syntax and usage of dhinfric are identical to those of hinfric. Restriction D 12 and D 21 must have full rank. See Also dhinflmi, hinfric dnorminf 9-28 9dnorminf Purpose Compute the random mean-squares (RMS) gain of discrete-time systems Syntax [gain,peakf] = dnorminf(g,tol) Description For sampled-data LTI systems with state-space equations Ex k+1 = Ax k + Bu k y k +1 = Cx k + Du k and transfer function G(z) = D + C(zE – A) –1 B, dnorminf computes the peak gain (9-6) of the response on the unit circle. When G(z) is stable, this peak gain coincides with the RMS gain or H ∞ norm of G, that is, The input g is a SYSTEM matrix containing the state-space data A, B, C, D, E. On output, dnorminf returns the RMS gain gain and the “frequency” peakf at which this gain is attained, i.e., the value of ω maximizing (9-6). The relative accuracy on the computed RMS gain can be adjusted with the optional input tol (the default value is 10 –2 ). Reference Boyd, S., V. Balakrishnan, and P. Kabamba, “A Bisection Method for Computing the H ∞ Norm of a Transfer Matrix and Related Problems,” Math. Contr. Sign. Syst., 2 (1989), pp. 207–219. Bruisma, N.A., and M. Steinbuch, “A Fast Algorithm to Compute the H ∞ -Norm of a Transfer Function Matrix,” Syst. Contr. Letters, 14 (1990), pp. 287–293. Robel, G., “On Computing the Infinity Norm,” IEEE Trans. Aut. Contr., AC–34 (1989), pp. 882–884. See Also norminf G ∞ sup σ max Ge jω ()()= ω 02π,[]∈ G ∞ sup y l 2 u l 2 -----------= ut 2 ∈ u0≠ evallmi 9-29 9evallmi Purpose Given a particular instance of the decision variables, evaluate all variable terms in the system of LMIs Syntax evalsys = evallmi(lmisys,decvars) Description evallmi evaluates all LMI constraints for a particular instance decvars of the vector of decision variables. Recall that decvars fully determines the values of the matrix variables X 1 , . . ., X K . The “evaluation” consists of replacing all terms involving X 1 , . . ., X K by their matrix value. The output evalsys is an LMI system containing only constant terms. The function evallmi is useful for validation of the LMI solvers’ output. The vector returned by these solvers can be fed directly to evallmi to evaluate all variable terms. The matrix values of the left- and right-hand sides of each LMI are then returned by showlmi. Observation evallmi is meant to operate on the output of the LMI solvers. To evaluate all LMIs for particular instances of the matrix variables X 1 , . . ., X K , first form the corresponding decision vector x with mat2dec and then call evallmi with x as input. Example Consider the feasibility problem of finding X > 0 such that A T XA – X + I < 0 where This LMI system is defined by: setlmis([]) X = lmivar(1,[2 1]) % full symmetric X lmiterm([1 1 1 X],A',A) % LMI #1: A'*X*A lmiterm([1 1 1 X],-1,1) % LMI #1: -X lmiterm([1 1 1 0],1) % LMI #1: I lmiterm([-2 1 1 X],1,1) % LMI #2: X lmis = getlmis To compute a solution xfeas, call feasp by [tmin,xfeas] = feasp(lmis) A 0.5 0.2– 0.1 0.7–?? ?? ?? .= evallmi 9-30 The result is tmin = -4.7117e+00 xfeas' = 1.1029e+02 -1.1519e+01 1.1942e+02 The LMI constraints are therefore feasible since tmin < 0. The solution X corresponding to the feasible decision vector xfeas would be given by X = dec2mat(lmis,xfeas,X). To check that xfeas is indeed feasible, evaluate all LMI constraints by typing evals = evallmi(lmis,xfeas) The left- and right-hand sides of the first and second LMIs are then given by [lhs1,rhs1] = showlmi(evals,1) [lhs2,rhs2] = showlmi(evals,2) and the test eig(lhs1-rhs1) ans = -8.2229e+01 -5.8163e+01 confirms that the first LMI constraint is satisfied by xfeas. See Also showlmi, setmvar, dec2mat, mat2dec feasp 9-31 9feasp Purpose Find a solution to a given system of LMIs Syntax [tmin,xfeas] = feasp(lmisys,options,target) Description The function feasp computes a solution xfeas (if any) of the system of LMIs described by lmisys. The vector xfeas is a particular value of the decision variables for which all LMIs are satisfied. Given the LMI system (9-7) xfeas is computed by solving the auxiliary convex program: Minimize t subject to . The global minimum of this program is the scalar value tmin returned as first output argument by feasp. The LMI constraints are feasible if tmin e 0 and strictly feasible if tmin < 0. If the problem is feasible but not strictly feasible, tmin is positive and very small. Some post-analysis may then be required to decide whether xfeas is close enough to feasible. The optional argument target sets a target value for tmin. The optimization code terminates as soon as a value of t below this target is reached. The default value is target = 0. Note that xfeas is a solution in terms of the decision variables and not in terms of the matrix variables of the problem. Use dec2mat to derive feasible values of the matrix variables from xfeas. Control Parameters The optional argument options gives access to certain control parameters for the optimization algorithm. This five-entry vector is organized as follows: options(1) is not used options(2) sets the maximum number of iterations allowed to be performed by the optimization procedure (100 by default) options(3) resets the feasibility radius. Setting options(3) to a value R > 0 further constrains the decision vector x = (x 1 , . . ., x N ) to lie within the ball N T LxNM T ≤ R x()M, N T L x()NM T R x()– M tI≤ feasp 9-32 In other words, the Euclidean norm of xfeas should not exceed R. The feasibility radius is a simple means of controlling the magnitude of solutions. Upon termination, feasp displays the f-radius saturation, that is, the norm of the solution as a percentage of the feasibility radius R. The default value is R = 10 9 . Setting options(3) to a negative value activates the “flexible bound” mode. In this mode, the feasibility radius is initially set to 10 8 , and increased if necessary during the course of optimization options(4) helps speed up termination. When set to an integer value J > 0, the code terminates if t did not decrease by more than one percent in relative terms during the last J iterations. The default value is 10. This parameter trades off speed vs. accuracy. If set to a small value (< 10), the code terminates quickly but without guarantee of accuracy. On the contrary, a large value results in natural convergence at the expense of a possibly large number of iterations. options(5) = 1 turns off the trace of execution of the optimization procedure. Resetting options(5) to zero (default value) turns it back on. Setting option(i) to zero is equivalent to setting the corresponding control parameter to its default value. Consequently, there is no need to redefine the entire vector when changing just one control parameter. To set the maximum number of iterations to 10, for instance, it suffices to type options=zeros(1,5) % default value for all parameters options(2)=10 Memory Problems When the least-squares problem solved at each iteration becomes ill conditioned, the feasp solver switches from Cholesky-based to QR-based linear algebra (see“Memory Problems” on page 9-74 for details). Since the QR mode typically requires much more memory, MATLAB may run out of memory and display the message ??? Error using ==> feaslv Out of memory. Type HELP MEMORY for your options. x i 2 i1= N ∑ R 2 < feasp 9-33 You should then ask your system manager to increase your swap space or, if no additional swap space is available, set options(4) = 1. This will prevent switching to QR and feasp will terminate when Cholesky fails due to numerical instabilities. Example Consider the problem of finding P > I such that (9-8) (9-9) (9-10) with data This problem arises when studying the quadratic stability of the polytope of matrices Co{A 1 , A 2 , A 3 } (see“Quadratic Stability” on page 3-6 for details). To assess feasibility with feasp, first enter the LMIs (9-8)–(9-10) by: setlmis([]) p = lmivar(1,[2 1]) lmiterm([1 1 1 p],1,a1,'s') % LMI #1 lmiterm([2 1 1 p],1,a2,'s') % LMI #2 lmiterm([3 1 1 p],1,a3,'s') % LMI #3 lmiterm([-4 1 1 p],1,1) % LMI #4: P lmiterm([4 1 1 0],1) % LMI #4: I lmis = getlmis Then call feasp to find a feasible decision vector: [tmin,xfeas] = feasp(lmis) A 1 T PPA 1 + 0< A 2 T PPA 2 + 0< A 3 T PPA 3 + 0< A 1 1– 2 13–?? ?? ?? = , A 2 0.8– 1.5 1.3 2.7–?? ?? ?? = , A 3 1.4– 0.9 0.7 2.0–?? ?? ?? = feasp 9-34 This returns tmin = -3.1363. Hence (9-8)–(9-10) is feasible and the dynamical system = A(t)x is quadratically stable for A(t) ∈ Co{A 1 , A 2 , A 3 }. To obtain a Lyapunov matrix P proving the quadratic stability, type P = dec2mat(lmis,xfeas,p) This returns It is possible to add further constraints on this feasibility problem. For instance, you can bound the Frobenius norm of P by 10 while asking tmin to be less than or equal to –1. This is done by [tmin,xfeas] = feasp(lmis,[0,0,10,0,0],-1) The third entry 10 of options sets the feasibility radius to 10 while the third argument -1 sets the target value for tmin. This yields tmin = -1.1745 and a matrix P with largest eigenvalue λ max (P) = 9.6912. Reference The feasibility solver feasp is based on Nesterov and Nemirovski’s Projective Method described in Nesterov, Y., and A. Nemirovski, Interior Point Polynomial Methods in Convex Programming: Theory and Applications, SIAM, Philadelphia, 1994. Nemirovski, A., and P. Gahinet, “The Projective Method for Solving Linear Matrix Inequalities,” Proc. Amer. Contr. Conf., 1994, Baltimore, Maryland, pp. 840–844. The optimization is performed by the C-MEX file feaslv.mex. See Also mincx, gevp, dec2mat x · P 270.8 126.4 126.4 155.1?? ?? ?? = getdg 9-35 9getdg Purpose Retrieve the D, G scaling matrices in problems Syntax D = getdg(ds,i) G = getdg(gs,i) Description The function mustab computes the mixed-μ upper bound over a grid of frequencies fs and returns the corresponding D, G scalings compacted in two matrices ds and gs. The values of D, G at the frequency fs(i) are extracted from ds and gs by getdg. See Also mustab getlmis 9-36 9getlmis Purpose Get the internal description of an LMI system Syntax lmisys = getlmis Description After completing the description of a given LMI system with lmivar and lmiterm, its internal representation lmisys is obtained with the command lmisys = getlmis This MATLAB representation of the LMI system can be forwarded to the LMI solvers or any other LMI-Lab function for subsequent processing. See Also setlmis, lmivar, lmiterm, newlmi gevp 9-37 9gevp Purpose Generalized eigenvalue minimization under LMI constraints Syntax [lopt,xopt] = gevp(lmisys,nlfc,options,linit,xinit,target) Description gevp solves the generalized eigenvalue minimization problem Minimize λ subject to: (9-11) (9-12) (9-13) where C(x) < D(x) and A(x) < λB(x) denote systems of LMIs. Provided that (9-11)–(9-12) are jointly feasible, gevp returns the global minimum lopt and the minimizing value xopt of the vector of decision variables x. The corresponding optimal values of the matrix variables are obtained with dec2mat. The argument lmisys describes the system of LMIs (9-11)–(9-13) for λ = 1. The LMIs involving λ are called the linear-fractional constraints while (9-11)–(9-12) are regular LMI constraints. The number of linear-fractional constraints (9-13) is specified by nlfc. All other input arguments are optional. If an initial feasible pair (λ 0 , x 0 ) is available, it can be passed to gevp by setting linit to λ 0 and xinit to x 0 . Note that xinit should be of length decnbr(lmisys) (the number of decision variables). The initial point is ignored when infeasible. Finally, the last argument target sets some target value for λ. The code terminates as soon as it has found a feasible pair (λ, x) with λ e target. Caution When setting up your gevp problem, be cautious to Always specify the linear-fractional constraints (9-13) last in the LMI system. gevp systematically assumes that the last nlfc LMI constraints are linear fractional Add the constraint B(x) > 0 or any other LMI constraint that enforces it (see Remark below). This positivity constraint is required for regularity and well-posedness of the optimization problem (see the discussion in “Well-Posedness Issues” on page 8-40). C x() D x()< 0 B x()< A x() λB x()< gevp 9-38 Control Parameters The optional argument options gives access to certain control parameters of the optimization code. In gevp, this is a five-entry vector organized as follows: options(1) sets the desired relative accuracy on the optimal value lopt (default = 10 –2 ). options(2) sets the maximum number of iterations allowed to be performed by the optimization procedure (100 by default). options(3) sets the feasibility radius. Its purpose and usage are as for feasp. options(4) helps speed up termination. If set to an integer value J > 0, the code terminates when the progress in λ over the last J iterations falls below the desired relative accuracy. By progress, we mean the amount by which λ decreases. The default value is 5 iterations. options(5) = 1 turns off the trace of execution of the optimization procedure. Resetting options(5) to zero (default value) turns it back on. Setting option(i) to zero is equivalent to setting the corresponding control parameter to its default value. Example Given consider the problem of finding a single Lyapunov function V(x) = x T Px that proves stability of = A i x (i = 1, 2, 3) and maximizes the decay rate . This is equivalent to minimizing α subject to (9-14) (9-15) A 1 1– 2 13–?? ?? ?? = , A 2 0.8– 1.5 1.3 2.7–?? ?? ?? = , A 3 1.4– 0.9 0.7 2.0–?? ?? ?? ,= x · dV x() dt ----------------– IP< A 1 T PPA 1 + αP< gevp 9-39 (9-16) (9-17) To set up this problem for gevp, first specify the LMIs (9-15)–(9-17) with α = 1: setlmis([]); p = lmivar(1,[2 1]) lemiterm([1 1 1 0],1) % P > I : I lemiterm([ 1 1 1 p],1,1) % P > I : P lemiterm([2 1 1 p],1,a1,'s') % LFC # 1 (lhs) lemiterm([ 2 1 1 p],1,1) % LFC # 1 (rhs) lemiterm([3 1 1 p],1,a2,'s') % LFC # 2 (lhs) lemiterm([ 3 1 1 p],1,1) % LFC # 2 (rhs) lemiterm([4 1 1 p],1,a3,'s') % LFC # 3 (lhs) lemiterm([ 4 1 1 p],1,1) % LFC # 3 (rhs) lmis = getlmis Note that the linear fractional constraints are defined last as required. To minimize α subject to (9-15)–(9-17), call gevp by [alpha,popt]=gevp(lmis,3) This returns alpha = -0.122 as optimal value (the largest decay rate is therefore 0.122). This value is achieved for Remark Generalized eigenvalue minimization problems involve standard LMI constraints (9-11) and linear fractional constraints (9-13). For well-posedness, the positive definiteness of B(x) must be enforced by adding the constraint B(x) > 0 to the problem. Although this could be done automatically from inside the code, this is not desirable for efficiency reasons. For instance, the set of constraints (9-12) may reduce to a single constraint as in the example above. In this case, the single extra LMI “P > I” is enough to enforce positivity of all A 2 T PPA 2 + αP< A 3 T PPA 3 + αP< P 5.58 8.35– 8.35– 18.64?? ?? ?? = gevp 9-40 linear-fractional right-hand sides. It is therefore left to the user to devise the least costly way of enforcing this positivity requirement. Reference The solver gevp is based on Nesterov and Nemirovski’s Projective Method described in Nesterov, Y., and A. Nemirovski, Interior Point Polynomial Methods in Convex Programming: Theory and Applications, SIAM, Philadelphia, 1994. The optimization is performed by the CMEX file fpds.mex. See Also dec2mat, decnbr, feasp, mincx hinfgs 9-41 9hinfgs Purpose Synthesis of gain-scheduled H ∞ controllers Syntax [gopt,pdK,R,S] = hinfgs(pdP,r,gmin,tol,tolred) Description Given an affine parameter-dependent plant where the time-varying parameter vector p(t) ranges in a box and is measured in real time, hinfgs seeks an affine parameter-dependent controller scheduled by the measurements of p(t) and such that K stabilizes the closed-loop system for all admissible parameter trajectories p(t) K minimizes the closed-loop quadratic H ∞ performance from w to z. The description pdP of the parameter-dependent plant P is specified with psys and the vector r gives the number of controller inputs and outputs (set r=[p 2 ,m2] if y ∈ R p2 and u ∈ R m2 ). Note that hinfgs also accepts the polytopic model of P returned, e.g., by aff2pol or sconnect. hinfgs returns the optimal closed-loop quadratic performance gopt and a polytopic description of the gain-scheduled controller pdK. To test if a P ? ? ? ? ? x · = A p()xB 1 p()wB 2 u++ z = C 1 p()xD 11 p()wD 12 u y = C 2 xD 21 wD 22 u++ K ζ · A K p()ζ B K p()y+= uC K p()ζ D K p()y+= ? ? ? P u y w z K hinfgs 9-42 closed-loop quadratic performance γ is achievable, set the third input gmin to γ. The arguments tol and tolred control the required relative accuracy on gopt and the threshold for order reduction (see hinflmi for details). Finally, hinfgs also returns solutions R, S of the characteristic LMI system. Controller Implementation The gain-scheduled controller pdK is parametrized by p(t) and characterized by the values K Π j of at the corners 3 j of the parameter box. The command Kj = psinfo(pdK,'sys',j) returns the j-th vertex controller K Π j while pv = psinfo(pdP,'par') vertx = polydec(pv) Pj = vertx(:,j) gives the corresponding corner 3 j of the parameter box (pv is the parameter vector description). The controller scheduling should be performed as follows. Given the measurements p(t) of the parameters at time t, 3 Express p(t) as a convex combination of the 3 j : p(t) = α 1 3 1 + . . .+ α N 3 N , α j ? 0, This convex decomposition is computed by polydec. 4 Compute the controller state-space matrices at time t as the convex combination of the vertex controllers K Π j : 5 Use A K (t), B K (t), C K (t), D K (t) to update the controller state-space equations. A K p()B K p() C K p()D K p() ?? ?? ?? α j i1= N ∑ 1= A K t()B K t() C K t()D K t() ?? ?? ?? α j i1= N ∑ K Π j .= hinfgs 9-43 Reference Apkarian, P., P. Gahinet, and G. Becker, “Self-Scheduled H ∞ Control of Linear Parameter-Varying Systems,” submitted to Automatica, October 1995. Becker, G., Packard, P., “Robust Performance of Linear-Parametrically Varying Systems Using Parametrically-Dependent Linear Feedback,” Systems and Control Letters, 23 (1994), pp. 205–215. Packard, A., “Gain Scheduling via Linear Fractional Transformations,” Syst. Contr. Letters, 22 (1994), pp. 79–92. Example See “Design Example” on page 7-10. See Also psys, pvec, pdsimul, polydec, hinflmi hinflmi 9-44 9hinflmi Purpose LMI-based H ∞ synthesis for continuous-time plants Syntax gopt = hinflmi(P,r) [gopt,K] = hinflmi(P,r) [gopt,K,x1,x2,y1,y2] = hinflmi(P,r,g,tol,options) Description hinflmi computes an internally stabilizing controller K(s) that minimizes the closed-loop H ∞ performance in the control loop The H ∞ performance is the RMS gain of the closed-loop transfer function F (P, K) from w to z. This function implements the LMI-based approach to H ∞ synthesis. The SYSTEM matrix P contains a state-space realization of the plant P(s) and the row vector r specifies the number of measurements and controls (set r = [p 2 m2] when y ∈ R p 2 and u ∈ R m 2 ). The optimal H ∞ performance gopt and an optimal H ∞ controller K are returned by [gopt,K] = hinflmi(P,r) Alternatively, a suboptimal controller K that guarantees ||F (P, K)|| ∞ < g is obtained with the syntax [gopt,K] = hinflmi(P,r,g) The optional argument tol specifies the relative accuracy on the computed optimal performance gopt. The default value is 10 –2 . Finally, [gopt,K,x1,x2,y1,y2] = hinflmi(P,r) also returns solutions R = x1 and S = y1 of the characteristic LMIs for γ = gopt (see “Practical Considerations” on page 6-18 for details). K(s) P(s) u y w z hinflmi 9-45 Control Parameters The optional argument options resets the following control parameters: options(1) is valued in [0, 1] with default value 0. Increasing its value reduces the norm of the LMI solution R. This typically yields controllers with slower dynamics in singular problems (rank-deficient D 12 ). This also improves closed-loop damping when p 12 (s) has jω-axis zeros. options(2): same as options(1) for S and singular problems with rank-deficient D 21 when options(3) is set to ε > 0, reduced-order synthesis is performed whenever ρ(R –1 S –1 ) ? (1 – ε) gopt 2 The default value is ε = 10 –3 Remark The LMI-based approach is directly applicable to singular plants without preliminary regularization. Reference Scherer, C., “H ∞ Optimization without Assumptions on Finite or Infinite Zeros,” SIAM J. Contr. Opt., 30 (1992), pp. 143–166. Gahinet, P., and P. Apkarian, “A Linear Matrix Inequality Approach to H ∞ Control,” Int. J. Robust and Nonlinear Contr., 4 (1994), pp. 421–448. Gahinet, P., “Explicit Controller Formulas for LMI-based H ∞ Synthesis,” in Proc. Amer. Contr. Conf., 1994, pp. 2396–2400. Iwasaki, T., and R.E. Skelton, “All Controllers for the General H ∞ Control Problem: LMI Existence Conditions and State-Space Formulas,” Automatica, 30 (1994), pp. 1307–1317. See Also hinfric, hinfmix, dhinflmi, ltisys, slft hinfmix 9-46 9hinfmix Purpose Mixed H 2 /H ∞ synthesis with pole placement constraints Syntax [gopt,h2opt,K,R,S] = hinfmix(P,r,obj,region,dkbnd,tol) Description hinfmix performs multi-objective output-feedback synthesis. The control problem is sketched in Figure 9.1. Figure 9-1: Mixed H 2 /H ∞ synthesis If T ∞ (s) and T 2 (s) denote the closed-loop transfer functions from w to z ∞ and z 2 , respectively, hinfmix computes a suboptimal solution of the following synthesis problem: Design an LTI controller K(s) that minimizes the mixed H 2 /H ∞ criterion subject to ||T ∞ || ∞ < γ 0 ||T 2 || 2 < ν 0 The closed-loop poles lie in some prescribed LMI region D. Recall that || . || ∞ and || . || 2 denote the H ∞ norm (RMS gain) and H 2 norm of transfer functions. More details on the motivations and statement of this problem can be found in “Multi-Objective H Synthesis” on page 5-15. On input, P is the SYSTEM matrix of the plant P(s) and r is a three-entry vector listing the lengths of z 2 , y, and u. Note that z ∞ and/or z 2 can be empty. The K(s) P(s) u y w z ∞ z 2 α T ∞ ∞ 2 β T 2 2 2 + hinfmix 9-47 four-entry vector obj = [γ 0 , ν 0 , α, β] specifies the H 2 /H ∞ constraints and trade-off criterion, and the remaining input arguments are optional: region specifies the LMI region for pole placement (the default region = [] is the open left-half plane). Use lmireg to interactively build the LMI region description region dkbnd is a user-specified bound on the norm of the controller feedthrough matrix D K (see the definition in “LMI Formulation” on page 5-16). The default value is 100. To make the controller K(s) strictly proper, set dkbnd = 0. tol is the required relative accuracy on the optimal value of the trade-off criterion (the default is 10 –2 ). The function hinfmix returns guaranteed H ∞ and H 2 performances gopt and h2opt as well as the SYSTEM matrix K of the LMI-optimal controller. You can also access the optimal values of the LMI variables R, S via the extra output arguments R and S. A variety of mixed and unmixed problems can be solved with hinfmix (see list in “The Function hinfmix” on page 5-20). In particular, you can use hinfmix to perform pure pole placement by setting obj = [0 0 0 0]. Note that both z ∞ and z 2 can be empty in such case. Example See the example in“H Synthesis” on page 5-10. Reference Chilali, M., and P. Gahinet, “H ∞ Design with Pole Placement Constraints: An LMI Approach,” to appear in IEEE Trans. Aut. Contr., 1995. Scherer, C., “Mixed H 2 H ∞ Control,” to appear in Trends in Control: A European Perspective, volume of the special contributions to the ECC 1995. See Also lmireg, hinflmi, msfsyn hinfpar 9-48 9hinfpar Purpose Extract the state-space matrices A, B 1 , B 2 , . . .from the SYSTEM matrix representation of an H ∞ plant Syntax [a,b1,b2,c1,c2,d11,d12,d21,d22] = hinfpar(P,r) b1 = hinfpar(P,r,'b1') Description In H ∞ control problems, the plant P(s) is specified by its state-space equations = Ax + B 1 w + B 2 u z = C 1 x + D 11 w + D 12 u y = C 2 x + D 21 w + D 22 u or their discrete-time counterparts. For design purposes, this data is stored in a single SYSTEM matrix P created by P = ltisys(a,[b1 b2],[c1;c2],[d11 d12;d21 d22]) The function hinfpar retrieves the state-space matrices A, B 1 , B 2 , . . . from P. The second argument r specifies the size of the D 22 matrix. Set r = [ny,nu] where ny is the number of measurements (length of y) and nu is the number of controls (length of u). The syntax hinfpar(P,r,'b1') can be used to retrieve just the B 1 matrix. Here the third argument should be one of the strings 'a', 'b1', 'b2',.... See Also hinfric, hinflmi, ltisys x · hinfric 9-49 9hinfric Purpose Riccati-based H ∞ synthesis for continuous-time plants Syntax gopt = hinfric(P,r) [gopt,K] = hinfric(P,r) [gopt,K,x1,x2,y1,y2,Preg] = hinfric(P,r,gmin,gmax,... tol,options) Description hinfric computes an H ∞ controller K(s) that internally stabilizes the control loop and minimizes the RMS gain (H ∞ performance) of the closed-loop transfer function T wz . The function hinfric implements the Riccati-based approach. The SYSTEM matrix P contains a state-space realization = Ax + B 1 w + B 2 u z = C 1 x + D 11 w + D 12 u y = C 2 x + D 21 w + D 22 u of the plant P(s) and is formed by the command P = ltisys(a,[b1 b2],[c1;c2],[d11 d12;d21 d22]) The row vector r specifies the dimensions of D 22 . Set r = [p 2 m2] when y ∈ R p2 and u ∈ R m2 . To minimize the H ∞ performance, call hinfric with the syntax gopt = hinfric(P,r) K(s) P(s) u y w z x · hinfric 9-50 This returns the smallest achievable RMS gain from w to z over all stabilizing controllers K(s). An H ∞ -optimal controller K(s) with performance gopt is computed by [gopt,K] = hinfric(P,r) The H ∞ performance can also be minimized within prescribed bounds gmin and gmax and with relative accuracy tol by typing [gopt,K] = hinfric(P,r,gmin,gmax,tol) Setting both gmin and gmax to γ > 0 tests whether the H ∞ performance γ is achievable and returns a controller K(s) such that ||T wz || ∞ < γ when γ is achievable. Setting gmin or gmax to zero is equivalent to specifying no lower or upper bound. Finally, the full syntax [gopt,K,x1,x2,y1,y2,Preg] = hinfric(P,r) also returns the solutions X ∞ = x2/x1 and Y ∞ = y2/y1 of the H ∞ Riccati equations for γ = gopt , as well as the regularized plant Preg used to compute the controller in singular problems. Given the controller SYSTEM matrix K returned by hinfric, a realization of the closed-loop transfer function from w to z is given by Twz = slft(P,K) Control Parameters hinfric automatically regularizes singular H ∞ problems and computes reduced-order controllers when the matrix I – γ –2 X ∞ Y ∞ is nearly singular. The optional input options gives access to the control parameters that govern the regularization and order reduction: options(1) controls the amount of regularization used when D 12 or D 21 is rank-deficient. This tuning parameter is valued in [0,1] with default value 0. Increasing options(1) augments the amount of regularization. This generally results in less control effort and in controller state-space matrices of smaller norm, possibly at the expense of a higher closed-loop RMS gain. options(2) controls the amount of regularization used when P 12 (s) or P 21 (s) has jω-axis zero(s). Such problems are regularized by replacing A by A + εI where ε is a small positive number. Increasing options(2) increases ε and improves the closed-loop damping. Beware that making ε too large may hinfric 9-51 destroy stabilizability or detectability when the plant P(s) includes shaping filters with poles close to the imaginary axis. In such cases the optimal H ∞ performance will be infinite (see “Practical Considerations” on page 6-18 for more details). when options(3) is set to ε > 0, reduced-order synthesis is performed whenever ρ(X ∞ Y ∞ ) ? (1 – ε) gopt 2 The default value is ε = 10 –3 Remark Setting gmin = gmax = Inf computes the optimal LQG controller. Example See the example in “H Synthesis” on page 5-10. Reference Doyle, J.C., Glover, K., Khargonekar, P., and Francis, B., “State-Space Solutions to Standard H 2 and H ∞ Control Problems,” IEEE Trans. Aut. Contr., AC–34 (1989), pp. 831–847. Gahinet, P., and A.J. Laub, “Reliable Computation of flopt in Singular H ∞ Control,” in Proc. Conf. Dec. Contr., Lake Buena Vista, Fl., 1994, pp. 1527– 1532. Gahinet, P., “Explicit Controller Formulas for LMI-based H ∞ Synthesis,” in Proc. Amer. Contr. Conf., 1994, pp. 2396–2400. See Also hinflmi, hinfmix, dhinfric, ltisys, slft lmiedit 9-52 9lmiedit Purpose Specify or display systems of LMIs as MATLAB expressions Syntax lmiedit Description lmiedit is a graphical user interface for the symbolic specification of LMI problems. Typing lmiedit calls up a window with two editable text areas and various pushbuttons. To specify an LMI system, 1 Give it a name (top of the window). 2 Declare each matrix variable (name and structure) in the upper half of the window. The structure is characterized by its type (S for symmetric block diagonal, R for unstructured, and G for other structures) and by an additional structure matrix similar to the second input argument of lmivar. Please use one line per matrix variable in the text editing areas. 3 Specify the LMIs as MATLAB expressions in the lower half of the window. An LMI can stretch over several lines. However, do not specify more than one LMI per line. Once the LMI system is fully specified, you can perform the following operations by pressing the corresponding pushbutton: Visualize the sequence of lmivar/lmiterm commands needed to describe this LMI system (view commands buttons) Conversely, display the symbolic expression of the LMI system produced by a particular sequence of lmivar/lmiterm commands (click the describe... buttons) Save the symbolic description of the LMI system as a MATLAB string (save button). This description can be reloaded later on by pressing the load button Read a sequence of lmivar/lmiterm commands from a file (read button). The matrix expression of the LMI system specified by these commands is then displayed by clicking on describe the LMIs... Write in a file the sequence of lmivar/lmiterm commands needed to specify a particular LMI system (write button) Generate the internal representation of the LMI system by pressing create. The result is written in a MATLAB variable with the same name as the LMI system lmiedit 9-53 Remark Editable text areas have built-in scrolling capabilities. To activate the scroll mode, click in the text area, maintain the mouse button down, and move the mouse up or down. The scroll mode is only active when all visible lines have been used. Example An example and some limitations of lmiedit can be found in “The LMI Editor lmiedit” on page 8-16. See Also lmivar, lmiterm, newlmi, lmiinfo lmiinfo 9-54 9lmiinfo Purpose Interactively retrieve information about the variables and term content of LMIs Syntax lmiinfo(lmisys) Description lmiinfo provides qualitative information about the system of LMIs lmisys. This includes the type and structure of the matrix variables, the number of diagonal blocks in the inner factors, and the term content of each block. lmiinfo is an interactive facility where the user seeks specific pieces of information. General LMIs are displayed as N' * L(x) * N < M' * R(x) * M where N,M denote the outer factors and L,R the left and right inner factors. If the outer factors are missing, the LMI is simply written as L(x) < R(x) If its right-hand side is zero, it is displayed as N' * L(x) * N < 0 Information on the block structure and term content of L(x) and R(x) is also available. The term content of a block is symbolically displayed as C1 + A1*X2*B1 + B1'*X2*A1' + a2*X1 + x3*Q1 with the following conventions: X1, X2, x3 denote the problem variables. Upper-case X indicates matrix variables while lower-case x indicates scalar variables. The labels 1,2,3 refer to the first, second, and third matrix variable in the order of declaration. Cj refers to constant terms. Special cases are I and I (I = identity matrix). Aj, Bj denote the left and right coefficients of variable terms. Lower-case letters such as a2 indicate a scalar coefficient. Qj is used exclusively with scalar variables as in x3*Q1. The index j in Aj, Bj, Cj, Qj is a dummy label. Hence C1 may appear in several blocks or several LMIs without implying any connection between the corresponding constant terms. Exceptions to this rule are the notations lmiinfo 9-55 A1*X2*A1' and A1*X2*B1 + B1'*X2'*A1' which indicate symmetric terms and symmetric pairs in diagonal blocks. Example Consider the LMI where the matrix variables are X of Type 1, Y of Type 2, and z scalar. If this LMI is described in lmis, information about X and the LMI block structure can be obtained as follows: lmiinfo(lmis) LMI ORACLE ------ This is a system of 1 LMI with 3 variable matrices Do you want information on (v) matrix variables (l) LMIs (q) quit ?> v Which variable matrix (enter its index k between 1 and 3) ? 1 X1 is a 2x2 symmetric block diagonal matrix its (1,1)-block is a full block of size 2 ------- This is a system of 1 LMI with 3 variable matrices Do you want information on (v) matrix variables (l) LMIs (q) quit ?> l Which LMI (enter its number k between 1 and 1) ? 1 This LMI is of the form 0 e 2X– A+ T YB B T Y T AI++XC C T XzI–?? ?? ?? lmiinfo 9-56 0 < R(x) where the inner factor(s) has 2 diagonal block(s) Do you want info on the right inner factor ? (w) whole factor (b) only one block (o) other LMI (t) back to top level ?> w Info about the right inner factor block (1,1) : I + a1*X1 + A2*X2*B2 + B2'*X2'*A2' block (2,1) : A3*X1 block (2,2) : x3*A4 (w) whole factor (b) only one block (o) other LMI (t) back to top level ------ This is a system of 1 LMI with 3 variable matrices Do you want information on (v) matrix variables (l) LMIs (q) quit ?> q It has been a pleasure serving you! Note that the prompt symbol is ?> and that answers are either indices or letters. All blocks can be displayed at once with option (w), or you can prompt for specific blocks with option (b). Remark lmiinfo does not provide access to the numerical value of LMI coefficients. See Also decinfo, lminbr, matnbr, decnbr lminbr 9-57 9lminbr Purpose Return the number of LMIs in an LMI system Syntax k = lminbr(lmisys) Description lminbr returns the number k of linear matrix inequalities in the LMI problem described in lmisys. See Also lmiinfo, matnbr lmireg 9-58 9lmireg Purpose Specify LMI regions for pole placement purposes Syntax region = lmireg region = lmireg(reg1,reg2,...) Description lmireg is an interactive facility to specify the LMI regions involved in multi-objective H ∞ synthesis with pole placement constraints (see msfsyn and hinfmix). Recall that an LMI region is any convex subset D of the complex plane that can be characterized by an LMI in z and , i.e., D = {z ∈ C : L + Mz + M T < 0} for some fixed real matrices M and L = L T . This class of regions encompasses half planes, strips, conic sectors, disks, ellipses, and any intersection of the above. Calling lmireg without argument starts an interactive query/answer session where you can specify the region of your choice. The matrix region = [L, M] is returned upon termination. This matrix description of the LMI region can be passed directly to msfsyn or hinfmix for synthesis purposes. The function lmireg can also be used to intersect previously defined LMI regions reg1, reg2,.... The output region is then the [L, M] description of the intersection of these regions. See Also msfsyn, hinfmix z z lmiterm 9-59 9lmiterm Purpose Specify the term content of LMIs Syntax lmiterm(termID,A,B,flag) Description lmiterm specifies the term content of an LMI one term at a time. Recall that LMI term refers to the elementary additive terms involved in the block-matrix expression of the LMI. Before using lmiterm, the LMI description must be initialized with setlmis and the matrix variables must be declared with lmivar. Each lmiterm command adds one extra term to the LMI system currently described. LMI terms are one of the following entities: outer factors constant terms (fixed matrices) variable terms AXB or AX T B where X is a matrix variable and A and B are given matrices called the term coefficients. When describing an LMI with several blocks, remember to specify only the terms in the blocks on or below the diagonal (or equivalently, only the terms in blocks on or above the diagonal). For instance, specify the blocks (1,1), (2,1), and (2,2) in a two-block LMI. In the calling of limterm, termID is a four-entry vector of integers specifying the term location and the matrix variable involved. where positive p is for terms on the left-hand side of the p-th LMI and negative p i s for terms on the right-hand side of the p-th LMI. Recall that, by convention, the left-hand side always refers to the smaller side of the LMI. The index p is relative to the order of declaration and corresponds to the identifier returned by newlimi. termID (1) +p -p ? ? ? = termID (2:3) 00,[]for outer factors ij,[]for terms in the ij,()-th block of the left or right inner factor ? ? ? ? ? = lmiterm 9-60 where x is the identifier of the matrix variable X as returned by lmivar. The arguments A and B contain the numerical data and are set according to: Note that identity outer factors and zero constant terms need not be specified. The extra argument flag is optional and concerns only conjugated expressions of the form (AXB) + (AXB) T = AXB + B T X (T) A T in diagonal blocks. Setting flag = 's' allows you to specify such expressions with a single lmiterm command. For instance, lmiterm([1 1 1 X],A,1,'s') adds the symmetrized expression AX + X T A T to the (1,1) block of the first LMI and summarizes the two commands lmiterm([1 1 1 X],A,1) lmiterm([1 1 1 X],1,A') Aside from being convenient, this shortcut also results in a more efficient representation of the LMI. Type of Term A B outer factor N matrix value of N omit constant term C matrix value of C omit variable term AXB or AX T B matrix value of A (1 if A is absent) matrix value of B (1 if B is absent) termID (4) 0 for outer factors x for variable terms AXB x– for variable terms AX T B? ? ? ? ? = lmiterm 9-61 Example Consider the LMI where X 1 , X 2 are matrix variables of Types 2 and 1, respectively, and x 3 is a scalar variable (Type 1). After initializing the LMI description with setlmis and declaring the matrix variables with lmivar, the terms on the left-hand side of this LMI are specified by: lmiterm([1 1 1 X2],2*A,A') % 2*A*X2*A' lmiterm([1 1 1 x3],-1,E) % -x3*E lmiterm([1 1 1 0],D*D') % D*D' lmiterm([1 2 1 -X1],1,B) % X1'*B lmiterm([1 2 2 0],-1) % -I Here X1, X2, X3 should be the variable identifiers returned by lmivar. Similarly, the term content of the right-hand side is specified by: lmiterm([-1 0 0 0],M) % outer factor M lmiterm([-1 1 1 X1],C,C','s') % C*X1*C'+C*X1'*C' lmiterm([-1 2 2 X2],-f,1) % -f*X2 Note that CX 1 C T + CC T is specified by a single lmiterm command with the flag 's' to ensure proper symmetrization. See Also setlmis, lmivar, getlmis, lmiedit, newlmi 2AX 2 A T x 3 E– DD T + B T X 1 X 1 T BI– ?? ?? ?? M T CX 1 C T CX 1 T C T + 0 0f– X 2 ?? ?? ?? M< X 1 T lmivar 9-62 9lmivar Purpose Specify the matrix variables in an LMI problem Syntax X = lmivar(type,struct) [X,n,sX] = lmivar(type,struct) Description lmivar defines a new matrix variable X in the LMI system currently described. The optional output X is an identifier that can be used for subsequent reference to this new variable. The first argument type selects among available types of variables and the second argument struct gives further information on the structure of X depending on its type. Available variable types include: type=1: Symmetric matrices with a block-diagonal structure. Each diagonal block is either full (arbitrary symmetric matrix), scalar (a multiple of the identity matrix), or identically zero. If X has R diagonal blocks, struct is an R-by-2 matrix where struct(r,1) is the size of the r-th block struct(r,2) is the type of the r-th block (1 for full, 0 for scalar, 1 for zero block). type=2: Full m-by-n rectangular matrix. Set struct = [m,n] in this case. type=3: Other structures. With Type 3, each entry of X is specified as zero or ±x n where x n is the n-th decision variable. Accordingly, struct is a matrix of the same dimensions as X such that struct(i,j)=0 if X(i, j) is a hard zero struct(i,j)=n if X(i, j) = x n struct(i,j)= n if X(i, j) = –x n Sophisticated matrix variable structures can be defined with Type 3. To specify a variable X of Type 3, first identify how many free independent entries are involved in X. These constitute the set of decision variables associated with X. If the problem already involves n decision variables, label the new free variables as x n+1 , . . ., x n+p . The structure of X is then defined in terms of x n+1 , . . ., x n+p as indicated above. To help specify matrix variables of Type 3, lmivar optionally returns two extra outputs: (1) the total number n of scalar decision variables used so far and (2) a matrix sX showing the entry-wise lmivar 9-63 dependence of X on the decision variables x 1 , . . ., x n . For more details, see Example 2 below and “Structured Matrix Variables” on page 8-33. Example 1 Consider an LMI system with three matrix variables X 1 , X 2 , X 3 such that X 1 is a 3 × 3 symmetric matrix (unstructured), X 2 is a 2 × 4 rectangular matrix (unstructured), X 3 = where ? is an arbitrary 5 × 5 symmetric matrix, δ 1 and δ 2 are scalars, and I 2 denotes the identity matrix of size 2. These three variables are defined by setlmis([]) X1 = lmivar(1,[3 1]) % Type 1 X2 = lmivar(2,[2 4]) % Type 2 of dim. 2x4 X3 = lmivar(1,[5 1;1 0;2 0]) % Type 1 The last command defines X 3 as a variable of Type 1 with one full block of size 5 and two scalar blocks of sizes 1 and 2, respectively. Example 2 Combined with the extra outputs n and sX of lmivar, Type 3 allows you to specify fairly complex matrix variable structures. For instance, consider a matrix variable X with structure where X 1 and X 2 are 2-by-3 and 3-by-2 rectangular matrices, respectively. You can specify this structure as follows: 1 Define the rectangular variables X 1 and X 2 by setlmis([]) [X1,n,sX1] = lmivar(2,[2 3]) ? 00 0 δ 1 0 00δ 2 I 2 ?? ?? ?? X X 1 0 0X 2?? ?? ?? = lmivar 9-64 [X2,n,sX2] = lmivar(2,[3 2]) The outputs sX1 and sX2 give the decision variable content of X 1 and X 2 : sX1 sX1 = 1 2 3 4 5 6 sX2 sX2 = 7 8 9 10 11 12 For instance, sX2(1,1)=7 means that the (1,1) entry of X 2 is the seventh decision variable. 2 Use Type 3 to specify the matrix variable X and define its structure in terms of those of X 1 and X 2 : [X,n,sX] = lmivar(3,[sX1,zeros(2);zeros(3),sX2]) The resulting variable X has the prescribed structure as confirmed by sX sX = 12300 45600 00078 000910 00011 12 See Also setlmis, lmiterm, getlmis, lmiedit, skewdec, symdec, delmvar, setmvar ltiss 9-65 9ltiss Purpose Extract the state-space realization of a linear system from its SYSTEM matrix description Syntax [a,b,c,d,e] = ltiss(sys) Description Given a SYSTEM matrix sys created with ltisys, this function returns the state-space matrices A, B, C, D, E. Example The system G(s) = is specified in transfer function form by sys = ltisys('tf', 2,[1 1]) A realization of G(s) is then obtained as [a,b,c,d] = ltiss(sys) a = -1 b = 1 c = -2 d = 0 Important If the output argument e is not supplied, ltiss returns the realization (E –1 A,E –1 B,C,D). While this is immaterial for systems with E = I, this should be remembered when manipulating descriptor systems. See Also ltisys, ltitf, sinfo 2– s1+ ------------ ltisys 9-66 9ltisys Purpose Pack the state-space realization (A, B, C, D, E) into a single SYSTEM matrix Syntax sys = ltisys(a) sys = ltisys(a,e) sys = ltisys(a,b,c) sys = ltisys(a,b,c,d) sys = ltisys(a,b,c,d,e) sys = ltisys('tf',num,den) Description For continuous-time systems, E = Ax + Bu, y = Cx + Du or discrete-time systems Ex k+1 = Ax k + Bu k , y k = Cx k + Du k , ltisys stores A, B, C, D, E into a single matrix sys with structure The upper right entry n gives the number of states (i.e., A ∈ R n × n ). This data structure is called a SYSTEM matrix. When omitted, d and e are set to the default values D = 0 and E = I. The syntax sys = ltisys(a) specifies the autonomous system = Ax while sys = ltisys(a,e) specifies E = Ax. Finally, you can specify SISO systems by their transfer function G(s) = The syntax is then sys = ltisys('tf',num,den) where num and den are the vectors of coefficients of the polynomials n(s) and d(s) in descending order. x · AjEI–()+ B n 0 CD 0 0 -Inf .. . x · x · ns() ds() ----------- ltisys 9-67 Examples The SYSTEM matrix representation of the descriptor system is created by sys = ltisys(-1,2,1,0,eps) Similarly, the SISO system with transfer function G(s) = is specified by sys = ltisys('tf',[1 4],[1 7]) See Also ltiss, ltitf, sinfo εx · x– 2u+= yx= ? ? ? s4+ s7+ ------------ ltitf 9-68 9ltitf Purpose Compute the transfer function of a SISO system Syntax [num,den] = ltitf(sys) Description Given a single-input/single-output system defined in the SYSTEM matrix sys, ltitf returns the numerator and denominator of its transfer function G(s) = The output arguments num and den are vectors listing the coefficients of the polynomials n(s) and d(s) in descending order. An error is issued if sys is not a SISO system. Example The transfer function of the system = –x + 2u, y = x – 1 is given by sys = ltisys(-1,2,1,-1) [n,d] = ltitf(sys) n = -1 1 d = 1 1 The vectors n and d represent the function See Also ltisys, ltiss, sinfo ns() ds() ----------- x · Gs() s– 1+ s1+ -----------------= magshape 9-69 9magshape Purpose Graphical specification of the shaping filters Syntax magshape Description magshape is a graphical user interface to construct the shaping filters needed in loop-shaping design. Typing magshape brings up a window where the magnitude profile of each filter can be specified graphically with the mouse. Once a profile is sketched, magshape computes an interpolating SISO filter and returns its state-space realization. Several filters can be specified in the same magshape window as follows: 1 First enter the filter names after Filter names. This tells magshape to write the SYSTEM matrices of the interpolating filters in MATLAB variables with these names. 2 Click on the button showing the name of the filter you are about to specify. To sketch its profile, use the mouse to specify a few characteristic points that define the asymptotes and the slope changes. For accurate fitting by a rational filter, make sure that the slopes are integer multiples of ±20 dB/dec. 3 Specify the filter order after filter order and click on the fit data button to perform the fitting. The magnitude response of the computed filter is then drawn as a solid line and its SYSTEM matrix is written in the MATLAB variable with the same name. 4 Adjustments can be made by moving or deleting some of the specified points. Click on delete point or move point to activate those modes. You can also increase the order for a better fit. Press fit data to recompute the interpolating filter. If some of the filters are already defined in the MATLAB environment, their magnitude response is plotted when the button bearing their name is clicked. This allows for easy modification of existing filters. Remark See sderiv for the specification of nonproper shaping filters with a derivative action. Acknowledg- ment The authors are grateful to Prof. Gary Balas and Prof. Andy Packard for providing the cepstrum algorithm used for phase reconstruction. See Also sderiv, frfit, mrfit matnbr 9-70 9matnbr Purpose Return the number of matrix variables in a system of LMIs Syntax K = matnbr(lmisys) Description matnbr returns the number K of matrix variables in the LMI problem described by lmisys. See Also decnbr, lmiinfo, decinfo mat2dec 9-71 9mat2dec Purpose Return the vector of decision variables corresponding to particular values of the matrix variables Syntax decvec = mat2dec(lmisys,X1,X2,X3,...) Description Given an LMI system lmisys with matrix variables X 1 , . . ., X K and given values X1,...,Xk of X 1 , . . ., X K , mat2dec returns the corresponding value decvec of the vector of decision variables. Recall that the decision variables are the independent entries of the matrices X 1 , . . ., X K and constitute the free scalar variables in the LMI problem. This function is useful, for example, to initialize the LMI solvers mincx or gevp. Given an initial guess for X 1 , . . ., X K , mat2dec forms the corresponding vector of decision variables xinit. An error occurs if the dimensions and structure of X1,...,Xk are inconsistent with the description of X 1 , . . ., X K in lmisys. Example Consider an LMI system with two matrix variables X and Y such that X is A symmetric block diagonal with one 2-by-2 full block and one 2-by-2 scalar block Y is a 2-by-3 rectangular matrix Particular instances of X and Y are and the corresponding vector of decision variables is given by decv = mat2dec(lmisys,X0,Y0) decv' ans = 1 3 -1 5 1 2 3 4 5 6 X 0 1300 31– 00 0050 0005 ?? ?? ?? ?? ,Y 0 123 456?? ?? ?? == mat2dec 9-72 Note that decv is of length 10 since Y has 6 free entries while X has 4 independent entries due to its structure. Use decinfo to obtain more information about the decision variable distribution in X and Y. See Also dec2mat, decinfo, decnbr mincx 9-73 9mincx Purpose Minimize a linear objective under LMI constraints Syntax [copt,xopt] = mincx(lmisys,c,options,xinit,target) Description The function mincx solves the convex program (9-18) where x denotes the vector of scalar decision variables. The system of LMIs (9-18) is described by lmisys. The vector c must be of the same length as x. This length corresponds to the number of decision variables returned by the function decnbr. For linear objectives expressed in terms of the matrix variables, the adequate c vector is easily derived with defcx. The function mincx returns the global minimum copt for the objective c T x, as well as the minimizing value xopt of the vector of decision variables. The corresponding values of the matrix variables is derived from xopt with dec2mat. The remaining arguments are optional. The vector xinit is an initial guess of the minimizer xopt. It is ignored when infeasible, but may speed up compu- tations otherwise. Note that xinit should be of the same length as c. As for target, it sets some target for the objective value. The code terminates as soon as this target is achieved, that is, as soon as some feasible x such that c T x e target is found. Set options to [] to use xinit and target with the default options. Control Parameters The optional argument options gives access to certain control parameters of the optimization code. In mincx, this is a five-entry vector organized as follows: options(1) sets the desired relative accuracy on the optimal value lopt (default = 10 –2 ). options(2) sets the maximum number of iterations allowed to be performed by the optimization procedure (100 by default). options(3) sets the feasibility radius. Its purpose and usage are as for feasp. minimize c T x subject to N T L x()NM T ≤ R x()M mincx 9-74 options(4) helps speed up termination. If set to an integer value J > 0, the code terminates when the objective c T x has not decreased by more than the desired relative accuracy during the last J iterations. options(5) = 1 turns off the trace of execution of the optimization procedure. Resetting options(5) to zero (default value) turns it back on. Setting option(i) to zero is equivalent to setting the corresponding control parameter to its default value. See feasp for more detail. Tip for Speed-Up In LMI optimization, the computational overhead per iteration mostly comes from solving a least-squares problem of the form where x is the vector of decision variables. Two methods are used to solve this problem: Cholesky factorization of A T A (default), and QR factorization of A when the normal equation becomes ill conditioned (when close to the solution typically). The message * switching to QR is displayed when the solver has to switch to the QR mode. Since QR factorization is incrementally more expensive in most problems, it is sometimes desirable to prevent switching to QR. This is done by setting options(4) = 1. While not guaranteed to produce the optimal value, this generally achieves a good trade-off between speed and accuracy. Memory Problems QR-based linear algebra (see above) is not only expensiveLMI solvers,memory shortagememory shortage in terms of computational overhead, but also in terms of memory requirement. As a result, the amount of memory required by QR may exceed your swap space for large problems with numerous LMI constraints. In such case, MATLAB issues the error ??? Error using ==> pds Out of memory. Type HELP MEMORY for your options. You should then ask your system manager to increase your swap space or, if no additional swap space is available, set options(4) = 1. This will prevent x min Ax b– mincx 9-75 switching to QR and mincx will terminate when Cholesky fails due to numerical instabilities. Example See “Example 8.2” on page 8-23. Reference The solver mincx implements Nesterov and Nemirovski’s Projective Method as described in Nesterov, Yu, and A. Nemirovski, Interior Point Polynomial Methods in Convex Programming: Theory and Applications, SIAM, Philadelphia, 1994. Nemirovski, A., and P. Gahinet, “The Projective Method for Solving Linear Matrix Inequalities,” Proc. Amer. Contr. Conf., 1994, Baltimore, Maryland, pp. 840–844. The optimization is performed by the C-MEX file pds.mex. See Also defcx, dec2mat, decnbr, feasp, gevp msfsyn 9-76 9msfsyn Purpose Multi-model/multi-objective state-feedback synthesis Syntax [gopt,h2opt,K,Pcl,X] = msfsyn(P,r,obj,region,tol) Description Given an LTI plant P with state-space equations msfsyn computes a state-feedback control u = Kx that Maintains the RMS gain (H ∞ norm) of the closed-loop transfer function T∞ from w to z ∞ below some prescribed value Maintains the H 2 norm of the closed-loop transfer function T 2 from w to z 2 below some prescribed value Minimizes an H 2 /H ∞ trade-off criterion of the form Places the closed-loop poles inside the LMI region specified by region (see lmireg for the specification of such regions). The default is the open left-half plane. Set r = size(d22) and obj = [γ 0 , ν 0 , α, β] to specify the problem dimensions and the design parameters γ 0 , ν 0 , α, and β. You can perform pure pole placement by setting obj = [0 0 0 0]. Note also that z ∞ or z 2 can be empty. On output, gopt and h2opt are the guaranteed H ∞ and H 2 performances, K is the optimal state-feedback gain, Pcl the closed-loop transfer function from w to , and X the corresponding Lyapunov matrix. The function msfsyn is also applicable to multi-model problems where P is a polytopic model of the plant: x · = Ax B 1 wB 2 u++ z ∞ = C 1 xD 11 wD 12 u++ z 2 = C 2 xD 22 u+ ? ? ? ? ? γ 0 0> υ 0 0> α T ∞ ∞ 2 β T 2 2 2 + z ∞ z 2 ?? ?? ?? msfsyn 9-77 with time-varying state-space matrices ranging in the polytope In this context, msfsyn seeks a state-feedback gain that robustly enforces the specifications over the entire polytope of plants. Note that polytopic plants should be defined with psys and that the closed-loop system Pcl is itself polytopic in such problems. Affine parameter-dependent plants are also accepted and automatically converted to polytopic models. Example See “Design Example” on page 4-13. See Also lmireg, psys, hinfmix x · = A t()xB 1 t()wB 2 t()u++ z ∞ = C 1 t()xD 11 t()wD 12 t()u++ z 2 = C 2 t()xD 22 t()u+ ? ? ? ? ? At() B 1 t() B 2 t() C 1 t()D 11 t()D 12 t() C 2 t() 0D 22 t() ?? ?? ?? Co A k B k B 2k C 1k D 11k D 12k C 2k 0D 22k?? ?? ?? : k 1 … K,,= ?? ?? ?? ?? ∈ mubnd 9-78 9mubnd Purpose Compute the upper bound for the μ structured singular value Syntax [mu,D,G] = mubnd(M,delta,target) Description mubnd computes the mixed-μ upper bound mu = ν ? (M) for the matrix M ∈ C m × n and the norm-bounded structured perturbation ? = diag(? 1 , . . ., ?n) (see “Structured Singular Value” on page 3-17). The reciprocal of mu is a guaranteed well-posedness margin for the algebraic loop Specifically, this loop remains well-posed as long as mu x σ max (? i ) < β i where β i is the specified bound on ? i . The uncertainty ? is described by delta (see ublock and udiag). The function mubnd also returns the optimal scaling matrices D and G arising in the computation of mu. To test whether where τ > 0 is a given scalar, set the optional argument target to τ . See Also mustab, muperf ? M w q mu τ≤ muperf 9-79 9muperf Purpose Estimate the robust H ∞ performance of uncertain dynamical systems Syntax [grob,peakf] = muperf(P,delta,0,freqs) [margin,peakf] = muperf(P,delta,g,freqs) Description muperf estimates the robust H ∞ performance (worst-case RMS gain) of the uncertain LTI system where ?(s) = diag(? 1 (s), . . ., ? n (s)) is a linear time-invariant uncertainty quantified by norm bounds σ max (? i (jω)) < β i (ω) or sector bounds ? i ∈ {a, b}. The robust performance γ rob is the smallest γ > 0 such that ||y|| L 2 < γ ||u|| L 2 for all bounded input u(t) and instance ?(s) of the uncertainty. It is finite if and only if the interconnection is robustly stable. muperf converts the robust performance estimation into a robust stability problem for some augmented ? structure (see “Robust Performance” on page 3-21). The input arguments P and delta contain the SYSTEM matrix of P(s) and the description of the LTI uncertainty ?(s) (see ublock and udiag). To compute the robust H ∞ performance grob for the prescribed uncertainty delta, call muperf with the syntax [grob,peakf] = muperf(P,delta) muperf returns grob = Inf when the interconnection is not robustly stable. Alternatively, [margin,peakf] = muperf(P,delta,g) ?(s) P(s) u y muperf 9-80 estimates the robustness of a given H ∞ performance g, that is, how much uncertainty ?(s) can be tolerated before the RMS gain from u to y exceeds g. Provided that σ max (? i (jω)) < margin × β i (ω) in the norm-bounded case and in the sector-bounded case, the RMS gain is guaranteed to remain below g. With both syntaxes, peakf is the frequency where the worst RMS performance or robustness margin is achieved. An optional vector of frequencies freqs to be used for the μ upper bound evaluation can be specified as fourth input argument. See Also mustab, norminf, quadperf ? i ab+ 2 ------------ ba– 2 ------------– margin, ab+ 2 -------------× ba– 2 ------------+ margin× ?? ?? ?? ∈ mustab 9-81 9mustab Purpose Estimate the robust stability margin of uncertain LTI systems Syntax [margin,peak,fs,ds,gs] = mustab(sys,delta,freqs) Description mustab computes an estimate margin of the robust stability margin for the interconnection where G(s) is a given LTI system and ? = diag(? 1 , . . ., ? n ) is a linear time-invariant uncertainty quantified by norm bounds σ max (?i(jω)) < β i (ω) or sector bounds ? i ∈ {a, b}. In the norm-bounded case, the stability of this interconnection is guaranteed for all ?(s) satisfying σ max (? i (jω)) < margin × β i (ω) See “Robust Stability Analysis” on page 3-19 for the interpretation of margin in the sector-bounded case. The value margin is obtained by computing the mixed-μ upper bound ν ? (G(jω i )) over a grid of frequencies ω i . The command [margin,peak] = mustab(sys,delta) returns the guaranteed stability margin margin and the frequency peakf where the margin is weakest. The uncertainty ?(s) is specified by delta (see ublock and udiag). The optional third argument freqs is a user-supplied vector of frequencies where to evaluate ν ? (G(jω)). To obtain the D, G scaling matrices at all tested frequencies, use the syntax [margin,peak,fs,ds,gs] = mustab(sys,delta) Here fs is the vector of frequencies used to evaluate ν ? and the D, G scalings at the frequency fs(i) are retrieved by ?(s) G(s) mustab 9-82 D = getdg(ds,i) G = getdg(gs,i) Caution margin may overestimate the robust stability margin when the frequency grid freqs is too coarse or when the μ upper bound is discontinuous due to real parameter uncertainty. See “Caution” on page 3-21 for more details. Example See “Example” on page 3-28. See Also mubnd, muperf, ublock, udiag newlmi 9-83 9newlmi Purpose Attach an identifying tag to LMIs Syntax tag = newlmi Description newlmi adds a new LMI to the LMI system currently described and returns an identifier tag for this LMI. This identifier can be used in lmiterm, showlmi, or dellmi commands to refer to the newly declared LMI. Tagging LMIs is optional and only meant to facilitate code development and readability. Identifiers can be given mnemonic names to help keep track of the various LMIs. Their value is simply the ranking of each LMI in the system (in the order of declaration). They prove useful when some LMIs are deleted from the LMI system. In such cases, the identifiers are the safest means of referring to the remaining LMIs. See Also setlmis, lmivar, lmiterm, getlmis, lmiedit, dellmi norminf 9-84 9norminf Purpose Compute the random mean-squares (RMS) gain of continuous-time systems Syntax [gain,peakf] = norminf(g,tol) Description Given the SYSTEM matrix g of an LTI system with state-space equations E = Ax + Bu y = Cx + Du and transfer function G(s) = D + C(sE – A) –1 B, norminf computes the peak gain of the frequency response G(jω). When G(s) is stable, this peak gain coincides with the RMS gain or H ∞ norm of G, that is, The function norminf returns the peak gain and the frequency at which it is attained in gain and peakf, respectively. The optional input tol specifies the relative accuracy required on the computed RMS gain (the default value is 10 –2 ). Example The peak gain of the transfer function is given by norminf(ltisys('tf',100,[1 0.01 100])) ans = 1.0000e+03 Reference Boyd, S., V. Balakrishnan, and P. Kabamba, “A Bisection Method for Computing the H ∞ Norm of a Transfer Matrix and Related Problems,” Math. Contr. Sign. Syst., 2 (1989), pp. 207–219. dx dt ------ G ∞ sup σ max Gjω()()= ω G ∞ sup y L 2 u L 2 -------------= uL 2 ∈ u0≠ Gs() 100 s 2 0.01s 100++ -------------------------------------------= norminf 9-85 Bruisma, N.A., and M. Steinbuch, “A Fast Algorithm to Compute the H ∞ -Norm of a Transfer Function Matrix,” Syst. Contr. Letters, 14 (1990), pp. 287–293. Robel, G., “On Computing the Infinity Norm,” IEEE Trans. Aut. Contr., AC–34 (1989), pp. 882–884. See Also dnorminf, muperf, quadperf norm2 9-86 9norm2 Purpose Compute the H 2 norm of a continuous-time LTI system Syntax h2norm = norm2(g) Description Given a stable LTI system driven by a white noise w with unit covariance, norm2 computes the H 2 norm (LQG performance) of G defined by where G(s) = C(sI – A) –1 B. This norm is computed as where P is the solution of the Lyapunov equation AP + PA T + BB T = 0 See Also norminf G: x · Ax Bw+= yCx= ? ? ? G 2 2 lim E 1 T ---- y T 0 T ∫ t()yt()dt ?? ?? ?? = 1 2π ------ G H ∞– ∞ ∫ jω()Gjω()dω= T ∞→ G 2 Trace CPC T ()= pdlstab 9-87 9pdlstab Purpose Assess the robust stability of a polytopic or parameter-dependent system Syntax [tau,Q0,Q1,...] = pdlstab(pds,options) Description pdlstab uses parameter-dependent Lyapunov functions to establish the stability of uncertain state-space models over some parameter range or polytope of systems. Only sufficient conditions for the existence of such Lyapunov functions are available in general. Nevertheless, the resulting robust stability tests are always less conservative than quadratic stability tests when the parameters are either time-invariant or slowly varying. For an affine parameter-dependent system E(p) = A(p)x + B(p)u y = C(p)x + D(p)u with p = (p 1 , . . ., p n ) ∈ R n , pdlstab seeks a Lyapunov function of the form V(x, p) = x T Q(p) –1 x, Q(p) = Q 0 + p 1 Q 1 + . . .p n Q n such that dV(x, p)/dt < 0 along all admissible parameter trajectories. The system description pds is specified with psys and contains information about the range of values and rate of variation of each parameter p i . For a time-invariant polytopic system E = Ax + Bu y = Cx + Du with (9-19) pdlstab seeks a Lyapunov function of the form V(x, α) = x T Q(α) –1 x, Q(α) = α 1 Q 1 + . . .+ α n Q n such that dV(x, α)/dt < 0 for all polytopic decompositions (9-19). x · x · AjE+ B CD?? ?? ?? α i AjE i + B i C i D i?? ?? ?? i1= n ∑ , α i 0≥ , α i i1= n ∑ 1= ,= pdlstab 9-88 Several options and control parameters are accessible through the optional argument options: Setting options(1)=0 tests robust stability (default) When options(2)=0, pdlstab uses simplified sufficient conditions for faster running times. Set options(2)=1 to use the least conservative conditions Remark For affine parameter-dependent systems with time-invariant parameters, there is equivalence between the robust stability of (9-20) and that of the dual system (9-21) However, the second system may admit an affine parameter-dependent Lyapunov function while the first does not. In such case, pdlstab automatically restarts and tests stability on the dual system (9-21) when it fails on (9-20). See Also quadstab, mustab Ep()x · Ap()x= Ep() T z · Ap() T z= pdsimul 9-89 9pdsimul Purpose Time response of a parameter-dependent system along a given parameter trajectory Syntax pdsimul(pds,'traj',tf,'ut',xi,options) [t,x,y] = pdsimul(pds,pv,'traj',tf,'ut',xi,options) Description pdsimul simulates the time response of an affine parameter-dependent system E(p) = A(p)x + B(p)u y = C(p)x + D(p)u along a parameter trajectory p(t) and for an input signal u(t). The parameter trajectory and input signals are specified by two time functions p=traj(t) and u=ut(t). If 'ut' is omitted, the response to a step input is computed by default. The affine system pds is specified with psys. The function pdsimul also accepts the polytopic representation of such systems as returned by aff2pol(pds) or hinfgs. The final time and initial state vector can be reset through tf and xi (their respective default values are 5 seconds and 0). Finally, options gives access to the parameters controlling the ODE integration (type help gear for details). When invoked without output arguments, pdsimul plots the output trajectories y(t). Otherwise, it returns the vector of integration time points t as well as the state and output trajectories x,y. Example See “Design Example” on page 7-10. See Also psys, pvec, gear x · popov 9-90 9popov Purpose Perform the Popov robust stability test Syntax [t,P,S,N] = popov(sys,delta,flag) Description popov uses the Popov criterion to test the robust stability of dynamical systems with possibly nonlinear and/or time-varying uncertainty (see “The Popov Criterion” on page 3-24 for details). The uncertain system must be described as the interconnection of a nominal LTI system sys and some uncertainty delta (see “Norm-Bounded Uncertainty” on page 2-27). Use ublock and udiag to specify delta. The command [t,P,S,N] = popov(sys,delta) tests the robust stability of this interconnection. Robust stability is guaranteed if t < 0. Then P determines the quadratic part x T Px of the Lyapunov function and D and S are the Popov multipliers (see p. ?? for details). If the uncertainty delta contains real parameter blocks, the conservatism of the Popov criterion can be reduced by first performing a simple loop transformation (see “Real Parameter Uncertainty” on page 3-25). To use this refined test, call popov with the syntax [t,P,S,N] = popov(sys,delta,1) Example See “Design Example” on page 7-10. See Also mustab, quadstab, pdlstab, ublock, aff2lft psinfo 9-91 9psinfo Purpose Inquire about polytopic or parameter-dependent systems created with psys Syntax psinfo(ps) [type,k,ns,ni,no] = psinfo(ps) pv = psinfo(ps,'par') sk = psinfo(ps,'sys',k) sys = psinfo(ps,'eval',p) Description psinfo is a multi-usage function for queries about a polytopic or parameter-dependent system ps created with psys. It performs the following operations depending on the calling sequence: psinfo(ps) displays the type of system (affine or polytopic); the number k of SYSTEM matrices involved in its definition; and the numbers of ns, ni, no of states, inputs, and outputs of the system. This information can be optionally stored in MATLAB variables by providing output arguments. pv = psinfo(ps,'par') returns the parameter vector description (for parameter-dependent systems only). sk = psinfo(ps,'sys',k) returns the k-th SYSTEM matrix involved in the definition of ps. The ranking k is relative to the list of systems syslist used in psys. sys = psinfo(ps,'eval',p) instantiates the system for a given vector p of parameter values or polytopic coordinates. For affine parameter-dependent systems defined by the SYSTEM matrices S 0 , S 1 , . . ., S n , the entries of p should be real parameter values p 1 , . . ., p n and the result is the LTI system of SYSTEM matrix S(p) = S 0 + p 1 S 1 + . . .+ p n S n For polytopic systems with SYSTEM matrix ranging in Co{S 1 , . . ., S n }, the entries of p should be polytopic coordinates p 1 , . . ., p n satisfying p j ? 0 and the result is the interpolated LTI system of SYSTEM matrix See Also psys, ltisys S p 1 S 1 … p n S n ++ p 1 … p n ++ ----------------------------------------------= psys 9-92 9psys Purpose Specify polytopic or parameter-dependent linear systems Syntax pols = psys(syslist) affs = psys(pv,syslist) Description psys specifies state-space models where the state-space matrices can be uncertain, time-varying, or parameter-dependent. Two types of uncertain state-space models can be manipulated in the LMI Control Toolbox: Polytopic systems E(t) = A(t)x + B(t)u y = C(t)x + D(t)u whose SYSTEM matrix takes values in a fixed polytope: where S 1 , . . ., S k are given “vertex” systems and Co{S 1 , . . ., S k } = denotes the convex hull of S 1 , . . ., S k (polytope of matrices with vertices S 1 , . . ., S k ) Affine parameter-dependent systems E(p) = A(p)x + B(p)u x · ?????????????????????? ?????????????????????? St() S 1 ?????????????????????? S k At() jE t()+ Bt() Ct() Dt() Co A 1 jE 1 + B 1 C 1 D 1 , . . ., A k jE k + B k C k D k ∈ α i S i i1= k ∑ : α i 0≥ , α i i1= k ∑ 1= ?? ?? ?? ?? x · psys 9-93 y = C(p)x + D(p)u where A( . ); B( . ), . . ., E( . ) are fixed affine functions of some vector p = (p 1 , . . ., p n ) of real parameters, i.e., where S 0 , S 1 , . . ., S n are given SYSTEM matrices. The parameters p i can be time-varying or constant but uncertain. Both types of models are specified with the function psys. The argument syslist lists the SYSTEM matrices S i characterizing the polytopic value set or parameter dependence. In addition, the description pv of the parameter vector (range of values and rate of variation) is required for affine parameter- dependent models (see pvec for details). Thus, a polytopic model with vertex systems S 1 , . . ., S 4 is created by pols = psys([s1,s2,s3,s4]) while an affine parameter-dependent model with 4 real parameters is defined by affs = psys(pv,[s0,s1,s2,s3,s4]) The output is a structured matrix storing all the relevant information. To specify the autonomous parameter-dependent system = A(p)x, A(p) = A 0 + p 1 A 1 + p 2 A 2 , type s0 = ltisys(a0) s1 = ltisys(a1,0) s2 = ltisys(a2,0) ps = psys(pv,[s0 s1 s2]) ?????????????????????? S 0 ?????????????????????? Sp() Ap() jE p()+ Bp() Cp() Dp() = ?????????????????????? S 1 ?????????????????????? S n A 0 jE 0 + B 0 C 0 D 0 + p 1 A 1 jE 1 + B 1 C 1 D 1 + . . . + p n A n jE n + B n C n D n x · psys 9-94 Do not forget the 0 in the second and third commands. This 0 marks the independence of the E matrix on p 1 , p 2 . Typing s0 = ltisys(a0) s1 = ltisys(a1) s2 = ltisys(a2) ps = psys(pv,[s0 s1 s2]) instead would specify the system E(p) = A(p)x, E(p) = (1+p 1 + p 2 )I, A(p) = A 0 +p 1 A 1 +p 2 A 2 Example See “Example 2.1” on page 2-21. See Also psinfo, pvec, aff2pol, ltisys x · pvec 9-95 9pvec Purpose Specify the range and rate of variation of uncertain or time-varying parameters Syntax pv = pvec('box',range,rates) pv = pvec('pol',vertices) Description pvec is used in conjunction with psys to specify parameter-dependent systems. Such systems are parametrized by a vector p = (p 1 , . . ., p n ) of uncertain or time-varying real parameters p i . The function pvec defines the range of values and the rates of variation of these parameters. The type 'box' corresponds to independent parameters ranging in intervals The parameter vector p then takes values in a hyperrectangle of R n called the parameter box. The second argument range is an n-by-2 matrix that stacks up the extremal values and of each p j . If the third argument rates is omitted, all parameters are assumed time-invariant. Otherwise, rates is also an n-by-2 matrix and its j-th row specifies lower and upper bounds and on : Set = Inf and = Inf if p j (t) can vary arbitrarily fast or discontinuously. The type 'pol' corresponds to parameter vectors p ranging in a polytope of the parameter space R n . This polytope is defined by a set of vertices V 1 , . . ., V n corresponding to “extremal” values of the vector p. Such parameter vectors are declared by the command pv = pvec('pol',[v1,v2, . . ., vn]) where the second argument is the concatenation of the vectors v1,...,vn. The output argument pv is a structured matrix storing the parameter vector description. Use pvinfo to read the contents of pv. Example Consider a problem with two time-invariant parameters p 1 ∈ [–1, 2], p 2 ∈ [20, 50] p j p j p j ≤≤ p j p j ν j ν j dp j dt -------- ν j dp j dt -------- ν j ≤≤ ν j ν j pvec 9-96 The corresponding parameter vector p = (p 1 , p 2 ) is specified by pv = pvec('box',[-1 2;20 50]) Alternatively, this vector can be regarded as taking values in the rectangle drawn in Figure 9.2. The four corners of this rectangle are the four vectors Hence, you could also specify p by pv = pvec('pol',[v1,v2,v3,v4]) Figure 9-2: Parameter box See Also pvinfo, psys v 1 1– 20?? ?? ?? = , v 2 1– 50?? ?? ?? = , v 3 2 20?? ?? ?? = , v 4 2 50?? ?? ?? = -1 2 20 50 p 1 p 2 pvinfo 9-97 9pvinfo Purpose Describe a parameter vector specified with pvec Syntax [typ,k,nv] = pvinfo(pv) [pmin,pmax,dpmin,dpmax] = pvinfo(pv,'par',j) vj = pvinfo(pv,'par',j) p = pvinfo(pv,'eval',c) Description pvec retrieves information about a vector p = (p 1 , . . ., p n ) of real parameters declared with pvec and stored in pv. The command pvinfo(pv) displays the type of parameter vector ('box' or 'pol'), the number n of scalar parameters, and for the type 'pol', the number of vertices used to specify the parameter range. For the type 'box': [pmin,pmax,dpmin,dpmax] = pvinfo(pv,'par',j) returns the bounds on the value and rate of variations of the j-th real parameter p j . Specifically, , For the type 'pol': pvinfo(pv,'par',j) returns the j-th vertex of the polytope of R n in which p ranges, while pvinfo(pv,'eval',c) returns the value of the parameter vector p given its barycentric coordinates c with respect to the polytope vertices (V 1 , . . .,V k ). The vector c must be of length k and have nonnegative entries. The corresponding value of p is then given by See Also pvec, psys pmin p j t() pmax≤≤ dpmin dp j dt --------- dpmax≤≤ p c i i1= k ∑ V i c i i1= k ∑ ----------------------------= quadperf 9-98 9quadperf Purpose Compute the quadratic H ∞ performance of a polytopic or parameter-dependent system Syntax [perf,P] = quadperf(ps,g,options) Description The RMS gain of the time-varying system (9-22) is the smallest γ > 0 such that (9-23) for all input u(t) with bounded energy. A sufficient condition for (9-23) is the existence of a quadratic Lyapunov function V(x) = x T Px, P > 0 such that Minimizing γ over such quadratic Lyapunov functions yields the quadratic H ∞ performance, an upper bound on the true RMS gain. The command [perf,P] = quadperf(ps) computes the quadratic H ∞ performance perf when (9-22) is a polytopic or affine parameter-dependent system ps (see psys). The Lyapunov matrix P yielding the performance perf is returned in P. The optional input options gives access to the following task and control parameters: If options(1)=1, perf is the largest portion of the parameter box where the quadratic RMS gain remains smaller than the positive value g (for affine parameter-dependent systems only). The default value is 0 Et()x · At()x= Bt()u, y C t()x= Dt()u++ y L 2 γ u L 2 ≤ u? L 2 ∈ , dV dt ------- y T y γ 2 u T u–+ 0< quadperf 9-99 If options(2)=1, quadperf uses the least conservative quadratic performance test. The default is options(2)=0 (fast mode) options(3) is a user-specified upper bound on the condition number of P (the default is 10 9 ). Example See “Example 3.4” on page 3-10. See Also muperf, quadstab, psys quadstab 9-100 9quadstab Purpose Quadratic stability of polytopic or affine parameter-dependent systems Syntax [tau,P] = quadstab(ps,options) Description For affine parameter-dependent systems E(p) = A(p)x, p(t) = (p 1 (t), . . ., p n (t)) or polytopic systems E(t) = A(t)x, (A, E) ∈ Co{(A 1 , E 1 ), . . ., (A n , E n )}, quadstab seeks a fixed Lyapunov function V(x) = x T Px with P > 0 that establishes quadratic stability (see “Quadratic Lyapunov Functions” on page 3-3 for details). The affine or polytopic model is described by ps (see psys). The task performed by quadstab is selected by options(1): if options(1)=0 (default), quadstab assesses quadratic stability by solving the LMI problem Minimize τ over Q = Q T such that A T QE + EQA T < τI for all admissible values of (A, E) Q > I The global minimum of this problem is returned in tau and the system is quadratically stable if tau < 0 if options(1)=1, quadstab computes the largest portion of the specified parameter range where quadratic stability holds (only available for affine models). Specifically, if each parameter p i varies in the interval p i ∈ [p i0 – δ i , p i0 + δ i ], quadstab computes the largest θ > 0 such that quadratic stability holds over the parameter box p i ∈ [p i0 – θδ i , p i0 + θδ i ] This “quadratic stability margin” is returned in tau and ps is quadratically stable if tau ? 1. Given the solution Q opt of the LMI optimization, the Lyapunov matrix P is given by P = . This matrix is returned in P. x · x · Q opt 1– quadstab 9-101 Other control parameters can be accessed through options(2) and options(3): if options(2)=0 (default), quadstab runs in fast mode, using the least expensive sufficient conditions. Set options(2)=1 to use the least conservative conditions options(3) is a bound on the condition number of the Lyapunov matrix P. The default is 10 9 . Example See “Example 3.1” on page 3-7 and “Example 3.2” on page 3-8. See Also pdlstab, mustab, decay, quadperf, psys ricpen 9-102 9ricpen Purpose Solve continuous-time Riccati equations (CARE) Syntax X = ricpen(H,L) [X1,X2] = ricpen(H,L) Description This function computes the stabilizing solution X of the Riccati equation (9-24) where E is an invertible matrix. The solution is computed by unitary deflation of the associated Hamiltonian pencil The matrices H and L are specified as input arguments. For solvability, the pencil H – λL must have no finite generalized eigenvalue on the imaginary axis. When called with two output arguments, ricpen returns two matrices X 1 , X 2 such that has or thonormal columns, and X = X 2 is the stabilizing solution of (9-24). If X 1 is invertible, the solution X is returned directly by the syntax X = ricpen(H,L). ricpen is adapted from care and implements the generalized Schur method for solving CAREs. Reference Laub, A. J., “A Schur Method for Solving Algebraic Riccati Equations,” IEEE Trans. Aut. Contr., AC–24 (1979), pp. 913–921. Arnold, W.F., and A.J. Laub, “Generalized Eigenproblem Algorithms and Software for Algebraic Riccati Equations,” Proc. IEEE, 72 (1984), pp. 1746– 1754. A T XE E T XA E T XFXE E( T XB– S )R 1– B( T XE S T ) Q++ + ++0= H λL– AFB Q– A– T S– S T B T R λ E00 0E T 0 000 –= X 1 X 2 X 1 1– ricpen 9-103 See Also care, dricpen sadd 9-104 9sadd Purpose Parallel interconnection of linear systems Syntax sys = sadd(g1,g2,...) Description sadd forms the parallel interconnection of the linear systems G 1 , . . ., G n . The arguments g1, g2,... are either SYSTEM matrices (dynamical systems) or constant matrices (static gains). In addition, one (and at most one) of these systems can be polytopic or parameter- dependent, in which case the resulting system sys is of the same nature. In terms of transfer functions, sadd performs the sum G(s) = G 1 (s) + G 2 (s) + . . .+ G n (s) See Also smult, sdiag, sloop, ltisys G 1 G 2 G n . . . + + + G sbalanc 9-105 9sbalanc Purpose Numerically balance the state-space realization of a linear system Syntax [a,b,c] = sbalanc(a,b,c,condnbr) sys = sbalanc(sys,condnbr) Description sbalanc computes a diagonal invertible matrix T that balances the state-space realization (A, B, C) by reducing the magnitude of the off-diagonal entries of A and by scaling the norms of B and C. The resulting realization is of the form (T –1 A T , T –1 B, CT) The optional argument condnbr specifies an upper bound on the condition number of T. The default value is 10 8 . Example Computing a realization of with the function tf2ss yields [a,b,c,d]=tf2ss([1 1 1],[1 1000 500^2]) a = -1000 -250000 1 0 b = 1 0 c = -999 -249999 d = 1 To reduce the numerical range of this data and improve numerical stability of subsequent computations, you can balance this realization by [a,b,c]=sbalanc(a,b,c) a = -1000.00 -3906.25 64.00 0 Ws() s 2 s1++ s500+() 2 --------------------------= sbalanc 9-106 b = 64.00 0 c = -15.61 -61.03 See Also ltisys sconnect 9-107 9sconnect Purpose Specify general control structures or system interconnections Syntax [P,r] = sconnect(inputs,outputs,Kin,G1in,g1,G2in,g2,...) Description sconnect is useful in loop-shaping problems to turn general control structures into the standard linear-fractional interconnection used in H ∞ synthesis. In this context, sconnect returns: The SYSTEM matrix P of the standard H ∞ plant P(s) The vector r = [p2 m2] where p2 and m2 are the number of input and outputs of the controller, respectively. More generally, sconnect is useful to specify complex interconnections of LTI systems (see Example 1 below). General control loops or system interconnections are described in terms of signal flows. Specifically, you must specify The exogenous inputs, i.e., what signals enter the loop or interconnection The ouput signals, i.e., the signals generated by the loop or interconnection How the inputs of each dynamical system relate to the exogenous inputs and to the outputs of other systems. The outputs of a system of name G are collectively denoted by G. To refer to particular outputs, e.g., the second and third, use the syntax G(2:3). The arguments of sconnect are as follows: The first argument inputs is a string listing the exogenous inputs. For instance, inputs='r(2), d' specifies two inputs, a vector r of size 2 and a scalar input d The second argument outputs is a string listing the outputs generated by the control loop. Outputs are defined as combinations of the exogenous inputs and the outputs of the dynamical systems. For instance, outputs='e=r-S; S+d' specifies two outputs e = r – y and y + d where y is the output of the system of name S. The third argument Kin names the controller and specifies its inputs. For instance, Kin='K:e' inserts a controller of name K and input e. If no name is specified as in Kin='e', the default name K is given to the controller. sconnect 9-108 The remaining arguments come in pairs and specify, for each known LTI system in the loop, its input list Gkin and its SYSTEM matrix gk. The input list is a string of the form system name : input1 ; input2 ; ... ; input n For instance, G1in='S: K(2);d' inserts a system called S whose inputs consist of the second output of the controller K and of the input signal d. Note that the names given to the various systems are immaterial provided that they are used consistently throughout the definitions. Remark One of the dynamical systems can be polytopic or affine parameter-dependent. In this case, sconnect always returns a polytopic model of the interconnection. Example 1 This simple example illustrates the use of sconnect to compute interconnections of systems without controller. The SYSTEM matrix of this series interconnection is returned by sys1 = ltisys('tf',1,[1 1]) sys2 = ltisys('tf',[1 0],[1 2]) S = sconnect('r','S2',[],'S1:r',sys1,'S2:S1',sys2) Note that inputK is set to [] to mark the absence of controller. The same result would be obtained with S = smult(sys1,sys2) r y 1 s + 1 1 s + 2 sconnect 9-109 Example 2 The standard plant associated with this simple tracking loop is given by g = ltisys('tf',1,[1 1]) P = sconnect('r','y=G','C:r-y','G:C',g) Example 3 If the SYSTEM matrices of the system G and filters W 1 , W 2 , W 3 , W d , W n are stored in the variables g, w1, w2, w3, wd, wn, respectively, the corresponding standard plant P(s) is formed by inputs = 'r;n;d' outputs = 'W1;W2;W3' Kin = 'K: e=r-y-Wn' W3in = 'W3: y=G+Wd' 1 s + 1 C(s) r e u y + ? r + ? K G W 1 W 2 W 3 W n W d d z 3 n + + + + z 2 z 1 e sconnect 9-110 [P,r] = sconnect(inputs,outputs,Kin,'G:K',g,'W1:e',w1,... 'W2:K',w2,W3in,w3,'Wd:d',wd,'Wn:n',wn) See Also ltisys, slft, ssub, sadd, smult sderiv 9-111 9sderiv Purpose Apply proportional-derivative action to some inputs/outputs of an LTI system Syntax dsys = sderiv(sys,chan,pd) Description sderiv multiplies selected inputs and/or outputs of the LTI system sys by the proportional-derivator ns + d. The coefficients n and d are specified by setting pd = [n , d] The second argument chan lists the input and output channels to be filtered by ns + d. Input channels are denoted by their ranking preceded by a minus sign. On output, dsys is the SYSTEM matrix of the corresponding interconnection. An error is issued if the resulting system is not proper. Example Consider a SISO loop-shaping problem where the plant P(s) has three outputs corresponding to the transfer functions S, KS, and T . Given the shaping filters w 1 (s) = , w 2 (s) = 100, w 3 (s) = 1 + 0.01s, the augmented plant associated with the criterion is formed by w1 = ltisys('tf',1,[1 0]) w2 = 100 paug = smult(p,sdiag(w1,w2,1)) paug = sderiv(paug,3,[0.01 1]) This last command multiplies the third output of P by the filter w 3 . See Also ltisys, magshape 1 s --- w 1 S w 2 KS w 3 T ?? ?? ?? ?? ∞ sdiag 9-112 9sdiag Purpose Append (concatenate) linear systems Syntax g = sdiag(g1,g2,...) Description sdiag returns the system obtained by stacking up the inputs and outputs of the systems g1,g2,... as in the diagram. If G 1 (s),G 2 (s), . . . are the transfer functions of g1,g2,..., the transfer function of g is The function sdiag takes up to 10 input arguments. One (and at most one) of the systems g1,g2,... can be polytopic or parameter-dependent, in which case g is of the same nature. . . . G 1 G n u 1 y 1 u n y n u y G Gs() G 1 s() 00 0G 2 s()0 00 ?? ?? ?? = . . . sdiag 9-113 Example Let p be a system with two inputs u 1 , u 2 and three outputs y 1 , y 2 , y 3 . The augmented system with weighting filters w 1 (s) = , w 2 (s) = 10, w 3 (s) = is created by the commands w1 = ltisys('tf',1,[1 0]) w3 = ltisys('tf',[100 0],[1 100]) paug = smult(p,sdiag(w1,10,w3)) See Also ltisys, ssub, sadd, smult, psys P w 1 w 2 w 3 u 1 u 2 y 1 y 2 y 3 y 1 ? y 2 ? y 3 ? P aug 1 s --- 100s s100+ ------------------ setlmis 9-114 9setlmis Purpose Initialize the description of an LMI system Syntax setlmis(lmi0) Description Before starting the description of a new LMI system with lmivar and lmiterm, type setlmis([]) to initialize its internal representation. To add on to an existing LMI system, use the syntax setlmis(lmi0) where lmi0 is the internal representation of this LMI system. Subsequent lmivar and lmiterm commands will then add new variables and terms to the initial LMI system lmi0. See Also getlmis, lmivar, lmiterm, newlmi setmvar 9-115 9setmvar Purpose Instantiate a matrix variable and evaluate all LMI terms involving this matrix variable Syntax newsys = setmvar(lmisys,X,Xval) Description setmvar sets the matrix variable X with identifier X to the value Xval. All terms involving X are evaluated, the constant terms are updated accordingly, and X is removed from the list of matrix variables. A description of the resulting LMI system is returned in newsys. The integer X is the identifier returned by lmivar when X is declared. Instantiating X with setmvar does not alter the identifiers of the remaining matrix variables. The function setmvar is useful to freeze certain matrix variables and optimize with respect to the remaining ones. It saves time by avoiding partial or complete redefinition of the set of LMI constraints. Example Consider the system = Ax + Bu and the problem of finding a stabilizing state-feedback law u = Kx where K is an unknown matrix. By the Lyapunov Theorem, this is equivalent to finding P > 0 and K such that (A + BK)P + P(A + BK) T + I < 0. With the change of variable Y := KP, this condition reduces to the LMI AP + PA T + BY + Y T B T + I < 0. This LMI is entered by the commands n = size(A,1) % number of states ncon = size(B,2) % number of inputs setlmis([]) P = lmivar(1,[n 1]) % P full symmetric Y = lmivar(2,[ncon n]) % Y rectangular lmiterm([1 1 1 P],A,1,'s') % AP+PA' x · setmvar 9-116 lmiterm([1 1 1 Y],B,1,'s') % BY+Y'B' lmiterm([1 1 1 0],1) % I lmis = getlmis To find out whether this problem has a solution K for the particular Lyapunov matrix P = I, set P to I by typing news = setmvar(lmis,P,1) The resulting LMI system news has only one variable Y = K. Its feasibility is assessed by calling feasp: [tmin,xfeas] = feasp(news) Y = dec2mat(news,xfeas,Y) The computed Y is feasible whenever tmin < 0. See Also evallmi, delmvar showlmi 9-117 9showlmi Purpose Return the left- and right-hand sides of an LMI after evaluation of all variable terms Syntax [lhs,rhs] = showlmi(evalsys,n) Description For given values of the decision variables, the function evallmi evaluates all variable terms in a system of LMIs. The left- and right-hand sides of the n-th LMI are then constant matrices that can be displayed with showlmi. If evalsys is the output of evallmi, the values lhs and rhs of these left- and right-hand sides are given by [lhs,rhs] = showlmi(evalsys,n) An error is issued if evalsys still contains variable terms. Example See the description of evallmi. See Also evallmi, setmvar sinfo 9-118 9sinfo Purpose Return the number of states, inputs, and outputs of an LTI system Syntax sinfo(sys) [ns,ni,no] = sinfo(sys) Description For a linear system E = Ax + Bu y = Cx + Du specified in the SYSTEM matrix sys, sinfo returns the order ns of the system (number of states) and the numbers ni and no of its inputs and outputs, respectively. Note that ns, ni, and no are the lengths of the vectors x, u, y, respectively. Without output arguments, sinfo displays this information on the screen and also indicates whether the system is in descriptor form (E | I) or not. See Also ltisys, ltiss, psinfo x · sinv 9-119 9sinv Purpose Compute the inverse G(s) –1 , if it exists, of a system G(s) Syntax h = sinv(g) Description Given a system g with transfer function G(s) = D + C(sE – A) –1 B with D square and invertible, sinv returns the inverse system with transfer function H(s) = G(s) –1 . This corresponds to exchanging inputs and outputs: y = G(s)u ? u = H(s)y See Also ltisys, sadd, smult, sdiag slft 9-120 9slft Purpose Form the linear-fractional interconnection of two time-invariant systems (Redheffer’s star product) Syntax sys = slft(sys1,sys2,udim,ydim) Description slft forms the linear-fractional feedback interconnection of the two systems sys1 and sys2 and returns a realization sys of the closed-loop transfer function from to . The optional arguments udim and ydim specify the lengths of the vectors u and y. Note that u enters sys1 while y is an output of this system.When udim and ydim are omitted, slft forms one of the two interconnections: w 1 z 1 u y w 2 z 2 sys1 sys2 w 1 w 2?? ?? ?? ?? z 1 z 2?? ?? ?? ?? slft 9-121 or depending on which system has the larger number of inputs/outputs. An error is issued if neither of these interconnections can be performed. slft also performs interconnections of polytopic or affine parameter-dependent systems. An error is issued if the result is neither polytopic nor affine parameter-dependent. u y sys1 sys2 w 1 z 1 uy sys2 sys1 z 2 w 2 slft 9-122 Example Consider the interconnection where ?(s), P(s), K(s) are LTI systems. A realization of the closed-loop transfer function from w to z is obtained as clsys = slft(delta,slft(p,k)) or equivalently as clsys = slft(slft(delta,p),k) See Also ltisys, sloop, sconnect uy P(s) K(s) w z ?(s) sloop 9-123 9sloop Purpose Form the feedback interconnection of two systems Syntax sys = sloop(g1,g2,sgn) Description sloop forms the interconnection The output sys is a realization of the closed-loop transfer function from r to y. The third argument sgn specifies either negative feedback (sgn = 1) or positive feedback (sgn = +1). The default value is 1. In terms of the transfer functions G 1 (s) and G 2 (s), sys corresponds to the transfer function (I – εG 1 G 2 ) –1 G 1 where ε =sgn. Example The closed-loop transfer function from r to y in is obtained by sys = sloop( ltisys('tf',1,[1 1]) , ltisys('tf',1,[1 0])) [num,den] = ltitf(sys) num = r y + sgn G 1 G 2 r y + ? 1/(s+1) 1/s sloop 9-124 0 1.00 0.00 den = 1.00 1.00 1.00 See Also ltisys, slft, sconnect smult 9-125 9smult Purpose Series interconnection of linear systems Syntax sys = smult(g1,g2,...) Description smult forms the series interconnection The arguments g1, g2,... are SYSTEM matrices containing the state-space data for the systems G 1 , . . ., G n . Constant matrices are also allowed as a representation of static gains. In addition, one (and at most one) of the input arguments can be a polytopic or affine parameter-dependent model, in which case the output sys is of the same nature. In terms of transfer functions, this interconnection corresponds to the product G(s) = G n (s) × . . . × G 1 (s) Note the reverse order of the terms. Example Consider the interconnection where W i (s), W 0 (s) are given LTI filters and G is the affine parameter-dependent system = θ 1 x + u y = x – θ 2 u parametrized by θ 1 , θ 2 . The parameter-dependent system defined by this interconnection is returned by y G 2 G 1 G n u ........ G G W i (s) W o (s) x · smult 9-126 sys = smult(wi,g,w0) While the SYSTEM matrices wi and w0 are created with ltisys, the parameter-dependent system g should be specified with psys. See Also sadd, sdiag, sloop, slft, ltisys splot 9-127 9splot Purpose Plot the various frequency and time responses of LTI systems Syntax splot(sys,type,xrange) splot(sys,T,type,xrange) Description The first syntax is for continuous-time systems E = Ax + Bu y = Cx + Du The first argument sys is the SYSTEM matrix representation of a system. The optional argument xrange is a vector of frequencies or times used to control the x-axis and the number of points in the plot. Finally, the string type consists of the first two characters of the diagram to be plotted. Available choices include: Frequency Response type plot 'bo' Bode plot 'sv' Singular value plot 'ny' Nyquist plot 'li' lin-log Nyquist plot 'ni' Black/Nichols chart Time Response type plot 'st' Step response 'im' Impulse response 'sq' Response to a square signal 'si' Response to a sinusoidal signal x · splot 9-128 The syntax splot(sys,T,type,xrange) is used for discrete-time systems, in which case T is the sampling period in seconds. Remark The Control System Toolbox is needed to run splot. Example Consider the third-order SISO system with transfer function To plot the Nyquist diagram of the frequency response G(jω), type g = ltisys('tf',1,[1 0.001 1 0]) splot(g,'ny') The resulting plot appears in Figure 9-3. Due to the integrator and poor damping of the natural frequency at 1 rd/s, this plot is not very informative. In such cases, the lin-log Nyquist plot is more appropriate. Given the gain/phase decomposition G(jω) = γ(ω)e jφ(ω) of the frequency response, the lin-log Nyquist plot is the curve described by x = ρ(ω) cos ?(ω), y = ρ(ω) sin ?(ω) where This plot is drawn by the command splot(g,'li') and appears in Figure 9-4. The resulting aspect ratio is more satisfactory thanks to the logarithmic scale for gains larger than one. A more complete contour is drawn in Figure 9-5 by splot(g,'li',logspace(-3,2)) Gs() 1 ss 2 0.001s 1++() -----------------------------------------------= p ω() γω if γω 1≥ 1log 10 γω if γω 1>+ ? ? ? = splot 9-129 Figure 9-3: splot(g,'ny') Figure 9-4: splot(g,'li') splot 9-130 Figure 9-5: splot(g,'li',logspace(-3,2)) See Also ltisys, bode, nyquist, sigma, step, impulse spol 9-131 9spol Purpose Return the poles of an LTI system Syntax poles = spol(sys) Description The function spol computes the poles of the LTI system of SYSTEM matrix sys. If (A, B, C, D, E) denotes the state-space realization of this system, the poles are the generalized eigenvalues of the pencil (A, E) in general, and the eigenvalues of A when E = I. See Also ltisys, ltiss sresp 9-132 9sresp Purpose Frequency response of a continuous-time system Syntax resp = sresp(sys,f) Description Given a continuous-time system G(s) of realization (A, B, C, D, E), sresp computes its frequency response G(jω) = D + C(jωE – A) –1 B at the frequency ω = f. The first input sys is the SYSTEM matrix of G. See Also ltisys, splot, spol ssub 9-133 9ssub Purpose Select particular inputs and outputs in a MIMO system Syntax subsys = ssub(sys,inputs,outputs) Description Given an LTI or polytopic/parameter-dependent system sys, ssub extracts the subsystem subsys mapping the inputs selected by inputs to the outputs selected by outputs. The resulting subsystem has the same number of states as sys even though some of these states may no longer be controllable/observable from the remaining inputs and outputs. Example If p is a system with 3 inputs and 6 outputs, a realization of the transfer function from the input channels 1,3 to the output channels 4,5,6 is obtained as: ssub(p,[1,3],4:6) The second and third arguments are the vectors of indices of the selected inputs/outputs. Inputs and outputs are labeled 1,2,3,... starting from the top of the block diagram: See Also ltisys, psys u 1 u 2 u 3 y 1 y 2 y 3 y 4 y 5 y 6 P ublock 9-134 9ublock Purpose Specify the characteristics of an uncertainty block Syntax block = ublock(dims,bound,type) Description ublock is used to describe individual uncertainty blocks ?i in a block-diagonal uncertainty operator ? = diag(? 1 , . . ., ? n ) Such operators arise in the linear-fractional representation of uncertain dynamical systems (see “How to Derive Such Models” on page 2-23). The first argument dims specifies the dimensions of ?i. Set dim = [m,n] if ? i is m-by-n, i.e., has m outputs and n inputs. The second argument bound contains the quantitative information about the uncertainty (norm bound or sector bounds). Finally, type is a string specifying the dynamical and structural characteristics of ? i . The string type consists of one- or two-letter identifiers marking the uncertainty block properties. Characteristic properties include: The dynamical nature: linear time invariant ('lti'), linear time varying ('ltv'), nonlinear memoryless ('nlm'), or arbitrary nonlinear ('nl'). The default is 'lti' The structure: 'f' for full (unstructured) blocks and 's' for scalar blocks of the form ? i = δ i x I r i Scalar blocks often arise when representing parameter uncertainty. They are also referred to as repeated scalars. The default is 'f' The phase information: the block can be complex-valued ('c') or real-valued ('r'). Complex-valued means arbitrary phase. The default is 'c'. For instance, a real-valued time-varying scalar block is specified by setting type = 'tvsr'. The order of the property identifiers is immaterial. Finally, the quantitative information can be of two types: Norm bound: here the second argument bound is either a scalar β for uniform bounds, ublock 9-135 ||?|| ∞ < β, or a SISO shaping filter W(s) for frequency-weighted bounds ||W –1 ?|| ∞ < 1, that is, σ max (?(jω)) < |W(jω)| for all ω Sector bounds: set bound = [a b] to indicate that the response of ?( . ) lies in the sector {a, b}. Valid values for a and b include Inf and Inf. Example See the example on p. 2-25 of this manual. See Also udiag, uinfo, slft, aff2lft udiag 9-136 9udiag Purpose Form block-diagonal uncertainty structures Syntax delta = udiag(delta1,delta2,....) Description The function udiag appends individual uncertainty blocks specified with ublock and returns a complete description of the block-diagonal uncertainty operator ? = diag(? 1 , ? 2 , . . .) The input arguments delta1, delta2,... are the descriptions of ? 1 , ? 2 , . . . . If udiag is called with k arguments, the output delta is a matrix with k columns, the k-th column being the description of ? k . See Also ublock, udiag, slft, aff2lft uinfo 9-137 9uinfo Purpose Display the characteristics of a block-diagonal uncertainty structure Syntax uinfo(delta) Description Given the description delta of a block-diagonal uncertainty operator ? = diag(? 1 , . . ., ? n ). as returned by ublock and udiag, the function uinfo lists the characteristics of each uncertainty block ? i . See Also ublock, udiag uinfo 9-138