Szidarovszky, F., Bahill, A.T. “Stability Analysis” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000 12 Stability Analysis 12.1 Introduction 12.2 Using the State of the System to Determine Stability 12.3 Lyapunov Stability Theory 12.4 Stability of Time-Invariant Linear Systems Stability Analysis with State-Space Notation?The Transfer Function Approach 12.5 BIBO Stability 12.6 Physical Examples 12.1 Introduction In this chapter, which is based on Szidarovszky and Bahill [1992], we first discuss stability in general and then present four techniques for assessing the stability of a system: (1) Lyapunov functions, (2) finding the eigenvalues for state-space notation, (3) finding the location in the complex frequency plane of the poles of the closed- loop transfer function, and (4) proving bounded outputs for all bounded inputs. Proving stability with Lyapunov functions is very general: it works for nonlinear and time-varying systems. It is also good for doing proofs. Proving the stability of a system with Lyapunov functions is difficult, however, and failure to find a Lyapunov function that proves a system is stable does not prove that the system is unstable. The next techniques we present, finding the eigenvalues or the poles of the transfer function, are sometimes difficult, because they require factoring high-degree polynomials. Many commercial software packages are now available for this task, however. We think most engineers would benefit by having one of these computer programs. Jamshidi et al. [1992] and advertisements in technical publications such as the IEEE Control Systems Magazine and IEEE Spectrum describe many appropriate software packages. The last technique we present, bounded-input, bounded-output stability, is also quite general. Let us begin our discussion of stability and instability of systems informally. In an unstable system the state can have large variations, and small inputs or small changes in the initial state may produce large variations in the output. A common example of an unstable system is illustrated by someone pointing the microphone of a public address (PA) system at a speaker; a loud high-pitched tone results. Often instabilities are caused by too much gain, so to quiet the PA system, decrease the gain by pointing the microphone away from the speaker. Discrete systems can also be unstable. A friend of ours once provided an example. She was sitting in a chair reading and she got cold. So she went over and turned up the thermostat on the heater. The house warmed up. She got hot, so she got up and turned down the thermostat. The house cooled off. She got cold and turned up the thermostat. This process continued until someone finally suggested that she put on a sweater (reducing the gain of her heat loss system). She did, and was much more comfortable. We modeled this as a discrete system, because she seemed to sample the environment and produce outputs at discrete intervals about 15 minutes apart. Ferenc Szidarovszky University of Arizona A. Terry Bahill University of Arizona ? 2000 by CRC Press LLC 12.2 Using the State of the System to Determine Stability The stability of a system is defined with respect to a given equilibrium point in state space. If the initial state x 0 is selected at an equilibrium state x of the system, then the state will remain at x for all future time. When the initial state is selected close to an equilibrium state, the system might remain close to the equilibrium state or it might move away. In this section we introduce conditions that guarantee that whenever the system starts near an equilibrium state, it remains near it, perhaps even converging to the equilibrium state as time increases. For simplicity, only time-invariant systems are considered in this section. Time-variant systems are discussed in Section 12.5. Continuous, time-invariant systems have the form (12.1) and discrete, time-invariant systems are modeled by the difference equation (12.2) Here we assume that f: X ? R n , where X í R n is the state space. We also assume that function f is continuous; furthermore, for arbitrary initial state x 0 ? X, there is a unique solution of the corresponding initial value problem x(t 0 ) = x 0 , and the entire trajectory x(t) is in X. Assume furthermore that t 0 denotes the initial time period of the system. It is also known that a vector x ? X is an equilibrium state of the continuous system, Eq. (12.1), if and only if f(x) = 0, and it is an equilibrium state of the discrete system, Eq. (12.2), if and only if x = f(x). In this chapter the equilibrium of a system will always mean the equilibrium state, if it is not specified otherwise. In analyzing the dependence of the state trajectory x(t) on the selection of the initial state x 0 nearby the equilibrium, the following stability types are considered. Definition 12.1 1.An equilibrium state x is stable if there is an e 0 > 0 with the following property: For all e 1 , 0 < e 1 < e 0 , there is an e > 0 such that if || x – x 0 || < e, then || x – x(t)|| < e 1 , for all t > t 0 . 2.An equilibrium state x is asymptotically stable if it is stable and there is an e > 0 such that whenever || x – x 0 || < e, then x(t) ? x as t ? ￥. 3.An equilibrium state x is globally asymptotically stable if it is stable and with arbitrary initial state x 0 ? X, x(t) ? x as t ? ￥. The first definition says an equilibrium state x is stable if the entire trajectory x(t) is closer to the equilibrium state than any small e 1 , if the initial state x 0 is selected close enough to the equilibrium state. For asymptotic stability, in addition, x(t) has to converge to the equilibrium state as t ? ￥. If an equilibrium state is globally asymptotically stable, then x(t) converges to the equilibrium state regard- less of how the initial state x 0 is selected. These stability concepts are called internal, because they represent properties of the state of the system. They are illustrated in Fig. 12.1. In the electrical engineering literature, sometimes our stability definition is called marginal stability, and our asymptotic stability is called stability. ˙ () (())xfxtt= xfx()(())tt+=1 FIGURE 12.1Stability concepts. (Source: F. Szi- darovszky and A.T. Bahill, Linear Systems Theory, Boca Raton, Fla.: CRC Press, 1992, p. 168. With permission.) ? 2000 by CRC Press LLC 12.3 Lyapunov Stability Theory Assume that x is an equilibrium state of a continuous or discrete system, and let W denote a subset of the state space X such that x ? W. Definition 12.2 A real-valued function V defined on W is called a Lyapunov function, if 1.V is continuous; 2.V has a unique global minimum at x with respect to all other points in W; 3.for any state trajectory x(t) contained in W, V(x(t)) is nonincreasing in t. The Lyapunov function can be interpreted as the generalization of the energy function in electrical systems. The first requirement simply means that the graph of V has no discontinuities. The second requirement means that the graph of V has its lowest point at the equilibrium, and the third requirement generalizes the well- known fact of electrical systems, that the energy in a free electrical system with resistance always decreases, unless the system is at rest. Theorem 12.1 Assume that there exists a Lyapunov function V on the spherical region (12.3) where e 0 > 0 is given; furthermore W í X. Then the equilibrium state is stable. Theorem 12.2 Assume that in addition to the conditions of Theorem 12.1, the Lyapunov function V(x(t)) is strictly decreasing in t, unless x(t) = x. Then the equilibrium state is asymptotically stable. Theorem 12.3 Assume that the Lyapunov function is defined on the entire state space X, V(x(t)) is strictly decreasing in t unless x(t) = x; furthermore, V(x) tends to infinity as any component of x gets arbitrarily large in magnitude. Then the equilibrium state is globally asymptotically stable. Example 12.1 Consider the differential equation The stability of the equilibrium state (1/w, 0) T can be verified directly by using Theorem 12.1 without computing the solution. Select the Lyapunov function where the Euclidian norm is used. This is continuous in x; furthermore, it has its minimal (zero) value at x = x . Therefore, to establish the stability of the equilibrium state we have to show only that V(x(t)) is decreasing. Simple differentiation shows that W= - <{}xxx*** ** e 0 ˙ xx= - ? è ? ? ? ÷ + ? è ? ? ? ÷ 0 0 0 1 w w V T ()()()xxxxx=- -=-** **xx 2 2 d dt Vt TT (()) ( ) ˙ ()()x x x x x x Ax b=-×=- +22 ? 2000 by CRC Press LLC with That is, with x = (x 1 , x 2 ) T , Therefore, function V(x(t)) is a constant, which is a (not strictly) decreasing function. That is, all conditions of Theorem 12.1 are satisfied, which implies the stability of the equilibrium state. Theorems 12.1, 12.2, and 12.3 guarantee, respectively, the stability, asymptotic stability, and global asymptotic stability of the equilibrium state, if a Lyapunov function is found. Failure to find such a Lyapunov function does not mean that the system is unstable or that the stability is not asymptotic or globally asymptotic. It only means that you were not clever enough to find a Lyapunov function that proved stability. 12.4 Stability of Time-Invariant Linear Systems This section is divided into two subsections. In the first subsection the stability of linear time-invariant systems given in state-space notation is analyzed. In the second subsection, methods based on transfer functions are discussed. Stability Analysis with State-Space Notation Consider the time-invariant continuous linear system (12.4) and the time-invariant discrete linear system (12.5) Assume that x is an equilibrium state, and let f(t,t 0 ) denote the fundamental matrix. Theorem 12.4 1.The equilibrium state x is stable if and only if f(t,t 0 ) is bounded for t 3 t 0 . 2.The equilibrium state x is asymptotically stable if and only if f(t,t 0 ) is bounded and tends to zero as t ? ￥. We use the symbol s to denote complex frequency, i.e., s = s + jw. For specific values of s, such as eigenvalues and poles, we use the symbol l. Theorem 12.5 1. If for at least one eigenvalue of A, Re l i > 0 (or *l i * > 1 for discrete systems), then the system is unstable. 2.Assume that for all eigenvalues l i of A, Re l i ￡ 0 in the continuous case (or *l i * ￡ 1 in the discrete case), and all eigenvalues with the property Re l i = 0 (or *l i * = 1) have single multiplicity; then the equilibrium state is stable. 3.The stability is asymptotic if and only if for all i, Re l i < 0 (or *l i * < 1). Ab= - ? è ? ? ? ÷ = ? è ? ? ? ÷ 0 0 0 1 w w and d dt Vt x x x x xx x xx x (()) , () x =- ? è ? ? ? ÷ -+ ? è ? ? ? ÷ =--+= 2 1 1 20 12 2 1 12 2 12 2 w w w ww ˙ xAxb=+ xAxb()()tt+= +1 ? 2000 by CRC Press LLC Remark 1.Note that Part 2 gives only sufficient conditions for the stability of the equilibrium state. As the following example shows, these conditions are not necessary. Example 12.2 Consider first the continuous system x · = Ox, where O is the zero matrix. Note that all constant functions x(t) o x are solutions and also equilibrium states. Since is bounded (being independent of t), all equilibrium states are stable, but O has only one eigenvalue l 1 = 0 with zero real part and multiplicity n, where n is the order of the system. Consider next the discrete systems x(t + 1) = Ix(t), when all constant functions x(t) o x are also solutions and equilibrium states. Furthermore, which is obviously bounded. Therefore, all equilibrium states are stable, but the condition of Part 2 of the theorem is violated again, since l 1 = 1 with unit absolute value having a multiplicity n. Remark 2.The following extension of Theorem 12.5 can be proven. The equilibrium state is stable if and only if for all eigenvalues of A, Re l i ￡ 0 (or *l i * ￡ 1), and if l i is a repeated eigenvalue of A such that Re l i = 0 (or *l i * = 1), then the size of each block containing l i in the Jordan canonical form of A is 1 3 1. Remark 3.The equilibrium states of inhomogeneous equations are stable or asymptotically stable if and only if the same holds for the equilibrium states of the corresponding homogeneous equations. Example 12.3 Consider again the continuous system the stability of which was analyzed earlier in Example 12.1 by using the Lyapunov function method. The characteristic polynomial of the coefficient matrix is therefore, the eigenvalues are l 1 = jw and l 2 = –jw. Both eigenvalues have single multiplicities, and Re l 1 = Re l 2 = 0. Hence, the conditions of Part 2 are satisfied, and therefore the equilibrium state is stable. The conditions of Part 3 do not hold. Consequently, the system is not asymptotically stable. If a time-invariant system is nonlinear, then the Lyapunov method is the most popular choice for stability analysis. If the system is linear, then the direct application of Theorem 12.5 is more attractive, since the eigenvalues of the coefficient matrix A can be obtained by standard methods. In addition, several conditions are known from the literature that guarantee the asymptotic stability of time-invariant discrete and continuous systems even without computing the eigenvalues. For examining asymptotic stability, linearization is an alternative approach to the Lyapunov method as is shown here. Consider the time-invariant continuous and discrete systems f(,) () tt e tt 0 0 == -O I f(,)tt tt tt 0 00 === -- AII ˙ x= - ? è ? ? ? ÷ + ? è ? ? ? ÷ 0 0 0 1 w w x j w w w()s s s s= - -- ? è ? ? ? ÷ =+det 22 ˙ () (())xtt=fx ? 2000 by CRC Press LLC and Let J(x) denote the Jacobian of f(x), and let x be an equilibrium state of the system. It is known that the method of linearization around the equilibrium state results in the time-invariant linear systems and where x d (t) = x(t) – x. It is also known from the theory of ordinary differential equations that the asymptotic stability of the zero vector in the linearized system implies the asymptotic stability of the equilibrium state x in the original nonlinear system. For continuous systems the following result has a special importance. Theorem 12.6 The equilibrium state of a continuous system [Eq. (12.4)] is asymptotically stable if and only if equation (12.6) has positive definite solution Q with some positive definite matrix M. We note that in practical applications the identity matrix is almost always selected for M. An initial stability check is provided by the following result. Theorem 12.7 Let j(l) = l n + p n–1 l n–1 + . . . + p 1 l + p 0 be the characteristic polynomial of matrix A. Assume that all eigenvalues of matrix A have negative real parts. Then p i > 0 (i = 0, 1,..., n – 1). Corollary.If any of the coefficients p i is negative or zero, the equilibrium state of the system with coefficient matrix A cannot be asymptotically stable. However, the conditions of the theorem do not imply that the eigenvalues of A have negative real parts. Example 12.4 For matrix the characteristic polynominal is j(s) = s 2 + w 2 . Since the coefficient of s 1 is zero, the system of Example 12.3 is not asymptotically stable. The Transfer Function Approach The transfer function of the time invariant linear continuous system (12.7) xfx()(())tt+=1 ˙ () ()()x Jxx dd tt= x Jxx dd ()()()tt+=1 AQ QA M T +=- A= - ? è ? ? ? ÷ 0 0 w w ˙ xAxBu yCx =+ = ? 2000 by CRC Press LLC and that of the time invariant linear discrete system (12.8) have the common form If both the input and output are single, then or in the familiar electrical engineering notation (12.9) where K is the gain term in the forward loop, G(s) represents the dynamics of the forward loop, or the plant, and H(s) models the dynamics in the feedback loop. We note that in the case of continuous systems s is the variable of the transfer function, and for discrete systems the variable is denoted by z. After the Second World War systems and control theory flourished. The transfer function representation was the most popular representation for systems. To determine the stability of a system we merely had to factor the denominator of the transfer function (12.9) and see if all of the poles were in the left half of the complex frequency plane. However, with manual techniques, factoring polynomials of large order is difficult. So engi- neers, being naturally lazy people, developed several ways to determine the stability of a system without factoring the polynomials [Dorf, 1992]. First, we have the methods of Routh and Hurwitz, developed a century ago, that looked at the coefficients of the characteristic polynomial. These methods showed whether the system was stable or not, but they did not show how close the system was to being stable. What we want to know is for what value of gain, K, and at what frequency, w, will the denominator of the transfer function (12.9) become zero. Or, when will KGH = –1, meaning, when will the magnitude of KGH equal 1 with a phase angle of –180 degrees? These parameters can be determined easily with a Bode diagram. Construct a Bode diagram for KGH of the system, look at the frequency where the phase angle equals –180 degrees, and look up at the magnitude plot. If it is smaller than 1.0, then the system is stable. If it is larger than 1.0, then the system is unstable. Bode diagram techniques are discussed in Chapter 11. The quantity KG(s)H(s) is called the open-loop transfer function of the system, because it is the effect that would be encountered by a signal in one loop around the system if the feedback loop were artificially opened [Bahill, 1981]. To gain some intuition, think of a closed-loop negative feedback system. Apply a small sinusoid at frequency w to the input. Assume that the gain around the loop, KGH, is 1 or more, and that the phase lag is 180 degrees. The summing junction will flip over the fed back signal and add it to the original signal. The result is a signal that is bigger than what came in. This signal will circulate around this loop, getting bigger and bigger until the real system no longer matches the model. This is what we call instability. The question of stability can also be answered with Nyquist diagrams. They are related to Bode diagrams, but they give more information. A simple way to construct a Nyquist diagram is to make a polar plot on the complex frequency plane of the Bode diagram. Simply stated, if this contour encircles the –1 point in the complex frequency plane, then the system is unstable. The two advantages of the Nyquist technique are (1) in xAxBu yCx ( ) () () () () ttt tt += + = 1 TF CI A B() ( )ss=- -1 TF Y U () () () s s s = TF G GH () () ()() s Ks Kss = +1 ? 2000 by CRC Press LLC addition to the information on Bode diagrams, there are about a dozen rules that can be used to help construct Nyquist diagrams, and (2) Nyquist diagrams handle bizarre systems better, as is shown in the following rigorous statement of the Nyquist stability criterion. The number of clockwise encirclements minus the number of counterclockwise encirclements of the point s = –1 + j 0 by the Nyquist plot of KG(s)H(s) is equal to the number of poles of Y(s)/U(s) minus the number of poles of KG(s)H(s) in the right half of the s-plane. The root-locus technique was another popular technique for assessing stability. It furthermore allowed the engineer to see the effects of small changes in the gain, K, on the stability of the system. The root-locus diagram shows the location in the s-plane of the poles of the closed-loop transfer function, Y(s)/U(s). All branches of the root-locus diagram start on poles of the open-loop transfer function, KGH, and end either on zeros of the open-loop transfer function, KGH, or at infinity. There are about a dozen rules to help draw these trajectories. The root-locus technique is discussed in Chapter 93.4. We consider all these techniques to be old fashioned. They were developed to help answer the question of stability without factoring the characteristic polynomial. However, many computer programs are currently available that factor polynomials. We recommend that engineers merely buy one of these computer packages and find the roots of the closed-loop transfer function to assess the stability of a system. The poles of a system are defined as all values of s such that sI – A is singular. The poles of a closed-loop transfer function are exactly the same as the eigenvalues of the system: engineers prefer the term poles and the symbol s, and mathematicians prefer the term eigenvalues and the symbol l. We will use s for complex frequency and l for specific values of s. Sometimes, some poles could be canceled in the rational function form of TF(s) so that they would not be explicitly shown. However, even if some poles could be canceled by zeros, we still have to consider all poles in the following criteria which is the statement of Theorem 12.5. The equilibrium state of the continuous system [Eq. (12.7)] with constant input is unstable if at least one pole has a positive real part, and is stable if all poles of TF(s) have nonpositive real parts and all poles with zero real parts are single. The equilibrium state is asymptotically stable if and only if all poles of TF(s) have negative real parts; that is, all poles are in the left half of the s-plane. Similarly, the equilibrium state of the discrete system [Eq. (12.8)] with constant input is unstable if the absolute value of at least one pole is greater than one, and is stable if all poles of TF(z) have absolute values less than or equal to one and all poles with unit absolute values are single. The equilibrium state is asymptotically stable if and only if all poles of TF(z) have absolute values less than one; that is, the poles are all inside the unit circle of the z-plane. Example 12.5 Consider again the system which was discussed earlier. Assume that the output equation has the form Then The poles are jw and –jw, which have zero real parts; that is, they are on the imaginary axis of the s-plane. Since they are single poles, the equilibrium state is stable but not asymptotically stable. A system such as this would produce constant amplitude sinusoids at frequency w. So it seems natural to assume that such systems would be used to build sinusoidal signal generators and to model oscillating systems. However, this is not the case, because (1) zero resistance circuits are hard to make; therefore, most function generators use other ˙ xx= - ? è ? ? ? ÷ + ? è ? ? ? ÷ 0 0 0 1 w w y=(,)11x TF()s s s = + + w w 22 ? 2000 by CRC Press LLC techniques to produce sinusoids; and (2) such systems are not good models for oscillating systems, because most real-world oscillating systems (i.e., biological systems) have energy dissipation elements in them. More generally, real-world function generators are seldom made from closed-loop feedback control systems with 180 degrees of phase shift, because (1) it would be difficult to get a broad range of frequencies and several waveforms from such systems, (2) precise frequency selection would require expensive high-precision compo- nents, and (3) it would be difficult to maintain a constant frequency in such circuits in the face of changing temperatures and power supply variations. Likewise, closed-loop feedback control systems with 180 degrees of phase shift are not good models for oscillating biological systems, because most biological systems oscillate because of nonlinear network properties. A special stability criterion for single-input, single-output time-invariant continuous systems will be intro- duced next. Consider the system (12.10) where A is an n ′ n constant matrix, and b and c are constant n-dimensional vectors. The transfer function of this system is which is obviously a rational function of s. Now let us add negative feedback around this system so that u = ky, where k is a constant. The resulting system can be described by the differential equation (12.11) The transfer function of this feedback system is (12.12) To help show the connection between the asymptotic stability of systems (12.10) and (12.11), we introduce the following definition. Definition 12.3 Let r(s) be a rational function of s. Then the locus of points is called the response diagram of r. Note that L(r) is the image of the imaginary line Re(s) = 0 under the mapping r. We shall assume that L(r) is bounded, which is the case if and only if the degree of the denominator is not less than that of the numerator and r has no poles on the line Re(s) = 0. Theorem 12.8 The Nyquist stability criterion.Assume that TF 1 has a bounded response diagram L(TF 1 ). If TF 1 has n poles in the right half of the s-plane, where Re(s) > 0, then H has r + n poles in the right half of the s-plane where Re(s) > 0 if the point 1/k + j · 0 is not on L(TF 1 ), and L(TF 1 ) encircles 1/k + j · 0 r times in the clockwise sense. Corollary.Assume that system (12.10) is asymptotically stable with constant input and that L(TF 1 ) is bounded and traversed in the direction of increasing n and has the point 1/k + j · 0 on its left. Then the feedback system (12.11) is also asymptotically stable. ˙ xAxb cx=+ =uy and T TFs s 1 1 () ( )=- - cIAb T ˙ ()xAxbcxAbcx=+ =+kk TT TF TF TF () () () s s ks = - 1 1 1 Lr a jba Rerjv b I rjv v() { (()), (()), =+= = ?* m R} ? 2000 by CRC Press LLC This result has many applications, since feedback systems have a crucial role in constructing stabilizers, observers, and filters for given systems. Fig. 12.2 illustrates the conditions of the corollary. The application of this result is especially convenient, if system (12.10) is given and only appropriate values k of the feedback are to be determined. In such cases the locus L(TF 1 ) has to be computed first, and then the region of all appropriate k values can be determined easily from the graph of L(TF 1 ). This analysis has dealt with the closed-loop transfer function, whereas the techniques of Bode, root-locus, etc. use the open-loop transfer function. This should cause little confusion as long as the distinction is kept in mind. 12.5 BIBO Stability In the previous sections, internal stability of time-invariant systems was examined, i.e., the stability of the state was investigated. In this section the external stability of systems is discussed; this is usually called the BIBO (bounded-input, bounded-output) stability. Here we drop the simplifying assumption of the previous section that the system is time-invariant: we will examine time-variant systems. Definition 12.4 A system is called BIBO stable if for zero initial conditions, a bounded input always evokes a bounded output. For continuous systems a necessary and sufficient condition for BIBO stability can be formulated as follows. Theorem 12.9 Let T(t, t) = (t ij (t, t)) be the weighting pattern, C(t)f(t, t)B(t), of the system. Then the continuous time-variant linear system is BIBO stable if and only if the integral (12.13) is bounded for all t > t 0 , i and j. Corollary.Integrals (12.13) are all bounded if and only if (12.14) FIGURE 12.2Illustration of Nyquist stability criteria. (Source: F. Szidarovszky and A. T. Bahill, Linear Systems Theory, Boca Raton, Fla.: CRC Press, 1992, p.184. With permission.) **ttd ij t t (,)tt 0 ò It tt d ij ji t t () (,)= ?? ò **tt 0 ? 2000 by CRC Press LLC is bounded for t 3 t 0 . Therefore, it is sufficient to show the boundedness of only one integral in order to establish BIBO stability. The discrete counterpart of this theorem can be given in the following way. Theorem 12.10 Let T(t, t) = (t ij (t, t)) be the weighting pattern of the discrete linear system. Then it is BIBO stable if and only if the sum (12.15) is bounded for all t > t 0 , i and j. Corollary.The sums (12.15) are all bounded if and only if (12.16) is bounded. Therefore it is sufficient to verify the boundedness of only one sum in order to establish BIBO stability. Consider next the time-invariant case, when A(t) o A, B(t) o B and C(t) o C. From the foregoing theorems and the definition of T(t, t) we have immediately the following sufficient condition. Theorem 12.11 Assume that for all eigenvalues l i of A, Re l i < 0 (or *l i * < 1). Then the time-invariant linear continuous (or discrete) system is BIBO stable. Finally, we note that BIBO stability is not implied by an observation that a certain bounded input generates bounded output. All bounded inputs must generate bounded outputs in order to guarantee BIBO stability. Adaptive-control systems are time-varying systems. Therefore, it is usually difficult to prove that they are stable. Szidarovszky et al. [1990], however, show a technique for doing this. This new result gives a necessary and sufficient condition for the existence of an asymptotically stable model-following adaptive-control system based on the solvability of a system of nonlinear algebraic equations, and in the case of the existence of such systems they present an algorithm for finding the appropriate feedback parameters. 12.6 Physical Examples In this section we show some examples of stability analysis of physical systems. 1. Consider a simple harmonic oscillator constructed of a mass and an ideal spring. Its dynamic response is summarized with In Example 12.3 we showed that this system is stable but not asymptotically stable. This means that if we leave it alone in its equilibrium state, it will remain stationary, but if we jerk on the mass it will oscillate forever. There is no damping term to remove the energy, so the energy will be transferred back and forth between potential energy in the spring and kinetic energy in the moving mass. A good approximation of such a harmonic oscillator is a pendulum clock. The more expensive it is (i.e., the smaller the damping), the less often we have to wind it (i.e., add energy). It tt ij t t () (,)= = - ? **t t 0 1 **tt ij jit t (,)t t ??? = - 0 1 ˙ xx= - ? è ? ? ? ÷ + ? è ? ? ? ÷ 0 0 0 1 w w u ? 2000 by CRC Press LLC 2. A linear second-order electrical system composed of a series connection of an input voltage source, an inductor, a resistor, and a capacitor, with the output defined as the voltage across the capacitor, can be characterized by the second-order equation For convenience, let us define and assume that z < 1. With these parameters the transfer function becomes Is this system stable? The roots of the characteristic equation are If z > 0, the poles are in the left half of the s-plane, and therefore the system is asymptotically stable. If z = 0, as in the previous example, the poles are on the imaginary axis; therefore, the system is stable but not asymptotically stable. If z < 0, the poles are in the right half of the s-plane and the system is unstable. 3. An electrical system is shown in Fig. 12.3. Simple calculation shows that by defining the state variables the system can be described by the differential equations FIGURE 12.3A simple electrical system. (Source: F. Szidarovszky and A. T. Bahill, Linear Systems Theory, Boca Raton, Fla.: CRC Press, 1992, p. 125. With permission.) V V LCs RCs out in = ++ 1 1 2 wz n LC RC L == 1 2 and V V s n nn out in = ++ w zw w 2 22 2 lzwwz 12 2 1 , =- ± - nn j xixv uv Lc s12 == =, , and ˙ ˙ x R L x L x L u x C x CR x 1 1 12 21 2 2 11 11 =- - + =- ? 2000 by CRC Press LLC The characteristic equation has the form which simplifies as Since R 1 , R 2 , L, and C are positive numbers, the coefficients of this equation are all positive. The constant term equals l 1 l 2 , and the coefficient of s 1 is –(l 1 + l 2 ). Therefore If the eigenvalues are real, then these relations hold if and only if both eigenvalues are negative. If they were positive, then l 1 + l 2 > 0. If they had different signs, then l 1 l 2 < 0. Furthermore, if at least one eigenvalue is zero, then l 1 l 2 = 0. Assume next that the eigenvalues are complex: Then and Hence l 1 + l 2 < 0 if and only if Re s < 0. In summary, the system is asymptotically stable, since in both the real and complex cases the eigenvalues have negative values and negative real parts, respectively. 4. The classical stick balancing problem is shown in Fig. 12.4. Simple analysis shows that y(t) satisfies the second-order equation If one selects L = 1, then the characteristic equation has the form So, the eigenvalues are -- ? è ? ? ? ÷ -- ? è ? ? ? ÷ +=s R L s CR LC 1 2 11 0 ss R LCR R LCR LC 2 1 2 1 2 11 0++ ? è ? ? ? ÷ ++ ? è ? ? ? ÷ = ll ll 12 12 00+< > and l 12, =±Res jIms ll 12 2+=Res ll 12 22 =+()()Res Ims ˙˙ ()y g L yu=- sg 2 0-= l 12, =±g ? 2000 by CRC Press LLC One is in the right half of the s-plane and the other is in the left half of the s-plane, so the system is unstable. This instability is understandable, since without an intelligent input to control the system, if the stick is not upright with zero velocity, it will fall over. 5. A simple transistor circuit can be modeled as shown in Fig. 12.5. The state variables are related to the input and output of the circuit: the base current, i b , is x 1 and the output voltage, v out , is x 2 . Therefore, The A matrix looks strange with a column of all zeros, and indeed the circuit does exhibit odd behavior. For example, as we will show, there is no equilibrium state for a unit step input of e s . This is reasonable, however, FIGURE 12.4Stick balancing. (Source: F. Szidarovszky and A. T. Bahill, Linear Systems Theory, Boca Raton, Fla.: CRC Press, 1992, p. 127. With permission.) FIGURE 12.5A model for a simple transistor circuit. (Source: F. Szidarovszky and A. T. Bahill, Linear Systems Theory, Boca Raton, Fla.: CRC Press 1992, p. 127. With permission.) ˙ (,)xxc= - ? è ? ? ? ? ? ? ÷ ÷ ÷ ÷ + ? è ? ? ? ? ? ÷ ÷ ÷ = h L h C L e ie fe s T 0 0 1 0 01 and ? 2000 by CRC Press LLC because the model is for mid-frequencies, and a unit step does not qualify. In response to a unit step the output voltage will increase linearly until the model is no longer valid. If e s is considered to be the input, then the system is If u(t) [ 1, then at the equilibrium state: That is, Since h fe /C 1 0, the second equation implies that x 1 = 0, and by substituting this value into the first equation we get the obvious contradiction 1/L = 0. Hence, with nonzero constant input no equilibrium state exists. Let us now investigate the stability of this system. First let ~ x(t)denote a fixed trajectory of this system, and let x(t) be an arbitrary solution. Then the difference x d (t) = x(t) – ~ x(t)satisfies the homogeneous equation This system has an equilibrium x d (t) = 0. Next, the stability of this equilibrium is examined by solving for the poles of the closed-loop transfer function. The characteristic equation is which can be simplified as ˙ xx= - ? è ? ? ? ? ? ? ÷ ÷ ÷ ÷ + ? è ? ? ? ? ? ÷ ÷ ÷ h L h C L u ie fe 0 0 1 0 - ? è ? ? ? ? ? ? ÷ ÷ ÷ ÷ ? è ? ? ? ? ÷ ÷ + ? è ? ? ? ? ? ÷ ÷ ÷ = ? è ? ? ? ? ÷ ÷ h L h C x x L ie fe 0 0 1 0 0 0 1 2 -+= h L x L ie 1 1 0 h C x fe 1 0= ˙ xx dd = - ? è ? ? ? ? ? ? ÷ ÷ ÷ ÷ h L h C ie fe 0 0 det -- - ? è ? ? ? ? ? ? ÷ ÷ ÷ ÷ = h L s h C s ie fe 0 0 ss h L ie2 00++= ? 2000 by CRC Press LLC The roots are Therefore, the system is stable but not asymptotically stable. This stability means that for small changes in the initial state the entire trajectory x(t) remains close to ~ x(t). Defining Terms Asymptotic stability: An equilibrium state x of a system is asymptotically stable if, in addition to being stable, there is an e > 0 such that whenever **x – x 0 ** < e, then x(t) ? x as t ? ￥. A system is asymptotically stable if all the poles of the closed-loop transfer function are in the left half of the s-plane (inside the unit circle of the z-plane for discrete systems). This is sometimes called stability. BIBO stability: A system is BIBO stable if for zero initial conditions a bounded input always evokes a bounded output. External stability:Stability concepts related to the input-output behavior of the system. Global asymptotic stability:An equilibrium state x of a system is globally asymptotically stable if it is stable and with arbitrary initial state x 0 ? X, x(t) ? x as t ? ￥. Internal stability:Stability concepts related to the state of the system. Instability:An equilibrium state of a system is unstable if it is not stable. A system is unstable if at least one pole of the closed-loop transfer function is in the right half of the s-plane (outside the unit circle of the z-plane for discrete systems). Stability:An equilibrium state x of a system is stable if there is an e 0 > 0 with the following property: for all e 1 , 0 < e 1 < e 0 , there is an e > 0 such that if **x – x 0 ** < e, then **x – x(t)** < e 1 for all t > t 0 . A system is stable if the poles of its closed-loop transfer function are (1) in the left half of the complex frequency plane, called the s-plane (inside the unit circle of the z-plane for discrete systems), or (2) on the imaginary axis, and all of the poles on the imaginary axis are single (on the unit circle and all such poles are single for discrete systems). Stability for a system with repeated poles on the j w axis (the unit circle) is complicated and is examined in the discussion after Theorem 12.5. In the electrical engineering literature, this definition of stability is sometimes called marginal stability and sometimes stability in the sense of Lyapunov. Related Topics 6.2 Applications?7.2 State Equations in Normal Form?100.2 Dynamic Response?100.7 Nonlinear Control Systems References A. T. Bahill, Bioengineering: Biomedical, Medical and Clinical Engineering, Englewood Cliffs, N.J.:Prentice-Hall, 1981, pp. 214–215, 250–252. R. C. Dorf, Modern Control Systems, 7th ed., Reading, Mass.: Addison-Wesley, 1996. M. Jamshidi, M. Tarokh, and B. Shafai, Computer-Aided Analysis and Design of Linear Control Systems, Engle- wood Cliffs, N.J.: Prentice-Hall, 1992. F. Szidarovszky and A. T. Bahill, Linear Systems Theory, Boca Raton, Fla.: CRC Press, 1992. F. Szidarovszky, A. T. Bahill, and S. Molnar, “On stable adaptive control systems,” Pure Math. and Appl., vol. 1, ser. B, no. 2–3, pp. 115–121, 1990. Further Information For further information consult the textbooks Modern Control Systems by Dorf [1996] or Linear Systems Theory by Szidarovszky and Bahill [1992]. ll 1 0==- and 2 h L ie ? 2000 by CRC Press LLC