? Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 1: Input/Output and State-Space Models 1 This lecture presents some basic de?nitions and simple examples on nonlinear dynam- ical systems modeling. 1.1 Behavioral Models. The most general (though rarely the most convenient) way to de?ne a system is by using a behavioral input/output model. 1.1.1 What is a signal? In these lectures, a signal is a locally integrable function z : R + ≤? R k , where R + denotes the set of all non-negative real numbers. The notion of “local integrability” comes from the Lebesque measure theory, and means simply that the function can be safely and meaningfully integrated over ?nite intervals. Generalized functions, such as the delta function ?(t), are not allowed. The argument t → R + of a signal function will be referred to as “time” (which it usually is). Example 1.1 Function z = z(·) de?ned by t ?0.9 sgn(cos(1/t)) for t > 0, z(t) = 0 for t = 0 1 Version of September 3, 2003 ? 2 is a valid signal, while 1/t for t > 0, z(t) = 0 for t = 0 and z(t) = ? ˙ (t) are not. The de?nition above formally covers the so-called continuous time (CT) signals. Dis- crete time (DT) signals can be represented within this framework as special CT signals. More precisely, a signal z : R + ≤? R k is called a DT signal if it is constant on every interval [k,k + 1) where k = 0, 1, 2,... . 1.1.2 What is a system? Systems are objects producing signals (called output signals), usually depending on other signals (inputs) and some other parameters (initial conditions). In most applications, mathematical models of systems are de?ned (usually implicitly) by behavior sets. For an autonomous system (i.e. for a system with no inputs), a behavior set is just a set B = {z} consisting of some signals z : R + ≤? R k (k must be the same for all signals from B). For a system with input v and output w, the behavior set consists of all possible input/output pairs z = (v(·),w(·)). There is no real di?erence between the two de?nitions, since the pair of signals z = (v(·),w(·)) can be interpreted as a single vector signal z(t) = [v(t); w(t)] containing both input and output stacked one over the other. Note that in this de?nition a ?xed input v(·) may occur in many or in no pairs (v,w) → B, which means that the behavior set does not necessarily de?ne system output as a function of an arbitrary system input. Typically, in addition to knowing the input, one has to have some other information (initial conditions and/or uncertain parameters) to determine the output in a unique way. Example 1.2 The familiar ideal integrator system (the one with the transfer function G(s) = 1/s) can be de?ned by its behavioral set of all input/output scalar signal pairs (v,w) satisfying t 2 w(t 2 ) ? w(t 1 ) = v(?)d?, ? t 1 ,t 2 → [0,?). t 1 In this example, to determine the output uniquely it is su?cient to know v and w(0). In Example 1.1.2 a system is characterised by an integral equation. There is a variety of other ways to de?ne the same system (by specifying a transfer function, by writing a di?erential equation, etc.) 3 1.1.3 What is a linear/nonlinear system? A system is called linear if its behavior set satis?es linear superposition laws, i.e. when for every z 1 ,z 2 → B and c → R we have z 1 + z 2 → B and cz 1 → B. Excluding some absurd examples 2 , linear systems are those de?ned by equations which are linear with respect to v and w. In particular, the ideal integrator system from Exam- ple 1.1.2 is linear. A nonlinear system is simply a system which is not linear. 1.2 System State. It is important to realize that system state can be de?ned for an arbitrary behavioral model B = {z(·}. 1.2.1 Two signals de?ning same state at time t. System state at a given time instance t 0 is supposed to contain all information relating past (t < t 0 ) and future (t > t 0 ) behavior. This leads us to the following de?nitions. De?nition Let B be a behavior set. Signals z 1 ,z 2 → B are said to commute at time t 0 if the signals ? z 1 (t) for t ∩ t 0 , z 12 (t) = z 2 (t) for t > t 0 and ? z 2 (t) for t ∩ t 0 , z 21 (t) = z 1 (t) for t > t 0 also belong to the behavior set. De?nition Let B be a behavior set. Signals z 1 ,z 2 → B are said to de?ne same state of B at time t 0 if the set of z → B commuting with z 1 at t 0 is the same as the set of z → B commuting with z 2 at t 0 . De?nition Let B be a behavior set. Let X be any set. A function x : R ×B ≤? X is called a state of system B if z 1 and z 2 de?ne same state of B at time t whenever x(t,z 1 (·)) = x(t,z 2 (·)). Example 1.3 Consider a system in which both input v and output w are binary signals, i.e. DT signals taking values from the set {0, 1}. De?ne the input/output relation by the following rules: w(t) = 1 only if v(t) = 1, and for every t 1 ,t 2 → Z + such that 2 Such as the (linear) system de?ned by the nonlinear equation (v(t) ? w(t)) 2 = 0 ? t 4 w(t 1 ) = w(t 2 ) = 1 and w(t) = 0 for all t → (t 1 ,t 2 ) ?Z, there are exactly two integers t in the interval (t 1 ,t 2 ) such that v(t) = 1. In other words, the system counts the 1’s in the input and, every time the count reaches three, the system resets its counter to zero, and outputs 1 (otherwise producing 0’s). It is easy to see that two input/output pairs z 1 = (v 1 ,w 1 ) and z 2 = (v 2 ,w 2 ) commute at a (discrete) time t 0 if and only if N(t 0 ,z 1 ) = N(t 0 ,z 2 ), where N(t 0 ,z) for z = (v,w) → B is the number of 1’s in v(t) for t → (t 0 ,t 1 ) ?Z, where t 1 means the next (after t 0 ) integer time t when w(t) = 1. Hence the state of the system can be de?ned by a function x : R + ×B ≤? {0, 1, 2}, x(t,z) = N(t,z). In this example, knowing a system state allows one to write down state space equations for the system: x(t + 1) = f(x(t),v(t)), w(t) = g(x(t),v(t)), (1.1) where f(x,v) = (x + v)mod3, and g(x,v) = 1 if and only if x = 2 and v = 1.