Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology1 1 c�Chapter 7 State-Space Mo dels 7.1 Intro duction A central question in dealing with a causal discrete-time (DT) system with input u, output y, is the following: Given the input at some time n, i.e. given u[n], how m uch information do we n e e d a b o u t past inputs, i.e. ab out u[k] for k � n , in order to determine the present output, namely y[n] � The same question can b e asked for continuous-time (CT) systems. This question addresses the issue of memory in the system. Why is this a central question� Some reasons: � The answer gives us an idea of the complexity, or numb e r o f degrees of freedom, asso-ciated with the dynamic b ehavior of the system. The more information we need ab out past inputs in order to determine the present output, the richer the variety of p ossible output b ehaviors. � In a control application, the answer to the ab ove question suggests the required degree ecau seof complexity of the controller, b the controller has to rememb er enough ab out the past to determine the e�ects of present c o n trol actions on the resp onse of the system. � For a computer algorithm that acts causally on a data stream, the answer to the ab ove question suggests how m uch memory will b e needed to run the algorithm. We n o w describ e the general structure of state-space m o dels, for which the preceding question has an immediate and transparent answer.7.2 General Description For a causal system with m inputs uj (t) and p outputs yi(t) (hence m + p manifest variables), an nth-order state-space description is one that intro duces n latent v ariables x`(t) called state variables in order to obtain a particular form for the constraints that de�ne the mo del. Letting 323232 u1(t) y1(t) x1(t) u(t) �64 . . . 75 � y(t) �64 . . . 75 � x(t) �64 . . 75 �. um(t) yp(t) xn(t) an nth-order state-space description takes the form x_ (t) � f (x(t)� u (t)� t ) (state evolution equations) (7.1) y(t) � g (x(t)� u (t)� t ) (instantaneous output equations) : (7.2) To s a ve writing the same equations over for b oth continuous and discrete time, we i n terpret dx(t) x_ (t) � � t 2 R or R + dt for CT systems, and x_ (t) � x(t + 1) � t 2 Z or Z+ for DT systems. We will only consider �nite-order (or �nite-dimensional, or lumped) state-space mo dels, although there is also a rather well develop ed (but much more subtle and technical) theory of in�nite-order (or in�nite-dimensional, or distributed) state-space mo dels. DT Mo dels The key feature of a state-space description is the following prop erty, w h i c h w e shall refer to as the state property. Given the present state vector (or \state") and present input at time t, w e can compute: (i) the present output, using (7.2)� and (ii) the next state using (7.1). It is easy to see that this puts us in a p osition to do the same thing at time t + 1, and therefore to continue the pro cess over any time interval. Extending this argument, we can make the following claim: State Prop erty of DT state-Space Mo dels Given the initial state x(t0) and input u(t) for t0 � t � t f (with t0 and tf arbitrary), we can compute the output y(t) for t0 � t � t f and the state x(t) for t0 � t � tf .Thus, the state at any time t0 summarizes everything ab out the past that is relevant to the future. Keeping in mind this fact | that the state variables are the memory variables (or, in more physical situations, the energy storage variables) of a system | often guides us quickly t o g o o d c hoices of state variables in any given context. CT Mo dels The same state prop erty turns out to hold in the CT case, at least for f ( : ) that are well b e h a ved enough for the state evolution equations to have a unique solution for all inputs of interest and over the entire time axis | these will typically b e t h e only sorts of CT systems of interest to us. A demonstration of this claim, and an elucidation of the precise conditions under which it holds, would require an excursion into the theory of di�erential equations b e y ond what is appropriate for this course. We can make this result plausible, however, by considering the Taylor series approximation �� dx(t) x(t0 + �) � x(t0) + � (7.3)dt t�t0 � x(t0) + f (x(t0)� u (t0 )� t 0) � (7.4) where the second equation results from applying the state evolution equation (7.1). This suggests that we can approximately compute x(t0 + �), given x(t0) and u(t0)� the error in the approximation is of order �2, and can therefore b e made smaller by making � smaller. For su�ciently well b ehaved f ( � ), we can similarly step forwards from t0 + � to t0 + 2 �, and so on, eventually arriving at the �nal time tf , taking on the order of �;1 step s in th e pro cess. The accumulated error at time tf is then of order �;1:�2 � �, and can b e made arbitrarily small by making � su�ciently small. Also note that, once the state at any time is determined and the input at that time is known, then the output at that time is immediately given by ( 7 . 2 ) , even in the CT case. The simple-minded Taylor series approximation in (7.4) corresp onds to the crudest of numerical schemes | the \forward Euler" metho d | for integrating a system of equations of the form (7.1). Far more sophisticated schemes exist (e.g. Runge-Kutta metho ds, Adams-Gear schemes for \sti� " systems that exhibit widely di�ering time scales, etc.), but the forward Euler scheme su�ces to make plausible the fact that the state prop erty highlighted ab ove applies to CT systems as well as DT ones. Example 7.1 RC Circuit This example demonstrates a �ne p oint in the de�nition of a state for CT systems. Consider an RC circuit in series with a voltage source u. Using KVL, we g e t the following equation describing the system: ;u + vR + RC v_C � 0 : It is clear that vC de�nes a state for the system as we describ ed b efore. Do es vR de�ne a state� If vR(t0) is given, and the input u(t)� t 0 � t � t f is known, thenone can compute vC (t0) and using the state prop erty vC (tf ) can …