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 8 Simulation/Realization 8.1 Intro duction Given an nth-order state-space description of the form x_ (t) � f (x(t)� u (t)� t ) (state evolution equations) (8.1) y(t) � g (x(t)� u (t)� t ) (instantaneous output equations) : (8.2) (which m a y b e CT or DT, dep ending on how w e i n terpret the symb o l x_ ), how do w e simulate the mo del, i.e., how do we implement it or realize it in hardware or software� In the DT case, where x_ (t) � x(t + 1), this is easy if we h a ve a vailable: (i) storage registers that can b e up dated at each time step (or \clo ck cycle") | these will store the state variables� and (ii) a means of evaluating the functions f ( � ) and g( � ) that app ear in the state-space description | in the linear case, all that we need for this are multipliers and adders. A s t r a i g h tforward realization is then obtained as shown in the �gure b e l o w. The storage registers are lab eled D for (one-step) delay, b ecause the output of the blo ck represents the data currently stored in the register while the input of such a blo ck represents the data waiting to b e read into the register at the next clo ck pulse. In the CT case, where x_ (t) � dx(t)�dt, the only di�erence is that the delay elements are replaced by i n tegrators. The outputs of the integrators are then the state variables. 8.2 Realization from I/O Representations In this section, we will describ e how a state space realization can b e obtained for a causal input-output dynamic system describ ed in terms of convolution. 8.2.1 Convolution with an Exp onential Consider a causal DT LTI system with impulse resp onse h[n] (which is 0 for n � 0):- -u[t] -x[t + 1] x[t] -y[t]f (:� :) Dg(:� :) 6 6 Figure 8.1: Simulation Diagram nX y[n] � h[n ; k]u[k] ;1 � n;1�X � h[n ; k]u[k] + h[0]u[n] (8.3) ;1 The �rst term ab ove, namely n;1X x[n] � h[n ; k]u[k] (8.4) ;1 represents the e�ect of the past on the present. This expression shows that, in general (i.e. if h[n] has no sp ecial form), the numb e r x[n] h a s to b e recomputed from scratch for each n. When we m o ve from n to n + 1, none of the past input, i.e. u[k] for k � n, can b e discarded, b ecause all of the past will again b e needed to compute x[n + 1]. In other words, the memory of the system is in�nite. Now lo ok at an instance where sp ecial structure in h[n] m a k es the situation much b etter. Supp ose h[n] � �n for n � 0, and 0 otherwise (8.5) Then n;1X x[n] � �n;k u[k] (8.6) ;1 and nX x[n + 1] � �n+1;k u[k] ;1 � n;1�X � � �n;k u[k] + �u[n] ;1 � �x[n] + �u[n] (8.7)- -(You will �nd it instructive to graphically represent the convolutions that are involved here, in order to understand more visually why the relationship (8.7) holds.) Gathering (8.3) and (8.6) with (8.7), we obtain a pair of equations that together constitute a state-space description for this system: x[n + 1] � �x[n] + �u[n] (8.8) y[n] � x[n] + u[n] (8.9) To realize this mo del in hardware, or to simulate it, we can use a delay-adder-gain system that is obtained as follows. We start with a delay element, whose output will b e x[n] when its input is x[n + 1 ] . Now the state evolution equation tells us how t o c o m bine the present output of the delay element, x[n], with the present input to the system, u[n], in order to obtain the present input to the delay element, x[n + 1]. This leads to the following blo ck diagram, in which we have used the output equation to determine how to obtain y[n] from the present state and input of the system: -u[n] -m -y[n] 6 x[n] x[n + 1]� � � D 8.2.2 Convolution with a Sum of Exp onentials Consider a more complicated causal impulse resp onse than the previous example, namely h[n] � �0�[n] + ( �1�n 2 + � � � + �L�n 1 + �2�nL ) (8.10) with the �i b eing constants. The following blo ck diagram shows that this system can b e considered as b eing obtained through the parallel interconnection of causal subsystems that are as simple as the one treated earlier, plus a direct feedthrough of the input through the gain �0 (each blo ck is lab eled with its impulse resp onse, with causality implying that these resp onses are 0 for n � 0): �0�[n] B B u[n] - -BBBN i y[n] -�1�n --1 ���� � � -�L�n -� L ...Motivated by the ab ove structure and the treatment o f the earlier, let us de�ne a state variable for each o f t h e L subsystems: n;1xi[n] � �n;k u[k] � i � 1 � 2� : : : � L (8.11)i ;1 With this, we immediately obtain the following state-evolution equations for the subsystems: xi[n + 1] � �ixi[n] + �iu[n] � i � 1 � 2� : : : � L (8.12) Also, after a little algebra, we directly �nd X LX y[n] � �1x1[n] + �2x2[n] + � � � + �LxL[n] + ( �i) u[n] (8.13) 0 We have thus arrived at an Lth-order state-space description of the given system. To write the ab ove state-space description in matrix form, de�ne the state vector at time n to b e 1 0 x[n] � BBBB@ x1[n] x2[n] . . . xL[n] CCCCA (8.14) Also de�ne the diagonal matrix A, column vector b, and row vector c as follows: 1010 �1 0 0 � � � 0 0 �1 A � BBBB@� 0 �2 0 � � � 0 0 . . .. . . . . . . . . . . . .. . CCCCA � b � BBBB@ �2 . . . CCCCA (8.15) 0 0 0 � � � 0 �L �L � c � �1 �2 � � � � � � � � � �L (8.16) Then our state-space mo del takes the desired matrix form, as you can easily verify: x[n + 1] � Ax[n] + bu[n] (8.17) y[n] � cx[n] + du[n] (8.18) where LX d � �i (8.19) 08.3 Realization from an LTI Di�erential/Di�erence equation In this section, we describ e how a realization can b e obtained from a di�erence or a di�erential equation. We b egin with an example. Example 8.1 (State-Space Mo dels for an LTI Di�erence Equation) Let us examine some ways of …