Page 379 - Schaum's Outline of Theory and Problems of Signals and Systems

366 STATE SPACE ANALYSIS [CHAP. 7

ments, the outputs of these memory elements can be chosen to be the state variables of the
system (Probs. 7.4 and 7.5). If the system is described by the difference or differential
equation, the state variables can be chosen as shown in the following sections.
Note that the choice of state variables of a system is not unique. There are infinitely
many choices for any given system.

7.3 STATE SPACE REPRESENTATION OF DISCRETE-TIME LTI SYSTEMS

A. Systems Described by Difference Equations:
Suppose that a single-input single-output discrete-time LTI system is described by an
Nth-order difference equation
y[n] +a,y[n - 11 + +aNy[n - N] =x[n] (7.1)

We know from previous discussion that if x[n] is given for n 2 0, Eq. (7.1) requires N
initial conditions y[- I], y[-21,. . ., y[-N] to uniquely determine the complete solution
for n > 0. That is, N values are required to specify the state of the system at any time.
Let us define N state variables q,[n], q2[n], . . . , qN[n] as

and y[n] = -aNq,[n] -aN- ,q2[n] - -. . -a,q~[~l +x[nl

In matrix form Eqs. (7.3~) and (7.36) can be expressed as

Now we define an N x 1 matrix (or N-dimensional vector) q[n] which we call the state