Page 381 - Schaum's Outline of Theory and Problems of Signals and Systems

368 STATE SPACE ANALYSIS [CHAP. 7

Equations (7.11a) and (7.11b) yield the same output y[n] for a given input x[n] with
different state equations. In matrix algebra Eq. (7.10a) is known as the similarity transfor-
mation and matrices A and are called similar matrices (App. A).

C. Multiple-Input Multiple-Output Systems:
If a discrete-time LTI system has m inputs and p outputs and N state variables, then a
state space representation of the system can be expressed as
q[n + 11 = Aq[n] + Bx[n]
y[nI = Cq[n] + Dx[n]
where

and
-

A =

- NXrn

7.4 STATE SPACE REPRESENTATION OF CONTINUOUS-TIME LTI SYSTEMS
A. Systems Described by Differential Equations:

Suppose that a single-input single-output continuous-time LTI system is described by
an Nth-order differential equation

One possible set of initial conditions is y(O), y(l)(O), . . . , Y(~-')(O), where y(k)(O =
dk y(t)/dt '. Thus, let us define N state variables ql( 0, q,( 0,. . . , qN(O as

q1W =y(t)
qz(1) = Y(')(~)