Page 431 - Schaum's Outline of Theory and Problems of Signals and Systems

STATE SPACE ANALYSIS [CHAP. 7

We assume that r, = 0 and q[Ol = 0. Then, by Eq. (7.63) we have

Now, by the Cayley-Hamilton theorem we can express e-A' as

Substituting Eq. (7.130) into Eq. (7.129) and rearranging, we get

Let

Then Eq. (7.131) can be rewritten as

For any given state q, we can determine from Eq. (7.132) unique Pk's (k = 0,1,. . . , N - I), and
hence x(t), if the coefficients matrix of Eq. (7.132) is nonsingular, that is, the matrix

has rank N.

7.52. Consider an Nth-order continuous-time LTI system with state space representation
q(t) = Aq(t) + bx(t)

~(t) c4t)
=
The system is said to be observable if any initial state q(t,) can be determined by
examining the system output y(t) over some finite period of time from to to t,. Show
that the system is observable if the observability matrix defined by

has rank N.

We prove this by contradiction. Suppose that the rank of M, is less than N. Then there
exists an.initia1 state q[O] = q, f 0 such that
Moq, = 0