Page 419 - Schaum's Outline of Theory and Problems of Signals and Systems

STATE SPACE ANALYSIS [CHAP. 7

is controllable if the controllability matrix defined by

M,= [b Ab -.- ~~-'b]
has rank N.
We assume that no = 0 and q[O] = 0. Then, by Eq. (7.23) we have

which can be rewritten as

Thus, if q[N] is to be an arbitrary N-dimensional vector and also to have a nonzero input
sequence, as required for controllability, the coefficient matrix in Eq. (7.122) must be nonsingu-
lar, that is, the matrix

must have rank N.

734. Consider an Nth-order discrete-time LTI system with state space representation

The system is said to be obseruable if, starting at an arbitrary time index nu, it is
possible to determine the state q[n,] = q, from the output sequence y[n,], y[nO +
I], . . . , y[n, + N - 11. Show that the system is observable if the obseruability matrix
defined by

has rank N.
We assume that n, = 0 and x[n] = 0. Then, by Eq. (7.25) the output y[n] for n =
0,1,. . . , N - 1, with x[n] = 0, is given by