Page 437 - Schaum's Outline of Theory and Problems of Signals and Systems

424 STATE SPACE ANALYSIS [CHAP. 7

7.62. Consider the discrete-time LTI system with the following state space representation:

(a) Find the system function H(z).
(b) Is the system controllable?
(c) IS the system observable?
1
1
Ans. (a) H(z)=
(2- 1)*
(b) The system is controllable.
(c) The system is not observable.

7.63. Consider the discrete-time LTI system in Prob. 7.55.

(a) Is the system asymptotically stable?
(b) Is the system BIBO stable?
(c) IS the system controllable?
(dl Is the system observable?

Am. (a) The system is asymptotically stable.
(b) The system is BIBO stable.
(c) The system is controllable.
(dl The system is not observable.

7.64. The controllability and observability of an LTI system may be investigated by diagonalizing the
system matrix A. A system with a state space representation

v[n + 11 = Av[n] + *bx[n]
y [n] = b[n]
(where A is a diagonal matrix) is controllable if the vector Ib has no zero elements, and it is
observable if the vector e has no zero elements. Consider the discrete-time LTI system in Prob.
7.55.
Let dn] = Tq[n]. Find the matrix T such that the new state space representation will have
a diagonal system matrix.
Write the new state space representation of the system.
Using the result from part (b), investigate the controllability and observability of the
system.

(c) The system is controllable but not observable.