Page 430 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 71 STATE SPACE ANALYSIS

In matrix form

where

Now

Thus, the eigenvalues of A are A, = - 1 and At = 2. Since Re{A,) > 0, the system is not
asymptotically stable.
(b) By Eq. (7.52) the system function H(s) is given by

(c) Note that there is pole-zero cancellation in H(s) at s = 2. Thus, the onIy pole of H(s)
is - 1 which is located in the left-hand side of the s-plane. Hence, the system is BIB0
stable.
Again it is noted that the system is essentially unstable if the system is not initially
relaxed.

7.51. Consider an Nth-order continuous-time LTI system with state equation

The system is said to be controllable if it is possible to find an input x(t) which will
drive the system from q(t,) = q, to q(t ,) = q, in a specified finite time and q, and q,
are any finite state vectors. Show that the system is controllable if the controllability
matrix defined by

has rank N.