Page 440 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 71 STATE SPACE ANALYSIS

(b) Using the spectral decomposition method.
-sin t cos t I
sin t
cos t
Am, eA' =e-'
Consider the matrix A in Prob. 7.69. Find e-A' and show that eWA' = [eA']- '.

Am. e-At=e cos t I
sin t

Find eA' for

(a) Using the diagonalization method.
(b) Using the Laplace transform method.

Consider the network in Prob. 7.65 (Fig. 7.26). Find u,(t) if x(t) = u(t) under an initially
relaxed condition.

Am. v,(t)= $0 +e-'sint -e-'cost), t>O

Using the state space method, solve the linear differentia! equation
yV(t) + 3y1(t) + 2y(t) = 0

with the initial conditions y(O) = 0, yl(0) = 1.
Am. y(t) = e-' - e-", t > 0

As in the discrete-time case, controllability and observability of a continuous-time LTI system
may be investigated by diagonalizing the system matrix A. A system with state space representa-
tion

where A is a diagonal matrix, is controllable if the vector b has no zero elements and is
observable if the vector C has no zero elements. Consider the continuous-time system in Prob.
7.50.
(a) Find a new state space representation of the system by diagonalizing the system matrix A.
(b) Is the system controllable?
(c) IS the system observable?
Am (a) ir(t)= [ -A ;]*(I) + [;]x(t)

y(t) = [2 - llv(t)
(b) The system is not controllable.
(c) The system is observable.