Page 389 - Schaum's Outline of Theory and Problems of Signals and Systems

376 STATE SPACE ANALYSIS [CHAP. 7

Now premultiplying both sides of Eq. (7.46a) by e--A', we obtain

+
)
ePA'q(t = L? -A'~q(t) eA'bx(t)
or ePA'q(t) - ePA'Aq(t) = e-A'bx(t)
From Eq. (7.59) Eq. (7.60) can be rewritten as

d
- [e-A'q(t)] = CA'bx(t)
dl

Integrating both sides of Eq. (7.61) from 0 to I, we get

Hence e-*'q(t) = q(0) + /'e-*'bx(r) di (7.62)
0

Premultiplying both sides of Eq. (7.62) by eA' and using Eqs. (7.55) and (7.561, we obtain

If the initial state is q(t,,) and we have x( t) for t 2 I,, then

which is obtained easily by integrating both sides of Eq. (7.61) from t, to t. The matrix
function eA' is known as the state-transition matrix of the continuous-time system.
Substituting Eq. (7.63) into Eq. (7.466), we obtain

D. Evaluation of eA':
Method 1: As in the evaluation of An, by the Cayley-Hamilton theorem we have

When the eigenvalues A, of A are all distinct, the coefficients b,, b,, . . . , bN-, can be
found from the conditions

For the case of repeated eigenvalues see Prob. 7.45.