Page 385 - Schaum's Outline of Theory and Problems of Signals and Systems

STATE SPACE ANALYSIS [CHAP. 7

from the conditions

For the case of repeated eigenvalues, see Prob. 7.25.
Method 2: The second method of finding An is based on the diagonalization of a matrix A. If
eigenvalues A, of A are all distinct, then An can be expressed as [App. A, Eq. (A.53)]

where matrix P is known as the diagonalization matrix and is given by [App. A, Eq.
(A.3611
p=[x, x, a.B x,] (7.30)

and x, (k = 1,2,. . . , N) are the eigenvectors of A defined by

Method 3: The third method of finding An is based on the spectral decomposition of a matrix A.
When all eigenvalues of A are distinct, then A can be expressed as

where A, (k = 1,2,. . . , N) are the distinct eigenvalues of A and E, (k = 1,2, . . . , N) are
called constituent matrices which can be evaluated as [App. A, Eq. (AH)]
N

Then we have

Method 4: The fourth method of finding An is based on the z-transform.

which is derived in the following section [Eq. (7.41)l.

C. The z-Transform Solution:
Taking the unilateral z-transform of Eqs. (7.22a) and (7.226) and using Eq. (4.51), we
get