670 Chapter 9 / Control Systems Analysis in State SpaceAside from computational methods, several analytical methods are available for the
computation of We shall present three methods here.
Computation of eAt: Method 1. If matrix Acan be transformed into a diagonal
form, then can be given by
(9–46)
wherePis a diagonalizing matrix for A. [For the derivation of Equation (9–46), see
ProblemA–9–11.]
If matrix Acan be transformed into a Jordan canonical form, then can be given by
whereSis a transformation matrix that transforms matrix Ainto a Jordan canonical
formJ.
As an example, consider the following matrix A:
The characteristic equation is
|lI-A|=l^3 -3l^2 +3l-1=(l-1)^3 =0
Thus, matrix Ahas a multiple eigenvalue of order 3 at It can be shown that matrix
Ahas a multiple eigenvector of order 3. The transformation matrix that will transform
matrixAinto a Jordan canonical form can be given by
The inverse of matrix Sis
Then it can be seen that
=C
1
0
0
1
1
0
0
1
1
S =J
S-^1 AS=C
1
- 1
1
0
1
- 2
0
0
1
SC
0
0
1
1
0
- 3
0
1
3
SC
1
1
1
0
1
2
0
0
1
S
S-^1 = C
1
- 1
1
0
1
- 2
0
0
1
S
S= C
1
1
1
0
1
2
0
0
1
S
l=1.
A= C
0
0
1
1
0
- 3
0
1
3
S
eAt=SeJt S-^1
eAt
eAt=PeDt P-^1 =PF
el^1 t
0
el^2 t
0
eln^ t
VP-^1
eAt
eAt.
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