Modern Control Engineering

Section 9–4 / Solving the Time-Invariant State Equation 665

If there is a multiplicity in the eigenvalues—for example, if the eigenvalues of Aare

l 1 ,l 1 ,l 1 ,l 4 ,l 5 ,p,ln,

then will contain, in addition to the exponentials terms like

and

Properties of State-Transition Matrices. We shall now summarize the important

properties of the state-transition matrix For the time-invariant system

for which

we have the following:

1.

2. or

3.

4.

5.

EXAMPLE 9–5 Obtain the state-transition matrix of the following system:

Obtain also the inverse of the state-transition matrix,

For this system,

The state-transition matrix is given by

Since

the inverse of (sI-A)is given by

= D

s+ 3

(s+1)(s+2)

• 2
  (s+1)(s+2)

1

(s+1)(s+2)

s

(s+1)(s+2)

T

(s I-A)-^1 =

1

(s+1)(s+2)

B

s+ 3

• 2

1

s

R

s I-A=B

s

0

0

s

R - B

0

- 2

1

- 3

R =B

s

2

- 1

s+ 3

R

(t)=eAt=l-^1 C(s I-A)-^1 D

(t)

A= B

0

- 2

1

- 3

R

-^1 (t).

B

x# 1

x# 2

R= B

0

- 2

1

- 3

RB

x 1

x 2

R

(t)

At 2 - t 1 B At 1 - t 0 B=At 2 - t 0 B=At 1 - t 0 B At 2 - t 1 B

C(t)Dn=(nt)

At 1 +t 2 B=eAAt^1 +t^2 B=eAt^1 eAt^2 =At 1 B At 2 B=At 2 B At 1 B

(t)=eAt=Ae-^ AtB-^1 =C(-t)D-^1 -^1 (t)=(-t)

(0)=eA^0 =I

(t)=eAt

x# =Ax

(t).

tel^1 t^ t^2 el^1 t.

(t) el^1 t, el^4 t, el^5 t,p,eln^ t,