Modern Control Engineering

Example Problems and Solutions 833

or

(10–158)

Equations (10–157) and (10–158) are in the observable canonical form.

A–10–10. For the system defined by

consider the problem of designing a state observer such that the desired eigenvalues for the

observer gain matrix are m 1 ,m 2 ,p,mn.

Show that the observer gain matrix given by Equation (10–61), rewritten as

(10–159)

can be obtained from Equation (10–13) by considering the dual problem. That is, the matrix Ke

can be determined by considering the pole-placement problem for the dual system, obtaining the

state-feedback gain matrix K, and taking its conjugate transpose, or Ke=K*.

Solution.The dual of the given system is

(10–160)

Using the state-feedback control

Equation (10–160) becomes

Equation (10–13), which is rewritten here, is

(10–161)

where

For the original system, the observability matrix is

Hence, matrix Tcan also be written as

Since we have

and

(T*)-^1 =(WN*)-^1

T=W N=WN

W=W*,

T=NW

CC*A* C*p(A*)n-^1 C*D=N

T=MW=CC*A* C*p(A*)n-^1 C*D W

K=Can-an  an- 1 - an- 1  p  a 2 - a 2  a 1 - a 1 D T-^1

z# =(A*-C* K) z

v=-Kz

n=B* z

z# =A* z+C* v

Ke=(WN*)-^1 F

an-an

an- 1 - an- 1







a 1 - a 1

V

y =Cx

x# =Ax+Bu

y=[1 1]B

- 1

1

0

1

RB

xˆ 1

xˆ 2

R =[0 1]B

xˆ 1

xˆ 2

R