Modern Control Engineering

Example Problems and Solutions 835

The closed-loop poles of the observed-state feedback control system with a minimum-order

observer consist of the closed-loop poles due to pole placement and the closed-loop poles due to

the minimum-order observer. (Therefore, the pole-placement design and the design of the

minimum-order observer are independent of each other.)

A–10–12. Consider a completely state controllable system defined by

(10–167)

where

Suppose that the rank of the following matrix

isn+1. Show that the system defined by

(10–168)

where

is completely state controllable.

Solution.Define

Because the system given by Equation (10–167) is completely state controllable, the rank of matrix

Misn. Then the rank of

isn+1.Consider the following equation:

(10–169)

Since matrix

is of rank n+1,the left-hand side of Equation (10–169) is of rank n+1.Therefore, the right-hand

side of Equation (10–169) is also of rank n+1.Since

= CAˆ BˆAˆ^2 BˆpAˆn BˆBˆD

= B

AB

- CB





A^2 B

- CAB

p

p

An B

• CAn-^1 B





B

0

R

B

AM

- CM

B

0

R= B

ACBABpAn-^1 BD

• CCBABpAn-^1 BD

B

0

R

B

A

- C

B

0

R

B

A

- C

B

0

RB

M

0

0

1

R = B

AM

- CM

B

0

R

B

M

0

0

1

R

M= CB  AB  p  An-^1 BD

Aˆ =B

A

- C

0

0

R, Bˆ = B

B

0

R , ue=u(t)-u(q)

e# =Aˆe+Bˆue

B

A

- C

B

0

R

(n+1)*(n+1)

C= 1 *n constant matrix

B=n*1 constant matrix

A=n*n constant matrix

y=output signal (scalar)

u=control signal (scalar)

x=state vector (n-vector)

y =Cx

x# =Ax+Bu