Closed-Loop System Analysis and Design - Control of Color Imaging Systems: Analysis and Design - page 220

Equation 5.26 can be used to solve for K by premultiplying both sides of this
equation by 0 ½ 0 01 . The resulting gain matrix K is given as

1 P ( A )

BABA 2 B A N 1 B

K ¼ ½ 00 01 ( 5 : 27 )

The above equation is known as Ackermann is formula for pole placement.

'

Example 5.2

Consider the dynamic system given by

x(k) u(k)

0 1 0

1

x(k þ 1) ¼ þ

0 : 72 1 : 7

Design a state feedback to place the closed-loop poles at

l 1 ¼ l 2 ¼ 0 : 2

S OLUTION
The characteristic polynomial of the closed-loop system is

2

P( l ) ¼ ( l l 1 )( l l 2 ) ¼ ( l 0 : 2)( l 0 : 2) ¼ l 0 : 4 l þ 0 : 04

The controllability matrix is

01
11

½ BAB ¼

: 7

Since the controllability matrix is full rank, the system is completely state control-
lable and pole placement is possible. The feedback gain vector is computed using
Equation 5.27

1 P(A) 1 (A 2 K ¼ ½ 01 BAB ½ ¼ ½ 01 BAB ½ 0 : 4A þ 0 : 04I)

1

01
11

0 : 68 1 : 3

¼ ½ 01 ¼ ½ 0 : 68 1 : 3

: 7 0 : 93 1 : 53

There are other algorithms that can be used for pole placement [1, 4]. The second
algorithm is summarized below.

a. For the desired poles, find the desired characteristic polynomial of the

closed-loop system

P c ( l ) ¼jlI A c j¼ ( l l 1 )( l l 2 ) ( l l N )

N N 1 N 2

¼ l þ a 1 l þ a 2 l þþa N (5 : 28)