Modern Control Engineering

Section 10–4 / Design of Servo Systems 741

Since the desired eigenvalues of A-BKare all in the left-half splane, the inverse of

matrixA-BKexists. Consequently,x(q) can be determined as

Also,u(q) can be obtained as

(See Example 10–4 to verify this last equation.)

EXAMPLE 10–4 Design a type 1 servo system when the plant transfer function has an integrator. Assume that the

plant transfer function is given by

The desired closed-loop poles are and s=–10.Assume that the system

configuration is the same as that shown in Figure 10–4 and the reference input ris a step function.

Obtain the unit-step response of the designed system.

Define state variables x 1 , x 2 ,andx 3 as follows:

Then the state-space representation of the system becomes

(10–26)

(10–27)

where

Referring to Figure 10–4 and noting that n=3,the control signal uis given by

(10–28)

where

The state-feedback gain matrix Kcan be obtained easily with MATLAB. See MATLAB

Program 10–4.

K=Ck 1 k 2 k 3 D

u=-Ak 2 x 2 +k 3 x 3 B+k 1 Ar-x 1 B=-Kx+k 1 r

A= C

0

0

0

1

0

- 2

0

1

- 3

S, B= C

0

0

1

S, C=[1 0 0]

y =Cx

x

=Ax+Bu

x 3 =x

2

x 2 =x

1

x 1 =y

s=- 2 ;j2 13

Y(s)

U(s)

=

1

s(s+1)(s+2)

u(q)=-Kx(q)+k 1 r= 0

x(q)=-(A-BK)-^1 Bk 1 r

MATLAB Program 10–4

A = [0 1 0;0 0 1;0 -2 -3];

B = [0;0;1];

J = [-2+j2sqrt(3) -2-j2sqrt(3) -10];

K = acker(A,B,J)

K =

160.0000 54.0000 11.0000