Modern Control Engineering

840 Chapter 10 / Control Systems Design in State Space

(b) for the minimum-order observer:

Also, compare the bandwidths of both systems.

Solution.We first determine the state-space representation of the system. By defining state

variablesx 1 andx 2 as

we obtain

For the pole-placement part, we determine the state feedback gain matrix K. Using MATLAB,

we find Kto be

K=[4 0.5]

(See MATLAB Program 10–28.)

Next, we determine the observer gain matrix Kefor the full-order observer. Using MATLAB,

we find Keto be

(See MATLAB Program 10–28.)

Ke=B

14

36

R

y =[1 0]B

x 1

x 2

R

B

x# 1

x# 2

R = B

0

0

1

- 2

RB

x 1

x 2

R + B

0

4

Ru

x 2 =y#

x 1 =y

x 1 (0)=1, x 2 (0)=0, e 1 (0)= 1

MATLAB Program 10–28

% Obtaining matrices K and Ke.

A = [0 1;0 -2];

B = [0;4];

C = [1 0];

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

L = [-8 -8];

K = acker(A,B,J)

K =

4.0000 0.5000

Ke = acker(A',C',L)'

Ke =

14

36

Now we find the response of this system to the given initial condition. Referring to Equation

(10–70), we have

This equation defines the dynamics of the designed system using the full-order observer. MATLAB

Program 10–29 produces the response to the given initial condition. The resulting response curves

are shown in Figure 10–52.

B

x#

e#

R = B

A-BK

0

BK

A-Ke C

RB

x

e

R

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