Modern Control Engineering

Example Problems and Solutions 837

Then, the state-space equations become

Define

The state feedback gain matrix Kand observer gain matrix Kecan be obtained easily by use of
MATLAB as follows:

(See MATLAB Program 10–26.)

Ke= B

14.4

0.3

0.6

15.7

R

K=[130.4444 - 41.5556 23.1000 15.4185]

A= E

0

0

- 36

18

0

0

36

- 18

1

0

- 0.6

0.3

0

1

0.6

- 0.3

U = C

Aaa

Aba

Aab

Abb

S, B= E

0

0



1

0

U = C

Ba



Bb

S

B

y 1

y 2

R =B

1

0

0

1

0

0

0

0

RD

x 1

x 2

x 3

x 4

T

D

x# 1

x# 2

x# 3

x# 4

T =D

0

0

- 36

18

0

0

36

- 18

1

0

- 0.6

0.3

0

1

0.6

- 0.3

TD

x 1

x 2

x 3

x 4

T+ D

0

0

1

0

Tu

MATLAB Program 10–26

A = [0 0 1 0;0 0 0 1;-36 36 -0.6 0.6;18 -18 0.3 -0.3];

B = [0;0;1;0];

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

K = acker(A,B,J)

K =

130.4444 -41.5556 23.1000 15.4185

Aab = [1 0;0 1];

Abb = [-0.6 0.6;0.3 -0.3];

L = [-15 -16];

Ke = place(Abb',Aab',L)'

place: ndigits= 15

Ke =

14.4000 0.6000

0.3000 15.7000

Response to Initial Condition: Next, we obtain the response of the designed system to the given
initial condition. Since

x =B

xa

xb

R = B

y

xb

R

u=-Kx

x# =Ax+Bu