Modern Control Engineering

826 Chapter 10 / Control Systems Design in State Space

A–10–6. A regulator system has a plant

Define state variables as

By use of the state-feedback control u=–Kx, it is desired to place the closed-loop poles at

Obtain the necessary state-feedback gain matrix Kwith MATLAB.

Solution.The state-space equations for the system become

Hence,

(Note that, for the pole placement, matrices CandDdo not affect the state-feedback gain

matrixK.)

Two MATLAB programs for obtaining state-feedback gain matrix Kare given in MATLAB

Programs 10–24 and 10–25.

C=[1 0 0], D=[0]

A= C

0

0

- 6

1

0

- 11

0

1

- 6

S, B= C

0

0

10

S

y =[1 0 0]C

x 1

x 2

x 3

S+0u

C

x# 1

x# 2

x# 3

S = C

0

0

- 6

1

0

- 11

0

1

- 6

SC

x 1

x 2

x 3

S +C

0

0

10

Su

s=- 2 +j2 13 , s=- 2 - j2 13 , s=- 10

x 3 =x# 2

x 2 =x# 1

x 1 =y

Y(s)

U(s)

=

10

(s+1)(s+2)(s+3)

MATLAB Program 10–24

A = [0 1 0;0 0 1;-6 -11 -6];

B = [0;0;10];

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

K = acker(A,B,J)

K =

15.4000 4.5000 0.8000

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