Modern Control Engineering

802 Chapter 10 / Control Systems Design in State Space

Next, let us obtain the response xof the regulator system to the initial condition x(0), where

With state feedback u=–Kx, the state equation for the system becomes

Then the system, or sys, can be given by

sys = ss(A-B*K, eye(3), eye(3), eye(3))

MATLAB Program 10–21 produces the response to the given initial condition. The response curves are shown in Figure 10–38.

x# =Ax+Bu=(A-BK) x

x(0)= C

1

0

S

EXAMPLE 10–13 Consider the system shown in Figure 10–39. The plant is defined by the following state-space

equations:

where

The control signal uis given by

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

A= C

0

1

0

- 2

0

1

- 3

S, B= C

0

1

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

y=Cx+Du

x# =Ax+Bu

MATLAB Program 10–21

% Response to initial condition.

A = [0 1 0;0 0 1;-35 -27 -9];

B = [0;0;1];

K = [0.0143 0.1107 0.0676];

sys = ss(A-B*K, eye(3),eye(3),eye(3));

t = 0:0.01:8;

x = initial(sys,[1;0;0],t);

x1 = [1 0 0]*x';

x2 = [0 1 0]*x';

X3 = [0 0 1]*x';

subplot(2,2,1); plot(t,x1); grid

xlabel('t (sec)'); ylabel('x1')

xlabel('t (sec)'); ylabel('x2)

xlabel('t (sec)'); ylabel('x3')

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