Modern Control Engineering

804 Chapter 10 / Control Systems Design in State Space

To get a fast response,q 11 must be sufficiently large compared with q 22 , q 33 ,andR. In this problem,

we choose

To solve this problem with MATLAB, we use the command

K = lqr(A,B,Q,R)

MATLAB Program 10–22 yields the solution to this problem.

q 11 =100, q 22 =q 33 =1, R=0.01

MATLAB Program 10–22

% ---------- Design of quadratic optimal control system ----------

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

B = [0;0;1];

Q = [100 0 0;0 1 0;0 0 1];

R = [0.01];

K = Iqr(A,B,Q,R)

K =

100.0000 53.1200 11.6711

Next we shall investigate the step-response characteristics of the designed system using the

matrixKthus determined. The state equation for the designed system is

and the output equation is

To obtain the unit-step response, use the following command:

[y,x,t] = step(AA,BB,CC,DD)

where

AA=A-BK, BB=Bk 1 , CC=C, DD=D

MATLAB Program 10–23 produces the unit-step response of the designed system. Figure 10–40

shows the response curves x 1 , x 2 ,andx 3 versuston one diagram.

y=Cx=[1 0 0]C

x 1

x 2

x 3

S

=(A-BK) x+Bk 1 r

=Ax+BA-Kx+k 1 rB

x# =Ax+Bu

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