Modern Control Engineering

Section 10–8 / Quadratic Optimal Regulator Systems 801

EXAMPLE 10–12 Consider the system given by

where

The performance index Jis given by

where

Obtain the positive-definite solution matrix Pof the Riccati equation, the optimal feedback gain

matrixK, and the eigenvalues of matrix A-BK.

MATLAB Program 10–20 will solve this problem.

Q=C

1

0

0

0

1

0

0

0

1

S, R=[ 1 ]

J=

3

q

0

(x¿ Qx+u¿ Ru)dt

A= C

0

0

- 35

1

0

- 27

0

1

- 9

S, B=C

0

0

1

S

x# =Ax+Bu

MATLAB Program 10–20

% ---------- Design of quadratic optimal regulator system ----------

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

B = [0;0;1];

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

R = [1];

[K,P,E] = lqr(A,B,Q,R)

K =

0.0143 0.1107 0.0676

P=

4.2625 2.4957 0.0143

2.4957 2.8150 0.1107

0.0143 0.1107 0.0676

E =

-5.0958

-1.9859 + 1.7110i

-1.9859 - 1.7110i