Modern Control Engineering

Section 10–8 / Quadratic Optimal Regulator Systems 797

from which we obtain the following three equations:

Solving these three simultaneous equations for p 11 , p 12 ,andp 22 ,requiringPto be positive definite,

we obtain

Referring to Equation (10–117), the optimal feedback gain matrix Kis obtained as

Thus, the optimal control signal is

(10–120)

Note that the control law given by Equation (10–120) yields an optimal result for any initial state

under the given performance index. Figure 10–37 is the block diagram for this system.

Since the characteristic equation is

ifm=1,the two closed-loop poles are located at

These correspond to the desired closed-loop poles when m=1.

Solving Quadratic Optimal Regulator Problems with MATLAB. In MATLAB,

the command

lqr(A,B,Q,R)

s=-0.866+j 0.5, s=-0.866-j 0.5

∑s I-A+BK∑=s^2 + 1 m+ 2 s+ 1 = 0

u=-Kx=-x 1 - 1 m+ 2 x 2

=C 1 1 m+ 2 D

=Cp 12 p 22 D

=[1][0 1]B

p 11

p 12

p 12

p 22

R

K=R-^1 B* P

P= B

p 11

p 12

p 12

p 22

R =B

1 m+ 2

1

1

1 m+ 2

R

m+2p 12 - p^222 = 0

p 11 - p 12 p 22 = 0

1 - p^212 = 0

ux 1

Plant

x 2



(^) m + 2
Figure 10–37
Optimal control of
the plant shown in
Figure 10–36.