Modern Control Engineering

796 Chapter 10 / Control Systems Design in State Space

EXAMPLE 10–9 Consider the system shown in Figure 10–36. Assuming the control signal to be

determine the optimal feedback gain matrix Ksuch that the following performance index is

minimized:

where

(m0)

From Figure 10–36, we find that the state equation for the plant is

where

We shall demonstrate the use of the reduced-matrix Riccati equation in the design of the

optimal control system. Let us solve Equation (10–118), rewritten as

Noting that matrix Ais real and matrix Qis real symmetric, we see that matrix Pis a real sym-

metric matrix. Hence, this last equation can be written as

This equation can be simplified to

B

0

p 11

0

p 12

R + B

0

0

p 11

p 12

R- B

p^212

p 12 p 22

p 12 p 22

p^222

R + B

1

0

0

m

R = B

0

0

0

0

R

• B

p 11

p 12

p 12

p 22

RB

0

1

R[1][0 1]B

p 11

p 12

p 12

p 22

R + B

1

0

0

m

R = B

0

0

0

0

R

B

0

1

0

0

RB

p 11

p 12

p 12

p 22

R +B

p 11

p 12

p 12

p 22

RB

0

0

1

0

R

A P+PA-PBR-^1 B P+Q= 0

A= B

0

0

1

0

R, B=B

0

1

R

x# =Ax+Bu

Q= B

1

0

0

m

R

J=

3

q

0

AxT Qx+u^2 Bdt

u(t)=-Kx(t)

ux 1

Plant

x 2

• K



Figure 10–36

Control system.

Openmirrors.com