Modern Control Engineering

848 Chapter 10 / Control Systems Design in State Space

Solving for the matrix P, we obtain

The performance index is then

(10–177)

To minimize J, we differentiate Jwith respect to k 2 and set equal to zero as follows:

Hence,

With this value of k 2 ,we have Thus, the minimum value of Jis obtained by substi-

tuting into Equation (10–177), or

The designed system has the control law

The designed system is optimal in that it results in a minimum value for the performance index J

under the assumed initial condition.

A–10–17. Consider the same inverted-pendulum system as discussed in Example 10–5. The system is shown

in Figure 10–8, where M=2kg,m=0.1kg, and l=0.5m. The block diagram for the system is

shown in Figure 10–9. The system equations are given by

j

=r-y=r-Cx

u=-Kx+kI j

y=Cx

x# =Ax+Bu

u=-4x 1 - 120 x 2

Jmin=

15

2

c^2

k 2 = 120

02 J 0 k^227 0.

k 2 = 120

0 J

0 k 2

=a

- 5

2k^22

+

1

8

b c^2 = 0

0 J 0 k 2

= a

5

2 k 2

+

k 2

8

b c^2

=[c 0 ]B

p 11

p 12

p 12

p 22

RB

c

0

R =p 11 c^2

J=xT( 0 ) Px( 0 )

P=B

p 11

p 12

p 12

p 22

R =D

5

2 k 2

+

k 2

8

1

8

1

8

5

8 k 2

T

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