Modern Control Engineering

Example Problems and Solutions 847

Consider only the case where the initial condition is

Choose the undamped natural frequency to be 2 radsec.

Solution.Substituting Equation (10–174) into Equation (10–173), we obtain

or

(10–175)

Thus,

Elimination of x 2 from Equation (10–175) yields

Since the undamped natural frequency is specified as 2 radsec, we obtain

Therefore,

is a stable matrix if k 2 >0.Our problem now is to determine the value of k 2 so that the
performance index

is minimized, where the matrix Pis determined from Equation (10–115), rewritten

Since in this system Q=IandR= 0 , this last equation can be simplified to

(10–176)

Since the system involves only real vectors and real matrices,Pbecomes a real symmetric matrix.
Then Equation (10–176) can be written as

B

0

1

- 4

• k 2

RB

p 11

p 12

p 12

p 22

R +B

p 11

p 12

p 12

p 22

RB

0

- 4

1

• k 2

R= B

- 1

0

0

- 1

R

(A-BK)* P+P(A-BK)=-I

(A-BK) P+P(A-BK)=-(Q+K RK)

J=

3

q

0

xT xdt=xT(0) P(0) x(0)

A-BK

A-BK= B

0

- 4

1

• k 2

R

k 1 = 4

x$ 1 +k 2 x# 1 +k 1 x 1 = 0

A-BK= B

0

• k 1

1

• k 2

R

= B

0

• k 1

1

• k 2

RB

x 1

x 2

R

B

x

1

x# 2

R= B

0

0

1

0

RB

x 1

x 2

R + B

0

1

RC-k 1 x 1 - k 2 x 2 D

x# =Ax-BKx

x(0)= B

c

0

R