Modern Control Engineering

856 Chapter 10 / Control Systems Design in State Space

B–10–9.Consider the inverted-pendulum system shown in

Figure 10–59. Assume that

M=2kg, m=0.5kg, l=1m

Define state variables as

and output variables as

Derive the state-space equations for this system.

It is desired to have closed-loop poles at

Determine the state-feedback gain matrix K.

Using the state-feedback gain matrix Kthus determined,

examine the performance of the system by computer simu-

lation. Write a MATLAB program to obtain the response of

the system to an arbitrary initial condition. Obtain the

response curves x 1 (t)versust, x 2 (t)versust, x 3 (t)versust,

andx 4 (t)versustfor the following set of initial condition:

x 1 (0)=0, x 2 (0)=0, x 3 (0)=0, x 4 (0)= 1 ms

s=- 4 +j4, s=- 4 - j4, s=-20, s=- 20

y 1 =u=x 1 , y 2 =x=x 3

x 1 =u, x 2 =u

, x 3 =x, x 4 =x#

where

Design a full-order state observer. The desired observer

poles are s=–5ands=–5.

B–10–11.Consider the system defined by

where

Design a full-order state observer, assuming that the desired

poles for the observer are located at

s=–10, s=–10, s=–15

B–10–12.Consider the system defined by

Given the set of desired poles for the observer to be

design a full-order observer.

B–10–13.Consider the double integrator system defined by

If we choose the state variables as

then the state-space representation for the system becomes

as follows:

y =[1 0]B

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

Ru

x 2 =y#

x 1 =y

y

$

=u

s=- 5 +j 513 , s=- 5 - j 513 , s=- 10

y=[ 1 0 0 ]C

x 1

x 2

x 3

S

+C

0

0

1.244

Su

C

x# 1

x# 2

x# 3

S = C

0

0

1.244

1

0

0.3956

0

1

- 3.145

SC

x 1

x 2

x 3

S

A= C

0

0

- 5

1

0

- 6

0

1

0

S, B= C

0

0

1

S, C=[1 0 0]

y =Cx

x# =Ax+Bu

A= B

- 1

1

1

- 2

R, C=[1 0]

0

M

P

z

u

mg

m

 sin u

x

x

 cos u



u

Figure 10–59

Inverted-pendulum system.

B–10–10.Consider the system defined by

y =Cx

x# =Ax

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