Modern Control Engineering

858 Chapter 10 / Control Systems Design in State Space

Determine the value of the parameter aso as to minimize

the following performance index:

Assume that the initial state x(0) is given by

B–10–18.Consider the system shown in Figure 10–62.

Determine the value of the gain Kso that the damping ratio

zof the closed-loop system is equal to 0.5. Then determine

also the undamped natural frequency vnof the closed-loop

system. Assuming that e(0)=1and evaluate

3

q

0

e^2 (t)dt

e#(0)=0,

x(0)= C

c 1

0

0

S

J=

3

q

0

xT xdt

B–10–21.Consider the inverted-pendulum system shown

in Figure 10–59. It is desired to design a regulator system

that will maintain the inverted pendulum in a vertical po-

sition in the presence of disturbances in terms of angle u

and/or angular velocity The regulator system is required

to return the cart to its reference position at the end of

each control process. (There is no reference input to the

cart.)

The state-space equation for the system is given by

where

We shall use the state-feedback control scheme

Using MATLAB, determine the state-feedback gain matrix

such that the following performance

indexJis minimized:

where

Then obtain the system response to the following initial

condition:

Plot response curves uversust, versust, xversust,and

versust.

u x#

D

x 1 (0)

x 2 (0)

x 3 (0)

x 4 (0)

T = D

0.1

0

0

0

T

Q= D

100

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1

T, R= 1

J=

3

q

0

Ax* Qx+u* RuBdt

K=Ck 1 k 2 k 3 k 4 D

u=-Kx

B= D

0

- 1

0

0.5

T, x=D

u

u

x

x#

T

A= D

0

20.601

0

- 0.4905

1

0

0

0

0

0

0

0

0

0

1

0

T

x# =Ax+Bu

u

.

+

• 
  

r= 0 e u c

K^5

(s+ 1) (2s+ 1)

Figure 10–62

Control system.

B–10–19.Determine the optimal control signal ufor the

system defined by

where

such that the following performance index is minimized:

B–10–20.Consider the system

It is desired to find the optimal control signal usuch that

the performance index

is minimized. Determine the optimal signal u(t).

J=

3

q

0

AxT Qx+u^2 Bdt, Q= B

1

0

0

m

R

B

x

1

x# 2

R = B

0

0

1

0

RB

x 1

x 2

R + B

0

1

Ru

J=

3

q

0

AxT x+u^2 Bdt

A= B

0

0

1

- 1

R, B=B

0

1

R

x# =Ax+Bu

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