Modern Control Engineering

646 Chapter 8 / PID Controllers and Modified PID Controllers

B–8–11.Consider the control system shown in Figure 8–79.

Assume that the PID controller is given by

It is desired that the unit-step response of the system exhibit

the maximum overshoot of less than 10%, but more than 2%

(to avoid an almost overdamped system), and the settling

time be less than 2 sec.

Using the computational approach presented in Section

8–4, write a MATLAB program to determine the values

ofKandathat will satisfy the given specifications. Choose

the search region to be

1 K4, 0.4a 4

Choose the step size for Kandato be0.05.Write the

program such that the nested loops start with the highest

values of Kandaand step toward the lowest.

Using the first-found solution, plot the unit-step

response curve.

B–8–12.Consider the same control system as treated in

ProblemB–8–11(Figure 8–79). The PID controller is given

by

It is desired to determine the values of Kandasuch that

the unit-step response of the system exhibits the maximum

Gc(s)=K

(s+a)^2

s

Gc(s)=K

(s+a)^2

s

overshoot of less than 8%, but more than 3%, and the settling

time is less than 2 sec. Choose the search region to be

2 K4, 0.5a 3

Choose the step size for Kandato be0.05.

First, write a MATLAB program such that the nested

loops in the program start with the highest values of Kand

aand step toward the lowest and the computation stops

when a successful set of Kandais found for the first time.

Next, write a MATLAB program that will find all pos-

sible sets of Kandathat will satisfy the given specifications.

Among multiple sets of Kandathat satisfy the given

specifications, determine the best choice. Then, plot the unit-

step response curves of the system with the best choice

ofKanda.

B–8–13.Consider the two-degrees-of-freedom control

system shown in Figure 8–80. The plant is given by

Design controllers and such that the

response to the unit-step disturbance input should have

small amplitude and settle to zero quickly (in approximately

2 sec). The response to the unit-step reference input should

be such that the maximum overshoot is 25%(or less) and

the settling time is 2 sec. Also, the steady-state errors in the

response to the ramp and acceleration reference inputs

should be zero.

Gc1(s) Gc2(s)

Gp(s)=

3(s+5)

s(s+1)As^2 +4s+ 13 B

Gp(s)

R(s) C(s)

PID

controller

1.2

(0.3s+ 1) (s+ 1) (1.2s+ 1)

+– Gc(s)

Figure 8–79

Control system.

Gc 1 (s) Gp(s)

Gc 2 (s)

U(s) Y(s)

D(s)

R(s)

B(s)

+

++ ++

Figure 8–80

Two-degrees-of-freedom control system.

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