Modern Control Engineering

574 Chapter 8 / PID Controllers and Modified PID Controllers

Next, let us examine the unit-step response of the system. The closed-loop transfer function

C(s)/R(s)is given by

The unit-step response of this system can be obtained easily with MATLAB. See MATLAB

Program 8–1. The resulting unit-step response curve is shown in Figure 8–8. The maximum

overshoot in the unit-step response is approximately 62%. The amount of maximum overshoot is

excessive. It can be reduced by fine tuning the controller parameters. Such fine tuning can be

made on the computer. We find that by keeping and by moving the double zero of the

PID controller to s=–0.65—that is, using the PID controller

(8–1)

the maximum overshoot in the unit-step response can be reduced to approximately 18%(see

Figure 8–9). If the proportional gain Kpis increased to 39.42, without changing the location of

the double zero (s=–0.65),that is, using the PID controller

Gc(s)=39.42a 1 + (8–2)

1

3.077s

+0.7692sb=30.322

(s+0.65)^2

s

Gc(s)= 18 a 1 +

1

3.077s

+0.7692sb=13.846

(s+0.65)^2

s

Kp= 18

C(s)

R(s)

=

6.3223s^2 +18s+12.811

s^4 +6s^3 +11.3223s^2 +18s+12.811

MATLAB Program 8–1

% ---------- Unit-step response ----------

num = [6.3223 18 12.811];

den = [1 6 11.3223 18 12.811];

step(num,den)

grid

title('Unit-Step Response')

Unit-Step Response

Time (sec)

024 6 8 10 12 14

Amplitude

0

0.8

1.8

1.2

0.6

0.2

1.4

1.6

1

0.4

Figure 8–8

Unit-step response

curve of PID-

controlled system

designed by use of

the Ziegler–Nichols

tuning rule (second

method).

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