Modern Control Engineering

612 Chapter 8 / PID Controllers and Modified PID Controllers

The response to the unit-step disturbance input is shown in Figure 8–42. The response curve seems

good and acceptable. Note that the closed-loop poles are located at s=–3_j2ands=–6.6051.

The complex-conjugate closed-loop poles act as dominant closed-loop poles.

Design of We now design to obtain the desired responses to the reference inputs.

The closed-loop transfer function Y(s)/R(s)can be given by

Zero placement. We place two zeros together with the dc gain constant such that the numera-

tor is the same as the sum of the last three terms of the denominator. That is,

By equating the coefficients ofs^2 terms and sterms on both sides of this last equation,

from which we get

Therefore,

Gc2(s)= 1 +1.2s (8–13)

Kˆp=1, Tˆd=1.2

47.63+5Kˆp=52.63

6.6051+5Kˆp Tˆd=12.6051

A6.6051+5KˆpT ˆdBs^2 +A47.63+5KˆpBs+85.8673=12.6051s^2 +52.63s+85.8673

=

A6.6051+5Kˆp TˆdBs^2 +A47.63+5KˆpBs+85.8673

s^3 +12.6051s^2 +52.63s+85.8673

=

c

1.321s^2 +9.526s+17.1735

s

+KˆpA 1 +Tˆd sBd

5

(s+1)(s+5)

1 +

1.321s^2 +9.526s+17.1735

s

5

(s+1)(s+5)

Y(s)

R(s)

=

AGc1+Gc2BGp

1 +Gc1 Gp

Gc2(s): Gc2(s)

y(d

t)

t (sec)

Unit-Step Response of Y(s)/D(s)

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.02

0.01

0

0 0.5 1 1.5 2 2.5 3

Figure 8–42

Response to unit-

step disturbance

input.

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