Modern Control Engineering

632 Chapter 8 / PID Controllers and Modified PID Controllers

the requirement can be satisfied ifH(s)is chosen to be

H(s)=ks

because

Thus, withH(s)=ks(velocity feedback), the inner loop becomes ineffective at both the low-

and high-frequency regions. It becomes effective only in the intermediate-frequency region.

A–8–12. Consider the control system shown in Figure 8–63. This is the same system as that considered in

Example 8–1. In that example we designed a PID controller , starting with the second method

of the Ziegler–Nichols tuning rule. Here we design a PID controller using the computational

approach with MATLAB. We shall determine the values of Kandaof the PID controller

such that the unit-step response will exhibit the maximum overshoot between 10%and 2%

(1.02maximum output1.10) and the settling time will be less than 3 sec. The search region is

2 K50, 0.05a 2

Let us choose the step size forKto be 1 and that forato be 0.05.

Write a MATLAB program to find the first set of variables Kandathat will satisfy the given

specifications. Also, write a MATLAB program to find all possible sets of variablesKandathat

will satisfy the given specifications. Plot the unit-step response curves of the designed system with

the chosen sets of variablesKanda.

Solution.The transfer function of the plant is

The closed-loop transfer functionC(s)/R(s)is given by

A possible MATLAB program that will produce the first set of variables Kandathat

will satisfy the given specifications is given in MATLAB Program 8–15. In this program we

C(s)

R(s)

=

Ks^2 +2Kas+Ka^2

s^4 +6s^3 +(5+K)s^2 +2Kas+Ka^2

Gp(s)=

1

s^3 +6s^2 +5s

Gc(s)=K

(s+a)^2

s

Gc(s)

(^) vlimSqG(jv)H(jv)=vlimSq
Kkjv
A 1 +jvT 1 BA 1 +jvT 2 B

= 0

(^) vlimS 0 G(jv)H(jv)=vlimS 0
Kkjv
A 1 +jvT 1 BA 1 +jvT 2 B

= 0

R(s) C(s)

PID

controller

1

s(s+ 1) (s+ 5)

+

• Gc(s)

Figure 8–63

Control system.

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