8

PID Controllers and

Modified PID
Controllers

8–1 INTRODUCTION

In previous chapters, we occasionally discussed the basic PID controllers. For example,
we presented electronic, hydraulic, and pneumatic PID controllers. We also designed
control systems where PID controllers were involved.
It is interesting to note that more than half of the industrial controllers in use today
are PID controllers or modified PID controllers.
Because most PID controllers are adjusted on-site, many different types of tuning
rules have been proposed in the literature. Using these tuning rules, delicate and fine tun-
ing of PID controllers can be made on-site. Also, automatic tuning methods have been
developed and some of the PID controllers may possess on-line automatic tuning
capabilities. Modified forms of PID control, such as I-PD control and multi-degrees-of-
freedom PID control, are currently in use in industry. Many practical methods for bump-
less switching (from manual operation to automatic operation) and gain scheduling are
commercially available.
The usefulness of PID controls lies in their general applicability to most control sys-
tems. In particular, when the mathematical model of the plant is not known and there-
fore analytical design methods cannot be used, PID controls prove to be most useful. In
the field of process control systems, it is well known that the basic and modified PID con-
trol schemes have proved their usefulness in providing satisfactory control, although in
many given situations they may not provide optimal control.
In this chapter we first present the design of a PID controlled system using Ziegler
and Nichols tuning rules.We next discuss a design of PID controller with the conventional

567
 frequency-response approach, followed by the computational optimization approach to
design PID controllers. Then we introduce modified PID controls such as PI-D control
and I-PD control.Then we introduce multi-degrees-of-freedom control systems, which can
satisfy conflicting requirements that single-degree-of-freedom control systems cannot.
(For the definition of multi-degrees-of-freedom control systems, see Section 8–6.)
In practical cases, there may be one requirement on the response to disturbance
input and another requirement on the response to reference input. Often these two re-
quirements conflict with each other and cannot be satisfied in the single-degree-of-
freedom case. By increasing the degrees of freedom, we are able to satisfy both. In this
chapter we present two-degrees-of-freedom control systems in detail.
The computational optimization approach presented in this chapter to design con-
trol systems (such as to search optimal sets of parameter values to satisfy given transient
response specifications) can be used to design both single-degree-of-freedom control sys-
tems and multi-degrees-of-freedom control systems, provided a fairly precice mathe-
matical model of the plant is known.

Outline of the Chapter. Section 8–1 has presented introductory material for the
chapter. Section 8–2 deals with a design of a PID controller with Ziegler–Nichols Rules.
Section 8–3 treats a design of a PID controller with the frequency-response approach.
Section 8–4 presents a computational optimization approach to obtain optimal param-
eter values of PID controllers. Section 8–5 discusses multi-degrees-of-freedom control
systems including modified PID control systems.

8–2 ZIEGLER–NICHOLS RULES FOR TUNING

PID CONTROLLERS

PID Control of Plants. Figure 8–1 shows a PID control of a plant. If a mathe-
matical model of the plant can be derived, then it is possible to apply various design
techniques for determining parameters of the controller that will meet the transient and
steady-state specifications of the closed-loop system. However, if the plant is so com-
plicated that its mathematical model cannot be easily obtained, then an analytical or
computational approach to the design of a PID controller is not possible. Then we must
resort to experimental approaches to the tuning of PID controllers.
The process of selecting the controller parameters to meet given performance spec-
ifications is known as controller tuning. Ziegler and Nichols suggested rules for tuning
PID controllers (meaning to set values Kp , Ti , and Td) based on experimental step
responses or based on the value of Kp that results in marginal stability when only pro-
portional control action is used. Ziegler–Nichols rules, which are briefly presented in
the following, are useful when mathematical models of plants are not known. (These
rules can, of course, be applied to the design of systems with known mathematical

1
+ Kp(1 + + Tds) Plant
– Tis
Figure 8–1
PID control
of a plant.

568 Chapter 8 / PID Controllers and Modified PID Controllers

 models.) Such rules suggest a set of values of Kp , Ti , and Td that will give a stable oper-
ation of the system. However, the resulting system may exhibit a large maximum over-
shoot in the step response, which is unacceptable. In such a case we need series of fine
tunings until an acceptable result is obtained. In fact, the Ziegler–Nichols tuning rules
give an educated guess for the parameter values and provide a starting point for fine tun-
ing, rather than giving the final settings for Kp , Ti , and Td in a single shot.

Ziegler–Nichols Rules for Tuning PID Controllers. Ziegler and Nichols pro-
posed rules for determining values of the proportional gain Kp , integral time Ti , and de-
rivative time Td based on the transient response characteristics of a given plant. Such
determination of the parameters of PID controllers or tuning of PID controllers can be
made by engineers on-site by experiments on the plant. (Numerous tuning rules for PID
controllers have been proposed since the Ziegler–Nichols proposal. They are available
in the literature and from the manufacturers of such controllers.)
There are two methods called Ziegler–Nichols tuning rules: the first method and the
second method. We shall give a brief presentation of these two methods.

First Method. In the first method, we obtain experimentally the response of the
plant to a unit-step input, as shown in Figure 8–2. If the plant involves neither integra-
tor(s) nor dominant complex-conjugate poles, then such a unit-step response curve may
look S-shaped, as shown in Figure 8–3. This method applies if the response to a step
input exhibits an S-shaped curve. Such step-response curves may be generated experi-
mentally or from a dynamic simulation of the plant.
The S-shaped curve may be characterized by two constants, delay time L and time
constant T. The delay time and time constant are determined by drawing a tangent line
at the inflection point of the S-shaped curve and determining the intersections of the
tangent line with the time axis and line c(t)=K, as shown in Figure 8–3. The transfer

Figure 8–2 1
Unit-step response Plant
of a plant. u(t) c(t)

c(t)
Tangent line at
inflection point

0 t
Figure 8–3
S-shaped response
L T
curve.

Section 8–2 / Ziegler–Nichols Rules for Tuning PID Controllers 569

 Table 8–1 Ziegler–Nichols Tuning Rule Based on Step Response
of Plant (First Method)
Type of
Controller Kp Ti Td

T
P q 0
L
T L
PI 0.9 0
L 0.3
T
PID 1.2 2L 0.5L
L

function C(s)/U(s) may then be approximated by a first-order system with a transport

lag as follows:
C(s) Ke-Ls
=
U(s) Ts + 1

Ziegler and Nichols suggested to set the values of Kp , Ti , and Td according to the formula
shown in Table 8–1.
Notice that the PID controller tuned by the first method of Ziegler–Nichols rules
gives

Gc(s) = Kp a 1 + + Td s b
1
Ti s

a1 + + 0.5Ls b
T 1
= 1.2
L 2Ls
1 2
as + b
L
= 0.6T
s

Thus, the PID controller has a pole at the origin and double zeros at s=–1/L.

Second Method. In the second method, we first set Ti = q and Td = 0. Using the
proportional control action only (see Figure 8–4), increase Kp from 0 to a critical value
Kcr at which the output first exhibits sustained oscillations. (If the output does not ex-
hibit sustained oscillations for whatever value Kp may take, then this method does not
apply.) Thus, the critical gain Kcr and the corresponding period Pcr are experimentally

r(t) u(t) c(t)

Figure 8–4 + Kp Plant
–
Closed-loop system
with a proportional
controller.

570 Chapter 8 / PID Controllers and Modified PID Controllers

 c(t)

Pcr

Figure 8–5
Sustained oscillation
with period Pcr . 0 t
(Pcr is measured in
sec.)

determined (see Figure 8–5). Ziegler and Nichols suggested that we set the values of
the parameters Kp , Ti , and Td according to the formula shown in Table 8–2.

Table 8–2 Ziegler–Nichols Tuning Rule Based on Critical Gain

Kcr and Critical Period Pcr (Second Method)
Type of
Controller Kp Ti Td

P 0.5Kcr q 0

1
PI 0.45Kcr P 0
1.2 cr

PID 0.6Kcr 0.5Pcr 0.125Pcr

Notice that the PID controller tuned by the second method of Ziegler–Nichols rules
gives

Gc(s) = Kp a 1 + + Td s b
1
Ti s

= 0.6Kcr a 1 + + 0.125Pcr s b
1
0.5Pcr s
4 2
as + b
Pcr
= 0.075Kcr Pcr
s

Thus, the PID controller has a pole at the origin and double zeros at s = -4兾Pcr .
Note that if the system has a known mathematical model (such as the transfer func-
tion), then we can use the root-locus method to find the critical gain Kcr and the fre-
quency of the sustained oscillations vcr , where 2p兾vcr = Pcr . These values can be found
from the crossing points of the root-locus branches with the jv axis. (Obviously, if the
root-locus branches do not cross the jv axis, this method does not apply.)

Section 8–2 / Ziegler–Nichols Rules for Tuning PID Controllers 571

 Comments. Ziegler–Nichols tuning rules (and other tuning rules presented in the
literature) have been widely used to tune PID controllers in process control systems
where the plant dynamics are not precisely known. Over many years, such tuning rules
proved to be very useful. Ziegler–Nichols tuning rules can, of course, be applied to plants
whose dynamics are known. (If the plant dynamics are known, many analytical and
graphical approaches to the design of PID controllers are available, in addition to
Ziegler–Nichols tuning rules.)

EXAMPLE 8–1 Consider the control system shown in Figure 8–6 in which a PID controller is used to control the
system. The PID controller has the transfer function

Gc(s) = Kp a 1 + + Td s b
1
Ti s

Although many analytical methods are available for the design of a PID controller for the pres-
ent system, let us apply a Ziegler–Nichols tuning rule for the determination of the values of pa-
rameters Kp , Ti , and Td . Then obtain a unit-step response curve and check to see if the designed
system exhibits approximately 25% maximum overshoot. If the maximum overshoot is excessive
(40% or more), make a fine tuning and reduce the amount of the maximum overshoot to ap-
proximately 25% or less.
Since the plant has an integrator, we use the second method of Ziegler–Nichols tuning rules.
By setting Ti = q and Td = 0, we obtain the closed-loop transfer function as follows:

C(s) Kp
=
R(s) s(s + 1)(s + 5) + Kp

The value of Kp that makes the system marginally stable so that sustained oscillation occurs can
be obtained by use of Routh’s stability criterion. Since the characteristic equation for the
closed-loop system is

s3+6s2+5s+Kp=0

the Routh array becomes as follows:

s3 1 5
s2 6 Kp
30 - Kp
s1
6
s0 Kp

R(s) 1 C(s)
+ Gc(s)
– s(s + 1)(s + 5)
Figure 8–6 PID
PID-controlled controller
system.

572 Chapter 8 / PID Controllers and Modified PID Controllers

 Examining the coefficients of the first column of the Routh table, we find that sustained oscilla-
tion will occur if Kp = 30. Thus, the critical gain Kcr is

Kcr=30

With gain Kp set equal to Kcr (= 30), the characteristic equation becomes

s3+6s2+5s+30=0

To find the frequency of the sustained oscillation, we substitute s=jv into this characteristic
equation as follows:

(jv)3+6(jv)2+5(jv)+30=0
or
6A5-v2 B+jvA5-v2 B=0

from which we find the frequency of the sustained oscillation to be v2 = 5 or v = 15 . Hence, the
period of sustained oscillation is

2p 2p
Pcr = = = 2.8099
v 15

Referring to Table 8–2, we determine Kp , Ti , and Td as follows:

Kp = 0.6Kcr = 18

Ti = 0.5Pcr = 1.405

Td = 0.125Pcr = 0.35124

The transfer function of the PID controller is thus

Gc(s) = Kp a 1 + + Td s b
1
Ti s

= 18 a 1 + + 0.35124s b
1
1.405s
6.3223(s + 1.4235)2
=
s

The PID controller has a pole at the origin and double zero at s=–1.4235. A block diagram of
the control system with the designed PID controller is shown in Figure 8–7.
Figure 8–7
Block diagram of the
system with PID
R(s) 6.3223 (s + 1.4235)2 1 C(s)
controller designed +
– s s(s + 1)(s + 5)
by use of the
Ziegler–Nichols PID controller
tuning rule (second
method).

Section 8–2 / Ziegler–Nichols Rules for Tuning PID Controllers 573

 Next, let us examine the unit-step response of the system. The closed-loop transfer function
C(s)/R(s) is given by
C(s) 6.3223s 2 + 18s + 12.811
=
R(s) s + 6s3 + 11.3223s2 + 18s + 12.811
4

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 Kp = 18 and by moving the double zero of the
PID controller to s=–0.65—that is, using the PID controller

(s + 0.65)2
Gc(s) = 18 a 1 + + 0.7692s b = 13.846
1
(8–1)
3.077s s

the maximum overshoot in the unit-step response can be reduced to approximately 18% (see
Figure 8–9). If the proportional gain Kp is increased to 39.42, without changing the location of
the double zero (s=–0.65), that is, using the PID controller
(s + 0.65)2
Gc(s) = 39.42 a 1 + + 0.7692s b = 30.322
1
(8–2)
3.077s s

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
1.8

1.6

1.4

1.2
Amplitude

0.8
Figure 8–8
Unit-step response 0.6
curve of PID- 0.4
controlled system
designed by use of 0.2
the Ziegler–Nichols
0
tuning rule (second 0 2 4 6 8 10 12 14
method). Time (sec)

574 Chapter 8 / PID Controllers and Modified PID Controllers