Vineet Kumar, B. C. Nakra and A.P. Mittal, “A Review of Classical and Fuzzy PID Controllers,” International Journal of Intelligent Control and Systems, vol. 16, no. 3, pp. 170-181, September 2011. (ISSN:0218-7965)

Abstract

The industrial evidence shows that a classical PID controller is the most popular controller due to its simplicity of operation and low cost. It has been realized that classical PID controllers are effective for linear systems but not suitable for nonlinear and complex systems. Scientists and researchers use fuzzy logic to enhance them due to its ability to translate the operator’s control action into the rule base. This paper presents a survey of classical and fuzzy PID controllers. Here, an attempt is made to present the history of the development of classical PID controllers and their enhancement using fuzzy logic theory.

Full text

A Review on Classical and Fuzzy PID Controllers
Vineet KUMAR, B. C. NAKRA and A. P. MITTAL
Abstract- The industrial evidence shows that a classical PID
controller is the most popular controller due to its simplicity of
operation and low cost. It has been realized that classical PID
controllers are effective for linear systems but not suitable for
nonlinear and complex systems. Scientists and researchers use fuzzy
logic to enhance them due to its ability to translate the operator’s
control action into the rule base. This paper presents a survey of
classical and fuzzy PID controllers. Here, an attempt is made to
present the history of the development of classical PID controllers
and their enhancement using fuzzy logic theory.
Index Terms— Classical PID, Fuzzy PID Controller, Formula-based
fuzzy PID controllers
1. INTRODUCTION
The conventional theory is well suited for applications
where the process can be reasonably described in advance.
However when plant dynamics is hard to characterize
precisely or is subject to environmental uncertainties, one
may encounter difficulties in using the conventional
controller design methodologies. For achieving high
degree of performance, the fine tuning of controller
parameters is a tedious task. Therefore, in recent years,
the control of systems with complexities, uncertain
dynamics and nonlinearities, has become a topic of
considerable importance in the literature and several
advanced strategies have been developed [21, 27, 51].
Intelligent process control systems with high degree of
autonomy should perform well under significant
uncertainties in the systems and environment for extended
period of time and they must be able to compensate for
certain system failures without external intervention. Such
control systems evolve from conventional control systems
by adding intelligent techniques and their development
requires interdisciplinary research [107, 108, 132].
Recently, excitement over the field of intelligent process
control has risen due to progress in the areas of fuzzy
control, neural networks, genetic algorithms, and expert
systems to name a few [24, 87, 88, 105, 109, 153]. Chiu
has highlighted the aspects of development of commercial
applications of intelligent control in [152].
Fuzzy control, occupying the boundary line between
artificial intelligence and control engineering, can be
considered as an obvious solution, which is confirmed by
engineering practice [72, 123]. According to the survey of
the Japanese control technology industry conducted by the
Japanese Society of Instrument and Control Engineering
[56], fuzzy and neural control constitutes one of the fastest-
growing areas of control technology development, and has
even better prospects for the future. Also, fuzzy logic
control has been suggested as an alternative approach for
complex systems with uncertain dynamics and those with
nonlinearities. Some progress has been made in both the
theoretical aspects and the implementation of the same for
application to industrial control systems [26, 64, 106, 128].
Actually, Fuzzy logic techniques represent application of
human knowledge and expertise for dealing effectively
with complex and nonlinear systems. Basically, it provides
an effective means of capturing the approximate and
inexact nature of the real world. Therefore, the essential
part of a fuzzy logic controller (FLC) is a set of linguistic
control strategies based on expert knowledge into an
automatic control strategy. FLC is considered as a good
methodology because it yields results superior to those
obtained by conventional control algorithms [41, 44, 69,
79, 86, 90, 145, 159].
Actually, fuzzy logic was first proposed by L. A. Zadeh
in 1965 [118] and it is based on the concept of fuzzy sets.
He gives more general ideas regarding the fuzzy logic in
[35, 112, 116, 117, 119 – 121]. Further, he introduces the
concept of “linguistic variables”, which in his article
equates to a variable defined as a fuzzy set [113–115].
Control engineering belongs to the most famous
application areas of fuzzy set theory and has attracted
most attention of researchers and scientists. In 1975, the
first successful application of fuzzy logic to the control of a
laboratory-scale process was reported by Mamdani and
Assilian in [31, 33, 36]. They suggested advances in
linguistic synthesis of a fuzzy controller in [32, 34].
Further, they published the analysis of a fuzzy controller in
[169]. Also, Kingt and Mamdani suggested the application
of fuzzy logic control systems to industrial processes [133].
The first industrial application of fuzzy logic was in the
area of fuzzy controllers. It was done by two Danish civil
engineers, Holmblad and Østergaard, who around 1980 at
the company F.L. Schmidt developed a fuzzy controller for
cement kilns. Their results were published in 1982 [122].
In 1990, Lee published two papers for the use of fuzzy
logic in control systems. He has given a survey about the
role of fuzzy logic in control systems. Also, he discussed
the fuzzy logic controller and its applications from
laboratory level to industrial process control [22, 23].
Fuzzy control is being applied to various systems in the
process industry [154, 160], consumer electronics [91, 136],
automatic train operation in Japan [158], traffic systems in
general [54], and in many other fields [91, 161]. An
excellent review of the Fuzzy Controller design, as well as
its relationship with classical control, is given in [53].
Another very good survey on analysis and design of model
based fuzzy control systems is given in [46].
Manuscript received March 2, 2011; revised July 8, 2011; accepted August
20, 2011
Vineet Kumar and A. P. Mittal are with Dept. of Instrumentation and
Control Engineering, Netaji Subhas Institute of Technology, Sector – 3,
Dwarka, New Delhi – 110078, India.(e-mail: vineetkumar27@gmail.com);
B. C. Nakra is with Dept. of Mechanical and Automobile Engineering,
I.T.M. University Gurgaon, Haryana , India.
INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS
VOL.16,NO.3,SEPTEMBER 2011,170-181

2. CLASSICAL AND FUZZY PID
CONTROLLERS
Classical proportional plus integral plus derivative (PID)
controllers are still the most widely adopted method in
industry for various control applications, due to their
simple structure, ease of design, and low cost in
implementation. Different sources estimate the share taken
by PID controllers at between 90 and 99% [43, 84, 89, 93 –
95, 100, 123, 124, 129, 157]. However, conventional PID
controllers are generally insufficient to control processes
with additional complexities such as time delays,
significant oscillatory behavior (complex poles with small
damping), parameter variations, nonlinearities, and Multi
Input and Multi Output (MIMO) plants [87, 105]. Also,
there are some practical implication with the conventional
structure of a PID controller, such as, proportional kick and
derivative kick, i.e., a sudden change in the PID controller
output resulting from proportional and derivative action
applied to error signal after a change in setpoint value. In
practice, this control signal could be driving an actuator
device like a motor or a valve, and the kick would create
serious problems for electronic circuitry used in the device.
Due to these industrial process problems, the classical PID
controller structure is modified to integral minus
proportional minus derivative (I–P–D) controller form [8,
11, 48, 71, 85, 99, 100, 125, 147, 156, 173]. The wide
application of PID control has stimulated and sustained
research and development to “get the best out of PID’’ [70],
and “the search is on to find the next key technology or
methodology for PID tuning” [134]. Therefore, one of the
research directions in advancing the existing PID
controllers is to combine fuzzy logic control technology
with the conventional PID controller to obtain behaviour
better than that of a regular PID controller.
2.1 Brief Review of Classical PID Controllers
In the 18th century, the most significant control
development was the steam engine governor. In 1788,
James Watt introduced a flyball governor into his steam
engine. It was the first mechanical feedback device with
only proportional control capabilities. The flyball
governor, acting as a proportional controller, controlled the
speed by releasing more steam to the engine when the
speed dropped lower than a set point, and vice versa [74,
131, 146, 147, 151, 162].
One of the first examples of PID-type controls that were
developed was by Sperry. In 1911, this type of systems
was used for automatic ship steering. Note that Sperry did
much work involving gyroscopic compasses as well.
Sperry's device compensated for disturbances in the water
as sea conditions changed. Although Sperry used a type of
PID control in 1911, the control law that we commonly
associate with the modern PID loop comes from Minorsky.
In 1922, he observed a helmsman controlling a ship and
came up with the proportional, integral, and derivative type
of control we know of today. Proportional is the control
required to steer the ship based on actual ship direction
compared to the desired course setpoint. Integral is the
amount of reset required to correct an amount of error. For
example, if the ship is off course by a small amount, and
correcting it to the left brings it back on bearing, then
turning the wheel all the way to the left is inappropriate.
Only a slight adjustment to the left is required. Derivative
is the attempt to see how far a process variable (ship course)
has been from the set point in the past, and anticipating
where the course correction will need to be in the future. In
1922, Minorsky in his paper on the “Directional stability of
automatically steered bodies” analyzed and discussed the
properties of the three-term controller [130, 146, 148].
In 1933, the Taylor Instrument Company introduced
Model 56R Fulscope controller, the first pneumatic with a
fully tunable proportional controller feature. However, a
proportional controller is not sufficient to control a process
variable thoroughly, as it amplifies error by multiplying it
by some proportional constant (Kc). The error generated is
eventually small, but not zero. In other words, it generates
a permanent error or offset or steady state error each time
the controller responds to the load [146, 147, 149, 150, 162,
163 ].
In the mid 1930s, control engineers found out that
steady state error/offset could be eliminated by resetting
the setpoint to some artificial higher or lower value, as
long as the error was nonzero. This resetting operation
integrates the error, and the result is added to the
proportional term; today this is known as Proportional-
Integral (PI) controller. In 1934-1935, Foxboro introduced
the first PI controller. Unfortunately, integral action does
not guarantee perfect feedback control. A PI controller can
cause closed-loop instability if the integral action is too
aggressive. The controller may over-correct for an error
and create a new one of even greater magnitude in the
opposite direction. When that happens, the controller
eventually starts driving its output back and forth between
fully on and fully off, a phenomenon known as hunting
[146, 147, 149, 150, 162].
In 1935, Taylor Instrument Companies introduced a
completely redesigned version of its “Fulscope” pneumatic
controller: this new instrument provided, in addition to
proportional and reset control actions, an action which the
Taylor Instrument Companies called “pre-act”. In the same
year the Foxboro Instrument Company added “Hyper-
reset’’ to the proportional and reset control actions
provided by their “Stabilog” pneumatic controller. Pre-act
and Hyper-reset actions each provided a control action
proportional to the derivative of the error signal. Reset
(also referred to as “floating”) provides a control action
proportional to the integral of the error signal and hence
both controllers offered PID control [146, 147, 149, 150,
162]. Compared to a two-term PI controller, a full PID
controller can even appear to anticipate the level of effort
that is ultimately required to maintain the process variable
at a new setpoint. On the other hand, dramatic swings in
the control effort can be troublesome in applications that
require slow and steady changes in the controller's output
[149, 150, 162].
Kumar et al: A Review on Classical and Fuzzy PID Controllers 171

Fig. 1 The classical PID control system
The block diagram of classical PID control system is
shown in Fig. 1.The output of the classical continuous-time
PID controller, as shown in Fig. 1, is given by
dt
de
Kdtte
K
teKtu DC
I
C
CPID
τ
τ
∫
++= )()()(
(1)
where e(t) is the error, KC is the proportional constant, τI is
the integral time constant , τD is the derivative time
constant and uPID(t) is the output of the classical PID
controller.
Derivative action also tends to add a dramatic spike or
"kick" to the controller's output in the case of an abrupt
change in the error due to a new setpoint [11, 37, 71, 85,
100, 125, 175]. This forces the controller to start taking
corrective action immediately without waiting for the
integral or proportional action to take effect. For such
cases it is advantageous to forego derivative action
altogether or calculate the derivative from the negative of
the process variable rather than directly from the error. If
the setpoint is constant, the two calculations will be
identical. If the setpoint only changes in a stepwise
manner, the two still remain identical except at the instant
when each step change is initiated. The negative derivative
of the process variable lacks the spike present in the
derivative of the error. Most modern controllers offer this
option for applications that cannot withstand “kicking”.
With an extra derivative action, problems such as
overshoot and hunting are reduced. However, issues like
finding the appropriate parameter of PID controllers were
yet to be solved [149, 150, 162].
Taylor engineers Ziegler and Nichols solved the
problem by developing the well-known "Ziegler-Nichols"
method of tuning, still in use today. The outcome of their
work was two papers published by them in 1942 and 1943
[75, 76]. In these papers Ziegler and Nichols showed how
optimal controller parameters could be chosen based first
on open-loop tests on the plant; and second on closed-loop
tests on the plant [147]. Further, Cohen and Coon [47, 147]
of the Taylor Instrument Companies during the 1950s,
proposed alternative choices of parameters accepted for
certain types of plants.
Fig. 2 The classical I – P – D control system
Their modified structure of classical PID controller, i.e.,
I – P – D controller is shown in Fig. 2. The output of the
classical I – P – D controller, as shown in Fig. 2, is given
by
dt
dy
KtyKdtte
K
tu
DCC
I
C
DPI
τ
τ
∫
−−=
−−
)()()(
(2)
where y(t) is the process variable and uI-P-D(t) is the output
of the I – P – D controller.
By the mid 1950’s, automatic controllers were firmly
established and adopted in various industries. A report
from the Department of Scientific and Industrial Research
of United Kingdom states, “Modern controlling units may
be operated mechanically, hydraulically, pneumatically or
electrically. The pneumatic type is technically the most
advanced and many reliable designs are available. It is
thought that more than 90 percent of the existing units are
pneumatic” [30]. The report indicated the need to
implement controllers in electrical and electronic form
[149, 150].
Young, in 1954, described six electronic PID controllers,
based on vacuum tube technology, developed by various
manufacturers around the world [4, 149, 150]. In 1957,
Williams [50] of George Kent commented that electronic
instruments were capable of performing all the functions
previously only available with pneumatic instruments and
that these included, in addition to PID, the ability to carry
out various mathematical operations [103, 149, 150]. He
also noted that the instrument manufacturers started to
realize the possibility of implementing the controllers
using transistors [149, 150]. In 1959, the first solid-state
electronic controller was introduced by Bailey Meter Co.
The advantage of using electronic instruments to
implement PID controllers was explored more deeply years
later. They are not only capable of including the functions
available in pneumatic instruments, but even more
complicated mathematical operations can be carried out as
well [50, 149, 150]. Electronic PID controllers have
become more common and more acceptable since then
[149, 150, 162].
During the 1960s, the digital computer became involved
in industrial process control. The catalytic polymerisation
unit plant at Texaco’s Port Arthur (Texas) was the first
plant, where closed loop control was implemented by a
digital computer on March 15, 1959. By 1960, many
control instrument companies responded to this new
technology and offered computer-based systems. “Analog
controllers should gradually evolve into digital devices,
providing accuracy at low cost. These controllers will be
relatively simple to combine into multipoint configurations,
which can be applied to optimize unit processes on a local
basis.” [7]. More discoveries concerning digitizing PID
controllers were made, and arguments for implementing
controllers on microprocessors were brought up as
microprocessors could handle calculations directly in
engineering units [28, 82, 149, 150, 162]. Further, various
172 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS,VOL.16,NO.3,SEPTEMBER 2011

structure of PID algorithms and tuning methods are
discussed in detail in [5, 6, 83].
Around 1990, it was realized that conventional PID
controllers were effective for simple linear systems, but
generally not suitable for nonlinear systems, higher order,
time-delayed systems, complex and vague systems that had
no precise mathematical models. For these reasons, various
types of modified conventional PID controllers such as
auto-tuning and adaptive PID controllers are proposed [92,
103, 155]. Also, during this period it was suggested that if
the process was too complex to achieve a good physical
description, conventional methods were not able to
guarantee the final control aims, and the controller
synthesis had to be based mainly on intuitions and heuristic
knowledge. So, expert control strategies are favored since
they are based on the process operator's experience and do
not need accurate models [97, 98, 101, 102, 104, 127, 142].
One of the most successful expert system techniques
applied to a wide range of control applications has been the
Fuzzy Set Theory, which has made possible the
establishment of "intelligent control". Its attraction, from
the Process Control Theory point of view, comes because
the fuzzy approach provides a good support for translating
the heuristic skilled operator's knowledge about the process
and control procedures expressed in imprecise linguistic
sentences into numerical algorithms [51, 87, 96, 97, 101,
102, 105, 106, 108, 127, 141].
2.2 Brief Review of Fuzzy PID Controllers
In the 1990s, scientists and researchers were trying to
use intelligent techniques, such as, fuzzy logic, to enhance
the capabilities of classical PID controllers and their family.
They were trying to combine fuzzy logic control
technology with a conventional PID controller to obtain
behavior similar to that of a regular PID controller [25, 26,
43, 178]. It is thus believed that by combining these two
techniques together a better control system can be achieved.
The majority of the research work on fuzzy PID
controllers focuses on the conventional two-input PI or PD
type controller proposed by Mamdani [31, 33, 36].
However, fuzzy PID controller design is still a complex
task due to the involvement of a large number of
parameters in defining the fuzzy rule base. Wang and
Kwok [139] have done the analysis and synthesis of an
intelligent control system based on fuzzy logic and the PID
principle. They propose the combination of fuzzy PD and
fuzzy I controllers. Li and Gatland [59] introduce a simple
fuzzy three-term controller with a small modification in a
fuzzy PI controller and normal two dimensional rule base
is used. Ketata et al. [144] introduce a new design concept
of fuzzy controllers. They studied the design and properties
of structures such as fuzzy control, commutation between
fuzzy and PID controllers and fuzzy supervision of a PID
controller. After having described two fuzzy controller
implementations (fuzzy PD and fuzzy PI controllers), the
comparison with a PID algorithm is a base for the design
of the parallel PID-fuzzy controller combination. The
proposed fuzzy supervisor leads to promising results
concerning the development of combined control structures.
Li and Gatland [57] propose a new methodology for
designing a fuzzy logic controller (e.g. Fuzzy PI). A phase
plane is used to bridge the gap between the time-response
and rule base. Further, they [58] introduce more systematic
analysis and design for the conventional fuzzy control. A
general robust rule base is proposed for fuzzy two-term
control, leaving the optimal tuning to the scaling gains,
which greatly reduces the difficulties of design and tuning.
Li [60] adds a new methodology for designing and tuning
the scaling gains of the conventional fuzzy logic controller
(FLC) based on its well-tuned linear counterpart. Mann et
al. [49] investigate different fuzzy PID controller structures,
including the Mamdani-type controller. By expressing the
fuzzy rules in different forms, each PID structure is
distinctly identified.
Huang and Yasunobu [174] propose a general practical
design method for fuzzy PID control from conventional
PID control. Based on the analysis of relationship between
conventional PID controller and fuzzy PID controller, they
propose a method on how to choose the type of fuzzy PID
controllers suitable for a plant. Li et al. [170] propose a
design of an enhanced hybrid fuzzy P+ID controller for a
mechanical manipulator. A function-based evaluation
approach is proposed by Hu et al. [10] for a systematic
study of fuzzy PID-like controllers. This approach is
applied for deriving process-independent design guidelines
from addressing two issues: simplicity and nonlinearity. To
examine the simplicity of fuzzy PID controllers, they
conclude that direct-action controllers exhibit simpler
design properties than gain-scheduling controllers. Further,
Michail et al. [135] introduce fuzzy PID control of a
nonlinear plant. Kumar et al. [167] evaluate the
performance of a fuzzy PI + fuzzy PD controller for a
liquid-flow process in real-time and find that a fuzzy
controller outperforms a classical controller.
The basic structure of a fuzzy logic controller is shown
in Fig. 3. Its fundamental components are fuzzification,
control rule base, inference mechanism and
defuzzzification.
Fig. 3 The structure of a fuzzy logic controller
The structure of a fuzzy PID controller is based upon
the classical PID controller as shown in Fig. 1. The output
of PID controller in an absolute form is expressed in Eq.
(1). Its discrete-time version is [49]
)()()()()( 0neTKTje
K
neKnu SDCS
n
j
I
C
CPID ∆++=
∑
=
τ
τ
(3)
Kumar et al: A Review on Classical and Fuzzy PID Controllers 173

Index n refers to time instant and TS sampling time. Further,
the output of PID controller in an incremental form may be
expressed as
)()()()()( 2neTKneT
K
neKnu SDCS
I
C
CPID ∆++∆=∆
τ
τ
(4)
where
)()1()( nununu
PIDPIDPID
∆+−=
(5)
Thus the basic structural elements of fuzzy PID controllers
are shown in Fig. 4.
Fig. 4 Fuzzy PID structure [49]
where error is e(n)=ySP(n)-y(n); error change Δe(n)=e(n)-
e(n-1); rate of error change Δ2e(n)= Δe(n)- Δe(n-1); and
sum-of-error
∑∑
=
=n
jjene 0)()(
with y(n) being the feedback
response signal, and ySP(n) the desired response or the
reference input at the nth sampling instant.
2.3 Brief Review of Analytical Formula-Based Fuzzy
PID Controllers
Another class of fuzzy PID controllers with analytical
formulas was proposed during the 1990s. Siler and Ying
[171] define a linear fuzzy PI controller with one input and
one output in terms of piecewise linear membership
function for fuzzification, control rules, and defuzzification
algorithm. Further, Ying et al. [65] prove analytically that
a simplest possible fuzzy controller is equivalent to a
proportional-integral controller when a linear
defuzzification algorithm is used or to a nonlinear
proportional-integral controller when a nonlinear
defuzzification algorithm is used. Buckley and Ying [78]
describe a fuzzy controller based on their general purpose
fuzzy expert system shell FLOPS. They take a decision
theoretic view in designing an optimal fuzzy controller.
Further, Chen and Ying [40] study the stability of
nonlinear fuzzy PI control systems. Malki et al. [52]
propose a new design method and stability analysis of a
fuzzy proportional-derivative control system. They have
derived the structure of fuzzy controllers, with simple
analytical formulas as the final results by considering two
fuzzy sets on each input variable and three fuzzy sets on
output variable in the fuzzification process, rule base with
four control rules, intersection T-norm, Lukasiewicz or T-
conorm, drastic product inference method, and center of
area (COA) defuzzification method. Further, Chen and
Malki [38] study the bounded-input and bounded-output
(BIBO) stability of a fuzzy PD controller using the small
gain theorem. Li et al. [55] conduct the performance
analysis of a fuzzy PD control system proposed by Malki
et al. [52]. Also, Lu et al. [81] have performed an
experiment, to evaluate the performance of a fuzzy PD
controller in real time.
Hsu et al. [172] propose a new fuzzy PD controller for
multi-link robot control and perform the stability analysis.
Chen et al. [45] study the fuzzy PI controller design and its
stability. Further, Chen and Ying [39] establish the BIBO
stability conditions for nonlinear fuzzy PI control systems
using the small gain theorem. The structure of nonlinear
fuzzy PI controller is similar to the structure of a fuzzy PD
controller as proposed by Malki et al. [52]. Chen has
reported that fuzzy logic based PID controllers have strong
capabilities of handling not only linear but also many
complex nonlinear, higher-order, time delayed, as well as
ill-defined systems [43].
Ying [66, 67] proposes the construction of nonlinear
variable gain controllers via the Takagi-Sugeno fuzzy
control and performs an analytical study on the structure,
stability and design of general Takagi-Sugeno fuzzy
control systems. Further, he [68] develops the theory and
application of a novel fuzzy PID controller using a
simplified Takagi-Sugeno rule scheme. Carvajal et al. [73]
introduce a three term fuzzy PID controller with three
dimensional rule base. The final version of the fuzzy PID
controller is a computationally efficient analytical scheme.
Lu et al. [80] propose the design of predictive fuzzy PID
control and perform simulation study. Ying [63]
investigates the analytical structure of TITO (two-input
two-output) Mamdani fuzzy PI/PD controllers with respect
to conventional PI/PD control and variable gain control.
Patel and Mohan [9] introduce an analytical structure
and analyze the simplest fuzzy PI controllers. The fuzzy PI
controllers employ two fuzzy numbers on the universe of
discourse (UOD) of each input variable, and three fuzzy
numbers on the UOD of an output variable. Analytical
structures of such controllers are derived using triangular
membership functions for fuzzification, different
combinations of T-norms and T-conorms, different
inference methods, and center of area (COA) method for
defuzzification. Moreover, sufficient conditions for BIBO
stability of fuzzy PI control systems are established using
the small gain theorem. Further, Mohan and Patel [19]
introduce an analytical structure and analyze the simplest
fuzzy PD controllers and perform stability analysis using
the small gain theorem.
Liu et al. [177] perform a study to control wing rock
using a fuzzy PD controller. Ding et al. [176] propose the
analytical structure and perform stability analysis of a
typical Takagi-Sugeno PI and PD controller. Further, Ying
[61, 62] proposes a general technique for deriving the
analytical structure of fuzzy controllers using arbitrary
trapezoidal input fuzzy sets and Zadeh’s AND operator and
then derives analytical input-output relationship for fuzzy
controllers using arbitrary input fuzzy sets and Zadeh’s
fuzzy AND operator. Alwadie et al. [1] study a practical
two-input two-output Takagi-Sugeno fuzzy controller. Haj-
Ali and Ying [2] study the input-output structural
relationship between fuzzy controllers using nonlinear
fuzzy input sets and PI or PD control. Further, Haj-Ali and
Ying [3] perform the simple analysis of fuzzy controllers
174 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS,VOL.16,NO.3,SEPTEMBER 2011

with nonlinear input fuzzy sets in relation to nonlinear PID
control with variable gains.
Mohan and Sinha [16] introduce an analytical structure
and analyze the stability of a simplest fuzzy PID controller.
Further, Mohan and Sinha [12] discuss the mathematical
models for the simplest fuzzy PID controllers which
employ two fuzzy sets for each of the three input variables
and four fuzzy sets for the output variable. Mathematical
models are derived via left and right trapezoidal
membership functions for each input, singleton or
triangular membership functions for output, algebraic
product triangular norm, different combinations of
triangular co-norms and inference methods, and center of
sums (COS) defuzzification method. For the structure
which is suitable for control, BIBO stability proof is
presented. Mohan and Sinha [15] in reference to the earlier
work [9] state that the analytical structure of the simplest
fuzzy PI controller, derived via algebraic product t-norm,
bounded sum t-conorm and Mamdani minimum inference,
is not suitable for control purpose. In [15] they show that
the above statement is incorrect, and the above analytical
structure is very much suitable for control. Moreover,
using the small gain theorem they establish sufficient
conditions for BIBO stability of feedback systems
containing the above controller as a subsystem.
Arya [143] proposes the analytical structure of a fuzzy
PD controller and performs the analysis of simplest fuzzy
PD controller with asymmetrical/symmetrical,
trapezoidal/triangular/singleton output membership
function. Mohan and Sinha [20] study the analytical
structures for fuzzy PID controllers. These fuzzy PID
controllers are derived by using triangular membership
functions for inputs, singletons, or triangular membership
functions for output, minimum triangular norm, maximum
or drastic sum triangular conorm, Mamdani minimum,
drastic or Larsen product inference, nonlinear control rules,
and center-of-sum defuzzification. It is shown that these
analytical structures are not suitable for control purpose.
They found that in this context, it is extremely important to
note that the analytical structures reported by Carvajal et al.
[73] are also not valid for control.
Further, Mohan and Sinha [17] introduce the analytical
structure for a fuzzy PID controller by employing two
fuzzy sets for each of the three input variables and four
fuzzy sets for the output variable. This structure is derived
via left and right trapezoidal membership functions for
inputs, trapezoidal membership functions for output,
algebraic product triangular norm, bounded sum triangular
co-norm, Mamdani minimum inference method, and center
of sums (COS) defuzzification method. Conditions for
BIBO stability are derived using the small gain theorem.
Mohan and Sinha [14] introduce a mathematical model of
the simplest fuzzy PID controller with asymmetric fuzzy
sets. Further, Mohan and Sinha [18] introduce
mathematical models of simplest fuzzy PI/PD controllers
with skewed input and output fuzzy sets. Also, Mohan and
Sinha [13] introduce mathematical models and BIBO
stability analysis of simplest fuzzy two-term controllers.
Kumar et al. [166] present the design, performance and
stability analysis of formula-based fuzzy PI (FPI)
controller. They use a large number of fuzzy sets for input
and output variables to obtain more formulae for corrective
action. The performance of FPI controller is evaluated for
control of outlet flow concentration of a nonlinear non-
thermic catalytic continuous stirred tank reactor (CSTR)
for setpoint tracking, disturbance rejection and noise
suppression. The performance of the proposed formula-
based FPI controller is considerably better than that of the
conventional FPI controller. Its computational time delay
is approximately 6µs which validates its use for very fast
process in real time. The analytical structure of simplest
FPI controller is presented next.
The discrete-time version of classical PI controller is
)()()()( neKnrKnu ICCPI
τ
+=∆
(6)
where r(n) is the rate of change of the error and ΔuPI(n)is
the incremental control output.
The analytical formula-based FPI controller is based on Eq.
(6). The error signal “(KC/τI)e(n)” and the rate of change of
the error signal “KCr(n)” are input to the FPI controller to
obtain the corresponding output “ΔuPI(n)”.
Fig. 5 Regions of FPI controller input IC values [41]
The structure of FPI controller is derived, with simple
analytical formulas as the final results by considering two
triangular fuzzy sets on each input variable and three
singleton fuzzy sets on output variable in the fuzzification
process, rule base with four control rules, intersection T-
norm, Lukasiewicz or T-conorm, drastic product inference
method, and center of area (COA) defuzzification method.
The value-ranges of two inputs, the error and the rate of
change of the error signal are decomposed into 20 adjacent
input combinations (IC) regions, as shown in Fig. 5. The
analytical formulas are obtained by the projection of
“(KC/τI)e(n)” and “KCr(n)” on different IC regions as in
[41]. The flowchart of FPI control operation is shown in
Fig. 6.
Kumar et al: A Review on Classical and Fuzzy PID Controllers 175

Fig. 6 Flowchart of FPI control operation [166]
2.4 Brief Review of Analytical Formula-Based Hybrid
Fuzzy PID Controllers
The hybrid architecture of fuzzy PID with analytical
formulas is reported in literature. Misir et al. [29] have
developed a fuzzy PID controller, which is a combination
of fuzzy PI and fuzzy D controller, with the same structure
as mentioned by Chen earlier [41, 44]. Here a derivative
function is performed on a process variable rather than
error signal. In the continuation of this, Tang et al. [111]
have developed an optimal fuzzy PID controller, having
the same structure as discussed earlier [29]. They
successfully implement it in to control a solar plant [110].
Chen [42] proposes a GA-optimized Fuzzy PD+I controller
for a nonlinear system. Sooraksa and Chen [137] develop a
fuzzy (PD+I)2 control scheme for both vibration
suppression and set-point tracking of a “shoulder-elbow-
like” single flexible link robot arm model with damping.
Simulations results show that fuzzy logic controller
perform very well.
Kim and Oh [77] have proposed another configuration
of fuzzy PID controllers (fuzzy PI + fuzzy ID) with the
analytical structure as discussed above [41, 44]. Sooraksa
et al. [138] perform a comparative study of a conventional
proportional-integral plus derivative controller versus a
fuzzy proportional-integral plus derivative controller for a
subsystem failure. Veeraiah et al. [126] have proposed a
fuzzy PI-PD controller and studied its performance.
Chatrattanawuth et al. [168] propose a hybrid fuzzy
controller with the structure suggested by Chen [41, 44].
Here integral function is performed on error signal while
proportional and derivative functions are performed on a
process/controlled variable. Li and Shen [140] propose a
new incremental fuzzy PD + fuzzy ID controller and
perform the comparative study with a conventional PID
controller.
Kumar and Mittal [165] have attempted a hybrid
structure of a fuzzy PID controller. It is a parallel formula-
based fuzzy P + fuzzy I + fuzzy D (FP + FI + FD)
controller. It is based upon the parallel architecture of a
conventional PID controller. It preserves the linear
structure of the corresponding conventional controller and
having simple analytical formulas. They evaluate the
performance of a formula-based FP + FI + FD controller
for some complex processes in simulation and real time
and find that its setpoint tracking and disturbance rejection
performance is much better than a classical PID controller.
Further, they [164] present the architecture, performance
assessment and stability analysis of a formula-based fuzzy
I – fuzzy P – fuzzy D (FI – FP – FD) controller. It is based
upon the architecture of a conventional I–P–D controller
with simple analytical formulas. The setpoint tracking and
disturbance rejection performance of formula-based FI –
FP – FD controller is evaluated for some complex
processes, such as, first- and second-order processes with
delay, inverse response processes with and without delay
and higher order processes in simulation. It is found that it
outperforms a classical PID controller. BIBO stability of
formula-based FI – FP – FD controller is performed using
the small gain theorem.
3. CONCLUSIONS
This paper has presented the review of the classical and
fuzzy PID controllers. Firstly, the history of the
development of classical PID controllers and their
anomalies are presented in the chronological order. Further,
the advancement of classical PID controllers using fuzzy
logic is presented. It has been observed that due to the
heuristic approach of fuzzy logic, it plays a significant role
in the enhancement of classical PID controllers. The
literature survey shows that fuzzy PID controllers perform
much better than classical PID controllers.
Analytical formula-based fuzzy PID controllers have
advantages due to their structure. Since the fuzzy control
law has analytical formulas, controller designers can
effectively implement these formula-based fuzzy PID
controllers in real-time systems, such as FPGA and
microcontroller, without any computational burden
because the computational delay is quite small. It strongly
validates its candidature for very fast processes, such as,
control of an electromagnetic shaker. Also, due to the self
tuning capabilities, these controllers are suitable for a non-
stationary process. It has been noted that due to the
advancement of fuzzy PID controllers, its acceptance is
rapidly increasing in various industrial applications.
176 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS,VOL.16,NO.3,SEPTEMBER 2011

For analytical formula-based fuzzy PID controllers, the
literature survey reveals that people have tried out the
triangular membership function due to its convenience in
calculating formulae for different input combination
regions. Hence, it is open to use another type of a
membership function in place of triangular membership
function. Also, fuzzy sets of input and output variables
should increase in order to have a better accuracy with
more corrective action formulae. Further, optimization
techniques, such as genetic algorithm and particle swan
optimization, may be used to perform optimal tuning of
controllers gain. We hope this survey will be useful to the
readers interested in classical PID controllers and its
enhancement using fuzzy logic.
ACKNOWLEDGEMENT
The authors would like to thank anonymous referees for
their kind encouragement and valuable suggestions to
improve the paper. Further, authors would like to thank
Netaji Subhas Institute of Technology (NSIT) for
providing excellent experimental facilities in the Advanced
Process Control Lab.
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180 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS,VOL.16,NO.3,SEPTEMBER 2011

Vineet Kumar received M.Sc. degree in
Physics with Electronics in 1991 from G. B.
Pant. Institute of Ag. & Technology, Pantnagar,
India and M. Tech. in Instrumentation from
Regional Engineering College, Kurukshetra,
India and currently pursuing Ph.D. degrees in
Intelligent Process Control from Delhi
University, Delhi, India.
He is an Assistant Professor of the Department of
Instrumentation and Control Engineering, Netaji Subhas Institute
of Technology (NSIT), Delhi University, Delhi, India since 2010.
He was a lecturer and Sr. Lecturer from 2000 to 2010 at NSIT.
He has also served industry more than 5 years. He has published
more than 25 papers in various international and national
conferences and journals.
His research interests include process dynamics and control,
intelligent control techniques and their applications, and
intelligent process control.
E-mail: vineetkumar27@gmail.com (Corresponding author)
B.C. Nakra is presently Professor of Eminence,
Mechanical and Automobile Engineering
Department at the I.T.M. University Gurgaon,
Haryana , India. He did his Ph.D from Imperial
College of Science and Technology, London,
and started his academic career at IIT Delhi.
He has also worked as Professor of Eminence
in Instrumentation & Control Engg. Division at N.S.I.T. Delhi.
He has been involved in teaching and research in Vibration
Engineering, System Dynamics, Instrumentation, Automatic
Controls, Mechatronics and Engineering Design for over four
decades.
A. P. Mittal received B.E. degree in
Electrical Engineering from M.M.M.
Engineering College Gorakhpur, India , M.
Tech. from University of Roorkee, India and
Ph. D. from IIT, Delhi, India.
At present he has worked as a Professor
& head of the division of Instrumentation
and Control Engineering, Netaji Subhas
Institute of Technology (NSIT), Delhi
University, Delhi, India since June, 2001. Earlier he was
working as Professor in the Electrical Engineering department at
CRSCE, Murthal, Haryana, India from July 1997 to June 2001.
Also, he served as Assitant Professor and Lecturer at REC,
Kurukshrtra and Hamirur, India. He has published more than 70
papers in various international and national journals and
conferences. He is a senior member of IEEE, USA and Fellow
of Institute of Engineers (India) etc.
His research interests include intelligent control, power
electronics, and electrical drives.
Kumar et al: A Review on Classical and Fuzzy PID Controllers 181