Information Sciences 123 (2000) 249270

www.elsevier.com/locate/ins

Fuzzy PID controller: Design, performance

evaluation, and stability analysis
James Carvajal a, Guanrong Chen b, Haluk Ogmen b,*

a
National Aeronautics and Space Administration, Johnson Space Center, Mail Code ER2,
Houston, TX 77058-3696, USA
b
Department of Electrical and Computer Engineering, University of Houston,
Houston, TX 77204-4793, USA
Received 8 November 1998; received in revised form 22 May 1999; accepted 29 October 1999

Abstract
This paper presents a design for a new fuzzy logic proportional-integral-derivative
(PID) controller. The main motivation for this design was to control some known
nonlinear systems, such as robotic manipulators, which violate the conventional as-
sumption of the linear PID controller. This controller is developed by rst describing the
discrete-time linear PID control law and then progressively deriving the steps necessary
to incorporate a fuzzy logic control mechanism into the modications of the PID
structure. The nal version of this new fuzzy PID controller is a computationally e-
cient analytic scheme suitable for implementation in a real-time closed-loop digital
control. Numerous computer simulations are included to demonstrate the eectiveness
of the controller for both linear and nonlinear systems. Finally, a brief analysis is
presented to prove that the controller has bounded-input/bounded-output (BIBO)
stability. 2000 Elsevier Science Inc. All rights reserved.

Keywords: Fuzzy control systems; PID controllers; Stability analysis

*
Corresponding author. Fax: +1-713-743-4444.
E-mail address: ogmen@uh.edu (H. Ogmen).

0020-0255/00/$ - see front matter 2000 Elsevier Science Inc. All rights reserved.
PII: S 0 0 2 0 - 0 2 5 5 ( 9 9 ) 0 0 1 2 7 - 9
250 J. Carvajal et al. / Information Sciences 123 (2000) 249270

1. Introduction

There are many systems, such as robotic manipulators, which have some
unique physical characteristics that are dicult to address mathematically. The
natural dynamics of a robotic manipulator are coupled nonlinear equations
subject to complicated friction and damping eects [4]. For example, the inertia
terms of a robot arm are functions of the conguration of each joint and are
coupled. The basic kinematic terms for revolute joints have coupled and
transcendental terms. The Coriolis and centrifugal acceleration terms produce
nonlinear velocity terms. Even the simplest Coulomb friction model is non-
linear and dicult to model mathematically. Moreover, the physical parame-
ters are time-varying as the components wear out or the sensors drift away
from their expected operating characteristics.
The main task of a controller is to nd a suitable set of commands that can
cause the system to smoothly reach the desired state with minimal deviations.
For many complex systems, the governing equations are coupled nonlinear
equations subject to various dampings. As such, the controlled system equa-
tions for the general case are complex, and, therefore, the controller must be
able to eectively incorporate nonlinear properties and unmodeled eects into
its basic design.
The most common industrial controllers are the proportional-integral-de-
rivative (PID) controllers [3]. They have well understood properties and mature
design methods. The classical PID control law provides the basis for the design
technique developed in this paper. Since the implementation of most modern
control systems is in a computer processor, the conventional PID control law
must be converted into a digital version in applications. The general continu-
ous-time PID controller has the expression
Z
_
utcmd KP et KI et dt KD et; 1:1

where et rt yt is the tracking error signal between the reference rt

and the controlled system output yt, and KP , KI , and KD are constant P, I, and
D control gains, respectively.
This is rst converted into the frequency domain to get
Es
U scmd KP Es KI KD sEs; 1:2
s
where the capital variable is used to indicate the corresponding Laplace
transform. This equation can then be converted into the discrete-frequency
domain with the variable z, using the bilinear transform

2 1 z1
s ; 1:3
T 1 z1
 J. Carvajal et al. / Information Sciences 123 (2000) 249270 251

_
where T > 0 is the sampling period, and the substitution of Ez for sEs, to
produce
T 1 z1 _
U zcmd KP Ez KI Ez KD Ez: 1:4
2 1 z1
Eliminating the denominator yields
T _
U zcmd 1 z1 KP Ez1 z1 KI 1 z1 Ez KD Ez
2
 1 z1 : 1:5
This equation can be converted back to the discrete-time domain using the
inverse z transformation to produce
T
unT cmd unT T cmd KP enT enT T KI enT
2
_
enT T KD enT
_
enT T ; 1:6
where n 0; 1; 2; . . . By rearranging terms, this equation can be expressed as
T T
unT cmd unT T cmd KP enT KI enT 2KI enT
2 2
T
KP enT T KI enT T
2
_
KD enT _
enT T : 1:7
Let
TKI
K~P KP ; 1:8
2

K~I KI T ; 1:9
so that Eq. (1.7) can be expressed as
unT cmd unT T cmd K~I enT K~P enT enT T
_
KD enT _
enT T : 1:10
This is the nal discrete controller equation to be used below. A special
fuzzy logic form of Eq. (1.10) will be developed later to yield a more powerful
and robust controller for various systems in which the plant is not actually
linear.
If a problem is not well understood and cannot be precisely described
mathematically, but has good general ``rules-of-thumb'' on how to control it, a
fuzzy logic controller often works well [9]. The design engineer rst determines
the membership functions and linguistic denitions to capture the desired
252 J. Carvajal et al. / Information Sciences 123 (2000) 249270

dynamics. Once these are determined, the implementation of the controller is

achieved by directly applying existing techniques. However, the determination
of these rules and denitions are not obvious for complex systems, but they are
critical to the performance of the controller.
Fuzzy logic controllers are intrinsically nonlinear, yet do allow direct insight
into their behavior [1]. Unlike other methods, such as neural networks, it is
usually easy to determine what action a fuzzy controller will take for a given
situation since fuzzy controllers generally have analytic structures. Fuzzy logic
is encoded in simple rules with a structure such as ``if this antecedent situation
is encountered, then take that consequence action''. The implication of the
relationship between the physical world and the fuzzy rules is dicult to
capture, but the nal action of the controller is easy to determine.
Fuzzy logic controllers are normally built with three distinct components, as
shown in Fig. 1 [5]. A fuzzy rule base is constructed in the linguistic form ``if x is
true then do y''. This rule base is built on general observations and knowledge of

Fig. 1. Structure of a typical fuzzy logic controller.

 J. Carvajal et al. / Information Sciences 123 (2000) 249270 253

the problem, and is usually straightforward to design. However, the other two
components are not as so. A fuzzication unit is used to transform the nu-
merical input signal into some fuzzy values, while a defuzzication step is used
to transform the nal fuzzy value into an output signal from the controller.
These two processes require heuristic rules and membership functions to encode
the desired system response characteristics and controller dynamics. It is not
obvious what these fuzzy transformations should be based upon only some
basic understanding of the physical system. This is a signicant problem in the
design of various fuzzy controllers, and is the basic justication for the reason of
using the well-known PID controller as the underlying structure for our new
design. This is to be further discussed in Section 2.
Diering from the existing fuzzy PI [11], PD [7], PI + D [8], PD + I [6], and
(PI + D)2 [10], herewith we develop a full-scale fuzzy PID controller of the same
type, with its control performance evaluation and stability analysis given al-
together in the paper.

2. Derivation of the nonlinear fuzzy PID controller

As stated previously, a fuzzy controller has fuzzication, rule base, and

defuzzication components. The rst problem is how to formulate the fuzz-
ication process using the common triangular and trapezoidal functions. Re-
call that the general PID controller equation is given by Eq. (1.10).
Let
eI K~I enT ; 2:1

eP K~P enT enT T ; 2:2

_
eD KD enT _
enT T 2:3
represent the tracking error, change in error, and change in error rate, re-
spectively. Eq. (1.10) is then written as
Ducmd eI eP eD ; 2:4
and the incremental control output is
Ducmd ucmd nT ucmd nT T : 2:5
The three error terms are comparable to the input signals shown in Fig. 1.
As presented in Fig. 2, the simplest input membership function used for eI ,
eP , and eD in the fuzzication process are the same, and have two straight lines
followed by constant hold when the values exceed some predetermined
threshold. The threshold parameter, L, is specied to dene the maximum and
minimum values for the fuzzication process. For example, for an input value
254 J. Carvajal et al. / Information Sciences 123 (2000) 249270

Fig. 2. The input membership functions.

greater than )L, the positive membership value function (designated by p)

starts at a small value, goes to a value of one half at 0, and nally goes to 1
when the input value is greater that L. The negative membership function
(designated by n) is the opposite.
The output membership functions are not as simple since there are more
possible outcomes. In our case, there are also two crossing straight lines fol-
lowed by constant holds, but there are two extra triangular membership
functions. These are shown in Fig. 3. As before, the parameter, L, denes the
minimum and maximum outputs, but there are two new output terms centered
at L=3, respectively.
Given these membership functions, it is now possible to present the infer-
ence composition rules. These can be expressed as follows:
R1 IF eP 2 eP  n & eI 2 eI  n & eD 2 eD  n Then Du 2 Du  n`
R2 IF eP 2 eP  n & eI 2 eI  n & eD 2 eD  p Then Du 2 Du  n

Fig. 3. The output membership functions.

 J. Carvajal et al. / Information Sciences 123 (2000) 249270 255

R3 IF eP 2 eP  p & eI 2 eI  n & eD 2 eD  n Then Du 2 Du  n

R4 IF eP 2 eP  p & eI 2 eI  n & eD 2 eD  p Then Du 2 Du  p
R5 IF eP 2 eP  n & eI 2 eI  p & eD 2 eD  n Then Du 2 Du  n
R6 IF eP 2 eP  n & eI 2 eI  p & eD 2 eD  p Then Du 2 Du  p
R7 IF eP 2 eP  p & eI 2 eI  p & eD 2 eD  n Then Du 2 Du  p
R8 IF eP 2 eP  p & eI 2 eI  p & eD 2 eD  p Then Du 2 Du  p`
where the & symbol represents the fuzzy ``and'' operation, and the 2 symbol
means ``is a member of'', and the  symbol represents the combination. Note
the ` symbol represents the ``large'' value in the corresponding direction of
either positive (p) or negative (n).
This notation can be interpreted as follows: if the antecedents memberships
having been combined with the ``and'' function are true to some degree, then
the consequences memberships must also be true to some degree. For example,
rule R1 can be stated as ``if the integral term is a member of the negative set,
and the proportional term is a member of the negative set, and the velocity
term is a member of the negative set, then the output is a member of the
negative large set''. The output from rule R2 is also a member of the negative
set, but not as large as rule R1, so its output will not be ``as true as'' rule R1.
For the defuzzication process, the most commonly used formula is the
center of gravity, or Sugeno method. This is expressed as
P
fmembershipinputi  outputi g
Ducmd i P ; 2:6
i fmembershipinputi g

where i is the number of rules. For our controller design, this formula reduces
to
lR1  n` lR2  n


lR7  p lR8  p`
Ducmd ;
lR1 lR2 lR3


lR6 lR7 lR8
2:7

where lR1 is the degree (membership value) from rule R1, and so on.
As there are three dierent input components in the control law corre-
sponding to the proportional, integral, and dierential terms, it is necessary to
view all the possible combinations as a cube with a limiting edge value of L.
Fig. 4 represents the three-dimensional cube with the boundaries drawn at the
value of L.
As shown in Fig. 5, the defuzzication rules will be constructed by dividing
the cube into 48 sectors with known characteristics (note that there are some
sectors on the back sides of the cube which cannot be seen from the front of the
gure). Fig. 6 shows a single sector of the fuzzy cube labeled Sector 1.
In Sector 1, the following boundaries can be seen:
0 6 eP 6 L; 2:8
256 J. Carvajal et al. / Information Sciences 123 (2000) 249270

Fig. 4. The fuzzy membership function domain cube.

Fig. 5. Sector denitions of the fuzzy membership cube.

0 6 eD 6 L; 2:9

0 6 eI 6 L: 2:10
Furthermore, the corresponding error relations are:
 J. Carvajal et al. / Information Sciences 123 (2000) 249270 257

Fig. 6. Sector 1 of the fuzzy membership cube.

eD 6 eP ; 2:11

eD 6 eI ; 2:12

eI 6 eP : 2:13
By using Fig. 2 and the mathematical formulas for the straight lines that
dene the memberships functions, the individual membership functions can be
expressed as
eP L
eP  n ; 2:14
2L
eI L
eI  n ; 2:15
2L
eD L
eD  n ; 2:16
2L
eP L
eP  p ; 2:17
2L
eI L
eI  p ; 2:18
2L
eD L
eD  p : 2:19
2L
258 J. Carvajal et al. / Information Sciences 123 (2000) 249270

By examining the regions dened by Sector 1 from Fig. 6, along with Fig. 2,
it is clear that all the negative terms are less than 1/2 and all the positive terms
are greater than 1/2. The positive membership relationships can now be ex-
pressed as
1=2 6 eD  p 6 eP  p; 2:20

1=2 6 eD  p 6 eI  p; 2:21

1=2 6 eI  p 6 eP  p: 2:22
Furthermore, by observing that a larger negative number subtracted from L is
less than a smaller negative number subtracted from L, the negative relation-
ships can be stated as
1=2 P eD  n P eP  n; 2:23

1=2 P eD  n P eI  n; 2:24

1=2 P eI  n P eP  n: 2:25
It is now possible to evaluate Eq. (2.7) for a control output. By using the
minimum function for the fuzzy ``and'' (t-norm) operation, the result from
each rule is expressed in Table 1.
By summing the equations in the third column of Table 1 for the denomi-
nator and the fth column for the numerator, the incremental control formula
for Sector 1 is obtained as
L4eP 2eD
Ducmd : 2:26
38L 4eP 2eI
As the maximum value that the eI , eP or eD term can have is L, the limiting
value for this control action is L. This is what would be expected from the
fuzzication functions.

Table 1
The fuzzy PID control values for Sector 1
Rule Minimum Membership Output Numerator
of rule equation
(R1) eP  n eP =2 L=2 L eP L=2 L2 =2
(R2) eP  n eP =2 L=2 L=3 eP L=6 L2 =6
(R3) eI  n eI =2 L=2 L=3 eI L=6 L2 =6
(R4) eI  n eI =2 L=2 L=3 eI L=6 L2 =6
(R5) eP  n eP =2 L=2 L=3 eP L=6 L2 =6
(R6) eP  n eP =2 L=2 L=3 eP L=6 L2 =6
(R7) eD  n eD =2 L=2 L=3 eD L=6 L2 =6
(R8) eD  p eD =2 L=2 L eD L=2 L2 =2
 J. Carvajal et al. / Information Sciences 123 (2000) 249270 259

Since this case was for Sector 1, the procedure needs to be repeated for all of
the cases similar to it. Altogether, there are eight cases out of 48 where the eP is
greater than either of the other two terms. Although similar, they are not quite
the same. Four more sectors are necessary to be discussed in detail, and the rest
can be omitted as they produce the same results as the others.
The next region to be analyzed is Sector 2. As seen in Fig. 7, the boundaries
are dened as
0 6 eP 6 L; 2:27

0 6 eD 6 L; 2:28

0 6 eI 6 L: 2:29
For Sector 2, the membership relationships are
0 6 eP  n 6 eD  n 6 eI  n 6 1=2; 2:30

1=2 6 eI  p 6 eD  p 6 eP  p 6 1: 2:31

By using these relationships, the result from each rule is expressed in Table 2.
By summing the formulas in the third column of Table 2 for the denomi-
nator and the fth column for the numerator, the control formula for Sector 2
is obtained as
L4eP 2eI
Ducmd : 2:32
38L 4eP 2eD

Fig. 7. Sector 2 of the fuzzy membership cube.

260 J. Carvajal et al. / Information Sciences 123 (2000) 249270

Table 2
The fuzzy PID control values for Sector 2
Rule Minimum Membership Output Numerator
of rule equation
(R1) eP  n eP =2 L=2 L eP L=2 L2 =2
(R2) eP  n eP =2 L=2 L=3 eP L=6 L2 =6
(R3) eD  n eD =2 L=2 L=3 eD L=6 L2 =6
(R4) eI  n eI =2 L=2 L=3 eI L=6 L2 =6
(R5) eP  n eP =2 L=2 L=3 eP L=6 L2 =6
(R6) eP  n eP =2 L=2 L=3 eP L=6 L2 =6
(R7) eD  n eD =2 L=2 L=3 eD L=6 L2 =6
(R8) eI  p eI =2 L=2 L eI L=2 L2 =2

The procedure will now be repeated for Sector 3. In this sector, the
boundaries and the membership relationships are dened as
0 6 eP 6 L; 2:33

0 6 eD 6 L; 2:34

L 6 eI 6 0; 2:35

0 6 eP  n 6 eD  n 6 eI  p 6 1=2; 2:36

1=2 6 eI  n 6 eD  p 6 eP  p 6 1: 2:37

By using these relationships, the result from each rule in Sector 3 is expressed in
Table 3.
For this sector, incremental control formula is obtained as
L4eP 2eI
Ducmd : 2:38
38L 4eP 2eD

Note that this is the same as Eq. (2.26) and a pattern is beginning to develop.
The procedure will now be repeated for Sector 4. In this sector, the
boundaries and membership relationships are dened as
0 6 eP 6 L; 2:39

0 6 eD 6 L; 2:40

L 6 eI 6 0; 2:41

0 6 eP  n 6 eI  p 6 eD  n 6 1=2; 2:42

1=2 6 eD  p 6 eI  n 6 eP  p 6 1: 2:43
 J. Carvajal et al. / Information Sciences 123 (2000) 249270 261

Table 3
The fuzzy PID control values for Sector 3
Rule Minimum Membership Output Numerator
of rule equation
(R1) eP  n eP =2 L=2 L eP L=2 L2 =2
(R2) eP  n eP =2 L=2 L=3 eP L=6 L2 =6
(R3) eD  n eD =2 L=2 L=3 eD L=6 L2 =6
(R4) eI  n eI =2 L=2 L=3 eI L=6 L2 =6
(R5) eP  n eP =2 L=2 L=3 eP L=6 L2 =6
(R6) eP  n eP =2 L=2 L=3 eP L=6 L2 =6
(R7) eD  n eD =2 L=2 L=3 eD L=6 L2 =6
(R8) eI  p eI =2 L=2 L eI L=2 L2 =2

By using these relationships, the result from each rule in Sector 4 is expressed in
Table 4.
For this case, the control formula is obtained as
L4eP 4eI 2eI
Ducmd : 2:44
38L 4eP 2eI

This equation is quite dierent than the previous ones.

The procedure will now be repeated for the other sectors. For Sector 5, the
incremental control formula is
L4eP 4eI 2eD
Ducmd ; 2:45
38L 4eP 2eI

for Sectors 6 and 7 is

L4eP 4eD 2eI
Ducmd ; 2:46
38L 4eP 2eD

Table 4
The fuzzy PID control values for Sector 4
Rule Minimum Membership Output Numerator
of rule equation
(R1) eP  n eP =2 L=2 L eP L=2 L2 =2
(R2) eP  n eP =2 L=2 L=3 eP L=6 L2 =6
(R3) eD  n eD =2 L=2 L=3 eD L=6 L2 =6
(R4) eD  p eD =2 L=2 L=3 eD L=6 L2 =6
(R5) eP  n eP =2 L=2 L=3 eP L=6 L2 =6
(R6) eP  n eP =2 L=2 L=3 eP L=6 L2 =6
(R7) eP  p eI =2 L=2 L=3 eI L=6 L2 =6
(R8) eI  p eI =2 L=2 L eI L=2 L2 =2
262 J. Carvajal et al. / Information Sciences 123 (2000) 249270

and for Sector 8 is

L4eP eD
Ducmd : 2:47
38L 4eP 2eI
As it turns out, the general formula for all sectors can be expressed in one of
two possible formulas, which are functions of the maximum, minimum, and
median of the magnitudes. If the maximum and the median have the same sign,
the incremental control formula is
L4MaxeP ; eI ; eD 2MineP ; eI ; eD
Ducmd ; 2:48
38L 4j MaxeP ; eI ; eD j 2j MedeP ; eI ; eD j
and if they have dierent signs,
L4MaxeP ; eI ; eD 4MedeP ; eI ; eD 2MineP ; eI ; eD
Ducmd :
38L 4j MaxeP ; eI ; eD j 2j MedeP ; eI ; eD j
2:49
Note that the Max, Min, and Med functions used here only examine the
magnitude of the input values to determine the functions value, but they do
retain the sign information. For instance, for the vector of 5; 4; 2, the Max is
5, the Min is 2, and the Med is 4. Also, the ranges of magnitudes of the eI , eP ,
or eD terms are limited to L to ensure that their maximum magnitudes are
constrained. This is not the standard mathematical denition of the three
functions.

3. Computer simulation results

This new fuzzy PID controller is now examined for its ability to control
linear and nonlinear plants, and to evaluate its performance in comparison
with the corresponding conventional PID controller tuned by trial and error.
The rst system to be tested is a third-order linear system with a transfer
function
s1
H s : 3:1
s3 9s2 26s 24
This system can be converted into a statespace representation as
2 3 2 3
0 1 0 0
x_ 4 0 0 1 5x 4 0 5u; 3:2
24 26 9 1

y 1 1 0 x 3:3
and simulated using the MATLAB language.
 J. Carvajal et al. / Information Sciences 123 (2000) 249270 263

To convert the continuous gains to discrete ones, the formulas

KP KPc KIc
T =2; 3:4

KI KIc
T ; 3:5

KD KDc =T 3:6
were employed to account for changes in the sampling time.
For this system, our best choice of continuous PID controller gains are 0.78
for the proportional term, 120 for the integral term, and 0.0002 for the de-
rivative term. The sampling time is 0.01 s and the desired setpoint value is 4:0.
Next, the fuzzy PID system is simulated using the same PID gains with a single
fuzzy controller gain of 4.5 and the threshold parameter L of 780 found by trial
and error. The rst plot shown in Fig. 8 presents the results from these two
simulations. As expected, both controllers produce excellent trajectories. To
investigate the robustness of the two controllers, the cases were redone with the
value for the integral gain is reduced by one tenth. This represents an imple-
mentation error in the hardware (Fig. 9). The fuzzy controller reached the
desired setpoint an order of magnitude faster than the linear controller, im-
plying that the fuzzy controller is more robust in terms of hitting the setpoint in
a reasonable amount of time.
Since the PID controller is known to perform well for regular lower-order
linear systems, an unstable third-order nonminimum phase system with a
transfer function of

Fig. 8. Simulation of a third-order system (demonstrates tracking).

264 J. Carvajal et al. / Information Sciences 123 (2000) 249270

Fig. 9. Simulation of a third-order system (demonstrates robustness).

s2 s 2
H s 3:7
s3 3s2 10s 24
was then examined. Simulated as before, our best choice of gains are 10.5 for
the proportional term, 20 000 for the integral term, and 0.0005 for the der-

Fig. 10. Simulation of a non-minimum phase system.

 J. Carvajal et al. / Information Sciences 123 (2000) 249270 265

rivative term. Since this is more sensitive to time changes, the time step is set at
T 0:001 s. The desired setpoint value is 10.0 and the threshold parameter L is
780. Fig. 10 presents the output from this simulation. As seen, the fuzzy PID
controller did produce a good trajectory. On the contrary, the conventional
linear controller could not produce reasonable results and is not presented in
the example.
Next, three nonlinear systems are simulated using trapezoidal integration.
These cases progress from simple functions to more complex ones, and the time
step is xed to be 0.1 s. For all the cases, the fuzzy PID controller does produce
good trajectories but no set of grains were found for the conventional PID
controllers which could track the setpoint, hence, no results are presented here.
The rst, and very simple nonlinear system is
_ 0:0001j yt j ut:
yt 3:8
For this equation, the desired setpoint value is 5:0 and the threshold pa-
rameter L 10. The best controller gains found by trial and error tuning are
0.7 for the proportional term, 1.3 for the integral term, and 0.01 for the der-
ivation term. Fig. 11 presents the output from this simulation.
Next, a nonlinear system is
p
_ yt j yt j ut:
yt 3:9
The desired setpoint value is 6.0 and the threshold parameter L 350. Using
trial and error tuning, the best controller gains for this simulation are 2 for the

Fig. 11. Simulation of absolute value non-linearity.

266 J. Carvajal et al. / Information Sciences 123 (2000) 249270

Fig. 12. Simulation of square-root non-linearity.

proportional term, 5 for the integral term, and 0.0002 for the derivative term.
Fig. 12 presents the output from this simulation. Note how this system con-
verges to the set point in very few iterations, and it was quite easy to nd gain
combinations that worked to produce an acceptable response. For this ex-
ample, the system was surprisingly insensitive to a range of gain combinations.
The last and most complex nonlinear case investigated is
p
_ yt sin2
yt j yt j ut: 3:10

The desired setpoint value is 4.0 and the threshold parameter L 500. The best
choice of controller gains for this simulation are 1.8 for the proportional term,
1.8 for the integral term, and 0.008 for the derivative term. Fig. 13 presents the
output from this simulation. Unlike the previous cases, nding a set of gains
that worked for this case was not easy, and the fuzzy PID system required
careful tuning to get the solution presented here.

4. Stability analysis

The BIBO stability of various types of fuzzy PID controllers is analysis by

numerous authors [3,7,8,12]. This analysis is based upon the response of the
control system to a bounded input and uses the small gain theorem to ensure
the output is also bounded.
 J. Carvajal et al. / Information Sciences 123 (2000) 249270 267

Fig. 13. Simulation of complex non-linearity.

Consider the nonlinear feedback system shown in Fig. 14. This system can
be expressed as
e1 u1 S2 e2 ; 4:1

e2 u2 S1 e1 ; 4:2
where the error terms are bounded, admissible, causal functions. This generally
requires that the innite integral of the function raised to some power is nite.
Suppose there exist constants L1 , L2 , M1 , and M2 , such that
kS1 e1 k 6 M1 L1 ke1 k; 4:3

kS2 e2 k 6 M2 L2 ke2 k; 4:4

R1 p 1=p
where kf k denotes either the function 0 j f t jdt with 1 6 p < 1, or the
function is the superium for p 1. The small gain theorem states that if the
product of L1 L2 is less than 1, then the following error bounds are true:
ke1 k 6 1 L1 L2 1 ku1 k L2 ku2 k M2 L2 M1 ; 4:5

Fig. 14. Non-linear feedback control system.

268 J. Carvajal et al. / Information Sciences 123 (2000) 249270

ke2 k 6 1 L1 L2 1 ku2 k L1 ku1 k M1 L1 M2 ; 4:6

so that a bounded input yields a bounded output.

Let the plant be dened as N. The fuzzy PID control system equations can
be expressed in the notation used previously as
e1 nT enT ; 4:7

e2 nT unT ; 4:8

u1 nT rnT ; 4:9

u2 unT T ; 4:10

S1 e1 nT KPID DunT ; 4:11

S2 e2 N e2 nT : 4:12

The controller equations are examined next. First, by examining the equa-
tions for Sector 1, with the condition of 0 6 eD 6 eI 6 eP 6 L, Eq. (4.11) can be
written as

L4eP 2eD

kS1 e1 nT k 6 KPID ; 4:13
38L 4e 2e
P I

and Eq. (4.12) as

kS2 e2 k kN kje2 nT j: 4:14

Next, let the values Me, Mr, and Ma be dened as the superium of the absolute
value of the error, the error rate, and the error acceleration signals. Namely,
Me sup j enT j; 4:15
nP0

Mr sup j enT enT T j; 4:16

nP0

_
Ma sup j enT _
enT T j: 4:17
nP0

These new values can be applied to the initial denitions of the controller terms
to produce the bounds
j eI j 6 K~I Me 6 K~I L; 4:18

j eP j 6 K~P Mr 6 2K~P L; 4:19

j eD j 6 KD Ma 6 2KD L; 4:20
 J. Carvajal et al. / Information Sciences 123 (2000) 249270 269

where the conditions from Eqs. (2.27)(2.29) have been used. Next, Eq. (4.9) is
rearranged to produce
 
 KPID L4K~P 
 
kS1 e1 nT k 6  
 38L 8K~P L 4K~I L 
 
4KD
 je1 nT j je1 nT T j L : 4:21
4K~P

Note that this equation can be written in a form similar to Eq. (4.3) with the
rst term dened as
 
 KPID K~P 
 
L1  ; 4:22
 62L 2K~P K~I 

and the rest of the constant terms grouped into M1 . Eq. (4.10) can also be
written as
L2 kN k: 4:23

By applying the small gain theorem, the stability condition for Sector 1 can
be found as
 
 KPID K~P 
 
 kN k < 1: 4:24
 62L 2K~P K~I 

This process is repeated for the other sectors to yield the combined condition
set as
 
 KPID K~P 
 
 kN k < 1; 4:25
 62L 2K~P minK~I ; KD 

 
 KPID K~I 
 
 kN k < 1; 4:26
 62L 2K~I minK~P ; KD 

 
 KPID KD 
 
 kN k < 1: 4:27
 62L 2KD minK~I ; K~P 

The value of the plant output kN k must also be bounded, so the given non-
linear system has a nite gain. Conditions (4.25)(4.27) together provide the
BIBO stability criteria for the fuzzy PID controller design for a given bounded
system.
270 J. Carvajal et al. / Information Sciences 123 (2000) 249270

5. Conclusions

The fuzzy PID controller derived in this paper successfully demonstrated

better performance than the conventional PID controller for many cases,
particularly for nonlinear plants. The fuzzy PID controller is also able to
tolerate many poor selections or inadequate implementations of the controller
gains which would make most conventional controllers unstable. Since non-
linear eects will be encountered in many complex systems, such as robotic
manipulators, the ability of the fuzzy PID controller to tolerate these
unmodeled nonlinear and gain-value variation factors is a substantial im-
provement over conventional linear PID controllers in real-world applications.

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