M. Santos*, S. Dormido**, J. M. de la Cruz*
*Dpto. de Informática y Automática. Facultad de Físicas. (UCM)
**Dpto. de Informática y Automática. Facultad de Ciencias. (UNED)
Ciudad Universitaria s/n. 28040-MADRID (Spain). FAX: (34)-1-3944687 e-mail: msantos@eucmax.sim.ucm.es
The synthesis of a control system includes both the controller selection and the adjustment of its parameters.
In some cases, the type of controller might be more complex or more general, like PID instead PI or PD, to improve the control system performance. In all cases, the tuning problem must be satisfactorily solved. On the other hand, Fuzzy Control has made possible the establishment of intelligent control. However, Fuzzy Logic Controllers
(FLC) are only used in simple configurations and their analytic knowledge is still poor. In this paper, a quantitative and qualitative study of fuzzy controllers is done for the most complete case of a Fuzzy-PID. The
FLC-PID analytic performance is summarized in terms of its three input variables, which allows us to obtain initial values for the FLC-PID scale factors in terms of the classical PID parameters. This initial tuning has been tested for several systems and a qualitative tuning has also been established. The advantages of the derivative term are also examined.
The need for simple advanced control alternatives especially arises in the Control Process area, where most of the real processes are generally complex and difficult to model [1]. The application of Fuzzy Logic to a wide range of control applications has made possible the establishment of intelligent control in these areas [4], [5].
Its appeal, from the Process Control Theory point of view, lies in the fact that this technique provides a good support for translating the heuristic knowledge of the skilled operator, expressed in linguistic terms, into computer algorithms. Fuzzy Control solves real problems, previously not tackled due to their complexity or to lack of information [9].
However, Fuzzy Logic Controllers (FLC) are usually applied with poor analytic knowledge of their behavior and only in simple configurations. In fact, they normally perform like PI or PD.
FLC-PI controllers are quite simple, though they are the most widely used in practice and provide similar results to conventional controllers. But in some applications it may be useful to employ more general controllers, which make it easier to reach the system specifications and improve their performance, though they can be also more difficult to tune.
The complete study of fuzzy controllers should involve all the terms of conventional controllers. The third control action must be included so as to consider the FLC-PID case. Though the derivative term is not commonly included -neither in the conventional case-, this allows us to complete the development of Fuzzy controllers in a similar way that of the classical ones. It also makes it possible to obtain certain conclusions about their stability and specifications.
But the main problem in the synthesis of a control system is not only the selection of a specific controller but also the adjustment of its parameters, to verify certain given specifications for the controlled process.
In this paper an analytic study of the FLC-PID is carried out in section 2, which allows us to establish an equivalence between the FLC-PID and a conventional
PID; thus a tuning method is proposed for these fuzzy controllers and is then evaluated. The qualitative analysis is done in section 3. The behavior of these controllers is compared with the FLC-PI type in section 4 and some general conclusions are summarized in the last section.
The aim of a controller is to reach or maintain a process in a specific state, by monitoring a set of variables and selecting the adequate control actions.
The Fuzzy-PID controller performs like its classical homonym, but both the input variables and the control action are given in linguistic terms. The analytic development of fuzzy controllers allows us to explain the influence of each tuning parameter on the system

response, as well as to compare them to the conventional one to obtain general results.
The incorporation of the derivative term provides a new control action to the controller. In the fuzzy case, increasing the number of input variables causes a rise in the dimension of the rule table and, therefore, in the complexity of the system; this makes its implementation more complicated and can make difficult its analytic study. For this reason, a PI is usually employed instead of a PID in most applications of fuzzy control.
Figure 1 shows the basic diagram of an incremental
Fuzzy-PID controller, where error e -the difference between the process output y, and the reference signal r -, the error change ce, and the second error derivative ac, are the input FLC variables, and the increment of the control action
∆ u is the FLC output. The parameters chosen to tune the FLC are the scale factors GE , GR , GA and GU , gains which weight the input and output variables respectively. r
-
+ e y
GE d/dt ce
GR d/dt
2 ac
GA
Fuzzy
Inference
∆ u
GU du
Figure 1. Fuzzy-PID controller and tuning parameters
The set of rules which describes the FLC-PID behavior have three antecedents and one consequent:
Ri: if GE .
e is Ei and GR.ce
is CEj and GA.ac
is Ak then
∆ u is Ui where GE .
e is the linguistic variable error ( e ) weighted by its gain GE , GR .
ce is the scaled change of the error and
GA .
ac is the weighted second derivative of the error. The conclusion is equal to the increment control action
∆ u . In order to simplify this analysis, two primary fuzzy sets are assigned to each one of the three input variables, corresponding to the labels P (positive) and N (negative); therefore, there are up to 8 control rules. The control output has three labels adding the linguistic term Z (zero).
The subscript could represent whichever one of the labels associated to each variable (Ep, En, Uz, etc.). The rules are shown in Table 1.
GE .
e(t) Ep En
GA .
ac(t) Ap An Ap An
GR .
ce(t) CEp Up Up Uz Un
Table 1. Control rules of the FLC-PID controller
The membership functions are defined in trapezoidal form but are symmetrical from their center, L . Therefore, the control action can be approximated by linear piece functions [8]. The final control action can be calculated by different defuzzification algorithms; two of them are used here: linear (L) and non-linear (Center of Area COA), which are given by the following expressions, where Uk is the conclusion value corresponding to each control label
(P, Z, N), Nu the number of control terms, pk the center of the membership associated to the control label k (
±
Lu or
0), and Ei, CEj, Ak are the membership value of the input variables.
### u
L
= Uk. pk k
Nu ∑
=
1
### u
COA
k
Nu ∑
=
1
Uk
= (Up - Un). Lu
Nu k
∑
= 1
Uk. pk
(Up
−
Un). Lu
(Up
+
Uz
+
Un)
(2)
Up =
(1)
or (min(Ep,CEp,Ap),min(Ep,CEp,An),min(Ep,CEn,Ap))
Un = or (min(En,CEp,An),min(En,CEn,Ap),min(En,CEn,An))
Uz = or (min (Ep, CEn, An), min (En, CEp, Ap))
After each one of the 8 rules has been evaluated applying the connective and as the minimum, and the
Lukasiewicz or , six different conclusions can be obtained for each rule. Therefore, there will be 48 zones of linear control. These zones are given by the different relations between the sign and the absolute value of each input. The controller output for these zones is given in terms of the input variables, according to the defuzzification algorithm employed, replacing the expression for the membership functions in each equation. The control output
∆ u(t) by any defuzzification method, with Ke, Kc and Ka being some equivalent coefficients obtained for each zone (see
Appendix ), is:
∆ u(t) = [Ke.
GE .
e(t) +Kc.
GR .
ce(t) +Ka.
GA .
ac(t) ] (3)
An in-depth study of these functions has allowed us to establish an equivalence in each linear control zone between the parameters of a conventional PID controller
(Ki, Kp, Kd), which also weight the variables e , ce and ac , and the FLC-PID output coefficients [6].

This analytic development depends on the starting configuration of the rule table since there are other sets of rules which also correctly describe the behavior of the
FLC, producing a smooth action or a stronger control
Although the equivalent parameters of the FLC are different for each one of the zones where the control is linear, their variation range is bounded, and it is possible to establish the limits of this variation both in terms of the
PID parameters or the FLC-PID parameters as shown in the following expressions [3], [7],:
Linear Defuzzification Method : The range of variation of the gains of the FLC is:
2 Ki /3 ### GE.GU
### 2 Ki
0 ### GR.GU
### 2 Kp
0 ### GA.GU
### 2 Kd
Based on the analysis of the output behavior in each different control zone, the initial FLC gains can be approximated by setting them to the following values:
GR = 2 Kp / GU
GE = Ki / GU
GA = 2 Kd / GU (4)
Non-Linear Defuzzification Method: The range of variation of the gains of the FLC is:
8 Ki /3 ### GE.GU
### 8 Ki
0 ### GR.GU
### 8 Kp
0 ### GA.GU
### 8 Kd
Therefore, a good approach to the initial parameters of the FLC-PID controller is:
GR = 4 Kp / GU
GE = 5.3
Ki / GU
GA = 8 Kd / GU (5)
Even though this approach has been developed for this specific FLC-PID controller, in the following section, we will prove that they are also valid for other different controllers and processes.
One of the main problems that arises in this type of regulator is the lack of systematic procedures for tuning.
The selection of the initial parameters of the FLC is usually carried out by trial and error, as we can read in the literature about the tuning of Fuzzy-PI or Fuzzy-PD controllers [2]. This is a tedious and time-consuming task, which makes it difficult to establish general results and notably increases the design time.
Therefore, if we have some starting values calculated by any systematic procedure, it makes it easier to analyze the FLC behavior, although those parameters may not be the best ones.
Performance index: The validity of these FLC-PID tuning parameters will be determined by inspecting the system response in the temporal domain. Both, a qualitative analysis of the output, and the evaluation of following performance indexes [5
]
will be considered:
- I1: quadratic error I1 = ∫
0 t e t
2 dt
- I2: normalized overshoot I2 = r
- I3: rise time
- I4: settling time
I3 = min t / y(t) = 90% r
I4=min t/ y(t)
[95% r , 105% r ]
These initial parameters have been tested for several systems. First of all, a typical 4º order system with monotonous response has been used [6]. It reflects the behavior of most industrial systems. After its process model has been estimated, the PID parameters (Kp, Ti and
Td) are calculated by any classical tuning technique in order to control the process with certain specifications; in this case, with the Ziegler-Nichols method, they are:
Kp = 1.6541; Ti = 3.7; Td = 0.925
With these parameters, the FLC-PID tuning parameters are calculated by equations (4) or (5), according to the defuzzification method (linear or non linear). These values and the response characteristics are shown in
Table 2, with their index values.
Contro ller
Gains Kp = 1.6541
Ti = 3.7
Td = 0.925 defuzzification
GE = 0.237
GR = 0.661
GA = 0.612 defuzzification
GE = 0.0447
GR = 0.3308
GA = 0.3060
I1
I2
22.0533 20.9341 23.1875
0.1227 0.4434 0.1916
( tp = 5.7) ( tp = 4.4) ( tp = 8)
I3 (sec) 4 2.9 4.8
I4 (sec) 7.4 - 10.6
Table 2. Response characteristics with a conventional PID and with a FLC-PID (* tp: pick time).
In order to verify the validity of the initial parameter formulas, the index values have been obtained for different systems. Table 3 shows simulated results for three plants.

System
Gains
1
( s
+
1 )
3
GE = 0.252
GR = 0.0875
GA = 0.42
1
( s
+
1 )( s
+
2 )
GE
GR
GA
= 0.3723
= 0.6
= 0.131 e
− s
+
1 s
GE = 0.3720
GR = 0.5454
GA = 0.10
I1
I2
14.1656 5.9647 12.255
0.2466
( tp = 4.9)
0.3840
( tp = 2)
1.5484
( tp = 0.9)
I3 (sec.) 3 1.2
I4 (sec.) 9.7 5 4.8
Table 3. Evaluation of initial tuning parameters for different systems
Although these gains depend on the estimated model, we are not looking for the most accurate parameters but some starting values from which one can obtain the specifications in few steps by qualitative tuning.
The FLC structure described in Figure 1 is the starting point for analyzing the qualitative behavior of the
FLC-PID controller. The influence of each gain (input and output scale factors) is considered for several systems.
This study allows us to produce some results that may be used as a guide in the adjustment of fuzzy controller parameters.
The fuzzy system consists of a plant whose transfer function is known or for which a good model exists, and a fuzzy incremental PID controller. The models represent the more usual dynamics of industrial processes.
In order to study the variations of the response with the
FLC scale factors, some initial values must be assigned to these parameters in a previous phase. They are then varied and the behavior of the system is observed. y(t) = F[ GE , GR , GA , GU , e(t ), u(t) ]
The controller has become more general. The number of primary terms of the variables has been increased to 3 labels for input and 7 labels for output: PG, PM, PP, Z,
NP, NM, NG where G, M and P are the modifiers of Big,
Medium and Small. The characterization of the membership functions is defined by trapezoidal functions, that have 0.5 degree of completeness over their corresponding universes.
Although this makes the analytic study quite complicated, the qualitative behavior can be analyzed in order to improve the performance when the application requires a more complex controller. The validity of the initial parameters is also proved for new cases, previously not contemplated in the mathematical study.
The rule base consists of 27 rules of three antecedents and one consequent that describes the behavior of the controller. As the number of rules has been increased, the difference between the results obtained depending on the different interpretation for the connectives becomes more evident, since they are several rules which generate the same output. This difference is accentuated with the non-linear defuzzification method. The next simulations are computed with the Lukasiewicz or function for greater simplicity.
The general effects on the index response of varying the scale factors or gains, which weigh both the input and output variables, can be summarized as follows
(k = constant, ### : increment):
GE
###
GR
k
GA GU Effects on the response
k k Faster, more oscillatory.
Improves the transient reducing the stationary error and rise time; increases the risk of instability k ### k k k k ### k
Faster; may reduce the overshoot in a narrow range of values; increases the quadratic error.
Makes the performance more sensitive around the set-point
Low dependence of the overshoot and quadratic error. Increases the rise time,reduces the settling time k k k ### Faster rise time, shorter integral squared error; increases the overshoot and settling time. The most destabilizing, significantly influences convergence.
We are going to show the specific influence of the gain of the derivative term on the system response for a given plant. This also makes it possible to prove the formulas of the initial parameters. The initial value of GA is 0.0306
(by (4)). As we can see (Figure 2), it is very close to the optimum value, according to the next graphs which show the behavior of that system when this gain changes within an interval between 0 and 3.

Fig. 2.1. Quadratic error Fig. 2.2. Overshoot x
10
-1 x 10
-1
Response (GA = 0 - 0.3)
GA =0.05
GA =0.1
r
GA =0
GA =0.3
GA =0.2
time.10
Reponse (GA: 0.3 - 3)
GA =0.7
GA =3
GA =1 r
GA =0.3
Fig. 2.3. Rise time Fig. 2.4. Settling time
The indexes of the system response show a complex performance. In general, they all are increasing functions of GA from a value near to the optimum point 0.2. The low dependence on the overshoot and the quadratic error with this factor is remarkable, as opposed to the great variation in the settling time. The response is hardly oscillatory except for very low values of this gain, or for values higher than 2, so the stability of the system is improved. Therefore, low values of GA (but not so low that this action would be canceled) give a response within the most usual specifications, however as this gain increases, the response becomes much slower.
Figure 3 shows the system response and the control action for different acceleration gain values. The rest of the variables gains are set to the initial values (4) and only this scale factor is changed.
The graph of the system response has been divided in two intervals to show its variation with more detail since its behavior is different for those intervals. The control graph has become unified. The gain values are:
GA = 0, 0.05, 0.1, 0.2, 0.3, 0.4, 0.7, 1, 2, 3. time.10
Figure 3.1. Response variation with the scale factor of the error acceleration
Control
G A =3
G A =0
G A =1
G A =0.4
time.10
Figure 3.2. Control variation with the scale factor of the error acceleration this could not only be applied in a first step to get some adequate parameters, but it also allows us to establish a finer adjustment of the initial parameters obtained by other analytic methods.

The derivative term is seldom employed even in classical control, mainly due to the fact that it increases sensibility to noise and that many times, a PI is good enough. Although most of the regulator structures incorporate this action, it is quite usual for the plant operators to inhibit this function. However, this third error variable gives a new control action to the fuzzy controller: the derivative action. This derivative term makes it possible:
- to complete the fuzzy controllers analysis in a similar way to the classical controllers and to establish some relationships between their parameters.
- to improve stability, since the derivative term can be a great help for stabilizing the control system, which remains one of the main problems of FLC.
- to give more flexibility to the FLC, since it allows us to expand the variation range of other controller scale factors within certain margins, making it easier to reach the response system specifications and improving its behavior.
It is also possible to prove that, in control laws, canceling the derivative action in control equations (by setting their coefficient GA = 0) does not produce a result like a PI, since the relationship between the inputs and the output is strongly non linear in spite of the fact that the rule table of the FLC-PID may be practically reduced to that of FLC-PI by removing the acceleration variable.
Response
FLC-PID FLC-PI
Control
FLC-PID
FLC-PI time.10
Figure 4.- Response of Fuzzy-PI and Fuzzy-PID
(with GA = 0) controllers
This result can be checked in Figure 4, where a comparison is made between the system response with a
Fuzzy-PI, and that with a Fuzzy-PID controller with quite similar rules except in the acceleration term, and where the derivative action gain has been canceled.
The complete study of the fuzzy controllers should involve all the terms that characterize the conventional ones.
The addition of the derivative term makes it possible to show the non-linear characteristic of the fuzzy controller, as well as to enlarge the variation range of the other input variables by means of their gains so as to improve the controller behavior.
Analytic tuning formulas for Fuzzy-PID controllers have been obtained. Therefore, initial parameters for these controllers have been proposed for the different defuzzification methods, and their validity has been evaluated by simulation examples with satisfactory results.
On the other hand, the qualitative study of these
FLC-PID has helped to produce some rules for a finer adjustment by means of the effects of these parameters on the system response.
However, the subject of the design and tuning of general fuzzy controllers is a problem that remains open.
This work has been partially supported by the Spanish
CICYT under project TAP94-0832-C02-01.
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Autosintonía de controladores borrosos utilizando técnicas clásicas basadas en reguladores PID. Proc. of III FLAT ,
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Dissertation, Department of Computer Science, Universidad
Complutense de Madrid, España, 1994.
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Engineering Applications of Fuzzy Logic , vol. 83, no. 3, pp.
407-422, 1994.

FLC-PID Control laws in the linear control zones
(Non-Linear Defuzzification Method)
###u(t)
4 L
−
2
4 L
−
[
[
. ( )
+
.
( )
]
.
( )
−
.
( )
. ( )
+
.
( )
]
−
2 .
( )
Zones
GE | e(t) | ###GA | ac(t) | ###
GR | ce(t) | sg(ce) = sg(ac)
GA | ac(t) | ###GE | e(t) | ###
GR | ce(t) | sg(e) = sg(ce)
4 L
−
2
[
. ( )
−
.
( )
]
. ( )
−
.
( )
GA | ac(t) | ###GR | ce(t) | ###
GE | e(t) | sg(e) = sg(ce)
[
. ( )
−
.
( )
]
4 L
−
2 . ( )
−
.
( )
[
GEe
+
2 GRce
+
GAac
]
GR | ce(t) | ###GA | ac(t) | ###
GE | e(t) | sg(e) = sg(ac)
4 L
−
2
[
.
+
.
+
.
4 L
−
4 L
−
[
.
−
.
.
−
2 .
. ( )
+
.
( )
]
−
2 .
( )
[
.
+
.
+
.
4 L
−
2
−
2 .
( )
]
4 L
−
[
. ( )
+
.
( )
]
.
( )
−
2 .
( )
]
GA | ac(t) | ###GR | ce(t) | ###
GE | e(t) | sg(e) ### sg(ce)
GA | ac(t) | ###GE | e(t) | ###
GR | ce(t) | sg(e) ### sg(ce)
GR | ce(t) | ###GE | e(t) | ###
GA | ac(t) | sg(e) = sg(ac)
GE | e(t) | ###GA | ac(t) | ###
GR | ce(t) | sg(ce) ### sg(ac)
GE | e(t) | ###GR | ce(t) | ###
GA | ac(t) | sg(ce) = sg(ac)
4 L
−
[
.
+
.
+
.
.
( )
−
2 .
( )
]
[ +
0 5Lu 2 .
.
+
.
4 L
− −
2 .
( )
[
GEe
+
GRce
+
2 GAac
]
]
4 L
−
2 . ( )
−
.
( )
GE | e(t) | ###GR | ce(t) | ###
GA | ac(t) | sg(ce) ### sg(ac)
GR | ce(t) | ###GE | e(t) | ###
GA | ac(t) | sg(e) ### sg(ac)
GR | ce(t) | ###GA | ac(t) | ###
GE | e(t) | sg(e) ### sg(ac)