2º SIMPÓSIO BRASILEIRO DE AUTOMAÇÃO INTELIGENTE CEFET-PR, 13 a 15 de Setembro de 1995 Curitiba Paraná

A Fuzzy Cootrol Metbod Ba.sed 00 Possibility Distributioos

Karl Heinz Kienitz

Divisão de Engenharia Eletrônica Instituto Tecnológico de Aeronáutica

Pça. Mal. Eduardo Gomes, 50 - Vila das Acácias 12.228-900 S. José dos Campos SP

Fax: (0123) 41-7069 Tel: (0123) 41-2211 kienitz@ita.cta.br

Abstract

A novel fuzzy control method is presented. lt uses the compositional rule of inference combined with an implication operator which has nol yet been considered for fuzzy controI. The defuzzification procedure proposed for this method is based on an interpretation of inference results using the concept of possibility distribution. The new method is suitable for the realization of faster controllers than those based on existing fuzzy control methods. A simple application example shows that the new method may outperfonn existing methods.

1. Introductioo

Fuzzy control has established itself as a useful heuristic control paradigm. Its origins lie in a more or less heuristic and experimentalcontext (Marndani, 1974). Its dissemination has been mainly due to the presentation of supponing application examples. Reports on less supponing experiments (such as that described by Buchholz, 1992) rarely found their way into main publications. Today there exists a number of successfuI fuzzy control methods. For a comparison of a few methods see Figueiredo et alo (1993). Fuzzy control methods can be divided into those which use the compositional rule of inference and those which do nol. Methods in the first class use the compositional rule of inference to solve the generalized modus ponens, ",hose simplest forro is

Prcmise 1: (V is M) Premise 2: IF (V is Al THEN ru is Bl Consequence: (U is C)

Mamdani's and Larsen's methods are the best known methods that belong to this class (Figueiredo ' et al., 1993). They differ in the choice ofthe definition for the implication operator IF ... THEN: • Mamdani's method uses the min operator. • Larsen's mehtod uses the algebraic product. The choice of these two operators was motivated by simplicity and by the good practical results attained with them. Most of the (theoretical) research on the analysis of different fuzzy implication operators and the resulting reasoning schemes (e.g., Miziunoto & Zimmermann, 1982) did not cany over into fuzzy control practice. Part of this research was presented to the control systems community at an early stage (Mizumoto et al., 1979). However, most of these results have never been applied to the improvement of speed and quality of fuzz\' controI. The "thorough studies carried out on the propenies of different fuzzy implication operators considered a more general context than fuzzy logic controlo In those studies several other implication definitions have shown more desirable propenies than the min or the algebraic product operators. The objective of this paper is to present a fuzzy logic controI method that empIoys an implication operator used for plausibIe approximate reasoning. Elsewhere it was shown that this operator has an fnteresting property: it exactly satisfies a series of common sense reasoning schemes (Mizumoto et al., 1979; Kienitz, 1990). In this contribution it is shown that the new method: • allows for the construction of even faster controllers than those possible wlth Larsen's or Mamdani's

methods • may qualitatively outperfonn existing methods in a practical application contexto

80 ~~I 2! SIMPÓSIO BRASILEIRO DE AUTOMAÇAO INTELIGENTE

2. Approximate reasoning and fuzzy control

The fuzzy controller considered in the scope of this paper essentially consists of a knowledge base, an inference mechanism and a defuzzification interface. It is schematically represented in Figure 1.

knowledge

base meas ürements

I output

inference measurement H defuzzification r- controI - - mechanism ~ pre-processmg interface -

Figure 1: Slruclure of lhe fuzzy contro//er

The knowledge base is composed of p roles of the type

IF (VI is Ali) AND (V2 is A2i) AND ... AND (Vn is Ani) THEN (U is Bi); i = 1, ... , P (1)

Measurements are available to the inference mechanism as

(2)

where in this case the Mi are singletons. The inference mechanism combines the knowIedge obtained from measurements with the knowIedge contained in the individual rules generating a set of conclusions {(U is Ci), i = 1, ... , p}, one conclusion for each role. On the basis of this set, the defuzzification interface comes up with a control value to be applied to the system. To be processed by the inference mechanism, linguistic expressions such as (1) and (2) are mathematically interpreted using translation roles. According to the translation role adopted for expressions such as (2) (Kandel, 1986), it induces the possibility distribution

where VI, ... , vn are the base variables for VI, ... , Vn and J.1Mi (Vi) is the membership function ofMi~ i = 1,

... , n. In this paper membership functions are supposed to be normal. The translation role for (1) depends on the definition chosen for the implication operator IF ... THEN. Various definitions were discussed in the literature. The implication definition adopted · here was originally studied by Mizumoto et ai. (1979) and was proposed for plausible approximate reasoning by Kienitz (1990). Severa! advantages of this definition are discussed in these references. According to the resulting transIation rule, (1) induces the possibility distribution

if J.1Ali (Vl)A. . .I\J.1Ani (vn) ~ J.1Bi (u)

if J.1Ali (Vt)A.·.I\J.1Ani (vn) > J.1Bi (u)

The possibility distributions induced by (1) and (2) are combined by the inference mechanism to generate a conclusion of the type (U is Ci). This is done using the compositional role of inference (Zadeh, 1979). As in the case of Mamdani's method, the use of the compositional role of inference in the new method is amenabIe to simple graphical visualization. Figure 2 exemplifies one role inference using Mamdani's method and the new method. Using the. proposed fuzzy control method, the inference result will be a rectangular membership function, as long as measurements are crisp data. The smalIer the basis of the rectangle under J.1Ci (u) , the more cIearly it restricts the choice of

the control value. In the limit case where at Ieast one of the measurements is such lhat J.1Mi (vimeasure) = O,

the rectangle will cover the whole universe of discourse of the control variable, i.e. with respect to th~ knowledge contained in the i-th rule alI available control values are equally suitable (possible) (Kandel, 1986).

.• ~

2! SIMPÓSIO BRASILEIRO DE AUTOMAÇAO INTELIGENTE 81

Marndani's (and also Larsen's) method results in membership functions which are interpreted differently. For example in the lirnit case J.lMi (vimeasme) = O, both methods yield a membership function identically zero

over the whole universe · of discourse of u. In the context of possibility distributions this means lhat no control value at alI is suitable (possible). ar course such an interpretation is not of interest in a practical setting. Thus the interpretation adopted in the scope of those fuzzy control methods regareis the membership value as a certainty value. This however is not in strict accordance with the theory of approximate reasoIÚng (Zadeh, 1979).

1

o

3. DefuzzificatioD

E!J Marndani's method

~ This method

Figure 2: Inference example. In this example the following are given:

rule: IF (V 1 is A li) ANO (V 2 is A2i) THEN (U is Bi) measurements: VI = vImeasure' V2 = v2measure

In applications, Marndani's and Larsen's methods have been used together with one of several defuzzification procedures. The most common of these procedures estabIishes the control which results from {(U is Ci), i = 1, ... , p} as the center of gravity of the area under J.1CI (u)v ... VJ.1Cp (u) . A variant of this procedure determines

the control value as the weightcd average ofthe centers of gravity ofthe areas under the J.lCi (u) .. The weights

are the associated areas. The defuzzification procedure proposed for use with the fuzzy control method advocated in this paper is different because the membership functions which result from the inference process are possibility distributiODS, as exposed above. The control value to be applied to the system is calculated as the weighted average of the centers of gravity of lhe rectangles under the J.1Ci (u) . However, lhe weights are given by the

values of max{O,I- a.A>{l where Ai is lhe area ofthe rectangle under J.1Ci (u), and a. is a scaling factor to be

chosen. (a. = lis suggested for use with a normalized universe of discourse, i.e. when -1 S U S 1.) Thus for rectangles which extend over the whole UIÚverse of discourse of u (no restriction on lhe control) . the weight is zero. On the other hand, for rectangles lhat are singletons in the limi~ lhe weight is one. The weights can be seen as confidences one has in lhe centers of gravity of ·J.1Ci (u) as correctly representing the control value for

the system: the larger the arca of the rectangle, lhe smaller lhe confidence.

4. Using the ne", method in a typical applicatioD

Fuzzy temperature controllers are commercially available from severa! sources. Thus, a heating device was selected to illustrate the applicaúon of lhe proposed method and to allow for a pracúcal comparison with Mamdanils method. The selected plant is a laboratory device described in lhe last reference given in secúon 7. In this plant air is blown through a tube. The air is heated at lhe entrance of the tube by a mesh of resistors.

82 2' SIMPÓSIO BRASILEIRO DE AUTOMAÇAO INTELIGENTE ~

.~.

These are excited by the controI u. The air temperature y is measured at the end of the tube by a termistor. The structure of the system with the proposed controller is detailed in Figure 3. For simplicity simulated data were used for comparison purposes with Mamdani's method. A discrete linear simulation model of the process was determined using the PC package described by Hemerly (1991). The sampling time used was lOO[ms]. The transfer function obtained is:

G(z) = O.l148z-2

+O.005243z-3

1-1.2275z-1 +0.3647z-2

~ ... _-- _ ..... .. ....................... ............... .... ....... .. ......... ................................................................................................................ _- ............. .. .... ..... .. ......... __ .. -:

E

knowledge

base

fuzzy controller

defuzzification

Figure 3: Temperature control using a fuzzy controJler

A very simple incremental controller was used. It has a knowledge base with 9 rules:

IF (E is PO) ANO (ÃE is PO) TIIEN (âUisONE) IF (E is PO) ANO (ÃE is ZE) TIIEN (âUisONE) IF (E is ZE) ANO (ÃE is PO) TIIEN (âU isONE) IF (E is NE) ANO (ÃE is NE) TIIEN (âU is -ONE) IF (E is NE) ANO (ÃE is ZE) TIIEN (âU is -ONE) IF (E isZE) AND (ÃE is NE) TIIEN (âU is -ONE) IF (E is PO) ANO (ÃE is NE) TIIEN (âU isZE) IF (E is ZE) AND (ÃE is ZE) TIIEN (âU isZE) IF (E is NE) AND (ÃE is PO) TIIEN (âU isZE)

The linguistic values used in these rules are defined by their membership functions, as follows:

NE stands for "negative":

PO stands for "positive":

ZE stands for "around O" :

ONE stands for "around 1":

-ONE stands for" around -I":

J.lneptive (x) = max[min(-x,I),O]

J.lpositive (x) = max[ min (x, 1), O]

J.laround~ (x) = max[O, I-Ixl] J.laroundone (x) = max[O, l-Ix -111 J.laroundminusane (x) = max[O, l-Ix + 1\]

measurement . noise

y

Two difIerent controllers were used. One used the new method and the other used Marndani's method. The value for KE was adjusted in order to allow E.KE to cover the range [-1, I], which is the normalized range expected in the definition of the rules. KÃE and KâU were exPerimentally adjusted. In both cases gaussian measurement noise with variance 0.0001 and zero mean was added.

2! SIMPÓSID BRASILEIRO DE AUTOMAÇAO INTELIGENTE 83

The step responses are found in Figure 4. From this figure it is seen that in this applicaúon the controller baseei on the new method is less sensitive to measurement noise than the controller based on Marndani's method. It was verified that even with other gain adjusttnents this statement rernained valid for this application.

5. Conclusions

The fuzzy controI mcthod prc$cntcd in this contribution uses the compositional rule of Ínference and an implication operator studied in the approximate reasoning literature. The defuzzification procedure used in the method has common points with the well known "center of gravity defuzzification". but considers the interpretation of approximate reasoning results as possibility distributions. The new method (inference and defuzzification) is of simple and efficient implementation. In comparison to Mamdani's and Larsen's methods the area and center of gravity detemúnation needed are much simpler because only rectangular membership functions need to be considered. As shown in Figure 2 the detemúnation ofthe ~C' (u) itselfis ofthe same complexity. However, the representation ofthe J..LC· (u) in the

. 1 1

new method is. For these reasons faster fuzzy controllers can be build with the new method. In the illustrative application example the new method showed smaller sensitivity to measurement noise than Mamdani's method. Unfortunately it is not possible to make a general statement on noise sensitivity. Additional research in this arca is necessary.

6. Acknowledgment The author thanks Prof. Elder M. Hemerly who supplied the transfer function of the plant discussed in section 4.

7. References

Buchholz, J.1 -A Bit Fuzzy, ARNO-l11 2809001-9, DLR Institut fiir Flugmechanik, Braunschweig, 1992. Hemerly, E.M. - PC-based packages for identification, optimization, and adaptive control, IEEE Control Systems Magazine, v. 11, No. 2, pp. 37-43, February 1991. Figueiredo, M.; Gomide, F.; Rocha, A.; Yager, R - Comparison of Yager's levei set method for fuzzy logic control with Mamdani's and Larsen's methods, IEEE Trans. Fuzzy Systems, v. 1, pp. 156-159, 1993. Kandel, A. - Fuzzy mathematical techniques with applications, Addison-Wesley, Reading, 1986. Kienitz, K. H. - PlausibIe approximate reasoning, Cyber. Syst., v. 21, pp. 647-654, 1990. Marndani, E.H. - "Application offuzzy algorithms for controI ofsimple dynamic pIant", Proc. IEE, v. 121, pp. 1585-1588, 1974. Mizumoto, M.; Fukami, S.; Tanaka, K. - Several methods for fuzzy conditional ·inference, Proc. 18th IEEE Conference on Decision and Control, pp. 777-787, 1979. Mizumoto, M. & Zimmermann, H.-J. - Comparison offuzzy reasoning methods, Fuzzy Sets and Systems, v. 8, pp.253-283, 1982. Zadeh, L.A. - A theory of approximate reasoning. In: Hayes, J.E.; Michie, D.; Mikulich, L.I. (Eds.) - Machine Intelligence 9, pp. 149-194, Ellis Hanvood, 1979. Feedback Instruments Ltd., Process Trainer PT326, Manual 326 Rev. 0977, Crowborough, Sussex, England.

84 ~2'~/ 2' SIMPÓSIO BRASILEIRO DE ~ AUTOMAÇAO INTELIGENTE

3~--~~------~------------

2.5

2

1.5

1

0.5

I I

o~----~------~----~----~ OL-----~------~----~------o 50 100 150 200 o 50 100 150

( a) (b)

Figure 4: Step response (a) and control input (b) of the

controlled system (solid: proposed methodi dotted:

200

Mamdani's method, dashed: reference) . Time is given

in nurnber of sampling periods.

KE = 1/2 K~E = 1/4 K~U 1/3