Automatica, Vol. 11, pp. 209-212. Pergamon Press, 1975. Printed in Great Britain

IFAC Report Round Table Discussion on the Estimation and Control

in Fuzzy Environments*t

Rapport IFAC. Discussion autour d'une Table, sur l'Estimation et le Contr61e dans un Environnement Flou

IFAC Bericht: Rundtischgespr~ich fiber die Schfitzung und Steuerung in unscharfer Umgebung

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M A D A N

Summary--The main contributions to the round table discussion on 'Estimation and Control in Fuzzy Environments' which took place during the 3rd IFAC Symposium on Identification and System Parameter Estimation are summarized. Some viewpoints describing the future trends of the subject are also presented.

It was a well-attended and lengthy session which lasted for about 4 hours. Considerable information was exchanged on this newly developed discipline.

Introductory lectures were given by a panel of five speakers (Gupta---Canada, Zadeh--U.S.A., Fu--U.S.A., Tamura--Japan, Sugeno---Japan) which lasted for about 2 hours. The need for these introductory lectures was felt by the panel as well as by the audience in view of a relative unfamiliarity with the subject by many of the audience. After a brief coffee break, the session was opened for dis- cussion and comments from the floor. A lively discussion took place and continued for another 2 hours. Even after this, the meeting proved to be too short, and Professor Zadeh was able to present some more results of his work the next day during the session on Adaptive Control.

The purpose of this brief report is to summarize the main contributions made in the course of this stimulating round table discussion, and to draw conclusions on which the future work should be d i rked . A partial list of the bibliography is also included.

1. Why fuzzy automata ? O ~ of the most innovative aspects of modern control engineering is undoubtedly the prevalence of rigorous mathematical theory. The design of deterministic or stochastic optimal control polici¢~ using optimal control theory is based upon the assumption that an exact matbe- matical model of the process to be controlled is available to the designer either in a deterministic or stochastic sense. However, as is well known, such a modal cannot be

* Report on a round table discussion held during the 3rd IFAC Symposium on Identification and System Parameter Estimation, The Hague/Delft, The Netherlands, Thursday, 14 June 1973. Received 25 April 1974; revised 24 June 1974.

1" This work is supporte~ by the National Research Council of Canada under Grants A-5625 and A-1080, and the Defence Research Board of Canada under Grants 4003-02 and 9781-04.

Systems and Adaptive Control Research Laboratory, College of Engineering, University of Saskatchewan, Saskatoon, Sask., Canada.

M. G U P T A ~

obtained for most processes. Situations of this type exist in many decision-making areas such as forecasting, economic planning, management, medical diagnostic and pattern recognition. This may, in fact, be the reason why the relation between theory and practice has recently become more tenuous and why control theory which made space mission highly successful is not directly applicable to many humanistic processes [59].

In general, mechanistic type systems are amenable to exact quantitative mathematical analysis, while humanistic type processes are too fuzzy to be amenable to such exact mathematical techniques. It is imperative, therefore, for complex fuzzily defined humanistic processes not to make use of exact models or conventional mathematical analysis as is being done today. The desirability of introducing mathematical concepts which reflect characteristics inherent to fuzzy humanistic processes has become apparent since the introduction of Fuzzy Sets in 1965 [48]. Researchers at many of the world's leading institutions are now work- ing on the creation of the foundations of a new and exciting field, viz. Fuzzy Automata. This new field has evolved from the pioneer work of Lotfi A. Zadeh.

These newly developed concepts of fuzzy automata seem to be extremely useful for humanistic type processes and are finding applications in many fields. For example, the concepts of fuzzy sets can be used to define a complex process more realistically than that which can be provided using the deterministic or probabilistic approach. Also the fuzzy automata can be used to generate decision policies for processes which are too complex or too ill-defined to be described exactly.

2. Invited contributions

The meeting was opened with a number of brief invited contributions followed by a general discussion. The first of these was given by the chairman, Gupta, who gave a brief introduction to the fuzzy sets and the concept of the fuzzy automata, and summarized the scope of this new field as compared to the deterministic and stochastic system theory. It was pointed out that the fuzzy automata may play a very important role in humanistic type of processes where a human element is involved in decision making. Examples of such processes are economic and management processes, computer language, medical diagnostic area, pattern classification, etc.

Professor L. A. Zadeh (University of California, Berkeley) gave a very illustrative presentation on 'A Linguistic Approach to Decision-making'. His talk is briefly summarized below:

'In the linguistic approach to decision-making, some of the variables entering into the decision process are allowed

xs 209

210 I F A C R e p o r t

to be linguistic, that is, are allowed to take on values which are not numbers but sentences in a natural or artificial language. For example, a linguistic variable x may have as its values: small, large, not small, not very small, extremely small, more or less small, etc. These values are, in effect, labels of fuzzy sets whose membership functions can be computed from the knowledge of the membership functions of the primary terms small and large, and the definition of the hedges such as very, extremely, more or less, etc.

'The use of linguistic variables is motivated by the fact that in many real-world decision processes the goals, the constraints, the interrelation between variables and the underlying probabilities are not known with sufficient precision to be susceptible of characterization in numerical terms. For example, we may know that an event is very likely without being able to specify whether its probability is 0.7 or 0.8 or 0.9. In this case, the probability may be assumed to be a linguistic variable with two possible values very likely and not very likely. In effect, the concept of a linguistic variable serves to provide a basis for an approxi- mate analysis of decision processes which are too complex or too ill-defined to be amenable to treatment by conven- tional numerically orientated techniques.'

In his discussion, he gave a detailed account of some of the main aspects of the linguistic approach. In particular, he showed how ill-defined goals, constraints and probabili- ties may be described in linguistic terms and how approxi- mate computations with linguistic variables may be performed. He showed that, in general, the results of computations is the membership function of a fuzzy set. This membership function is approximated by a linguistic label whose meaning can be computed by a procedure described in his paper [61]. However, a general approach to this approximation problem has not yet been developed.

The next invited speaker was Professor K. S. Fu (Purdue University, Indiana), who gave an illustrative talk entitled 'The kth Optimal Policy Algorithm for Decision Making in Fuzzy Environments'. The summary* of his lucid presentation is:

'The kth optimal policy algorithm is a natural generalization of the optimal routing problem. We consider a finite-state automaton characterized by a transition-cost matrix and definite initial and final states. The kth optimal policy is the kth best valued sequence of the transitions from the initial state to the final state, assuming that the total cost is the sum of transition costs between intermediate states in the policy. This problem has been studied by several authors, and at this stage the Bellman-Kalaba algorithm appears superior to all the others for larger networks. However, the total amount of computation is still excessive for practical problems due to its two-stage process, namely, a direct extension of the dynamic programming approach and the backward sub- traction routine.'

In his talk, Professor Fu proposed a new algorithm which incorporates the Dijkstra labelling method with his optimal policy elimination routine. In essence, the algorithm is recursive and is started with the 1st optimal policy, then an automaton with the 1st optimal policy eliminated is constructed so that the next 1st optimal policy of this new automaton will be the 2nd optimal policy of the original automaton. Other suboptimal policies can be generated in the same way. Advantages and drawbacks of the recursive scheme were discussed. On the other hand, it was shown that the algorithm can be extended to .fuzzy automata.

Practical applications of the kth optimal policy algorithm were described. In the first step, one considers as many uncertainties in the model as one can using the fuzzy set approach or the risk function approach. Then he asks the decision maker to provide a range of allowable deviations (from the nominally optimal value) in which he would choose a policy of his preference or his best judgement.

* Reference [19] is a full paper on this subject.

The kth optimal policy algorithm is then employed to enumerate all possible optimal policies inside the prescribed range. In this manner, human factors in decision-making processes can best be considered.

In the fuzzy set approach to decision making, the problem of assigning meaningful membership functions to the variables in the model is of utmost significance. Professor Fu then discussed the possibility of using the sensitivity analysis; this was first studied by Bellman and Kalaba on the basis of the kth optimal policy algorithm, for the identification of membership functions in a fuzzy decision-making model.

The talk by Dr. H. Tamura (Osaka University, Japan) was on 'Adaptive Optimal Controller and Control Responses of Man'. A summary of his interesting talk is given here:

'The control movement of man in a manual control system can be classified in several modes of control depending on the controller, controlled plant and training. The differences and the changes of mode from one to another are considered to be made by subjects to choose an appropriate signal for the response identifications.

"The manual control system under consideration is furnished with CRT display, a hand-driven controller and a linear controlled plant of various order, G(s) = 1/(s + 1) s ~. The task to be implemented by the subject is compensatory tracking to step input, which is applied to the system with a sufficiently long interval of time.

"The control action to a step input can be divided into three processes, the initial process, the control process and the stabilizing process.

"The control mode of a man in the control and stabilizing process changes with the order and type of the plant to be controlled and with the controller.

"In the control process, two types of control modes are observed. The one, named the linear mode (L), is observed for the plants of the lower order (lst and 2nd). For the higher order plants (2nd and 3rd), the bang-bang mode (BB) is dominant.

" In the stabilizing process, four modes are observed. Firstly, the damped bang-bang mode (DB), secondly the steady-state bang-bang mode (SB), thirdly, the bang-zero mode (BO) and, lastly, the spike mode (SP).

"The L mode is observed with untrained subjects, while the DB mode is observed with well-trained ones. The SB mode is observed when the subject is about to lose the system stability. The SP mode is very effective in con- trolling the higher order plants, but this mode is only seen when the controller is furnished with the sensor for the correct position of zero output.

"The limit of control was determined by the paramelers in G(s), at which the subject failed to maintain stability. The limit of control thus determined is dependent mainly on the mode of control that the subject adopted. So far as the mode is the same, the limit of control does not differ very much.

" In the conclusion it was emphasized that control output of man should not be formulated as the response to the input supplied from the display. This formulation implies that the subject is passively controlled by the display signal. Conversely, one has to take notice of the fact, that he is actively controlling the unknown, fuzzy, environment.

"The control behavior has to be understood to have two roles. The one is the command signal to govern the plant, and the second is the test signal to identify the response characteristics. With increase in the order of the plant, the latter aspect becomes essential. Thus the plant is controllable only when the subject adopts the mode of control which is more powerful in identifying the response to his command.

"The above features are worth considering in the design of an adaptive optimal controller."

The last speaker on the panel was Dr. M. Sugeno (Tokyo Institute of Technology, Japan) who talked on "An Approach to the Identification of Human Characteristics by Applying the Fuzzy Integrals". A brief summary of his interesting talk is:

IFAC Report 211

"Lately, it has become of general interest to control the complicated systems such as social, biological and economical systems. These systems have so many uncertain elements that it is difficult to make their mathe- matical models. Of course, it is possible to make the stochastic models by using the probability measure, ff plenty of data on such systems and the environment surrounding them are available.

"However, it is doubtful that the stochastic models are suitable for the mechanisms of a human decision or evaluation process including many unclear factors, because these factors mainly depend on man's subjectivity".

Dr. Sugeno proposed the concept of fuzzy measure and fuzzy integral as a means of measuring the fuzziness in such systems as described above.

There were three more invited contributions which could not be discussed due to the authors' absence. These were:

(i) A. G. Ivakhnenko, "Self-organizing Theory on the Basis of Direct Complex System Modelling after the Experimental Data".

(ii) K. Asai and H. Tanaka, "On Fuzzy Mathematical Programming".

(iii) K. Tanaka, "Some Behaviors of Composite Fuzzy Automata in Random Environment".

3. General Discussion

In opening the general discussion, Professor Zadeh commented on the importance of this developing area. Then to illustrate the importance of decision making in a fuzzy environment, he gave a demonstration in which a blindfolded person was directed from a given initial position to a final destination purely with the help of fuzzy commands, or algorithms.

In addition to the panel, there were many other persons who participated in the discussion although the time allocated for the session was limited, but the panel and the audience were so enthusiastic that the discussion was con- tinued until 1.00 p.m. A partial list of the persons who showed enthusiasm in fuzzy automata and participated in the discussion is given here.

P. A. Devijver (M.B.L.E. Research Laboratory, Brussels, Belgium).

G. J. Gaalman (University of Technology, Twente, The Netherlands).

J. gt~pan (Novoborska, Praha). M. Installe (University of Louvain, Belgium). A. A. Van Rede (University of Technology, Eindhoven,

The Netherlands). P. A. A. Van Boxtel (University of Technology, Twente,

The Netherlands). S. F. Thelliez (Laboratoire Automatique, C.N.A.M.,

Paris). F. Donati (I.E.N.G.F., C.SO M.D.'Azeglio, Italy). L. Pun (Laboratoire d'Electronique Appliqu~e, University

of Bordeaux, Talence, France). J. Gordesch (Interforkult6res Rechenzentrum, Wien). H. Abele (Institut fiir Wirtchaftswissenschaften, Wien). E. Drossel-Meijer (Gesellschaft fiir Kernforschung,

Germany).

4. Summary

This new direction on fuzzy automata was pioneered by Zadeh in 1965, and since then a new field has evolved having many promising future developments. As stated by Professor Zadeh, the concepts of linguistic variables may serve to provide a basis for developing fuzzy algorithms in decision-making processes when processes are too-complex or too ill-defined.

It was concluded that in terms of future development, the most dramatic results very likely will emerge from the implementation of fuzzy algorithms in the control of industrial processes, such as chemical processes and power systems, also in pattern classification, in designing decision processes for social sciences, in economic processes and management, and in the medical diagnostic area. It was envisaged that the fuzzy automata which underline much

of human thinking promises to have a revolutionary impact on the future of systems engineering.

5. List of the panel (1) Dr. M. M. Gupta (Chairman), Systems and Adap-

tive Control Research Laboratory, University of Saskatchewan, Saskatoon, Canada.

(2) Professor L. A. Zadeh, Department of Electrical Engineering, Computer Science and Electronics, University of California, Berkeley, California 94720, U.S.A.

(3) Professor K. S. Fu, School of Electrical Engineering, Purdue University, West Lafayette, Indiana47907, U.S.A.

(4) Dr. H. Tamura, Faculty of Engineering Science, Osaka University, Toyonaka, Osaka, Japan.

(5) Dr. M. Sugeno, Department of Control Engineering, Faculty of Engineering, Tokyo Institute of Technology, Tokyo, Japan.

(6) Dr. K. Kanai (Secretary), Systems and Adaptive Con- trol Research Laboratory, University of Saskatchewan, Saskatoon, Sask., Canada.

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