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Feedback Control Applications of Fuzzy Set Theory: A Survey
FUZZY CONTROL
Copyright © IFAC Control Sri cncc and Technology 18th Triennial World Congress) K\,oto. Japan , 1981
FEEDBACK CONTROL APPLICATIONS OF FUZZY SET THEORY: A SURVEY M. M. Gupta Cybernetz'cs Research Laboratory, College of Engl'neerz'ng, Unz'versz'ty of Saskatchewan, Saskatoon, Saskatchewan, S7N WO, Canada
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Abstract. The development of feedback control theory for applications to industrial processes is one of the fundamental issues facing the control scientist. This issue arises because of the complexity and vagueness in the description of such processes. Some attempts have been made to use the fuz zy set theory in the design of feedback control algorithms for industrial applications and this development is still in its infancy . In this paper we gi ve a bri e f survey of some of the work that is being done in this fi e ld. Keywords. Automata Theory, Computer Control, Fuzzy Controll er, Feedback Control, Fuzzy Lo gic Control, Linguistic Control. terms in contrast to precise numeric data as demanded by computers. Much of this difference lies in th e insufficiently understood ch aracte ristics of the human brain, '.e., a f a culty of global reasoning and pattern recognition at man y l evels. As the system becomes complex," . . our ability to make precise and yet significant statements about its behavior diminishes, until a threshold is reached beyond which precision and significance (or relevance) become mutually exclusive characteristics". L.A. Zadeh [53].
INTRODUCTIO N Th e development of feedback control theory for applications to industrial process which are vague, imprecise and complex i s one o f the most fundamental i ss ues and challengin g problems facing the control scientist [1, 46~ At present, several attempts have been made to use the modern control theory in the analysis and design of such feedback control systems but with very little successes. ~Iod ern control theory has had tremendous success in areas where the system is well defined either deterministically or stochasticall y . Aerospace systems, missile and space vehicl e guidance systems which can be described with a mathematical precision are some examples where the modern control th eory has proved to be very successful. Control scientists att empted to appl y these theories to modern complex plants such as in manufacturing technology, chemical process, pulp and paper mills, power plants, steel making industry, cement klein, etc., but they faced nothing but failures and frsutrations; these theories are not capable of coping with the vagueness and imprecision inh erent in an y practical plant despite the development of a huge body of mathematical knowledge.
The essential featur es of humans in decis ion-making prompts the question 'can we make comparable use of ambiguity , imprecision and human language in the analysis and design of feedback controllers for complex industrial processes?' Fuzz y set theory was founded by L.A. Zadeh in 1965 [SO]. This theory is a mathematical concept that gives some mathematical precision to imprecise and ambiguous statements and the vagueness inherent in modern complex processes. Its essential concept is that of the graded membership. In ordinary set theory , a member is either in or it is out of a subset. In Boolean logic a proposition is e ither true or false. The introduction o f graded membership allows us an infinite extension to the basic concepts in set theory and logic in a way that appears to be very natural. An underlying philosophy of the fuzzy set theory is to provide a strict mathematical framework. It provides a gradual transition from the realm of vigorous, quantitative and precise phenomena to that of vague, qualitative, and imprecise conceptions. This theory enables one to characterize impreclslon in terms of "fuzziness", a concept to which one can assign many meanings [11, 12 , 18, 21, 40, 60].
There are undoubtedl y many reasons for these failures, but primaril y it is the lack of detailed and precise structural knowledge of the process demanded by th e modern control theory. Despite this, these processes are often satisfactorily controlled by a human operator who has the ability to reason and act on information which is imprecise. He has a remarkable adaptibility to interpret linguistic statements ab out the process and to reason in a qualitati ve fashion. Vaguenes s , imprecisi on, and ambigui t. ' have not deterred human beings from taking actions. Communication between humans is in linguistic
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information and, hence, precision is lost. Th e precis may have what seemed to be the mo s t needed ar.d relevant information, yet Some other aspects are lost in such modeling. For many purposes, a very approximate characteri za tion or mode ling of a collection of data is sufficient because most of the basic tasks performed by humans do not require a high degree of precision in their executio~ For very highly precise and detailed modelling, one needs equally an elaborate system of mathematical description and measurements . In any real life industrial situation, this is difficult to achieve. Supervisory control by a human operator with a set of linguistically described rules can achieve successful control in an industrial situation. Supervisory personnel with years of experience can express their control process effectively in linguistic terms but not so effectively in mathematical terms. These complex processes are often controlled with great skill by a human operator who makes decisions on the basis of intuition and linguistic measures of the process state. Unfortunately, the speed and capacity of the human operator is limited to handle the multivariable situation at the speed modern processes run. SUBJECTIVITY, INTUITION, VAGUENESS, PRECISION AND MODERN CONTROL THEORY Feedback control theory has evolved during the last fifteen years from an empirically oriented approximate method such as Bode, Nyquist, Nichols and root-locus methods, to a strongly mathematically based modern control theory approach. Modern control theory is very precise and has well-defined control concepts, and needs a precise knowledge of the process model to be controlled and exact measurement (either in deterministic or probabilistic sense) of input and output data. In spite of the precision of the modern control theory, both in the modelling and design, the degree of vagueness and subjectivity did not alter much. For example, in modelling a process to a large extent the individual's experience and mathematical convenience were primarily responsible in determining the process structure and its parameters. So was the situation in the design of a feedback controller to give a 'good' system performance. The large spread in the design criteria found in each design was mainly dependent upon the designer's personal views and his experiences. Thus, subjectivity, inexactness and vagueness still play a major role even in these modern control tools which are mathematically precise. There is a question, however, which remains unanswered. 'How good are these modern mathematical control tools which are too precise for the applications to these modern processes which are complex, too ill-defined, and for which a precise model almost does not exist?' These processes under consideration are non-
linear, time-varying and ar e subjec t to s t ochastic disturbance s i n input and output measurements, and s tochas t ic behavioral pe rturbations in parameters. Surprisingly, it has been reported that in many cases a human operator is able t o control the se processes much more successfully than an automati c controller designed using the modern co ntrol theory. There are example s known, for example in a power industry, that a simple controller designed using classical approach is being adopted s ucc ess full y. The app lic ation of modern control theory t o these pro cesses gave nothing worth reporting but th e failures and frustration s . In some cases, successful applicat ions of adaptive methods, which do not rely heavily upon the ' exac tness' of the pro cess model, have been reported (47). WHY FAILURES? It is clear from the various real-life app lication -oriented publi ca t io ns th at have appeared during the l as t decade, and fr om the brief remarks made above tha t the ' vagueness ' associated with the modelling of too compl ex and ill-defined processes is not compati bl e with the 'precision' of the mode rn control tools. In many pro cess si tuations such as in steel making furnaces, cement kiln s , and processes in glass industry, it i s almost impossible to construct a r easonab ly good model structure and estimate its parameters whi ch are compatible with the preci s ion of modern control theory. Of course, no one will deny that modern control theory has resulted into control algorithms whi ch have been very successful on computer s imul ation models of industrial proces ses , but designers have failed to convenience the industry on the use fulness of these a lgorithms in practice . For many years we have had the convi ction that the most fruitful areas for the growth in the applications f i eld of automatic control are those which have been neglected as a no-man' s land between the various es tablished field s. It is clear, therefore , that if the proces s cannot be modelled accurately as demanded by the modern control theory, we must work to develop alternate approaches which a re compatible to the inaccuracies associated with the modelling of the se processes. QUALITATIVE DES CR IPTION OF THE PROCESSES AND FUZZY CONTROLLER It is well known that for these complex processes a sufficient amount of qualitati ve ~ priori information is available to the designer. The design criteria may also be formed based upon the operator's experience. This qualitative information and the de sign specifications can be described only linguistically, which inherently is s ubjected to the subjectivity of the human operat or and that of the designer. An approach which can handle this imprecision created by human experience and intuition will be most desirab l e [20,21,23,24]. The approach must possess
Feedback Control Applications some of the followin g characteristics: - mOdelling capab ility the inexactness associated with human subjectivity, - linguistic apDroach to link the operator wi th the computer f or on -l ine computer control. It is the cont ention of thi s paper that the sor t of complexity and uncertainty that ar e being generated in modellin g the processes is better captured by the 'notion of fuzzine ss ' than that of the probability . The notion of fuzziness is explained in a number of r epo rts, papers and books [16, 20, 3S, 54 J by many world researchers in the field.
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impl ementation of th e fuzzy algorithm using either hardware or a computer program. This will need a numeric a l characterization of the linguistics . It consists of several problems, all related t o the overall problem of realization of fuzzy a l gor ithm for a nonfuzz y machin e vi z . a digital computer. Some of th e associated problems are the de terminati on of the range of values of the control~ the choice of quantization levels for the controller inputs and outputs, the numerical definition of the primary fu zzy sets on the quantized spaces, and th e choice of the sampling interval. All th ese problems are interdependent and mus t be considered in keeping a particular process and control objectives in vi ew . Some Process App l ications
APPLICATIONS TO PROCESS CONTROL PROBLH1S: A REVIEW Since the introdu cti on of fuzzy set theory there have been numerous attempts to forward th e th eo r eti cal basis and their applications to areas such as cluster analysis, decision ana l ys is, medical diagnosis , and in particu lar to th e control of industrial processes. In thi s paper , we briefly review some of the work that is direc tl y related to control problems . Several att emp t s have be en made to mode l the operat or ' s s trategy in a feedback control loop for the control processes using fuzzy mathematics. The main advantage of this fuzz y approach lies in the fact that th e notion of fuzzy se t th eory helps to implement th e operator's experience, heuristics and intuition on a computer, and also that this approach do es not require a detailed and accurate description (mode l) of th e process as demanded by the modern control theory . However, in spi te of several successful test cases reported in the literature , the development of the applications i s s till at its embryo nic l eve l, and we have to work t o advance such concepts as that of the s t ability , optimality, etc . This does not necessaril y mean that one needs to develop the definition and concepts in analogous f as hion to that of th e modern control theory. A control scientist, when attempting to design a f eedback control scheme using fuzzy a l go rithms, is faced with two basic problems . 1. The fir s t problem is the determination of th e primary fuzzy variab l es f or the pro cess , and the tran s l atio n of fuz zy rules into fuzzy con trol algorithms . This problem is more related to th e determination of input-output variables an d modelling of the process structure. A use of the a priori kn ow l edge of the pr ocess base d upon the operator's experience and intuition will be mos t helpful as well as the understanding of physical, chemical and engineering principles. 2.
The second problem is concerned with th e
A selected bibliography of th e industrial control applic ations of fuzzy set theory is given towards the end of this paper. The first application to industrial processes was reported by Mamdani, Assilian and Baak lini [2, 3, 26, 30]. This, followed by some other significant work by King, MamOOn~ Kickert, Ton g , 0ster gaard and Pappis [26, 27, 30, 32,42,43,45], their findings on pi lot plant are Significant. Here we list th e names of some of the important groups which were or are involved with the fu zzy logics in process contro l applications: De lft Technical Hi gh School, Delft, Holland: Professor Van Nau ta and W.J.M. Kickert (now at the Technical Univers ity of Eindhoven) control of a warm water plant. Danish Technical High School, Lyngby, Denmark: Professor P.M. Larsen and Dr. J.J. 0stergaard - fuzzy l og ic control of a pilot exchanger process. Hct·laster Uni ve rsity, Canada: Professor N.K . Sinha and J. W. I'Irigh t - heat exchanger process. Umist, l~ anche s t er , England: Dr. D.A. Rutherford - studies on an industrial plant, si nter making plant (in cooperation with G.A Carter and M.J. Hague of British Steel Corporation, 1.liddleborough) . British Steel Corporation, Battersea, England: (also at the Universi t y of Cambridge, England), Dr . R. Tong - studies on basic oxygen steel making progess. Warren Spring Laborato r y, Stevenage, England: Dr. P.J . Ki ng (jointl y with Drs. E.il. 11amdani and W.J .l'~. Kickert of Queen nary College) - pilot study on batch chem:ical process. Mamdani and Assilian [2, ~] were concerned with the control of a small laboratory steam engine. The control problem being to regulat e engine speed and boiler steam pressure by means of the heat applied to the boiler and the throttl e setting on the engine. The
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difficulties with the process are that it is nonlinear, noisy, strongly coupled and rather hard to control manually. However, their algorithm was found to perform better than a well tuned DDC algorithm and it appears to be less sensitive to changes in the operating conditions. At the same time, but independently of this work, Kickert with van Nauta Lemke [24] examined the performance of a fuzzy control algorithm for an experimental warm water plant. Here the problem is to regulate the temperature of water leaving a tank at a constant flow rate by altering the flow of hot water in a heat exchanger contained in the tank. A secondary control task was to ensure a fast response to step changes in outlet water temperature set point, the basic difficulties being that the process is nonlinear and noisy, and has asymmetric gain characteristics and a pure time delay. The performance of their basic algorithm was compared with an optimal PI controller and showed a faster step response, settling in approximately 0.3 min. compared with 0.7 min. for a 10DC set point change. An improved version of the algorithm also had a stead~ state performance as comparable to that of the PI controller. The success of Mamdani and Assilian's work led King and Mamdani [27] to attempt the control of temperature in a pilot scale batch chemical reactor. The control input to this process is the heating/cooling applied to the batch kettle, the main difficulties in control being a nonlinear timevarying gain and a pure time delay. Further work by Kickert [23] showed that the fuzzy controller was at least as good as a human controller. A sophisticated PI controller, with which it was also compared, did better but was more sensitive to process changes. A fourth application is reported by Rutherford and Carter [43] in which a fuzzy algorithm is used to control the raw mix permeability of a full scale sinter strand. This is a single - input single-output process in which the control variable is the flow of water to the process. The principal difficulties here are a nonlinear time-varying gain, noise and a pure time delay. These results show that the fuzzy algorithm performed better than a human process controller, reducing the standard deviation of permeability by over 40% and just as well as a conventional PI controller . The insensitivity to process parameter changes reported by others was also observed. 0stergaard [36] reports the results of experiments with a fuzzy algorithm for controlling a small scale heat exchanger. This is a two-input, two-output process in which the control problem is to regulate the cold water outlet temperature and the hot water inlet temperature by means of the hot water flow and the power used to heat it, the difficulty being that the process is highly
nonlinear and strongly coupled. The results show that the fuzzy controller did just about as well as a two loop PlO controller. Tong's experiments [44,46) were concerned with the control of a pressurized tank containing liquid. The control problem is to regulate the total pressure and the level of liquid inside the tank by altering the rates of flow of the liquid into the tank and the pressurizing air. The main features of this process which make control difficult are that it is nonlinear, strongly coupled, and has two very different time constants of approximately 3 and 3000 sec. Good control was achieved although this was not as good as that obtained by a controller designed using the characteristic locus technique. CONCLUSIONS The brief survey given above shows that in spite of some encouraging results in the area of process controls via fuzzy set theory, the development is still at an early stage, and it is difficult to draw any solid conclusion~ Theoretical work is growing at an exponential rate, but we need some theoretically suppolted applications-oriented work for the advancement of the fuzzy field. As we have described, modern processes are too complex to structure a fairly good model that has a compatibility with the 'exactness' of the modern control theory. However, these processes are often controlled with great skill by a human operator who makes decisions on the basis of intuition and linguistic measures of the process state. Some of the works described in this paper show that qualitative information can be for malized via fuzzy set theory for their processing by a computer, and that this information can be used in fuzzy control algorithm to control a process without having an explicit model of the process to be controlled. There are, however, many directions in which further work is required before we can convince the process industry on the useful applications of these fuzzy algorithms. We know that the learning and adaptive capabilities of a controller in the conventional control scheme have played a very important part in handling parameter uncertainties and other ignorances associated with the process and its measurements. We have an intuitive feeling that a fuzzy controller with heuristic learning and adaptive capabilities will have great potential in the process industry control applications, and this direction must be followed closely in the future development area. Also, it is felt that one does not have to follow too rigidly to develop the fuzzy concepts and fuzzy mathematics in an analogous manner to that of the modern theory.
Feedback Control Applications GENERAL BIBLIOGRAPHY ON FUZZY SET THEORY 1. Aizmerman,
~I.A. (1976), Fuzzy sets, fuzzy proofs and some unsolved problems in the theory of automatic control, Special Interest Discussion Session Fuzzy Automata and Decision Processes, 6th IFAC World Congress, Boston, Mass. 2. Assilian, S. (1974). "Artificial intelligence in the control of real dynamic systems", Ph.D. Thesis, London University. 3. Assilian, S. and I--Iamdani, E.H. (1974), "Learning Control Al gori thms in Real Dynamic Systems", Proc. 4th Int. IFAC/IFIP Conf. on Digital Computer Applications to Process Control, Zurich, Harch. 4. Bainbridge, L. (1975), "The Process Controller" in the Study of Real Skills, ed. W.T. Singleton, Academic Press, N.Y. 5. Bellman, R.E. and Zadeh, L.A. (1970), Decision Making in a Fuzzy Environment, Management Science, Vol. 17, pp. B14lB.164. 6. Bellman, R.E. and Zadeh, L.A. (1976), "Local and Fuzzy Logics", ERL rIemorandum M-584, University of California, Berkeley. Appeared in Modern Uses of Multiple-Valued Logic, D. Epstein, ed., D. Reidel, Dordrecht. 7. Carter, G.A. and Hague, M.J. (1976), "Fuzzy Control of Raw t-lix Permeabili ty at a Sinter Plant", Proc. Workshop on Discrete Systems and Fuzzy Reasoning, Queen Mary College, London 8. Gaines, B.R. (1976), General fuzzy logics, Proc. 3rd European Meeting on Cybernetics and Systems Research, Vienna. 9. Gaines, B.R. (1976), Fuzzy reasoning and the logic of uncertainty, Proc. 6th Int. Symp. Multiple-values Logic, Utah IEEE 76CH llll-4C, 179-188. 10. Gaines, B.R. (1976), "Understanding Uncertainty", Proc. Workshop on Discrete Systems and Fuzzy Reasoning, Queen Mary College, London. 11. Gaines, B.R. (1977), "Foundations of Fuzzy Reasoning", (appeared in 16). 12. Gaines, B.R. and Kohout, L.J. (1977), ''The Fuzzy Decade: A Bibliography of Fuzzy Sets" (appeared in 16). 13. Gupta, ~U1. (1975), IFAC Report: Round Table Discussion on the Estimation and Control ina Fuzzy Environment, Automa tica, Vol. 11, 209-212. -----14. Gupta, ~1.~1. (1975), Fuzzy automata and decision processes: The First Decade, Sixth Trienniel World IFAC Congress, Boston/Cambridge, August 24-30. 15. Gupta, ~U1. and t.lamdani, E.H. (1975), IFAC Report on the second IFAC Round Table Discussion on Fuzzy Automata and Decision processes, held at the Sixth Trienniel IFAC World Congress, Boston/ Cambridge, August. Also in Automatica, Vol. 12, pp. 241-296. 16 Gupta, M.M., Saridis, G.N. and Gaines, B. (1977), "Fuzzy Automata and Decision Processes", Edited Volume, Elsevier
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North-Holland. 17. Gupta, M.M., Ragade, R.K., Yager, R. (1977), "Report of the IEEE Special Symposium on Fuzzy Set Theory and Applications", at the 1977 CDC, New Orleans, December, 1978. 18. Gupta, M.M. (1977), "Fuzzy-ism", the First Decade", (appeared in 16). 19. Gupta, 11.~1., Ragade, R.K. (1978) "Fuzzy Set Theory and its Applications", (invited paper), IFAC Symposium on l-lultivariable Systems, New Brunswick, July. 20 Gupta, M.~I., Nikiforuk, P.N. (1978), (a) "Feedback Control of Industrial Problems Via Fuzzy Set Theory, Some Remarks", Joint Automata Control Conference, Philadelphia, pp. 1-16, Session 27. 20 Gupta, '1.14., Ragade, R.K., and Yager, R. (b) (1979), "Advances in Fuzzy Set Theory and Applications", North Holland. 21. Kaufman, A. (1975), Introduction to the Theory of Fuzzy Subsets, Vol. 1, New York Academic Press. 22. Kickert, iLJ.M. (1975 a) "Analysis of a fuzzy logic controller" Internal Report, Queen ~lary College London. 23. Kickert, W.J .M. (1975 b), "Further Analysis and Application of Fuzzy Logic Control", Internal Report, Queen Mary College, London. 24. Kickert, W.J.M. and Van Nauta Lemke, H.R. (1976), "Application of a Fuzzy Controller in a Warm Water Plant", Automatica, Vol 12, 301-308. 25. Kling, R. (1974), Fuzzy Planner: Reasoning with inexact concepts in a procedural problem-solving language, J. Cybernetics, 4, 105-122. 26. Kickert, W.J.M., Mamdani, E.H. (1978), "Analysis of Fuzzy Logic Controller", Int. J. Fuzzy Sets and Systems, pp. 29-44. 27. King, P.J. and Mamdani, E.H. (1976), "The Application of Fuzzy Control Systems to Industrial Processes", Proc. Workshop on Discrete Systems and Fuzzy Reasoning, Queen Mary College, London. 28. Lee, E.T. and Chang, C.L. (1971), Some properties of fuzzy logic, Information and Control, 19, 417-431. 29. Mamdani, E.H. and Assilian, S. (1975), "An Experiment in linguistic synthesis with a fuzzy logic controller", Int. J. Man-Machine Studies, 7, 1-13. 30. fo.1amdani, E.H. and Baaklini, N. (1975), "Prescripti ve method for deriving control policy in a fuzzy-logic controller", Electronics Letters, Vol. 11, pp. 625. 31. Hamdani, E.H. (1976), Application of fuzzy logic to approximate reasoning using linguistic synthesis, Proc. 6th Int. Symp. Multiple-values Logic, Utah, IEEE 76CH llll-4C, 192-202. 32. Mamdani, E.H., Procyk, T.J., and Baaklini, N. (1976), "Application of Fuzzy Logic Controller Design based on Linguistic Protocol", Proc. Workshop on Discrete Systems and Fuzzy
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Reasoning, 0ueen Hary College, London. 33. Mamdani, E.H. (1977), "Applications of fuzz y set theory to control systems: A survey", (appeared in 16). 34. Marks, P. (1976), "A Fuzzy Logic Control So ftware", Internal Report, Queen Nary College,London 35. Negoita, C.V. (1975), Introduction to Fuzzy Set Theory for Systems Analysis, New York, John Wiley & Sons Ltd. 36. ~stergaard, J.J. (1976), "Fuzzy Logic Con tro1 of a heat exchanger proces sI!, Internal Report, Electric Power Engineering Department, Danis Technical Highshcool, Lyngb y , Denmark (also appeared in 16). 37. Pappis, C.P. and 11amdani, E.H. (1976), "A Fuzzy Logic Controller for a Traffic Junction", Research Report, Department of Electrical Engineering, Queen Mary College, London. 38. Peray, K.E. and Waddell, J.J. (1972), The Rotary Cement Kiln, The Chemical Publishing Co., New York. 39. Procyk, T.J. (1976), "A proposal for a Learning Sys tem", Internal Report, Queen Mary College, London. 40. Ragade, R.K. and Gupta, M.M. (1977), "Fuzzy Set Theory: An Introduction", (appeared in 16). 41. Rutherford, D.A. and Bloore, G.C. (1975), "The Implementation of Fuzzy Algorithm for Control", Control System Centre Report No. 279, UMIST, Manches ter, (to appear in Proc. IEEE). 42. Rutherford, D.A. (1976), "The Implementation and Evaluation of a Fuzzy Control Algorithm for a Sinter Plant", Proc. Workshop on Discrete Systems and Fuz zy Reasoning, Queen Mary Co lIege, London. 43. Rutherford, D. and Carter, G.A. (1976), A Heuristic Adaptive Controller for a Sinter Plant, Proc. 2nd IFAC Symp. Automation in Mining, ~1ineral and Metal Processing. 44. Tong, R.Il. (1976), "An Assessment of a Fuzzy Control Algorithm for a NonLinear Multi-Variable System", Proc. Workshop on Discrete Systems an-d--Fuzzy Reasoning, Queen Mary College, London. 45. Tong, R.M. (1976), Some Problems with the Design and Implementation of Fuzzy Controllers. Internal Report CUED-F-CAMS TR127, Cambridge University Engineering Department. 46. Tong. R.I~. (1977), A Control Engineering Review of Fuzzy Systems", Automatica, Vol. 13, pp. 559-569. 47. Tsypkin, Ya. Z. (1968), Is there a theory of optimal adaptive systems? Automat. Remote Control, (USSR), 29 (1). 48. Zadeh, L.A. (1958), What is optimal? IRE Trans. Inform. Theory, IT-4 (1), 3. 49. Zadeh, L.A. (1963), On the definition of adaptivity , Proc. IEEE, 51, 469. 50. Zadeh, L.A. (1965), Fuzzy sets, Information and Control, 8, 338-353.
51. Zadeh, L.A. (1972), A new approach to system analysis. In Marois, M. (ed.) Man and Computer, Amsterdam: NorthHolland, 55-94. 52. Zadeh, L.A. (1973), OUtline of a new approach to the anal ysis of complex systems and decision processes, IEEE Trans. Syst. Man Cybern., SMC-l,---28-44. 53. Zadeh, L.A. (1973), "Outline of a new approach to the analysis of complex systems and decision processes", IEEE Trans., S, C-3, 28. 54. Zadeh, L.A., Fu, K.S., Tanaka, K., and Shimura, ~1. (1975), Fuzzy Sets and their Application to Cognitive and Decision Processes, New York: Academic Press. 55. Zadeh, L.A. (1975), "Fuzz y Logic and Approximate Reasoning (In ~1emory of Grigore Moisil)", Synthese, Vol. 30, pp. 407-428. 56. Zadeh, L.A. (1975), "The Concept of a Linguistic Variable and its Application to Approximate Reasoning", Inf. Sci., Part I, Vol. 8, pp. 199-24g;Part II, Vol. 8, pp. 301-357; Part IIr, Vol. 9, pp. 43-80. 57. Zadeh, L.A. (1976), "The Linguistic Approach and Its Application to Decision Analysis", in Directions in Large Scale Systems, Y.C. Ho and S.K. I~itter, eds., pp. 339-370. Plenum Press, New York. 58. Zadeh, L.A. (1976), "A Fuzzy Algorithmic Approach to the Definition of Complex or Imprecise Concepts", Inter. Jour. Man-Machine Studies, Vol. 8, pp. 249291. 59. Zadeh, L.A. (1976), "Semantic Inference from Fuzzy Premises", Proc. 6th Inter. Symp. on Multiple-Values Logic, Utah State University, Logan, pp. 217-218. 60. Zadeh, L.A. (1977), "Fuzzy Set Theory: A Perspective", (appeared in 16).
Copyright © IFAC Control Sri cncc and Technology 18th Triennial World Congress) K\,oto. Japan , 1981
FEEDBACK CONTROL APPLICATIONS OF FUZZY SET THEORY: A SURVEY M. M. Gupta Cybernetz'cs Research Laboratory, College of Engl'neerz'ng, Unz'versz'ty of Saskatchewan, Saskatoon, Saskatchewan, S7N WO, Canada
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Abstract. The development of feedback control theory for applications to industrial processes is one of the fundamental issues facing the control scientist. This issue arises because of the complexity and vagueness in the description of such processes. Some attempts have been made to use the fuz zy set theory in the design of feedback control algorithms for industrial applications and this development is still in its infancy . In this paper we gi ve a bri e f survey of some of the work that is being done in this fi e ld. Keywords. Automata Theory, Computer Control, Fuzzy Controll er, Feedback Control, Fuzzy Lo gic Control, Linguistic Control. terms in contrast to precise numeric data as demanded by computers. Much of this difference lies in th e insufficiently understood ch aracte ristics of the human brain, '.e., a f a culty of global reasoning and pattern recognition at man y l evels. As the system becomes complex," . . our ability to make precise and yet significant statements about its behavior diminishes, until a threshold is reached beyond which precision and significance (or relevance) become mutually exclusive characteristics". L.A. Zadeh [53].
INTRODUCTIO N Th e development of feedback control theory for applications to industrial process which are vague, imprecise and complex i s one o f the most fundamental i ss ues and challengin g problems facing the control scientist [1, 46~ At present, several attempts have been made to use the modern control theory in the analysis and design of such feedback control systems but with very little successes. ~Iod ern control theory has had tremendous success in areas where the system is well defined either deterministically or stochasticall y . Aerospace systems, missile and space vehicl e guidance systems which can be described with a mathematical precision are some examples where the modern control th eory has proved to be very successful. Control scientists att empted to appl y these theories to modern complex plants such as in manufacturing technology, chemical process, pulp and paper mills, power plants, steel making industry, cement klein, etc., but they faced nothing but failures and frsutrations; these theories are not capable of coping with the vagueness and imprecision inh erent in an y practical plant despite the development of a huge body of mathematical knowledge.
The essential featur es of humans in decis ion-making prompts the question 'can we make comparable use of ambiguity , imprecision and human language in the analysis and design of feedback controllers for complex industrial processes?' Fuzz y set theory was founded by L.A. Zadeh in 1965 [SO]. This theory is a mathematical concept that gives some mathematical precision to imprecise and ambiguous statements and the vagueness inherent in modern complex processes. Its essential concept is that of the graded membership. In ordinary set theory , a member is either in or it is out of a subset. In Boolean logic a proposition is e ither true or false. The introduction o f graded membership allows us an infinite extension to the basic concepts in set theory and logic in a way that appears to be very natural. An underlying philosophy of the fuzzy set theory is to provide a strict mathematical framework. It provides a gradual transition from the realm of vigorous, quantitative and precise phenomena to that of vague, qualitative, and imprecise conceptions. This theory enables one to characterize impreclslon in terms of "fuzziness", a concept to which one can assign many meanings [11, 12 , 18, 21, 40, 60].
There are undoubtedl y many reasons for these failures, but primaril y it is the lack of detailed and precise structural knowledge of the process demanded by th e modern control theory. Despite this, these processes are often satisfactorily controlled by a human operator who has the ability to reason and act on information which is imprecise. He has a remarkable adaptibility to interpret linguistic statements ab out the process and to reason in a qualitati ve fashion. Vaguenes s , imprecisi on, and ambigui t. ' have not deterred human beings from taking actions. Communication between humans is in linguistic
In aggregation and modelling problems, some 761
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information and, hence, precision is lost. Th e precis may have what seemed to be the mo s t needed ar.d relevant information, yet Some other aspects are lost in such modeling. For many purposes, a very approximate characteri za tion or mode ling of a collection of data is sufficient because most of the basic tasks performed by humans do not require a high degree of precision in their executio~ For very highly precise and detailed modelling, one needs equally an elaborate system of mathematical description and measurements . In any real life industrial situation, this is difficult to achieve. Supervisory control by a human operator with a set of linguistically described rules can achieve successful control in an industrial situation. Supervisory personnel with years of experience can express their control process effectively in linguistic terms but not so effectively in mathematical terms. These complex processes are often controlled with great skill by a human operator who makes decisions on the basis of intuition and linguistic measures of the process state. Unfortunately, the speed and capacity of the human operator is limited to handle the multivariable situation at the speed modern processes run. SUBJECTIVITY, INTUITION, VAGUENESS, PRECISION AND MODERN CONTROL THEORY Feedback control theory has evolved during the last fifteen years from an empirically oriented approximate method such as Bode, Nyquist, Nichols and root-locus methods, to a strongly mathematically based modern control theory approach. Modern control theory is very precise and has well-defined control concepts, and needs a precise knowledge of the process model to be controlled and exact measurement (either in deterministic or probabilistic sense) of input and output data. In spite of the precision of the modern control theory, both in the modelling and design, the degree of vagueness and subjectivity did not alter much. For example, in modelling a process to a large extent the individual's experience and mathematical convenience were primarily responsible in determining the process structure and its parameters. So was the situation in the design of a feedback controller to give a 'good' system performance. The large spread in the design criteria found in each design was mainly dependent upon the designer's personal views and his experiences. Thus, subjectivity, inexactness and vagueness still play a major role even in these modern control tools which are mathematically precise. There is a question, however, which remains unanswered. 'How good are these modern mathematical control tools which are too precise for the applications to these modern processes which are complex, too ill-defined, and for which a precise model almost does not exist?' These processes under consideration are non-
linear, time-varying and ar e subjec t to s t ochastic disturbance s i n input and output measurements, and s tochas t ic behavioral pe rturbations in parameters. Surprisingly, it has been reported that in many cases a human operator is able t o control the se processes much more successfully than an automati c controller designed using the modern co ntrol theory. There are example s known, for example in a power industry, that a simple controller designed using classical approach is being adopted s ucc ess full y. The app lic ation of modern control theory t o these pro cesses gave nothing worth reporting but th e failures and frustration s . In some cases, successful applicat ions of adaptive methods, which do not rely heavily upon the ' exac tness' of the pro cess model, have been reported (47). WHY FAILURES? It is clear from the various real-life app lication -oriented publi ca t io ns th at have appeared during the l as t decade, and fr om the brief remarks made above tha t the ' vagueness ' associated with the modelling of too compl ex and ill-defined processes is not compati bl e with the 'precision' of the mode rn control tools. In many pro cess si tuations such as in steel making furnaces, cement kiln s , and processes in glass industry, it i s almost impossible to construct a r easonab ly good model structure and estimate its parameters whi ch are compatible with the preci s ion of modern control theory. Of course, no one will deny that modern control theory has resulted into control algorithms whi ch have been very successful on computer s imul ation models of industrial proces ses , but designers have failed to convenience the industry on the use fulness of these a lgorithms in practice . For many years we have had the convi ction that the most fruitful areas for the growth in the applications f i eld of automatic control are those which have been neglected as a no-man' s land between the various es tablished field s. It is clear, therefore , that if the proces s cannot be modelled accurately as demanded by the modern control theory, we must work to develop alternate approaches which a re compatible to the inaccuracies associated with the modelling of the se processes. QUALITATIVE DES CR IPTION OF THE PROCESSES AND FUZZY CONTROLLER It is well known that for these complex processes a sufficient amount of qualitati ve ~ priori information is available to the designer. The design criteria may also be formed based upon the operator's experience. This qualitative information and the de sign specifications can be described only linguistically, which inherently is s ubjected to the subjectivity of the human operat or and that of the designer. An approach which can handle this imprecision created by human experience and intuition will be most desirab l e [20,21,23,24]. The approach must possess
Feedback Control Applications some of the followin g characteristics: - mOdelling capab ility the inexactness associated with human subjectivity, - linguistic apDroach to link the operator wi th the computer f or on -l ine computer control. It is the cont ention of thi s paper that the sor t of complexity and uncertainty that ar e being generated in modellin g the processes is better captured by the 'notion of fuzzine ss ' than that of the probability . The notion of fuzziness is explained in a number of r epo rts, papers and books [16, 20, 3S, 54 J by many world researchers in the field.
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impl ementation of th e fuzzy algorithm using either hardware or a computer program. This will need a numeric a l characterization of the linguistics . It consists of several problems, all related t o the overall problem of realization of fuzzy a l gor ithm for a nonfuzz y machin e vi z . a digital computer. Some of th e associated problems are the de terminati on of the range of values of the control~ the choice of quantization levels for the controller inputs and outputs, the numerical definition of the primary fu zzy sets on the quantized spaces, and th e choice of the sampling interval. All th ese problems are interdependent and mus t be considered in keeping a particular process and control objectives in vi ew . Some Process App l ications
APPLICATIONS TO PROCESS CONTROL PROBLH1S: A REVIEW Since the introdu cti on of fuzzy set theory there have been numerous attempts to forward th e th eo r eti cal basis and their applications to areas such as cluster analysis, decision ana l ys is, medical diagnosis , and in particu lar to th e control of industrial processes. In thi s paper , we briefly review some of the work that is direc tl y related to control problems . Several att emp t s have be en made to mode l the operat or ' s s trategy in a feedback control loop for the control processes using fuzzy mathematics. The main advantage of this fuzz y approach lies in the fact that th e notion of fuzzy se t th eory helps to implement th e operator's experience, heuristics and intuition on a computer, and also that this approach do es not require a detailed and accurate description (mode l) of th e process as demanded by the modern control theory . However, in spi te of several successful test cases reported in the literature , the development of the applications i s s till at its embryo nic l eve l, and we have to work t o advance such concepts as that of the s t ability , optimality, etc . This does not necessaril y mean that one needs to develop the definition and concepts in analogous f as hion to that of th e modern control theory. A control scientist, when attempting to design a f eedback control scheme using fuzzy a l go rithms, is faced with two basic problems . 1. The fir s t problem is the determination of th e primary fuzzy variab l es f or the pro cess , and the tran s l atio n of fuz zy rules into fuzzy con trol algorithms . This problem is more related to th e determination of input-output variables an d modelling of the process structure. A use of the a priori kn ow l edge of the pr ocess base d upon the operator's experience and intuition will be mos t helpful as well as the understanding of physical, chemical and engineering principles. 2.
The second problem is concerned with th e
A selected bibliography of th e industrial control applic ations of fuzzy set theory is given towards the end of this paper. The first application to industrial processes was reported by Mamdani, Assilian and Baak lini [2, 3, 26, 30]. This, followed by some other significant work by King, MamOOn~ Kickert, Ton g , 0ster gaard and Pappis [26, 27, 30, 32,42,43,45], their findings on pi lot plant are Significant. Here we list th e names of some of the important groups which were or are involved with the fu zzy logics in process contro l applications: De lft Technical Hi gh School, Delft, Holland: Professor Van Nau ta and W.J.M. Kickert (now at the Technical Univers ity of Eindhoven) control of a warm water plant. Danish Technical High School, Lyngby, Denmark: Professor P.M. Larsen and Dr. J.J. 0stergaard - fuzzy l og ic control of a pilot exchanger process. Hct·laster Uni ve rsity, Canada: Professor N.K . Sinha and J. W. I'Irigh t - heat exchanger process. Umist, l~ anche s t er , England: Dr. D.A. Rutherford - studies on an industrial plant, si nter making plant (in cooperation with G.A Carter and M.J. Hague of British Steel Corporation, 1.liddleborough) . British Steel Corporation, Battersea, England: (also at the Universi t y of Cambridge, England), Dr . R. Tong - studies on basic oxygen steel making progess. Warren Spring Laborato r y, Stevenage, England: Dr. P.J . Ki ng (jointl y with Drs. E.il. 11amdani and W.J .l'~. Kickert of Queen nary College) - pilot study on batch chem:ical process. Mamdani and Assilian [2, ~] were concerned with the control of a small laboratory steam engine. The control problem being to regulat e engine speed and boiler steam pressure by means of the heat applied to the boiler and the throttl e setting on the engine. The
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difficulties with the process are that it is nonlinear, noisy, strongly coupled and rather hard to control manually. However, their algorithm was found to perform better than a well tuned DDC algorithm and it appears to be less sensitive to changes in the operating conditions. At the same time, but independently of this work, Kickert with van Nauta Lemke [24] examined the performance of a fuzzy control algorithm for an experimental warm water plant. Here the problem is to regulate the temperature of water leaving a tank at a constant flow rate by altering the flow of hot water in a heat exchanger contained in the tank. A secondary control task was to ensure a fast response to step changes in outlet water temperature set point, the basic difficulties being that the process is nonlinear and noisy, and has asymmetric gain characteristics and a pure time delay. The performance of their basic algorithm was compared with an optimal PI controller and showed a faster step response, settling in approximately 0.3 min. compared with 0.7 min. for a 10DC set point change. An improved version of the algorithm also had a stead~ state performance as comparable to that of the PI controller. The success of Mamdani and Assilian's work led King and Mamdani [27] to attempt the control of temperature in a pilot scale batch chemical reactor. The control input to this process is the heating/cooling applied to the batch kettle, the main difficulties in control being a nonlinear timevarying gain and a pure time delay. Further work by Kickert [23] showed that the fuzzy controller was at least as good as a human controller. A sophisticated PI controller, with which it was also compared, did better but was more sensitive to process changes. A fourth application is reported by Rutherford and Carter [43] in which a fuzzy algorithm is used to control the raw mix permeability of a full scale sinter strand. This is a single - input single-output process in which the control variable is the flow of water to the process. The principal difficulties here are a nonlinear time-varying gain, noise and a pure time delay. These results show that the fuzzy algorithm performed better than a human process controller, reducing the standard deviation of permeability by over 40% and just as well as a conventional PI controller . The insensitivity to process parameter changes reported by others was also observed. 0stergaard [36] reports the results of experiments with a fuzzy algorithm for controlling a small scale heat exchanger. This is a two-input, two-output process in which the control problem is to regulate the cold water outlet temperature and the hot water inlet temperature by means of the hot water flow and the power used to heat it, the difficulty being that the process is highly
nonlinear and strongly coupled. The results show that the fuzzy controller did just about as well as a two loop PlO controller. Tong's experiments [44,46) were concerned with the control of a pressurized tank containing liquid. The control problem is to regulate the total pressure and the level of liquid inside the tank by altering the rates of flow of the liquid into the tank and the pressurizing air. The main features of this process which make control difficult are that it is nonlinear, strongly coupled, and has two very different time constants of approximately 3 and 3000 sec. Good control was achieved although this was not as good as that obtained by a controller designed using the characteristic locus technique. CONCLUSIONS The brief survey given above shows that in spite of some encouraging results in the area of process controls via fuzzy set theory, the development is still at an early stage, and it is difficult to draw any solid conclusion~ Theoretical work is growing at an exponential rate, but we need some theoretically suppolted applications-oriented work for the advancement of the fuzzy field. As we have described, modern processes are too complex to structure a fairly good model that has a compatibility with the 'exactness' of the modern control theory. However, these processes are often controlled with great skill by a human operator who makes decisions on the basis of intuition and linguistic measures of the process state. Some of the works described in this paper show that qualitative information can be for malized via fuzzy set theory for their processing by a computer, and that this information can be used in fuzzy control algorithm to control a process without having an explicit model of the process to be controlled. There are, however, many directions in which further work is required before we can convince the process industry on the useful applications of these fuzzy algorithms. We know that the learning and adaptive capabilities of a controller in the conventional control scheme have played a very important part in handling parameter uncertainties and other ignorances associated with the process and its measurements. We have an intuitive feeling that a fuzzy controller with heuristic learning and adaptive capabilities will have great potential in the process industry control applications, and this direction must be followed closely in the future development area. Also, it is felt that one does not have to follow too rigidly to develop the fuzzy concepts and fuzzy mathematics in an analogous manner to that of the modern theory.
Feedback Control Applications GENERAL BIBLIOGRAPHY ON FUZZY SET THEORY 1. Aizmerman,
~I.A. (1976), Fuzzy sets, fuzzy proofs and some unsolved problems in the theory of automatic control, Special Interest Discussion Session Fuzzy Automata and Decision Processes, 6th IFAC World Congress, Boston, Mass. 2. Assilian, S. (1974). "Artificial intelligence in the control of real dynamic systems", Ph.D. Thesis, London University. 3. Assilian, S. and I--Iamdani, E.H. (1974), "Learning Control Al gori thms in Real Dynamic Systems", Proc. 4th Int. IFAC/IFIP Conf. on Digital Computer Applications to Process Control, Zurich, Harch. 4. Bainbridge, L. (1975), "The Process Controller" in the Study of Real Skills, ed. W.T. Singleton, Academic Press, N.Y. 5. Bellman, R.E. and Zadeh, L.A. (1970), Decision Making in a Fuzzy Environment, Management Science, Vol. 17, pp. B14lB.164. 6. Bellman, R.E. and Zadeh, L.A. (1976), "Local and Fuzzy Logics", ERL rIemorandum M-584, University of California, Berkeley. Appeared in Modern Uses of Multiple-Valued Logic, D. Epstein, ed., D. Reidel, Dordrecht. 7. Carter, G.A. and Hague, M.J. (1976), "Fuzzy Control of Raw t-lix Permeabili ty at a Sinter Plant", Proc. Workshop on Discrete Systems and Fuzzy Reasoning, Queen Mary College, London 8. Gaines, B.R. (1976), General fuzzy logics, Proc. 3rd European Meeting on Cybernetics and Systems Research, Vienna. 9. Gaines, B.R. (1976), Fuzzy reasoning and the logic of uncertainty, Proc. 6th Int. Symp. Multiple-values Logic, Utah IEEE 76CH llll-4C, 179-188. 10. Gaines, B.R. (1976), "Understanding Uncertainty", Proc. Workshop on Discrete Systems and Fuzzy Reasoning, Queen Mary College, London. 11. Gaines, B.R. (1977), "Foundations of Fuzzy Reasoning", (appeared in 16). 12. Gaines, B.R. and Kohout, L.J. (1977), ''The Fuzzy Decade: A Bibliography of Fuzzy Sets" (appeared in 16). 13. Gupta, ~U1. (1975), IFAC Report: Round Table Discussion on the Estimation and Control ina Fuzzy Environment, Automa tica, Vol. 11, 209-212. -----14. Gupta, ~1.~1. (1975), Fuzzy automata and decision processes: The First Decade, Sixth Trienniel World IFAC Congress, Boston/Cambridge, August 24-30. 15. Gupta, ~U1. and t.lamdani, E.H. (1975), IFAC Report on the second IFAC Round Table Discussion on Fuzzy Automata and Decision processes, held at the Sixth Trienniel IFAC World Congress, Boston/ Cambridge, August. Also in Automatica, Vol. 12, pp. 241-296. 16 Gupta, M.M., Saridis, G.N. and Gaines, B. (1977), "Fuzzy Automata and Decision Processes", Edited Volume, Elsevier
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North-Holland. 17. Gupta, M.M., Ragade, R.K., Yager, R. (1977), "Report of the IEEE Special Symposium on Fuzzy Set Theory and Applications", at the 1977 CDC, New Orleans, December, 1978. 18. Gupta, M.M. (1977), "Fuzzy-ism", the First Decade", (appeared in 16). 19. Gupta, 11.~1., Ragade, R.K. (1978) "Fuzzy Set Theory and its Applications", (invited paper), IFAC Symposium on l-lultivariable Systems, New Brunswick, July. 20 Gupta, M.~I., Nikiforuk, P.N. (1978), (a) "Feedback Control of Industrial Problems Via Fuzzy Set Theory, Some Remarks", Joint Automata Control Conference, Philadelphia, pp. 1-16, Session 27. 20 Gupta, '1.14., Ragade, R.K., and Yager, R. (b) (1979), "Advances in Fuzzy Set Theory and Applications", North Holland. 21. Kaufman, A. (1975), Introduction to the Theory of Fuzzy Subsets, Vol. 1, New York Academic Press. 22. Kickert, iLJ.M. (1975 a) "Analysis of a fuzzy logic controller" Internal Report, Queen ~lary College London. 23. Kickert, W.J .M. (1975 b), "Further Analysis and Application of Fuzzy Logic Control", Internal Report, Queen Mary College, London. 24. Kickert, W.J.M. and Van Nauta Lemke, H.R. (1976), "Application of a Fuzzy Controller in a Warm Water Plant", Automatica, Vol 12, 301-308. 25. Kling, R. (1974), Fuzzy Planner: Reasoning with inexact concepts in a procedural problem-solving language, J. Cybernetics, 4, 105-122. 26. Kickert, W.J.M., Mamdani, E.H. (1978), "Analysis of Fuzzy Logic Controller", Int. J. Fuzzy Sets and Systems, pp. 29-44. 27. King, P.J. and Mamdani, E.H. (1976), "The Application of Fuzzy Control Systems to Industrial Processes", Proc. Workshop on Discrete Systems and Fuzzy Reasoning, Queen Mary College, London. 28. Lee, E.T. and Chang, C.L. (1971), Some properties of fuzzy logic, Information and Control, 19, 417-431. 29. Mamdani, E.H. and Assilian, S. (1975), "An Experiment in linguistic synthesis with a fuzzy logic controller", Int. J. Man-Machine Studies, 7, 1-13. 30. fo.1amdani, E.H. and Baaklini, N. (1975), "Prescripti ve method for deriving control policy in a fuzzy-logic controller", Electronics Letters, Vol. 11, pp. 625. 31. Hamdani, E.H. (1976), Application of fuzzy logic to approximate reasoning using linguistic synthesis, Proc. 6th Int. Symp. Multiple-values Logic, Utah, IEEE 76CH llll-4C, 192-202. 32. Mamdani, E.H., Procyk, T.J., and Baaklini, N. (1976), "Application of Fuzzy Logic Controller Design based on Linguistic Protocol", Proc. Workshop on Discrete Systems and Fuzzy
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Reasoning, 0ueen Hary College, London. 33. Mamdani, E.H. (1977), "Applications of fuzz y set theory to control systems: A survey", (appeared in 16). 34. Marks, P. (1976), "A Fuzzy Logic Control So ftware", Internal Report, Queen Nary College,London 35. Negoita, C.V. (1975), Introduction to Fuzzy Set Theory for Systems Analysis, New York, John Wiley & Sons Ltd. 36. ~stergaard, J.J. (1976), "Fuzzy Logic Con tro1 of a heat exchanger proces sI!, Internal Report, Electric Power Engineering Department, Danis Technical Highshcool, Lyngb y , Denmark (also appeared in 16). 37. Pappis, C.P. and 11amdani, E.H. (1976), "A Fuzzy Logic Controller for a Traffic Junction", Research Report, Department of Electrical Engineering, Queen Mary College, London. 38. Peray, K.E. and Waddell, J.J. (1972), The Rotary Cement Kiln, The Chemical Publishing Co., New York. 39. Procyk, T.J. (1976), "A proposal for a Learning Sys tem", Internal Report, Queen Mary College, London. 40. Ragade, R.K. and Gupta, M.M. (1977), "Fuzzy Set Theory: An Introduction", (appeared in 16). 41. Rutherford, D.A. and Bloore, G.C. (1975), "The Implementation of Fuzzy Algorithm for Control", Control System Centre Report No. 279, UMIST, Manches ter, (to appear in Proc. IEEE). 42. Rutherford, D.A. (1976), "The Implementation and Evaluation of a Fuzzy Control Algorithm for a Sinter Plant", Proc. Workshop on Discrete Systems and Fuz zy Reasoning, Queen Mary Co lIege, London. 43. Rutherford, D. and Carter, G.A. (1976), A Heuristic Adaptive Controller for a Sinter Plant, Proc. 2nd IFAC Symp. Automation in Mining, ~1ineral and Metal Processing. 44. Tong, R.Il. (1976), "An Assessment of a Fuzzy Control Algorithm for a NonLinear Multi-Variable System", Proc. Workshop on Discrete Systems an-d--Fuzzy Reasoning, Queen Mary College, London. 45. Tong, R.M. (1976), Some Problems with the Design and Implementation of Fuzzy Controllers. Internal Report CUED-F-CAMS TR127, Cambridge University Engineering Department. 46. Tong. R.I~. (1977), A Control Engineering Review of Fuzzy Systems", Automatica, Vol. 13, pp. 559-569. 47. Tsypkin, Ya. Z. (1968), Is there a theory of optimal adaptive systems? Automat. Remote Control, (USSR), 29 (1). 48. Zadeh, L.A. (1958), What is optimal? IRE Trans. Inform. Theory, IT-4 (1), 3. 49. Zadeh, L.A. (1963), On the definition of adaptivity , Proc. IEEE, 51, 469. 50. Zadeh, L.A. (1965), Fuzzy sets, Information and Control, 8, 338-353.
51. Zadeh, L.A. (1972), A new approach to system analysis. In Marois, M. (ed.) Man and Computer, Amsterdam: NorthHolland, 55-94. 52. Zadeh, L.A. (1973), OUtline of a new approach to the anal ysis of complex systems and decision processes, IEEE Trans. Syst. Man Cybern., SMC-l,---28-44. 53. Zadeh, L.A. (1973), "Outline of a new approach to the analysis of complex systems and decision processes", IEEE Trans., S, C-3, 28. 54. Zadeh, L.A., Fu, K.S., Tanaka, K., and Shimura, ~1. (1975), Fuzzy Sets and their Application to Cognitive and Decision Processes, New York: Academic Press. 55. Zadeh, L.A. (1975), "Fuzz y Logic and Approximate Reasoning (In ~1emory of Grigore Moisil)", Synthese, Vol. 30, pp. 407-428. 56. Zadeh, L.A. (1975), "The Concept of a Linguistic Variable and its Application to Approximate Reasoning", Inf. Sci., Part I, Vol. 8, pp. 199-24g;Part II, Vol. 8, pp. 301-357; Part IIr, Vol. 9, pp. 43-80. 57. Zadeh, L.A. (1976), "The Linguistic Approach and Its Application to Decision Analysis", in Directions in Large Scale Systems, Y.C. Ho and S.K. I~itter, eds., pp. 339-370. Plenum Press, New York. 58. Zadeh, L.A. (1976), "A Fuzzy Algorithmic Approach to the Definition of Complex or Imprecise Concepts", Inter. Jour. Man-Machine Studies, Vol. 8, pp. 249291. 59. Zadeh, L.A. (1976), "Semantic Inference from Fuzzy Premises", Proc. 6th Inter. Symp. on Multiple-Values Logic, Utah State University, Logan, pp. 217-218. 60. Zadeh, L.A. (1977), "Fuzzy Set Theory: A Perspective", (appeared in 16).