Adaptive Tuning of Fuzzy Logic Controllers

ADAPTIVE TUNING OF FUZZY LOGIC CONTROLLERS E. K. JUUSO, J. MYLLYNEVA and K. LEIVISKA Univer.~ty of 011.111., Department of ProcelJlJ Engineering, Control Engineering Laboratory Linnanmaa, 90570 Oulu, Finland Abstract. A Linguistic Equation Framework developed for adaptive expert systems provides a flexible enviro nment for tuning fuzzy logic controllers. Simulation results, expert knowledge and process experiments can be combined in the development procedure. Controllers represented by compact matrix equations are easi ly combined with corresponding linguistic process models. The controllers are tuned by adjusting the meanings of the linguistic variab les to different working areas. The results are used in real control practice by transferring them to automation systems, or to a fuzzy logic controller FuzzyCon where the rules and the membership functions can be changed online. FuzzyCon is connected to processes with a data acquisition card , and the data is transferred throug h DOE-links . Key Words. Adaptive systems; fuzzy systems; process control; expert systems : process models : simulation: nonlinear systems; knowledge engineering 1. I:'-l"TRODUCTION lating the models and in defining the estimates of some membership functions. Operator's control actions are used together with fuzzy and linguistic models in tuning the system. The theory of fuzzy logic provides a method for con\"erting the control knowledge of an operator into a control strategy. Usually, a fuzzy logic controller models the operator rat.her than the process. This is a quite useful procedure if t here really is a lack of a well-posed mathematical model, or if the process is highly nonlinear and sensitive in the operation region. The method also prO\ides an intuitively appealing form of rules which arc more readily customizable in nat.urallanguage terms than conventional cont rollers. 2. LINGUISTIC Sn.lULATIO:'-J Rule-based programming is commonly used III the development of expert systems. However , this paradigm leads to serious problems in practical applications. Maintaining massive rulebased systems is practically impossible. Actually, it is not even possible to reliably test the system in the first place. Therefore , linking the rule-based systems to more efficient modelling methods is essential for practical systems. However, there are also serious problems in designing a fuzzy logic controller. Acquiring the knowll'dge from the human operator is a tedious and time-consuming task at least if the ruleb<1.'ied procedure is used. Also a trial-and-error h<1.<;ed tuning is non-trivial and time consuming, and therefore, far from acceptable. The optimality and stability of the FLC is also hard to prow . Especially for more complicated applications, the designing procedure of FLC must be improved. Actually, it is better to use FLC together with conventional control systems. 2.1 . Linguistic rules The linguistic simulation was originally ba..<;ed on linguistic rules, n M if Z; then Y' (1) j=l where Z· and Y' are linguistic vectors, and there was a clear distinction between input and output variables (Juuso and Leiviska., 1990) . In the present tuning system, the linguistic rules used in previous systems are replaced by linguistic relations and linguistic equations (Juuso and Leiviska, 1991). In order to get flexibility, the rules are usually used in final applications. In this paper, the emphasis is laid on the methods which could be useful in combining existing simulation models and new ideas of describing qualitative models for t.he development of combined control system . Expert's experience and control engineering knowledge is used in formu- 82 Linguistic Equations Procedure I Interaction Matrix I I Fuzzy Constraints Matrix Equation I Rule-based knowledge I I .! Rules 1Relations Tuning I Simulation I I Expert Knowledge I IProcess Experiments I 1Membership Functions Relations Rule.s Adaptive Expert System I Application I Figure 1: Development of adaptive expert systems. 2.2. Linguistic relations The linguistic process model is described by groups of linguistic relations: each group can be ba.<;ed on a single fuzzy model, or several fuzzy equations can be aggregated into a single group of linguistic relations (Juuso and Leiviskii., 1991). The variables of the relations are chosen in such a way that the directions of the changes are balanced, e.g. the change-of-control output, flu, de('fea.<;es with increasing error, e, and increasing change-of-error, fle. E r r o PB §]~ PS §J §] ~ ZO §]~ NS ~§J] KB §] §] §] §] §] r A fuzzy PI controller is usually represented by relations control(x,y,z) where x, y and z are the linguistic values for the error, e, the change-of-error, fle, and the change-ofcontrol output, flu, respectively. Each relation describes which linguistic values belong togt'ther, e.g. control(normal, normal, normal), control (negative_small, positive_small, normal). The complete set shown in Fig. 2 consists of 25 linguistic relations if each variable has five linguistic values: negative_big, negative_small, zero, positire_small, positive_big. Also finer partit ions are used for control purposes. NB NS §] §] ZO PS PB Derivative fle Figure 2: The rule base of a Fuzzy PI Controller. guistic values depends on the working point of the process. This presentation is ea.<;ily generalized for finer fuzzy partitions and transferred between the programming systems (Juuso and Leiviskii., 1993). A set of linguistic relations can be changed into a compact equation 2.3. Lmguzstic Equations m 2: A Linguistic Equation approach developed for expert systems provides a flexible environment for combining expertise. The knowledge base of the expert system is represented by linguistic relations which can be changed into matrix equations. The reasoning is based on these equations or on the aggregated sets of linguistic relations obtained by solving the equations. The system is adaptive since the meaning of the lin- AijXj = 0, (2) j=l where Xj is a linguistic level for the variable j, j = l...m, i.e. the linguistic values veryJow, low, normal, high, and very_high are replaced by numbers -2, -1, 0, 1 and 2. The direction of the interaction is represented by coefficients Aij E {-1,0, I}. If an interaction is not present, Aij 83 = O. 3. 2 1 E r r 0 0 r -1 -2 00000 00000 00000 00000 08]8]00 -2 -1 1 0 Derivative CO~TRLE TU~I:"JG Membership functions are tuned by simulation experiments with multilayer szmulatzon systems. The simulation system can also be (partly) replaced by experts or by experiments with real systems (Fig. 1). Both analytical and IH'uristic knowledge can be used simultaneously. As many rules as possible are replaced by linguistic relations. However, some of them are needed , alid the system provides a flexible environment for combining these rules with more efficient modelling methods. In the linguistic equation approach, the relations are developed gradually: only a small part of the problem is taken into account at a time. 2 ~e Figure 3: The rule base of a Fuzzy PI Controller in the matrix form. 3.1 Controller Rules In the general case, a set of fuzzy inference rules is represented by a single linguistic equation (Juuso, 1993a) Fuzzy PI Controller. The rule base shown in Fig. 2 can be represented in a matrix form if the linguistic values, negativcbig, negative_small, zero, positivcsmall, positivcbig, are replaced by numbers -2, -I, 0, 1 and 2 (Fig. 3). All these rules can be obtained from a single linguistic equation ~u = e+ ~e, m U where u is a control action, alHi Xj a linguistic level for the variable j obtained hy the mea<;urements. is also applicable on more detailed fuzzy partitIOns. This procedure produces always a rule set which is complete, consistent. alld continuous. If a noncomplete set is satisfactory, a part of the rules Call he rejected already before tuning. (3) IT. Fuzzy PD Controller. The table of rules shown in Fig. 2 can used also for fuzzy PD Controller represented by a single linguistic equation e+ ~e, (5) j=1 which is a special case of Equation 2 with the interaction matrix A = [1 1 -1 l, and variables X = [e ~e ~u u = = LAijXj , As all these control equations are only special cases of the model above, the system can easily take into account the principles of the model reference control by combining the control equations and the process model into a single set of equations. (4) which is a special ca'>e of Equation 2 with the interaction matrix A = [1 1 -1], and variables X = [e ~e u jT. The PI and PD contro11('rs shown above can also be combined into a single matrix equation, i.e. there are two output variables. For an industrial project, the control system wa<; originally developed on the basis of operator's control actions. Eight variables wa'> used, and the resulting rule ba<;e of 22 rules was challged into five sets of rules which corresponded to five linguistic equations. After solving the equation, a more complicated system was handled by 27 rules. The equation system can be used in developing a rule base for finer partitions as well. Several equa.tions. Several sets of linguistic relations can be combined by matrix presentation AX = O. In order to solve this problem, a sufficient number of these variables should be known or variated. Because of nearly singular matrices, some of these combinations cannot be used. However, only the integer solutions are required, and exactly the same set of solutions is obtained by any combination. In the Linguistic Equation Method, interactions are benefical: they reduce the number of rules necessary for halldling the system. It is also possible to identify interactions on the basis of experimental data. In the equation form, large systems are handled quite easily compared to conventional fuzzy methods. For some systems, a really drastic reduction of rules is achieved (Juuso and Leiviskii, 1993). The result is an aggregated set of those linguistic relations which are relevant if the process constraints described by the complete set of linguistic equations are taken into account. As noninteger alternatives correspond to the solutions in finer fuzzy partition, they can be excluded. 84 3.2 Membership Functions 3.3 Controller Testing The controller tuning is started by defining the working area for variables, e, and ~e, by fuzzy trapezoidal numbers, several experts are used if available. Fuzzy differential constraints can be used in a similar way as in the DSS applications (Juuso et al. 1993). Alternatively, the feasible range can be defined on the basis of experimental data. Actually, the approach is chosen for each variable separately. The feasible range corresponds normally to labels -1, 0 and 1 (Fig. 4). In the tuning system, crosspoint ratio between the membership functions of the neighbouring labels is one for both antecedent and COllSequent variables. Therefore, the fuzzijication of the crisp input values on the basis of trapezoidal membership functions is implemented very efficiently. The knowledge base and the inference engine are implemented in two alternative ways, i.e. traditional rule-based controller and mat rix controller. Actually, the rule-ba.'ied controller is also running in a matrix form similar to one used in FuzzyCon. The final membership functions of the labels are obtained by a polynomial regression model (Fig. 4) which takes into account the sequence of the lahels, e.g. positive big is bigger than positive etc. The polynomial model produces more labels close to the working point. The system generates membership functions for finer partition levels. Error ...A". ___ .- 8' -5r Enor , 58 ~ __ •. _ . k ~ 4 i q .' / \ 1 -I -2 11 i 2 8 ..... 8 _,..---1- 11 8 ' ... _ / 8 .9 8 8 -2 Cha"!fe-of-controJ output 1 28 - · -- - - - - - - I, -1 -2 _ ~ 8· - -2 - -1 - -)il : 8 .. --. -28 28 8 \/ /\ 48 / \/I! \// n '. x \ ' 1;\ I. -1 8 = -25 . 79 x = 1.eBL = -1.124 x = 2 Change-or-contra I output 1 .. ~ \ .:' }; 8.~ 2 i, ), I' t. !\i', 0.& "!fe-of -I!!'ror ; . -><- --y:: -2~ )t----. • I /\\ Ch ..nge-or-error 8: \/ y\/ The defuzzijication module is ba.'ied 011 the Center-of-Area method (Fig. 5), in the litterature also referred to as Center of Gravity method . In a general ca..,e, this method is rather complex and slow. However, our implemetation is quite fast since it is specialized to the membership functions with the crosspoint ratio is one. The resulting control surface (Fig. 6) is very smoothly varying compared to trial-anderror based fuzzy controllers . As the Fig. 5 shows, diagnostical features are a essential part of the system (Juuso 1994). 8 -28 V ) !, .\ -18 ., , r 8 18 Error 58 1 : 8t-··· ....··_··,,· -5,1t8- -2 8 I, . .. _ ._-,,-.j: 1 - -- - - - - - ' -1 1 :le Contra I output 2 Control output 2 2 8'j 8 -28 , 2.2413 , ,. , I X,\/' ff,\ /\Y \ ' \ ) \ I, 8 28 48 -111 Cha . . ~ot-c"r Figure 4: Labels and membership functions for ~Ul' and U2 . e, ~e, 11 18 48 I output 1 Figure 5: Control Example of a Fuzzy PI Controller with diagnostical features. The membership functions of the process state variables, E, and ~e, are used in developing scenarios for experts (or simulations system), and the response produces data for the estimation of membership functions for the labels of the change-of-control-output, ~u (Fig. 4). For some variables, the scenarios and the expert response is replaced by experimental data. In this case, selected input variables are classified by fuzzification routines, i.e. several relations must be taken into account simultaneously. The same methodology is also used when the system is adapted to several working points close to each other. 3.3 FuzzyCon For adaptive tuning of fuzzy logic controllers, it is necessary to have a FLC where the rules and the membership functions can be changed online. Although a wide variety of fuzzy logic tools available on a PC enviroment were examined, none of them could come up with the requirements of on-line tuning. The knowledge about the systems was limited, and transfering to new computer enviroments would have been difficult. 85 Some additional labels are required for the systems consisting of several sets of rules: A:\Ylabel in the premise part of the rule means that the input has no influence on the rule, and a star in some output in the consequent part of the rule means that the rule has no influence on that output. Rules are stored in number form which makes their treatment ea,>ier . , .. 18 .- .- 5 5: ~ .- • .~ 8 C Fuzzification is made by calculating the valm's of membership functions of all the fuzzy sets for input value in question (Viot 1993). :\lamdani's min-max method was selected because it provided reasonably good results and wa<; fastest and easiest to calculate (Lee 1990). FuzzyCon does not require similar restrictions for membership functions as the tuning system dpscribed above, and therefore, the fast algorthm for the Center-of-Area method cannot be used . TIIP defuzzification module is based on the Center-ofSums method which is one of the most common defuzzification techniques in control. r• -5 .. I 1 u -18 ' 48 Figure 6: Control Surface of a Fuzzy PI Controller. User interface. FuzzyCon provides valuable Ollline tools for the controller tuning . The llser interface is easy to use and several windows can be open at the same time for showing differellt kind of data. FuzzyCon can also save definitions, membership functions and rules on the file system in matrix forms which are closely related to the matrices used in the tuning system. Usually, fuzzy logic development tools permit changes in rules and membership functions in simulation mode , but when the codp is compiled alld the program is running, changes call not be done whithout compiling the code again. The calculation tim e in adaptive FLC is a little bit longer thall in strictly bound FLC (Brubaker 1993), but the speed is enough for controlling processes in process industry. For users, the rules are shown in linguistic form and the membership functions in numeric and graphical form. Users can add and edit rules in rule window and edit membership functions in membership function windows, or the rules and membership functions created in some other way can be read to FuzzyCon from the file or traI1Sfer with DDE-links. The rules and membership functions can changed on-line by both methods. FllzzyCon allows adaptive tuning and offers good tools for tuning , and is specially designed for fuzzy control in process industry (Juuso et al. 1994). As a \Vindows application created in Visual Ba<;ic 3.0, it uses the advantages of Windows: graphical interface, windows and DDElinks. The links are created after applications arp started and there is no need to know the mpmory addresses beforphand. The data acquisition application makes conversions, if needed, to crisp input data and sends them to FuzzyCon. At the moment FuzzyCon can handle ten inputs and five outputs, but the expansion is ea,>y. FuzzyCon offprs several aids to examine control (Juuso et al. 1994) . The completeness of the rulC' base can be tested by simulation. During control and simulation there is chance for monitoring the firing of the rules and the forming of degrees of membership for inputs and outputs. The data acquisition card ha'> a \Vindows support, so thp data acquisition program can use DLL-library commands for reading measured data from the card. 4. ADAPTIVE FUZZY CONTROLLERS Fuzzy logic controller. The knowledge base contains membership functions and rules. The membership functions are trapezoidal or triangular defined by four points (Fig. 4). The maximum number of membership functions is nine for each input and output variable. In the general form, each rule contains several inputs and outputs: at the moment, FuzzyCon can handle ten inputs and five outputs. The Control System is adaptive since the meaIling of the linguistic values depends on the working point of the process. Only five parameters are needed for reconstructing the set membership functions for any variable on any level of fuzzy partition (Fig. 4). In control applications, this level does not usually change, aIld it is better to use a set of corner points of the membership functions also shown in Fig. 4. 86 The adaptivity is pretuned, i.e. the tuning of the membrrship functions is performed for different working point areas defined by some suitable statr variables, e.g. flow velocity, temperat1lre etc. The resulting membership functions, and tlH' corresponding control surface, depend on the working point, i.e. the cube containing the control surface has rubber likr dimensions. The selection of alternative tuning area.., corresponds to the table shown in Fig. 2, and each variable of the control rules requires an own table. Juuso, E. K. 1993. "Linguistic Simulation in Process Control." In Applied Simulation in Industry. Proceedings of the 35th SIMS Simulation Conference (Kongsberg, NOT'way, 9 - 11 June), T. Iversen, ed., 107-113. Juuso, E. K. 1994. "Fault Diagnosis Based on Linguistic Equation Framework." To be presented in SAFEPROCESS'94 (E8poo , Finland, 13-15 June, 1994). Juuso, E. K., J .C. Bennavail, and M.G. Singh 1993. "Hybrid Knowledge-Based System for Managerial Decision :'1aking in l" lIcertainty Environment." In Qualitatzve Reasoning and Decision Technologies, Proceedings of the QUARDET'93 (Ba.rcelona, June 16 - 18), ~. Piera Carrete and :M. G. Singh, eds., CI~NE, Barcelona, 234-243 . These tables are used together with linguistic equation models, if available, to obtain the appropriate definitions for the sets of membership functions. This procedure a1lowes gradual changes in any part of the set of membership functions, and therefore, it is more flexible than ::\ ormalization-Denormalization procedure. By this approach, adaptive control can be realized ill FuzzyCon or in automation systems a.., well. Thr membership function can be regenerated by the database produced by the tuning system. Juuso, E. K. and K. Leiviska. 1990. "Expert Systems Combined with a Multilayer Simulation System." In MATHEMATICAL and INTELLIGENT models in system simulation, R. Ha.J.1Us and P . Kool and S. Tzafestas, eds., Baltzer, Scientific Publishing, 325-330. If the databa..'le defining the membership functions is changed online, the adaptation should he rrstricted to an appropriate range depending on the working point. Otherwise, the normal problrms of adaptive systems may arise. Neural nrts are planned to use in fine tuning especially when extending the operating area of the adaptive fuzzy controller. 5. CO~L Juuso, E. K. and K. Leiviska. 1991. "Adaptive Expert Systems for Metallurgical Processes." In Expert Systems in Mineral and Metal Processing, Proceedings of the IFAC Workshop (Espoo, Finland, August 26-28, 1991), IFAC Workshop Series, 1992, Number 2, S.-L. Jamsa-Jounela and A. J. Niemi, eds., Pergamon, Oxford, UK, 119-124. USIONS In the linguistic equation approach, the relations are developed gradually: only for a small part of the problem is taken into account at a time. By the matrix method, it is very easy to dewlope and tune adaptive fuzzy control applications. The linguistification methods have a vital importance in connect.ing this methodology to the real practice: the relations are tuned by adjusting the meanings of the linguistic variables. Juuso, E. K. and K. Leiviska. 1993. "Linguistic Equation Approach for Adaptive Expert Systems." In Proceedings of the EUFIT'93 (Aachen, September 7 - 10), H.J. Zimmermannn, ed., Augustinus, Aachen, Vol. 3, 1550-1556. The resuits of the adaptive tuning method are used in real control practice by transferring them to FuzzyCon which is a fuzzy logic controller wllt're the rules and the membership functions can be changed on-line. FuzzyCon is connected to processes with a data acquisition card, and the data is transferred through DDE-links. Juuso, E. K., J. Myllyneva, and K. Leiviska I! "Fuzzy Logic Controller and Adaptive Tuning." To be presented in ESM'94 (Barcelona, Spain, 1-3 June, 1994). Lee, C. C. 1990. " Fuzzy Logic in Control Systems: Fuzzy Logic Controller - Part I & n." IEEE Transactions on Systems, Man, and Cybernetics, 20:404-435. 6. REFERENCES Viot, G. 1993. "Fuzzy logic in C, creating a fuzzy-based inference engine." Dr Dobb's Journal, 18(2):40-49. Brubaker, D. 1993. "An accelerated kernel for fuzzy systems." AI Expert 8, no. 3:3944. 87