Pergamon 0967-0661(94)E0007-3

ControlEag. Practice, VoL 2, No. 3, pp. 431-437, 1994 Copyright © 1994 Elseviex Science Lid

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MULTIVARIABLE FUZZY CONTROLLER DESIGN

R.E. King*, G.D. Magoulas* and A.A. Stathaki**

*Advanced Control Research Group, Division of Systems and Control, Department of Electrical Engineering, University of P atras, P atras, Greece

**HITEC S.A.o Athens, Greece and formerly with AMBER S.,4., the HERACLES Group, Athens, Greece

Abstract. The control of large multivariable industrial processes whose dynamics are either unknown or vague poses a very serious challenge in process control. Over the last fifteen years or so, the process industry has made serious progress in putting fuzzy logic to practice. Indeed numerous successful applications have been reported to date. The design of fuzzy controllers today is very much an art and there are presently few effective design tools available for their development. This fact has, regrettably, Iimited their application considerably. To this end, this paper describes an integrated development system which permits the controller designer to test hypotheses, examine rule validity and rule conflict, examine the effect of changes in the controller parameters and perform a complete off-line simulation of a proposed multivariable fuzzy controller.

Keywords. Multivariable control, fuzzy logic.

1. INTRODUCTION

The process industry has many control problems which conventional techniques have been unable to resolve. Advanced Process Control, viewing these problems from a different perspective, offers a host of new tools with which to resolve these problems. A few examples of these are: serf-tuning, predictive, fuzzy and neural control. The first two which fall in the category of hard computing use macroscopic models of the process to be controlled, obtained from tests on the process, whereas the last two model the human operator controlling the plant. The latter two being linguistic fall into the category of soft computing. Fuzzy control, based on fuzzy logic theory due to Zadeh (1973), which is one of the principal soft computing techniques, has proved very effective in offering solutions to many rule-based problems. To-date numerous industrial fuzzy controllers have been successfully commissioned in a

variety of process environments (Holmblad and Ostergaard., 1981; King & Mamdani., 1977; King & Kar6ni~, 1986, 1988; King, 1992; Mamdani, 1974).

Two problems encountered in practice in the process of developing a rule-based fuzzy controller, other than that of eliciting the rules by which a process is manually controlled, are rule validity and rule conflict. Different human operators quite often specify different control actions for the same process conditions. This conflict, usually brought about by an incomplete knowledge of process behaviour, very often has adverse effects on the operation of the process and a consequent loss of production. It is a commonly observed phenomenon that productivity can vary dramatically from shift to shift and operator to operator which is indicative of the differences in opinion on how a process should be controlled. Other problems common to fuzzy controller design are:

431

432 R.E. King et al.

• Debugging, modification and testing of the control logic, which can be extremely time- consuming. From a purely pragmatic viewpoint, it makes sense to consider more-efficient ways of implementing the heuristic part of the algorithm (AstrOm et al. 1986). Making reasonable assumptions on the fuzzy sets (e.g. assuming a symmetric triangular form), often simplify the control algorithm and speed up computation significantly.

The necessity of implementing extensive error- handling and error-recovery code, making the software robust and commercially viable.

Although some of these design considerations can be met imperatively, others are amenable to trade-offs. It is generally impossible, or at least impractical, for the controller designer to characterise exactly what is meant by an optimal design at the outset. The term optimal in this context therefore, can only be viewed in a heuristic sense. Except in the simplest cases, achieving anything close to optimal would be impossible without the repeated intervention of the controller designer because multiple performance trade-offs are bound to exist in practice. Accordingly, flexibility and a user-friendly man- machine interface are essential.

Even if a fuzzy controller is found to perform correctly and efficiently on-line, a user may still have certain qualms for the following reasons:

,, the controller may be difficult to modify,

parameters or re-examine specifications interactively with a view to arriving at a compromise solution. Such interaction is an integral part of any iterative design procedure and one which is effectively provided by the development system presented here.

A fuzzy controller typically involves a set of conditional if-then production rules whose antecedents and consequents completely specify the desired linguistic actions to be taken. The proposed multivariable fuzzy controller permits a maximum of three antecedents (i.e. inputs) and up to five consequents (outputs). This number has been found to be sufficient for most industrial controllers. Consequents from different rules are numerically combined by means of the union operator. Both max product (Larsen, 1980) and min max (Mamdani, 1974) inference is provided to permit comparison of the results of the two methods.

If more than one rule is fired simultaneously, a conflict resolution method (i.e. de-fuzzification) such as the centre of gravity (COG) or the mean of max (MOM) method can be selected to infer the unique controller decision (Mamdani, 1974; Kosko, 1992; Harris et al. 1993).

This paper describes an integrated microcomputer- based fuzzy controller development system for the design, testing and evaluation of industrial multivariable fuzzy controllers. The complete software package is written in C. In an attempt to simplify real-world controller design, a two-module structure has been adopted:

• the controller may be difficult to use or easy to misuse by the plant operators,

the controller software may be machine- dependent or difficult to integrate into an existing production management and control system of the plant.

This is the problem of controller evaluation which is essential in determining whether a given controller meets the requirements and objectives specified.

the Definition Module, which allows for the definition of: • the names of the control and process variables as

well as their nominal values and permissible ranges,

• the fuzzy sets for the various linguistic operators and

• the conditional rules which constitute the knowledge base, and

a Test Module, which can perform both: • static and • dynamic simulation

2. THE MULTIVARIABLE FUZZY CONTROLLER DEVELOPMENT SYSTEM

When dealing with fuzzy controllers it cannot be expected that the rules provided by the process operators or the production manager will lead to a feasible controller whose performance best matches that of a human operator. Indeed to achieve this end a fair amount of experimentation (or 'trimming') is required, not unlike that necessary in commissioning a conventional controller. In doing so, the controller designer will usually wish to modify controller

under a variety of operating conditions, allowing the process engineer to examine the decision-making capabilities of any proposed fuzzy controller with a view to improving its performance.

The flexibility of the man-machine interface of the development system allows the user to modify the parameters of the fuzzy controller easily. Thus the fuzzy membership functions can readily be modified in order to observe the effect of changes in their shapes and their locations in the universe of discourse on the control decisions. Likewise, the

Multivariable Fuzzy Controller Design 433

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Fig. 1. The fuzzy sets option

conditional ~f-then rules, which establish the controller knowledge base, can be inserted, deleted and edited at will. An important feature of the development system is the ability offered process operators to visualise the rules by which they control a process, not as a series of linguistic /f-then statements which are very difficult to interpret globally, but chromatically as a multicoloured display of rule tiles, the colours of which are coded so as to give a visoalisation of the consequents of the various rules. Confronted with a visual presentation of the rules, knowledge engineers and process operators can immediately examine the validity of the conditional rules and identify any rule-conflict which arises. Rules can be changed or eliminated at will. The ease with which rule-validity and rule- conflict can be resolved is a principal feature of the development system.

3. THE DEVELOPMENT SYSTEM FUNCTIONS

The basic functions of the Definition Module of the development system are given below and are subsequently defined in greater detail: • Input-Output Variable Name Definitions • Permissible Range of Variables • Fuzzy Set Distribution Functions • Rule Base In the first option of the Definition Module, the controller and the process output variable names are specified so that all subsequent references to these variables will be stated in their engineering terms.

The second option of the DM allows for specification of the permissible ranges of the controller and output variables since the fuzzy controller is incremental by nature. This is necessary since the linguistic terms ~ery high' (VII) and ~¢ery low' (VL) must be interpreted numerically from transducer measurements and must in practice be translated into numerical variables so that they may be transmitted to the appropriate control actuators. The controller output at the (k+l)st time instant is then e(k+l) = e(k) + q(k) where q(k)=gt(y(k)) is the incremental output of the controller. The operator 9t specifies the functional mapping between the incremental inputs and outputs of the fuzzy controller and is uniquely determined by the rule base, the fuzzification and defuzzification procedures and the fuzzy sets.

Five distinct fuzzy sets, one for each linguistic level (VL for very low, LO for low, OK for acceptable, HI for high and VII for very high respectively) must be specified in the third option of the Definition Module. Experience has shown that in an industrial environment this number of linguistic levels usually suffices to describe the production rules by which the fuzzy controller infers its control action. Rarely, in any case, do human operators describe their control actions in more linguistic terms. These fuzzy sets are discretized into 13 singletons over the universe of discourse and are presented in bar-graph form as in Fig. I. Each fuzzy set can be specified independently and for each input. For convenience the height of each singleton is quantized into 10 distinct levels as finer quantization does not appear to be necessary.

434 R.E. King et al.

f i l e : L E P O L R U L . K B

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Indeed recent research has shown that the controller decision is insensitive to small changes in the height of each singleton (Boverie et al., 1981).

The knowledge (or rule) base by which the fuzzy controller infers its decisions are also specified in the Definition Module. The number of inputs to the controller has been limited to three, a number that has been found sufficient for most industrial processes. If more than three controller input variables are necessary then a hierarchical structure (Saridis, 1979) is adopted. At the lower level of the hierarchy a series of parallel three-input-multi- output fuzzy controllers are established, and are co- ordinated by a higher level supervisory fuzzy controller (King e t al. 1988). The knowledge base of each fuzzy controller is presented as an exploded 5 x 5 x 5 three-dimensional Rubik cube i.e. as a sequence of 5 x 5 a le s not every one of which need be specified, as shown in Fig. 2. This chromatic tile display of the rule base is a feature of the development system which is unique and which has proved to be extremely valuable in editing rules, investigating rule conflict and rule consistency visually.

For lack of colour, the tiles are shown in shades of grey instead of chromat ica l ly as on a computer screen. Each tile represents an element of the control space the size of which is defined by the permissible range of each variable divided by the number of elements used to quantise the control space. It is evident from a cursory glance at this rule-pattern that any rule inconsistency can be detected immediately.

The very first matter with which the fuzzy controller designer must contend, after eliciting coding and storing the rules in the knowledge base, is rule- consistency. It is unlikely, for instance, that a rule transition from, say, VL to VH will occur in adjacent rule tiles as this would imply a radical control profile. Likewise it is unlikely that more than one linguistic level change from tile to tile for this would imply too coarse a quantization of the control space. With experience, a number of such meta rules can be developed to help the designer examine and question the validity of the rules elicited from the plant operators. Very often, in fact, plant operators must

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Multivafiable Fuzzy Controller Design 435

be questioned again on their decisions in the light of rule inconsistencies. Indeed when faced with the obvious inconsistencies which the colour-coded rule- tiles offer, operators immediately appreciate their errors, modify and learn the right rules and become improve their performance as a result of this experience.

The Test Module has two functions:

• Static Simulation and • Dynamic Simulation

In the first function of this module, that of static simulation, the numerical values of the input variables are specified as a percentage of the permissible deviation of each variable and the fuzzy controller is then instructed to execute and display its decision, yielding the controller inputs and outputs as bar graphs as shown in Fig. 3. The development system being integrated, it is a simple matter to return to the knowledge base or fuzzy sets options, make changes and re-execute the controller to observe the changes in controller performance as a result.

The second option of the Test Module, that of dynamic simulation, is similar to the first, but differ in that now the controller is instructed to execute the controller algorithm each time a random number generator produces a set of uniformly distributed random values corresponding to the input variables. Two display forms are possible: the first is identical to the static simulation option, except that the bars change continuously in response to each execution of the control algorithm, and the second is a moving plotter-like display as shown in Fig. 4.

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4. APPLICATION EXAMPLE

By way of example, the design of a multivariable fuzzy controller for a cement mill shown schematically in Fig. 5. was studied with a view to commissioning. Following discussions with the process manager, the process output and control variables were first specified as were their permissible deviations about their nominal values. The next, and most critical stage was knowledge elicitation, a task that often proves to be difficult, time-consuming and frustrating. Eight mill operators were supplied with questionnaires requesting their decisions to all possible combinations of the process outputs. The process, having three control variables, each of which was described by five linguistic levels, required operator responses to a total of 5><5 x 5=125 linguistic rules.

Not unremarkably, the responses so elicited, shows a wide discrepancy of control actions, at times widely different, even opposing, for the same plant conditions. The results for any given operator are fraught with conflict and erratic control actions show up immediately in the colour-coded rule-tile displays. When the control actions of different operators are placed side by side and compared it is often impossible to find a common action. Taking the average of all plant operator control actions for each plant state often proves futile due to the widely differing and often contradictory decisions. The overall results in this case often average out to a 'do nothing' response! Taking the majority action as the basis of the rule base, on the other hand, yields somewhat more acceptable results but here again rule conflict may still be very much in evidence.

Faced with conflicting operator decisions, a procedure of rule editing in which a systematic step-by-step procedure is followed, leads to a final logical rule-base where conflicts are resolved. Here it would be useful to establish meta-rules, i.e. rules for processing rules in order to specify the final or logical rule-base. This is an area of ongoing research. One useful meta-rule which has been applied successfully is: there can be at most two linguistic level differentials in adjoining rule-tiles.

Following a small number of visual passes of each tile-floor, it rapidly becomes evident which rules must be edited in order to resolve rule-conflict and arrive at a logical rule-base. In fact it becomes evident that knowledge elicitation becomes infinitely easier, faster and more precise when operators state their decisions directly in tile form on the development system.

436 R.E. King et al.

KW

+

RET

Io, i, ~ CONTROLLER • 11,,, •

I

SDPR

RDPR

i /

Fig. 5. Schematic of cement mill and fuzzy controller variables: UP=Under_.pressure, KW=Mill__power, RET=Returns, FEED=Feed_rate, SRPM=Separatorspeed, RDPR=Returns_dampersetting

5. CONCLUSIONS

The microcomputer-based fuzzy controller development system outlined in this paper provides an integrated environment in which the controller designer can develop and experiment with industrial multivariable fuzzy controllers. The development system has been used with success for the design of a number of industrial multivariable fuzzy controllers which have subsequently been commissioned at a major cement plant in Greece. The development system is particularly effective at establishing and clarifying the rules by which human operators control a process and powerful as a test bench for comparing the decision-making process of a fuzzy controller with that of a human operator.

6. ACKNOWLEDGEMENT

The work described in this paper was partially funded by the Hellenic Ministry of Industry by way of a VALOREN development grant to the 'HERACLES' General Cement Co. of Athens, Greece.

7. REFERENCES

AstrOm, K.J., J.J. Anton and K,E. Arzen (1986). Expert control, Automatica, 22, 277-286.

Boverie, S., B. Demaya and A. Titli (1991). Fuzzy logic control compared with other automatic control approaches, Proc. 30th CDC, Brighton, 1212-1216.

Hams, C.J., C.G. Moore and M. Brown (1993). Intelligent Control, World Scientific. London.

Holmblad, L.P. and J.J. Ostergaard (1981). Fuzzy logic control: operator experience applied to automatic process control, Zement, Kalk, Gips, 34.

King P.J. and E.H. Mamdani (1977). The Application of fuzzy control systems to industrial processes, Automatica, 13, 235-242.

King, R.E. and F.C. Kar6nis (1986). Rule-based systems in the process industry, Proc. 25th IEEE CDC Conference, Athens.

Multivariable Fuzzy Controller Design 437

King, R.E. and F.C. Kar6nis (1988). Multi-level control of a large scale industrial process, In Fuzzy Computing (M.M. Gupta and T. Yamakawa Eds.), Elsevier Science Publishers, New York.

King, R.E. (1992). Expert supervision and control of a large-scale plant", Journal of Intelligent Systems and Robotics, 5, pp. 167-176.

fuzzy logic control, Int. J. Man-Machine Studies, 12, 3-10.

Mamdani, E.H., (1974). An application of fuzzy algorithms for the control of a dynamic plant, Proc. IEE, 121.

Saridis, G.N., (1979). Toward the realization of Intelligent control, Proc.IEEE, 87.

Kosko, B. (1992), Neural Networks and Fuzzy Systems, Prentice Hall, New York.

Larscn, P.M., (1980). Industrial applications of

Zadeh, L.A., (1973). Outline of a New Approach to the Analysis of Complex systems and Decision processes, IEEE Trans. on Systems, Man & Cybernetics, SMC-3, 28-44.