Parameter Arrangement in Multivariable FLC Tuning Alberto SORIA LÓPEZ, Jean-Claude LAFONT Institut National des Télécommunications. Evry -France email :[email protected] Claude BARRET Centre d’Études Mécaniques d’Ile-de-France - Université d’Evry Val d’Essonne e-mail: [email protected] Abstract We study how FLC parameter sets (input membership functions, output membership functions and scaling factors) arrangement, influences the final controller’s performance. In this extended abstract we will present the parameters associated with the FLC, its structure, the optimisation algorithm, the simulated system and finally the results we have obtained in simulation and its application to control the real system.

Keywords : Fuzzy Logic Controller Tuning. Multivariable Systems. Optimisation. Introduction In a Fuzzy Logic Controller (FLC), difficulties for choosing the multiple parameters associated with this control paradigm has led researchers to examine systematic methods to find FLC parameters. The FLC parameters that must be chosen are rule base, input and output variable's membership functions (type, width and position) and their universes of discourse. The most frequently the tuning is achieved by a manual trial and error method. In this case the rule base is obtained thanks to an operators experience. The membership functions are changed to get the desired performance. The universe of discourse is given by a priori knowledge on the process. However when this information is not readily at hand it’s necessary to go trough many trials to set correctly the parameters. Systematic tuning methods can help to eliminate the trial and error method thus reducing the time required to implement a fuzzy controller. Great interest has been given to the issue and a methodology capable of handling the problem in the majority of fuzzy control applications has not yet been established by the fuzzy control community. It still remains a major research topic in fuzzy control. Recently a growing interest has been given to the application of fuzzy control to Multivariable systems. The tuning problem becomes more difficult caused by an increasing dimensionality. Three major trends can be distinguished within the tuning techniques: 1) Neuro-Fuzzy techniques: These techniques combine the advantages of the Neural and Fuzzy paradigms. Fuzzy systems have a defined structure and represent knowledge linguistically. Neural systems can be trained to approximate non-linear functions and work as black boxes. See for example the work by Linkens and Nie in [1]. 2) Optimisation based techniques (i.e. GA’s and Simulated Annealing) : these techniques use an optimisation algorithm to change FLC parameters. This is the case of the work done by Isaka & Sebald in [2] using simulated annealing. An example of the use of GA’s can be found in Ng & Yun [3]. 3) Other tuning techniques: there is a large number of works using specific techniques that may not be classified in the two above. An example is the pioneering work done by Mamdani & Procyk with the self-organising controller [4] ; See also the work based on techniques taken from the classical PID control such as done by Chen et al. [5] To find the FLC parameters we apply here an optimisation technique using a genetic algorithm in an off-line task. This algorithm is capable of handling the tuning problem and has shown it’s applicability in a number of studies. In Figure 1 we can see the tuning architecture principle. In this case the objective function block is taken to evaluate the FLC performance. The optimiser is then used to find FLC optimal parameters. The system block is a model simulating the real system that can be for example a neural network [6].

Objective Function Optimizer Set Point

e

Fuzzy Logic u Controller

System

y

Figure 1 Structure of the optimisation technique. In this case, one of the problems encountered is what parameters must be optimised and in what order. We could for example optimise all fuzzy controllers parameters at the same time: input/output membership function width and position; rule base and universe of discourse. We could also optimise all these parameters but in a specific sequence: first the input membership functions, then the output membership functions, afterwards rules and finally the universe of discourse scaling factors. We are here interested to investigate the effect of different tuning of policies. We want to find an optimal parameter arrangement sequence. We apply an optimisation scheme leaving some parameters fixed, while changing others. In Figure 5 we can see the arrangement/combination sequences. For example GI-O, stands for an optimisation run with scaling factors (Gains) together with input membership functions (Input) first and then output membership function (Output).

Structure of the fuzzy controller We have chose to use a Mamdani PD-Fuzzy type controllers (output is dui ) as shown in Figure 2.

Objective Function Set Point 1

e1

GErr1

Fuzzy Controller 1 Mux

de1/dt SetPoint 2

e2

du1 1/s

GDErr1 GErr2

Gdu2

GDErr2

Level - h(t) 2

u1 u2

Fuzzy Controller 2 Mux

de2/dt

Gdu1

1/s du2

1 Fobj

3 Temperature . To(t) Stirred Tank

Figure 2. Controller Structure We employ two controllers, one for each control loop. We use three Gaussian input membership functions (MF’s) for error and change in error (named Negative, Zero and Positive) and five Gaussian membership functions (named Negative Large, Negative Small, Zero, Positive Small and Positive Large) for the output. Each of these MF's are controlled by two parameters, one dealing with its position and the other with its width. In this study the membership position for zero remains fixed to help to maintain the logical coherence of the rules [7]. The number of parameters for the MF’s of one controller adds up to nineteen.

The universes of discourse are in the interval [−1, 1] . Six scaling factors, that we call the "gains", are given by the two element matrixes GErr, GDErr and GdU that correspond to the errors, change in errors and the output variables dui . The rules used , that remain fixed during the optimisation, are shown below in Figure 3.

Error/

Negative

Zero

Positive

Negative

Negative Medium

Negative Small

Zero

Zero

Negative Small

Zero

Positive Small

Positive

Zero

Derr

Positive Small Figure 3. Rule base

Positive Medium

Optimisation Algorithm Some studies have shown that the Genetic Algorithms (GA's) are suitable for FLC tuning, such as developed by Karr in [8]. For our study we use a binary type coding to obtain a concatenated string that represents each individual in the population. One point crossover with crossover probability set to 0.8. Mutation rate is set to 0.01. Population size is set to 100 for an evolution to 100 generations. We take up the stochastic universal selection operator proposed by Baker in [9]. We choose an elitist selection scheme that passes the best individuals to the next generation before the application of genetic operators. In addition a niching procedure proposed by A. Petrowski in [10] is employed to treat the tuning problem as a multimodal function optimisation, in view of choosing from a family of solutions, since the FLC tuning problem does not necessarily lead to a unique solution. The objective function used is the integral of squared errors given by : tf

f obj =

∫ (k e

1 1

t0

2

2

)

+ k 2 e2 dt

Simulated System Model The real process concerned is a stirred tank schematically shown in Figure 4, that must be controlled both in level and in temperature. The system is characterised by two-input variables to govern and two-output variables.

H eat E xchanger

K

q i ( t ) In p u t flo w T i ( t ) In p u t T e m p h ( t ) L ev el S en so r T O ( t ) T e m p .S e n s o r

V a lv e ; α

a

O u tp u t F lo w q Pum p

Vo

v(t)

o

H e a t In p u t

U p p er T a n k S u rfac e S L o w e r T a n k V o lu m e

Figure 4. Stirred Tank.

()

Inputs are the liquid flow rate qi t and heating element power

v( t ) ;the outputs are liquid level h( t ) and upper

tank temperature To ( t ) . The equations for this system are given by :

q ( t ) − α h( t ) h&(t ) = i S

T&o (t ) =

α h( t )T0 ( t ) − qi T ( t ) T& ( t ) = V0 − Sh( t )

 ka  Ti ( t ) = T ( t )∗1 −  ρcqi ( t ) 

1  v(t )   qi (t )(Ti ( t ) − To (t )) +  Sh(t )  cρ 

To ( t ) is coupled to h( t ) witch is governed by a non-linear relation. We have to add a non-linear threshold effect due to the pump characteristics as well as a singularity when h( t ) approaches zero. In these equations

Results In Figure 5 we present the optimisation results with different parameter arrangement and combinations. The minimal value is obtained with the GO-I combination. It presents a very good control for h( t ) , but an irregular control for To ( t ) . The next better value was that of G-I-O. The results of control using these controllers, are shown in Figure 6 (simulated)and in Figure 7(real plant). The corresponding control surfaces are shown in Figure 8 .

1250 Objective function value 1230

1210

1190

1170

1150 GIO

G-IO

GI-O

G-I-O

GO-I

G-O-I

I-G-O

I-OG

I-O-G

O-GI

O-G-I

OI-G

Arangement/Combination G=Scaling factor parameter set;I =Input parameter set;O=Output parameter set

Figure 5. Optimisation results with different parameter arrangement and combinations.

O-I-G

Level Output - Simulated System 4 Output (volts)

Temperature Output - Simulated System 2 Output (volts) 1.8

Set Point h(t)

3.5

1.6 3

1.4

2.5

1.2 1

2

0.8

1.5

Set Point To(t)

0.6 1

0.4

0.5 0

0.2

0

5000

10000

0

15000

0

10000

5000

15000

Time (sec)

Time (sec)

Figure 6. Simulated System output Level Output - Real

4

Temperature Output - Real

Output (volts)

Output (volts)

Set Point h(t)

3.5

1.8 1.6

3

1.4

2.5

1.2 1

2

Set Point To(t)

0.8 1.5 0.6 1

0.4 0.2

0.5

0

0 0

5000

10000

15000

2000

4000

6000

Time (sec)

8000 Time (sec)

10000

12000

14000

Figure 7. Real System output

FCTR1

FCTR2

0.6

0.4

0.4

0.2

0.2 du

du

0.8 0.6

0

0 -0.2

-0.2

-0.4

-0.4

-0.6

-0.6 1

1 0.5

1

1

-1

Error

0

-0.5

-0.5 -1

0.5

0

0

-0.5 DError

0.5

0.5

0

DError

Figure 8. Control surfaces

-0.5 -1

-1

Error

Conclusion The arrangement and combination has an important impact for tuning the FLC when a priori information on the system remains unknown. In our example the arrangements where the scaling factor is present in the first optimisation sequence( GIO, GIO,G-I-O and GO-I) turned out to be better than when done afterward (I.-G-O,I-OG,I-O-G,O-GI,O-G-I,OI-G and O-IG). It points out that it is better to optimise first scaling factors and then other FLC parameters. The first optimisation sequences (G-IO,G-I-O and GO-I) having a close value present quite close performances. However, the sequence GIO turned out with a much less capacity to control effectively the process. Numerical differences in the objective function are small but the performances vary significantly from one case to the other. The numerical value of the objective function is taken in account to acknowledge controller performance, but it’s necessary to consider also the actual control curves. Real system control is satisfactory, despite considerable variations in steady sate error, because the model employed is an approximation of the real system. Differences are caused by static friction in the real pump provoking changes in the gains; real temperature hating input power is provided to the system by varying the time (proportional to the input voltage) during witch current to the heating resistance is switched on. Despite these differences, we have successfully synthetized a FLC capable to control the real system.

References [1 ] LINKENS, Derek A. & NIE, Junhong [2 ] ISAKA, Satoru & SEBALD, V.

[3 ] NG, Kim Chwee & YUN, Li [4 ] Mamdani, E.H. & Procyk, T.J. [5 ] CHEN, Jen-Yang ; CHEW, Jen-Moon ; LIN, Ying-Hao & TSI, Pvo-San

Fuzzy-Neural Control. U.K.: Prentice Hall, 1995. 239 p. "An Optimisation Approach for Fuzzy Controller Design". IEEE Transactions on Systems, Man & Cybernetics. Vol. 22, Nº 6, Nov. 1992. p. 1469-1473. Design of Sophisticated Fuzzy Logic Controllers using Genetic Algorithms. U.K.: University of Glasgow. 1996. 24 p. "A Linguistic Self-Organizing Process Controller". Automatica. Vol. 15, 1979. p. 15-30.

"Fuzzy Self-Tuning via Gray Predict of PID Controllers". [Procs.] Third European Congress on Intelligent Techniques and Soft Computing (EUFIT '95). Aachen, September 4-8, 1995. p. 1095-1100. "Identification and Control of Dynamical Systems Using Neural [6 ] NARENDRA, K S. & PARTHASARATHY, K. Networks". IEEE Transactions on Neural Networks. Vol.1, No 1. March 1990. [7 ] LAUTAUD-BRUNET, MICHELE. Process Identification and control by Neuro-Fuzzy Networks. PhD. Dissertation. France: Université d'Evry Val d'Essonne. 1996. 185 p. (In French). [8 ] KARR, CHARLES L. & GENTRY EDWARD J. "Fuzzy Control of pH Using Genetic Algorithms". IEEE Transactions on Fuzzy Systems. Vol.1 Nº 1. February 1993. p. 46-53. "Reducing Bias and Inefficiency in the Selection Algorithm". [9 ] BAKER, J.E. Genetic Algorithms and their Applications : [Procs.] 2nd International Conference on Genetic Algorithms. 1987 . p. 41-49. [10 ] PETROWSKI A. "A Clearing Procedure as a Niching Method for Genetic Algorithms". [Procs.] IEEE 3rd International Conference on Evolutionary Computation (ICEC'96). may 20-22,1996. Nagoya.