Heng-Ming Tai - Robust Fuzzy Controllers

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Robust Fuzzy ControllersHeng-Ming Tai and Sujeet Shenoi

Center for Intelligent SystemsKeplinger Hall, University of Tulsaemail. ta@ohm. ee .utulsa.edu, s u j e e e u l e r .mcs utulsa.eduTulsa, OK 74104-3189, U.S.A.

Abstract: Robustness is an important requirementplaced on welldesigned uzzy controllers A robust fuzzycontroller can deal with significant and unpredictableprocess changes When designing a fuzzy controller it ispartmlarly important to ensure that reasonableperformance is obtained for parameter vanations within--nd outside the design ranges This paper presents asystematic design methodology for tuning a class of fuzzycontro llers using cell maps of dynamic systems Cell mapinformation is incorporated in a genetic search for tuningthe mle-consequent function parame ters of Taka@-Stigeno h z z y controllers The cffectiveness androbustness of the designed fuzzy controller are evaluatedurmg the inverted pendulum control problem

I INTRODUCTIONFwq- modeling and control techniques have seenwide comm ercial and industria l application. Thesemethods can incorporate linguistic human operatorexperience and engineering knowledge in problemsolving. The main problems in designing fuzzy con-trollers are: (1 ) the identification of fuzzy rules and the

tuning of the associated membership functions; and 2)the realization of an o ptimal control policy an d Stability.To solve the first problem, many automatic designmethodsbased on neural networks 191, genetic algorithms[ 1 I ?121, or gradient descent method [13] have proveduseful. For example. when using genetic algorithm s todetermine membership functions and rule-consequents,the f u z q system parameters are genetically coded andfitness functions expressing the control objectives areselected. The performance of the design method and thatof the designed fizzy controllers can be improved bytaking advantage of the robustness of genetic algorithmsand the ir ability to efficienty search large problem spaces.This paper focuses on designing a stable fuzzy controllerwhich is time-optimal and robust. This goal is acluevedusing the cell mapping technique [7] and geneticalgorithms [41.

A cell map conveys important feedback controlknowledge about system dynarmcs of a closed-loopsystem, including system trajectories and the region ofstability. This information is readily incorporated into agenetic search procedure for designing t i m e q t i m a l fuzzycantrollers with high levels ofperformance.Chen and Tsao [3] first used cell maps to analyzefuzzy systemsand o btain global evaluation s of the stabilityand performance of fuzzy controllers. However, they didnot actually use cell maps in f i c r ; q controller design.Other researchers, notably Smith and Comer [14,15] andVachtsevan os and co-workers [10 ,18], have applied cellmapping techniques to the design of optimal f k u ycontrollers. Sm ith and Comer's work 114,151 focuses onthe synthesis of optimal Takagi-Sugeno fuzzy controllers[171. They employ the least-mean-squ are algorithm [191to fine-tune the control policy embodied in a cell spaceapproximation of an optimal control solution 161. Cellmap accuracy is increased using a n o n -W o rmquantization scheme and a variable time intervalprocedure for avoiding spurious equilibrium cells. Thedesigned technique yields a minimum-time fuzqcontroller with good controllability and stability.

The optimal design methodology developed byVachtsevanos and his co-workers [10,18] does notconstruct an explicit fuzzy controller as a function of statevariables. Instead , control action information is embodiedin a cell map. In their design, both state and controlvariables are partitioned into sub-intervals called cell-groups. The evolution of cell-groups in a cell map isobtained by simulation. A search procedure then isapplied to determine the optimal control a h o n for eachcell in the cell space. The methodolog is ve y versatile inthat the designed controller articulates the optimalstrategy encoded in the search. It can be used to design avariety of optimal uzzy controllers such as minimum-energy, minimum-time and minimum-squareerrorcontrollers.This paper examines the robustness of fuzzycontrollers designed using cell map information in agenetic search . Section 11 reviews the Takagi-Sugeno

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fuu?. controller formulation. a modrfied simple geneticalgorithm. and Ihe control information incorporated incell maps. Section I presents the genetic-algorithm-based design methodology utilizing cell maps of closed-loop systems to realize time-optimal TS uzzy controllers.The method is applied to the inverted pendulum controlproblem. T he robustness of the designed TS controller istested using plant parameter variations and control forceslying outside the design ranges.11. CELL MAPPING AND GENETIC ALCdRITHMS

The goal of this work is to develop an automatedmethodology for designing optimal and robust TS fuzzycontrollers. This section describes the TS fuzzy controllerformulation, a modified sim ple genetic algorithm, and thecell map information to be incorporated in the geneticsearch..4. Iakagi-SugenoFuzzy Controllercontroller [171 has he form:IF error x1 is Negative BigTHEN control action is g ( x r x2) = cJlxl+ c I 2 x 2+cJ3 .

The coefficients cj , in the consequent function can beadjusted in the o p t sense to m m he control rule.The final output of the TS fuuy controller,f, s computedas he weighted averag e of the outputs of rules in the rulebase

A typical control rule in a Takagi-Sugeno (TS) uzzyand error-change ( x 2 ) is Zero

c; H] ;PE;=, ),r . -. .1

where r is the number of rules and wJ is the degree ofmatch of the premise of thejth rule.A key advantage of the TS controller formulation isthat the rule consequents are crisp (usually linear)functions of the input variables. It also is easier tooptimize the consequents of the rules. i.e., to optimallytune the con troller using optimization techniques such asthe least-mean-square method [19]. The main task in TScontroller design is to adjiist the coeflicients. c,,: to obtainthe desired pnrfbrniance.i. \ f (Jd ed .?IT?/LJ/CJ~ ~ t ? f l e ~ i C;/@?/*it/ltnS

(jenetic algarittims (G A S ) can Fe used to efficienllvs e a m large paraiiieter spaces for opiiiiial or near optimalsolutions. A simple genetic search [J] is implemented byrandomly geiicrating sets o r parameter values. Eachparameter set, representing a possible solution to thesearch. is coded as a bi na p string called a chromosome.

The chromosome is obtained by concatenating codedparameter values. A chromosome population is a set ofsolutions of varying quality. Populations evolve byreproduction, crossover and mutation. Reproductioninvolves the selection of strings on a probabilistic basis inproportion to their relative fitness. Two selected strings(parents) in the mating pool then undergo one-pointcrossover. Two new strings (offspring) are cra te d byswapping parent bits at the crossover point. Mutationsoccur as occasional random alterations of bits. Thecrossover and mutation rates are typically in the ranges,0 .6 4 .9 nd 0.001-0.01, respectwely [4].A GA is particularly suited to synthesizing optimalfuzzy controllers. It is an efficient search techniqueusing a chromosome population enables the search toproceed in parallel. The genetic operators bring aboutinformation exchange between strings. This reduces thelikelihood of termination at a local optimum. A GA alsois extremely flexible as it only uses candidate solutionquality to guide its search. Auxiliary problem -specificinformaton, e.g., derivatives or continuity properties, isnot required.The simple GA model is modified in this work byrepresenting chromosomes as integer strings instead ofbinary strings. Using chromosomes with integer allelespositively impacts codingldecoding and mutation. Theinteger representation reduces the coding/decoding timeand yields a faster genetic search. Mutation ofchromosomes with integer alleles is implemented simplyby incrementing or decrementing integer alleles by one.C,ontroller design experiments indicate that this mutationtechnique produces better results than changing bit valuesin a binary string. Moreover, an elitist strategy isemployed. It retains the best string in the currentpopulation during the reproduction process. Thisprocedure guarantees asymptotic convergence to a globaloptimum [161.C Cell M ap Information

Cell mapping is a powerfid technique for analyzingthe g lobal behavior of nonlinear dynamical systems [5,7].It approximates a continuous phase space with a discretecell space comprising a finite number of disjoint, equal-iz e d cells. Regions lying outside the domain of interestare lumped as a single sink cell. Cells are processedcxhaustively using an unraveling algorithm [S ] tocompute cell transitioris The algorithm involves calling~ i p ach cell in turn aiid processing it to determine itsglobal properties. It is assumed that d l he states in a celltransit to states in a single image cell within a fisedtime period. and that all cell states can be approximatedby the center-point of the cell. The unraveling algorithm

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identifies all the period motions. domains of attractionand cell space trajectories in a cell map.Each non-sink cell z in a cell map is characte rized bya group number G z), a step number S(z), and aperiodicity number P(z). Cells in the periodic motion ordomain of attraction of cell z belong to the same groupG(z). The step number S(z) indicates the number of celltransitions needed to transit from cell 1: to cell in aperiodic motion. This information is employed toconstruct the fitness function used in the genetic search.For detailed information about cell mappings and theunraveling algorithm readers are referred to the originalwork by Hsu [5.71

111. FUZZY CONTROLLER SYNTHESISThis section describes how cell map infomation canbe employed in a genetic algorithm for tuning TS fuzzycontrollers. Unlike other cell-map-based approach es

114,181, the GA-based design technique only uses asimple, intuitive cell-map-based objective function for thegenetic search.

GA; ControlPlant

PFigure 1 Fuzzy controller design meth odolon

A schematic diagram of the q controller designmethodology is sh own in Figure 1. An initial populationof chromosomes representing coded TS parameter sets -is randomly generated. These chromosomes are decodedlo yield the actual parameter values. The parameters thenare supplied to the corresponding fuzzy control rules andto the governing equations of the system underconsideration. A cell map is constructed for eachparameter set, and its fitness value computed. The fitnessvalues are utilized along with three genetic operatorsselection, crossover and mutation to produce ;L newgeneration of chromosomes. This procedure is repeateduntil the new population has converged to some bestfitness value producing an optimal parameter set or a pre-set limit is reached on the generation number.To synthesize high-perfonnance uzzy controllers it isnecessary to incorporate cell map information in thefitness function for the genetic search. This functionmeasures solution quality and provides environmentalfeedback to the search algorithm . It must correctlyarticulate the controi goals and their impact on systemdynamics. The fitness function J used in this work isgiven by:

2 zJ a Ccon z)d FfZ ) J t ) + P - c m r ) A s s t a b r p ~ )1+r nzwhere nz is the number of controllable cells. These cellssatisfy the predicate 2). d FjZ),Zr) s the Euclidean&stance in the cell space between the final image cell

F z ) of cell z and the target cell Zt. The predicatek a b z ) is satisfied by each cell z whose final image cellF j z ) lies within 6 cell of the target cell. S(z) is thenumber of time steps taken fiom cell z to its final imagecell. a and f are positive constan ts. Interested readersare referred to [8] for additional details of the fitnessfunction and its impact on controller performance.Maximizing J favors solutions containing themaximum number of controllable initial cells which canbe driven to the target cell (equilibrium state). on theaverage, in minimum time. Therefore time-optimalbehavior can be obtained by realizing a cell map with theshortest cell trajectories. Maxima l controllability isrealized in a cell map con taining the largest stable region.

IV. RESULTS AND DISCUSSIONThe effectiveness and robustness of the uzzycontroller design methadology are assessed for theinverted pendulum control problem, a benchmark problemin classical and intelligent control. Simulationexperiments indicate that the cell-map-based uzzycontroller designed with a GA out-performs controllersdesigned by other techniques [2,9,14,15]. The controller

uses less control effort, reaches its goal more directly, andconsistently produces less overshoot despite large inituldeviations. This paper considers its ability to deal withplant parameters and control force 1yng outside thedesign ranges.The control objective of the inverted pendulumproblem is to bring the p i e to the equilibrium verticalposition starting from an initial non-vertical poleinclination. Many ap pk atio ns of intelligent and learningcontrol to this problem use time to failure to gauge thequality of the controller [1,2]. However. in t h i s paper.three m easures are used to characterize the quality of agood controller. They are (1) the time taken to achievethe control objective, (2) the range of controllable initialstates, and 3) the robustness of the designed controller toparameter variations. The first two goals are achievedusing the design methodology and are described in detailin 181. The robustness property is evaluated bvconsidering variations in the nominal parameter set.Since the initial state of the pole is more importantthan that of the cart in determining the inclination of the

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pole, i t is necessary to focus on balancing the pole withoutregard to the position and velocity of the cart. The stateequations governing the pole ang le (xl = 8 ) nd angularvelocity (x2 = 6 ) or the inverted pendulum are given by13,151

r, = x2

21 m3 mp m c

2 =~ ICOS ( I l l

where ge = 9.8 m/sec2 is the gravitational constant, mp =0.1 kg is the mass of the pole, me = 0.9 kg is the mass ofthe cart, = 0.5 m is the half length of the pole, and f sthe force applied to the cart.The required TS kq controller is constructed withx, and x2 as the conditional variables and the force g asthe control variable. The ranges used for the variables xl.x2, an df ar e [-1.5,1.5] radian, [-8,8] rad/= and [-50,501New tons, respectively . The cell spac e is :Formed bypartitioning the phase space into 101 x 101 = 1,0201cellswith q u a l divisions.

-1 0 1Normalized Universe of Discourse

Figure 2. Membership functions of pole angleand angu lar velocity.

Figure 3. Control force surface.

Triangular membership functions which overlap intheir respective domains are employed (Figure 2). Theresulting TS uzzy controller thus has nine linguistic rulesand 27 parameters. Each controller parameter is codedusing five integer alleles. The last two digits represent thefractional part. Each chromosome in the population haslength 135 = 27 x 5 ) . Parameters are oded and decodedby adding and subtracting 500.00 from the actual andcoded values. The population size is fixed at 50. Thecrossover and mutation rates used are 0.8 and 0.1,respectively The chromosome with the highest fitnessvalue is automatically advanced to the next generation.The best TS fuzzy controller derived from cell mapinformation is obtained in the 70th GA generation. Theoutputfof the TS fuzzy controller is directly served as th econtrol force. The final control surface is shown n Fig. 3.The ability of the designed controller to surviveparameter variations successfully is tested using differentpole lengths. Figure 4 shows that the TS controllerdesigned for a pole length of 0.5 m can adequatelystabilize small poles. Note that most existing approachestawards inverted pendulum control consider bung-bungoptimal control schemes, where only right and left forcesof fixed magnitude can be applied. As c n be seen inFigure 4(a), that the control force has a variable mag-nitude, i.e., it is a continuous function of time. A majoradvantage of the cell-map designed controller over othercontrollers (e.g., [2,9,11]) is that the range of initial statesstabilized by the controller is explicitly displayed in thecell map. Graph s of the time-optimal cell and real trajec-tories are shown in Figure 5(a). Both trajectories start atthe initial state x 1 = 1.49 rad and x2 = -7.94 rad/sec.The maximum control force delivered is a goodmeasure for assessing the quality of a control solution tothe inverted pendulum problem because it determines therange of controllable initial states. Figures 5(a) and So>)show the cell map regions in which the contrdler is ableto stabilize the pole for maximum deliverable forces of 50Newtons nd 10 Newtons, respectwely. In general, thelarger the maximum force delivered by the actuator, thelarger the controllable region. As expected, thecontrollable shaded) egion in the cell map in Figure 33)is reduced in size. The designed controller is robust inthat it can stabilize the pole even when the maximumforce delivered is less than the designed value. Timeresponses of the inverted pendulum system to variousmaximum deliverable forces are shown in Figure 6. Forthe initial state x1 = 1.36 rad and x2 = 0.0 rad/=, thedesigned controller with a maximum force larger than 50Newtons (default value) is able to stabilize the pole.However, the controller is not successful when themaximum deliverable force is restricted to 30 Newtons.

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This phenomenon is seen in Figure 5 ( b ) where the initialstate is outside the controllable region. Figure 6 alsoshows that the larger the maximum deliverable force, thefaster the control.v CONCLUSIONS

Incorporating cell map in fo m uo n in a genetic searchalgorithm for tuning the TS fuzzy controller results in atime-optimal controller with maximum stability.Simulation results show that this fuzzy controller alsoadequately sustains desired performance for parametervariations within and outside the design ranges. Cell-map designed controllers offer an important feature rarelyobtained with other design methodologies all the initialstates that can be stabilized by the designed controller in atime-optimal manner.ACKNOWLEDGMENT

T h s research is supported by OCAST Grants AR9-010 and AR2-002. and by a grant rom the OklahomaCenter for Integrated Design and Manufacturing.REFERENCES

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C. W. nderson, Learning to control an invertedpendulum using neural networks, IEEE ControlSystemsMagazine, vol. 10, no. 3, pp. 3 1-36: 1989.H. R. Berenji and P. Khedkar, Learning and tuningf k z y logic controllers through reinforcements,IEEE Transaction s on Neural N etworks, vol 7, pp.Y. Y. Chen and T. C. Two, " A description of thedynamical behavior of fuzzy systems, lEEETransactions on Svstems, Man and Cyb ernetics , vol.19. pp. 745-755; 1989.D. Goldberg, Genetic Algorithms in Search, Optimi-zation and Machine Learning, Addison-Wesley,Reading, Massachusetts, 1989.C . S. Hsu and R. Cuttalu. An unravelling algorithmfor global analysis of dynamical systems: Anapplication of cell-to-cell mappings, ASME JoumalofAp plied hfechanics.vol. 47: pp. 93 1-939, 1980.C. S Hsu. A &%rete method of optimal controlbased upon the cell state space concept, Journal qfOptimization Theow and Applications. vol. 46. pp.

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C. S . Hsu, Cell-to-Cell Mapping, Springer-Veriag,New York, 1987.H. T. Hu, H. M. Tai and S . Shenoi, Incorporatingcell map information in fuzzy controller des ign, toappear in Proceedings of the Third IEEEInternational Conference on Fuzzy Sjstem s, 1994.J. S. ang, Self-learning fuzzy controllers based ontemporal back-propagation, IEEE Transactions onNeural Networks, vol. 3, pp. 714-723, 1992.H. Kang and G. Vachtsevanos, Non-linear uzzycontrol based on the vector field of the phase portraitassignment algorithm, Proceedings of the AmericanControl Conference,pp. 1179-1484, 1990.C. L. Karr and E. J. Gentry, F v ontrol of pHusing genetic algorithms,' IEEE Transactions onFuzzy Systems, vol. 1, pp. 4 6 - 5 3 , 1993.M. Lee and H. Takagi, Integrating design stages offuzzy systems using genetic algorithms,Proceedings o the Second IEEF InternationalConfrence onFuzty Systems, pp. 612-617, 1993.H. Nomura, I. Hayashi and N. Wakami, A self-tuning method of fuzzy control by descent method,Proceedings of the 4th IFSA World Congress, vol.Engineering, pp. 155-158, 1991.S. M. Smith and D . J Comer, Automated calibra-tion of a fuzzy logic controller using a cell statespace algorithm, IEEE Control Systems Magazine,vol. 11, no. 8, pp. 18-28, 1991.S M. Smith and D. J. Comer, An algorithm forautomated fuzzy logic controller tuning,Proceedings o the First IEEE InternationalConfireme on Fuzzy Systems, pp. 61S-622, 1992.G. Rudolph, Convergence analysis of canonicalgenetic algorithms, IEEE Transactions on NeuralNetworks, vol. 5 , pp 96-101, 1994.T. Takagi and M . Sugeno, 'Fuzzy identification ofsystems and its applications to modeling andcontrol, IEEE Transactions on Systems, Man andCybernefics,vol. 15, pp. 116-132, 1985.G. Vachtsevanos, S. S Farinwata and D. K.Pirovolou, Fuzzy logic control of an automotiveengine, IEEE Control Systems Magazine, vol. 13,no. 3, pp. 62-68 1993,B. Widrow and S Steams, Adaptive Signal Pro-cessing, hentice-H all, Englewood Cliffs, NJ, 1985.

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U

I50 . . . . . . . . .I.. .................. -

c, j* j.: jI . -j ...............150

0 0.5 1

.......... ................................

............. * . :

......................0 05 1

( c ) PolrMglc(nd) (d) time (sec)Figure 4. Time responses for the inverted pendulum system. (a) Force: (bl Pole angle: (c) Phase plane trajectories:(d) Angular velocity. (Solid line I = 0 5 m: Dashdotted line: = 0.25 m: Dashed line: 1 = 0.125 m.)

b)

Figure 5 . Controllable regions for different maximum force magnitudes. ((a) 50 newtons: b) 10 Newtons.)50

.......05 1

(a) r m sec)

li&

.......... .......................

. . . . . . . . .....

................... v .-.0 05 1

b) t im sec)

1 0 I 0 0.5 1c ) pokaIlg* rad) ( d ) l ime =)Figure 6. Time responses for the m er te d pendulum system (a) Force, (b) Pole angle; (c) Phase plane trajectories:

d) Angular velocity (Solid line:./ = 50 Nenlons: Dashdotted line: f = 70Newtons: Dashed line.f 0 Newtons.)

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Heng-Ming Tai - Robust Fuzzy Controllers

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