We introduce the idea of the fuzzy control of chaos: we show how fuzzy logic
can be applied to the control of chaos, and provide an example of fuzzy control
used to control chaos in Chua's circuit

Calvo, Oscar

Cartwright, Julyan H. E.

English

arXiv

IWe introduce the idea of the fuzzy control of chaos: we show how fuzzy logic can be applied to the control of chaos, and provide an example of fuzzy control used to control chaos in Chua’s circuit.Instituto de Investigaciones en Electrónica, Control y Procesamiento de SeñalesComisión de Investigaciones Científicas de la provincia de Buenos Aire

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

International Journal of Bifurcation and Chaos, Vol. 8, No. 8 (1998) 1743–1747c World Scientific Publishing CompanyFUZZY CONTROL OF CHAOSOSCAR CALVO∗CICpBA, L.E.I.C.I., Departamento de Electrotecnia, Facultad de Ingenieŕıa,Universidad Nacional de La Plata, 1900 La Plata, ArgentinaJULYAN H. E. CARTWRIGHT†Departament de F́ısica & Centre de Càlcul i Informatització,Universitat de les Illes Balears, 07071 Palma de Mallorca, SpainReceived July 31, 1997; Revised January 15, 1998We introduce the idea of the fuzzy control of chaos: we show how fuzzy logic can be applied tothe control of chaos, and provide an example of fuzzy control used to control chaos in Chua’scircuit.1. IntroductionChaos control exploits the sensitivity to initial con-ditions and to perturbations that is inherent inchaos as a means to stabilize unstable periodic or-bits within a chaotic attractor. The control canoperate by altering system variables or systemparameters, and either by discrete corrections or bycontinuous feedback. Many methods of chaos con-trol have been derived and tested [Chen & Dong,1993; Lindner & Ditto, 1995; Ogorza lek, 1993].Why then consider fuzzy control of chaos?A fuzzy controller works by controlling a con-ventional control method. We propose that fuzzycontrol can become useful together with one of theseother methods — as an extra layer of control — inorder to improve the effectiveness of the control interms of the size of the region over which control ispossible, the robustness to noise, and the ability tocontrol long period orbits.In this paper, we put forward the idea of fuzzycontrol of chaos, and we provide an example show-ing how a fuzzy controller applying occasional pro-portional feedback to one of the system parameterscan control chaos in Chua’s circuit.2. Fuzzy ControlFuzzy control [Driankov et al., 1993; Terano et al.,1994] is based on the theory of fuzzy sets and fuzzylogic [Yager & Zadeh, 1991; Bezdek, 1993]. Theprinciple behind the technique is that imprecisedata can be classified into sets having fuzzy ratherthan sharp boundaries, which can be manipulatedto provide a framework for approximate reason-ing in the face of imprecise and uncertain infor-mation. Given a datum, x, a fuzzy set A is saidto contain x with a degree of membership µA(x),where µA(x) can take any value in the domain[0, 1]. Fuzzy sets are often given descriptive names(called linguistic variables) such as FAST; the mem-bership function µFAST(x) is then used to reflectthe similarity between values of x and a contex-tual meaning of FAST. For example, if x repre-sents the speed of a car in kilometres per hour,and FAST is to be used to classify cars travellingfast, then FAST might have a membership func-tion equal to zero for speeds below 90 km/h andequal to one for speeds above 130 km/h, with acurve joining these two extremes for speeds betweenthese values. The degree of truth of the statement∗E-mail: calvo@athos.fisica.unlp.edu.ar†E-mail: julyan@hp1.uib.es, URL: http://formentor.uib.es/∼julyan17431744 O. Calvo & J. H. E. Cartwrightthe car is travelling fast is then evaluated by read-ing off the value of the membership function corre-sponding to the car’s speed.Logical operations on fuzzy sets require an ex-tension of the rules of classical logic. The threefundamental Boolean logic operations, intersection,union, and complement, have fuzzy counterpartsdefined by extension of the rules of Boolean logic. Afuzzy expert system uses a set of membership func-tions and fuzzy logic rules to reason about data.The rules are of the form “if x is FAST and y isSLOW then z is MEDIUM”, where x and y are in-put variables, z is an output variable, and SLOW,MEDIUM, and FAST are linguistic variables. Theset of rules in a fuzzy expert system is known as therule base, and together with the data base of inputand output membership functions it comprises theknowledge base of the system.A fuzzy expert system functions in four steps.The first is fuzzification, during which the member-ship functions defined on the input variables are ap-plied to their actual values, to determine the degreeof truth for each rule premise. Next under inference,the truth value for the premise of each rule is com-puted, and applied to the conclusion part of eachrule. This results in one fuzzy set to be assigned toeach output variable for each rule. In composition,all of the fuzzy sets assigned to each output vari-able are combined together to form a single fuzzyset for each output variable. Finally comes defuzzi-fication, which converts the fuzzy output set to acrisp (nonfuzzy) number.A fuzzy controller may then be designed usinga fuzzy expert system to perform fuzzy logic opera-tions on fuzzy sets representing linguistic variablesin a qualitative set of control rules (see Fig. 1).As a simple metaphor of fuzzy control in prac-tice, consider the experience of balancing a stickDATA BASE RULE BASEKNOWLEDGE BASEINFERENCEKERNELFUZZIFIER DEFUZZIFIERCONTROLLEDSYSTEMSETPOINT+-CONTROLLEDVARIABLEFig. 1. Fuzzy logic controller block diagram.vertically on the palm of one’s hand. The equationsof motion for the stick (a pendulum at its unstablefixed point) are well-known, but we do not inte-grate these equations in order to balance the stick.Rather, we stare at the top of the stick and carryout a type of fuzzy control to keep the stick in theair: We move our hand slowly when the stick leansby a small angle, and fast when it leans by a largerangle. Our ability to balance the stick despite theimprecision of our knowledge of the system is at theheart of fuzzy control.3. Techniques for Fuzzy ChaosControlTo control a system necessitates perturbing it.Whether to perturb the system via variables or pa-rameters depends on which are more readily acces-sible to be changed, which in turn depends on whattype of system is to be controlled — electronic, me-chanical, optical, chemical, biological, etc. Whetherto perturb continuously or discretely is a questionof intrusiveness — it is less intrusive to the system,and less expensive to the controller, to perturb dis-cretely. Only when discrete control is not effectivemight continuous control be considered.Ott, Grebogi and Yorke [Ott et al., 1990]invented a method of applying small feedbackperturbations to an accessible system parameter inorder to control chaos. The OGY method uses thedynamics of the linearized map around the orbitone wishes to control. Using the OGY method,one can pick any unstable periodic orbit that existswithin the attractor and stabilize it. The control isimposed when the orbit crosses a Poincaré sectionconstructed close to the desired unstable periodicorbit. Since the perturbation applied is small, it issupposed that the unstable periodic orbit is unaf-fected by the control.Occasional proportional feedback [Hunt, 1991;Lindner & Ditto, 1995] is a variant of the originalOGY chaos control method. Instead of using theunstable manifold of the attractor to compute cor-rections, it uses one of the dynamical variables, ina type of one-dimensional OGY method. This feed-back could be applied continuously or discretely intime; in occasional proportional feedback it is ap-plied discretely. Occasional proportional feedbackexploits the strongly dissipative nature of the flowsoften encountered, enabling one to control themwith a one-dimensional map. The method is easyFuzzy Control of Chaos 1745to implement, and in many cases one can stabi-lize high period unstable orbits by using multiplecorrections per period. It is a suitable method onwhich to base a fuzzy logic technique for the controlof chaos, since it requires no knowledge of a systemmodel, but merely an accessible system parameter.4. An Example: Fuzzy Controlof Chaos in Chua’s CircuitChua’s circuit [Matsumoto, 1984; Kennedy, 1993]exhibits chaotic behavior that has been extensivelystudied, and whose dynamics is well known [Madan,1993]. Recently, occasional proportional feedbackhas been used to control the circuit [Johnson et al.,1993]. The control used an electronic circuit to sam-ple the peaks of the voltage across the negative re-sistance and if it fell within a window, centred abouta by a set-point value, modified the slope of thenegative resistance by an amount proportional tothe difference between the set point and the peakZEROPOSITIVESMALLNEGATIVESMALLPOSITIVEBIGNEGATIVEBIG1MEMBERSHIPFUNCTIONSOF FUZZYSETSe∆e∆a-1 10-0.5 0.5Fig. 2. Membership functions of the input and outputvariables e, ∆e, and ∆a.value. The nonlinear nature of this system and theheuristic approach used to find the best set of pa-rameters to take the system to a given periodic or-bit suggest that a fuzzy controller that can includeknowledge rules to achieve periodic orbits may pro-vide significant gains over occasional proportionalfeedback alone.We have implemented a fuzzy controller to con-trol the nonlinearity of the nonlinear element (athree segment nonlinear resistance) within Chua’scircuit. The block diagram of the controller isFig. 3. The whole controller and control system in the form of a block diagram, including the fuzzy controller, the peakdetector, the window comparator, and the Chua’s circuit system being controlled.Table 1. Quantification levels and membership functions.Error, e −1 −0.75 −0.5 −0.25 0 0.25 0.5 0.75 1Change in error, ∆e −1 −0.75 −0.5 −0.25 0 0.25 0.5 0.75 1Control, ∆a −1 −0.75 −0.5 −0.25 0 0.25 0.5 0.75 1Quantification level −4 −3 −2 −1 0 1 2 3 4Linguistic Variables Membership FunctionsPositive Big, PB 0 0 0 0 0 0 0 0.5 1Positive Small, PS 0 0 0 0 0 0.5 1 0.5 0Approximately Zero, AZ 0 0 0 0.5 1 0.5 0 0 0Negative Small, NS 0 0.5 1 0.5 0 0 0 0 0Negative Big, NB 1 0.5 0 0 0 0 0 0 01746 O. Calvo & J. H. E. CartwrightTable 2. Rule table for the linguistic vari-ables in Table 1.e NB NS AZ PS PB∆eNB NB NS NS AZ AZNS NS AZ AZ PSAZ NS AZ PS PSPS AZ AZ PS PBPB AZ PS PS PBshown in Fig. 1. It consists of four blocks: knowl-edge base, fuzzification, inference and defuzzifica-tion. The knowledge base is composed of a database and a rule base. The data base consists of theinput and output membership functions (Fig. 2).It provides the basis for the fuzzification, defuzzi-fication and inference mechanisms. The rule baseis made up of a set of linguistic rules mapping in-puts to control actions. Fuzzification converts theinput signals e and ∆e into fuzzified signals withmembership values assigned to linguistic sets. Theinference mechanisms operate on each rule, apply-ing fuzzy operations on the antecedents and bycompositional inference methods derives the con-sequents. Finally, defuzzification converts the fuzzyoutputs to control signals, which in our case controlthe slope of the negative resistance ∆a in Chua’sFig. 4. The fuzzy controller stabilizes a previously unsta-ble period-1 orbit. The control is switched on at time 20.The lower trace shows the correction pulses applied by thecontroller.circuit (Fig. 3). The fuzzification maps the error e,and the change in the error ∆e, to labels of fuzzysets. Scaling and quantification operations are ap-plied to the inputs. Table 1 shows the quantifiedlevels and the linguistic labels used for inputs andoutput. The knowledge rules (Table 2) are repre-sented as control statements such as “if e is NEG-ATIVE BIG and ∆e is NEGATIVE SMALL then∆a is NEGATIVE BIG”.0.0 100.0−4.0−2.00.02.04.0(a)0.0 100.0−4.0−2.00.02.04.0(b)Fig. 5. Trajectory traces show higher period orbits stabilized by the controller. As before, the lower trace shows the correctionpulses applied by the fuzzy control.Fuzzy Control of Chaos 1747The normalized equations representing the cir-cuit areẋ = α(y − x− f(x)) ,ẏ = x− y + z ,ż = −βy ,(1)where f(x) = bx+ 12(a− b)(|x+ 1| − |x− 1|) repre-sents the nonlinear element of the circuit. Changesin the negative resistance were made by changing aby an amount∆a = Fuzzy Controller Output×Gain× a . (2)We have performed numerical simulations, bothin C and in Simulink, of Chua’s circuit controlledby the fuzzy logic controller. Figure 3 shows thewhole control system in the form of a block dia-gram, including Chua’s circuit, the fuzzy controller,the peak detector, and the window comparator.Figure 4 gives a sample output of the fuzzy con-troller stabilizing an unstable period-1 orbit by ap-plying a single correction pulse per cycle of oscilla-tion. By changing the control parameters we canstabilize orbits of different periods. In Fig. 5 weillustrate more complex higher period orbits stabi-lized by the controller. One can tune the fuzzy con-trol over the circuit to achieve the type of responserequired in a given situation by modifying some orall of the rules in the knowledge base of the system.Of course, in the case of Chua’s circuit the sys-tem equations are available and fuzzy logic is thusnot necessary for control, but this simple examplepermits us to see the possibilities that fuzzy controlprovides, by allowing a nonlinear gain implementedin the form of knowledge based rules.5. ConclusionsWe have introduced the idea of using fuzzy logicfor the control of chaos. Fuzzy logic controllers arecommonly used to control systems whose dynamicsis complex and unknown, but for expositional clar-ity here we have given an example of its use with awell-studied chaotic system. We have shown that itis possible to control chaos in Chua’s circuit usingfuzzy control. Further work is necessary to quantifythe effectiveness of fuzzy control of chaos comparedwith alternative methods, to identify ways in whichto systematically build the knowledge base for fuzzycontrol of a particular chaotic system, and to applythe fuzzy controller to future chaotic systems.ReferencesBezdek, J. C. [1993] “Fuzzy models — What are they,and why?” IEEE Trans. Fuzzy Syst. 1, 1–6.Chen, G. & Dong, X. [1993] “From chaos to order —Perspectives and methodologies in controlling chaoticnonlinear dynamical systems,” Int. J. Bifurcation andChaos 3, 1363–1409.Driankov, D., Hellendoorn, H. & Reinfrank, M. [1993]An Introduction to Fuzzy Control (Springer).Hunt, E. R. [1991] “Stabilizing high period orbits in achaotic system,” Phys. Rev. Lett. 67, 1953–1955.Johnson, G. A., Tigner, T. E. & Hunt, E. R. [1993]“Controlling chaos in Chua’s circuit,” J. Circuits Syst.Comput. 3, 109–117.Kennedy, M. P. [1993] “Three steps to chaos — Part II:A Chua’s circuit primer,” IEEE Trans. Circuits Syst.40, 657–674.Lindner, J. F. & Ditto, W. L. [1995] “Removal, suppres-sion, and control of chaos by nonlinear design,” Appl.Mech. Rev. 48, 795–808.Madan, R. N. (ed.) [1993] Chua’s Circuit: A Paradigmfor Chaos (World Scientific, Singapore).Matsumoto, T. [1984] “A chaotic attractor from Chua’scircuit,” IEEE Trans. Circuits Syst. 31, 1055–1058.Ogorza lek, M. J. [1993] “Taming chaos: Part II — con-trol,” IEEE Trans. Circuits Syst. 40, 700–706.Ott, E., Grebogi, C. & Yorke, J. A. [1990] “Controllingchaos,” Phys. Rev. Lett. 64, 1196–1199.Terano, T., Asai, K. & Sugeno, M. (eds.) [1994] AppliedFuzzy Systems (Academic Press).Yager, R. R. & Zadeh, L. A. [1991] An Introductionto Fuzzy Logic Applications in Intelligent Systems(Kluwer).

Fuzzy Control of Chaos

5 pages, 5 figures, 2 tables.-- ArXiv pre-print available at: http://arxiv.org/abs/chao-dyn/9905033Full-text version available Open Access at: http://www.eecs.berkeley.edu/~chua/papers/Calvo98.pdfWe introduce the idea of the fuzzy control of chaos: we show how fuzzy logic can be applied to the control of chaos, and provide and example of fuzzy control used to control chaos in Chua's circuit.Peer reviewe

Calvo, Óscar

Digital.CSIC

International Journal of Bifurcation and Chaos, Vol. 8, No. 8 (1998) 1743{1747c© World Scientic Publishing CompanyFUZZY CONTROL OF CHAOSOSCAR CALVOCICpBA, L.E.I.C.I., Departamento de Electrotecnia, Facultad de Ingeniera,Universidad Nacional de La Plata, 1900 La Plata, ArgentinaJULYAN H. E. CARTWRIGHTyDepartament de Fsica & Centre de Calcul i Informatitzacio,Universitat de les Illes Balears, 07071 Palma de Mallorca, SpainReceived July 31, 1997; Revised January 15, 1998We introduce the idea of the fuzzy control of chaos: we show how fuzzy logic can be applied tothe control of chaos, and provide an example of fuzzy control used to control chaos in Chua’scircuit.1. IntroductionChaos control exploits the sensitivity to initial con-ditions and to perturbations that is inherent inchaos as a means to stabilize unstable periodic or-bits within a chaotic attractor. The control canoperate by altering system variables or systemparameters, and either by discrete corrections or bycontinuous feedback. Many methods of chaos con-trol have been derived and tested [Chen & Dong,1993; Lindner & Ditto, 1995; Ogorza lek, 1993].Why then consider fuzzy control of chaos?A fuzzy controller works by controlling a con-ventional control method. We propose that fuzzycontrol can become useful together with one of theseother methods | as an extra layer of control | inorder to improve the eectiveness of the control interms of the size of the region over which control ispossible, the robustness to noise, and the ability tocontrol long period orbits.In this paper, we put forward the idea of fuzzycontrol of chaos, and we provide an example show-ing how a fuzzy controller applying occasional pro-portional feedback to one of the system parameterscan control chaos in Chua’s circuit.2. Fuzzy ControlFuzzy control [Driankov et al., 1993; Terano et al.,1994] is based on the theory of fuzzy sets and fuzzylogic [Yager & Zadeh, 1991; Bezdek, 1993]. Theprinciple behind the technique is that imprecisedata can be classied into sets having fuzzy ratherthan sharp boundaries, which can be manipulatedto provide a framework for approximate reason-ing in the face of imprecise and uncertain infor-mation. Given a datum, x, a fuzzy set A is saidto contain x with a degree of membership A(x),where A(x) can take any value in the domain[0, 1]. Fuzzy sets are often given descriptive names(called linguistic variables) such as FAST; the mem-bership function FAST(x) is then used to reflectthe similarity between values of x and a contex-tual meaning of FAST. For example, if x repre-sents the speed of a car in kilometres per hour,and FAST is to be used to classify cars travellingfast, then FAST might have a membership func-tion equal to zero for speeds below 90 km/h andequal to one for speeds above 130 km/h, with acurve joining these two extremes for speeds betweenthese values. The degree of truth of the statementE-mail: calvo@athos.sica.unlp.edu.aryE-mail: julyan@hp1.uib.es, URL: http://formentor.uib.es/julyan17431744 O. Calvo & J. H. E. Cartwrightthe car is travelling fast is then evaluated by read-ing o the value of the membership function corre-sponding to the car’s speed.Logical operations on fuzzy sets require an ex-tension of the rules of classical logic. The threefundamental Boolean logic operations, intersection,union, and complement, have fuzzy counterpartsdened by extension of the rules of Boolean logic. Afuzzy expert system uses a set of membership func-tions and fuzzy logic rules to reason about data.The rules are of the form \if x is FAST and y isSLOW then z is MEDIUM", where x and y are in-put variables, z is an output variable, and SLOW,MEDIUM, and FAST are linguistic variables. Theset of rules in a fuzzy expert system is known as therule base, and together with the data base of inputand output membership functions it comprises theknowledge base of the system.A fuzzy expert system functions in four steps.The rst is fuzzication, during which the member-ship functions dened on the input variables are ap-plied to their actual values, to determine the degreeof truth for each rule premise. Next under inference,the truth value for the premise of each rule is com-puted, and applied to the conclusion part of eachrule. This results in one fuzzy set to be assigned toeach output variable for each rule. In composition,all of the fuzzy sets assigned to each output vari-able are combined together to form a single fuzzyset for each output variable. Finally comes defuzzi-cation, which converts the fuzzy output set to acrisp (nonfuzzy) number.A fuzzy controller may then be designed usinga fuzzy expert system to perform fuzzy logic opera-tions on fuzzy sets representing linguistic variablesin a qualitative set of control rules (see Fig. 1).As a simple metaphor of fuzzy control in prac-tice, consider the experience of balancing a stickDATA BASE RULE BASEKNOWLEDGE BASEINFERENCEKERNELFUZZIFIER DEFUZZIFIERCONTROLLEDSYSTEMSETPOINT+-CONTROLLEDVARIABLEFig. 1. Fuzzy logic controller block diagram.vertically on the palm of one’s hand. The equationsof motion for the stick (a pendulum at its unstablexed point) are well-known, but we do not inte-grate these equations in order to balance the stick.Rather, we stare at the top of the stick and carryout a type of fuzzy control to keep the stick in theair: We move our hand slowly when the stick leansby a small angle, and fast when it leans by a largerangle. Our ability to balance the stick despite theimprecision of our knowledge of the system is at theheart of fuzzy control.3. Techniques for Fuzzy ChaosControlTo control a system necessitates perturbing it.Whether to perturb the system via variables or pa-rameters depends on which are more readily acces-sible to be changed, which in turn depends on whattype of system is to be controlled | electronic, me-chanical, optical, chemical, biological, etc. Whetherto perturb continuously or discretely is a questionof intrusiveness | it is less intrusive to the system,and less expensive to the controller, to perturb dis-cretely. Only when discrete control is not eectivemight continuous control be considered.Ott, Grebogi and Yorke [Ott et al., 1990]invented a method of applying small feedbackperturbations to an accessible system parameter inorder to control chaos. The OGY method uses thedynamics of the linearized map around the orbitone wishes to control. Using the OGY method,one can pick any unstable periodic orbit that existswithin the attractor and stabilize it. The control isimposed when the orbit crosses a Poincare sectionconstructed close to the desired unstable periodicorbit. Since the perturbation applied is small, it issupposed that the unstable periodic orbit is unaf-fected by the control.Occasional proportional feedback [Hunt, 1991;Lindner & Ditto, 1995] is a variant of the originalOGY chaos control method. Instead of using theunstable manifold of the attractor to compute cor-rections, it uses one of the dynamical variables, ina type of one-dimensional OGY method. This feed-back could be applied continuously or discretely intime; in occasional proportional feedback it is ap-plied discretely. Occasional proportional feedbackexploits the strongly dissipative nature of the flowsoften encountered, enabling one to control themwith a one-dimensional map. The method is easyFuzzy Control of Chaos 1745to implement, and in many cases one can stabi-lize high period unstable orbits by using multiplecorrections per period. It is a suitable method onwhich to base a fuzzy logic technique for the controlof chaos, since it requires no knowledge of a systemmodel, but merely an accessible system parameter.4. An Example: Fuzzy Controlof Chaos in Chua’s CircuitChua’s circuit [Matsumoto, 1984; Kennedy, 1993]exhibits chaotic behavior that has been extensivelystudied, and whose dynamics is well known [Madan,1993]. Recently, occasional proportional feedbackhas been used to control the circuit [Johnson et al.,1993]. The control used an electronic circuit to sam-ple the peaks of the voltage across the negative re-sistance and if it fell within a window, centred abouta by a set-point value, modied the slope of thenegative resistance by an amount proportional tothe dierence between the set point and the peakZEROPOSITIVESMALLNEGATIVESMALLPOSITIVEBIGNEGATIVEBIG1MEMBERSHIPFUNCTIONSOF FUZZYSETSe∆e∆a-1 10-0.5 0.5Fig. 2. Membership functions of the input and outputvariables e, e, and a.value. The nonlinear nature of this system and theheuristic approach used to nd the best set of pa-rameters to take the system to a given periodic or-bit suggest that a fuzzy controller that can includeknowledge rules to achieve periodic orbits may pro-vide signicant gains over occasional proportionalfeedback alone.We have implemented a fuzzy controller to con-trol the nonlinearity of the nonlinear element (athree segment nonlinear resistance) within Chua’scircuit. The block diagram of the controller isFig. 3. The whole controller and control system in the form of a block diagram, including the fuzzy controller, the peakdetector, the window comparator, and the Chua’s circuit system being controlled.Table 1. Quantication levels and membership functions.Error, e −1 −0:75 −0:5 −0:25 0 0.25 0.5 0.75 1Change in error, e −1 −0:75 −0:5 −0:25 0 0.25 0.5 0.75 1Control, a −1 −0:75 −0:5 −0:25 0 0.25 0.5 0.75 1Quantication level −4 −3 −2 −1 0 1 2 3 4Linguistic Variables Membership FunctionsPositive Big, PB 0 0 0 0 0 0 0 0.5 1Positive Small, PS 0 0 0 0 0 0.5 1 0.5 0Approximately Zero, AZ 0 0 0 0.5 1 0.5 0 0 0Negative Small, NS 0 0.5 1 0.5 0 0 0 0 0Negative Big, NB 1 0.5 0 0 0 0 0 0 01746 O. Calvo & J. H. E. CartwrightTable 2. Rule table for the linguistic vari-ables in Table 1.e NB NS AZ PS PBeNB NB NS NS AZ AZNS NS AZ AZ PSAZ NS AZ PS PSPS AZ AZ PS PBPB AZ PS PS PBshown in Fig. 1. It consists of four blocks: knowl-edge base, fuzzication, inference and defuzzica-tion. The knowledge base is composed of a database and a rule base. The data base consists of theinput and output membership functions (Fig. 2).It provides the basis for the fuzzication, defuzzi-cation and inference mechanisms. The rule baseis made up of a set of linguistic rules mapping in-puts to control actions. Fuzzication converts theinput signals e and e into fuzzied signals withmembership values assigned to linguistic sets. Theinference mechanisms operate on each rule, apply-ing fuzzy operations on the antecedents and bycompositional inference methods derives the con-sequents. Finally, defuzzication converts the fuzzyoutputs to control signals, which in our case controlthe slope of the negative resistance a in Chua’sFig. 4. The fuzzy controller stabilizes a previously unsta-ble period-1 orbit. The control is switched on at time 20.The lower trace shows the correction pulses applied by thecontroller.circuit (Fig. 3). The fuzzication maps the error e,and the change in the error e, to labels of fuzzysets. Scaling and quantication operations are ap-plied to the inputs. Table 1 shows the quantiedlevels and the linguistic labels used for inputs andoutput. The knowledge rules (Table 2) are repre-sented as control statements such as \if e is NEG-ATIVE BIG and e is NEGATIVE SMALL thena is NEGATIVE BIG".0.0 100.0−4.0−2.00.02.04.0(a)0.0 100.0−4.0−2.00.02.04.0(b)Fig. 5. Trajectory traces show higher period orbits stabilized by the controller. As before, the lower trace shows the correctionpulses applied by the fuzzy control.Fuzzy Control of Chaos 1747The normalized equations representing the cir-cuit are_x = (y − x− f(x)) ;_y = x− y + z ;_z = −y ;(1)where f(x) = bx+ 12(a− b)(jx+ 1j − jx− 1j) repre-sents the nonlinear element of the circuit. Changesin the negative resistance were made by changing aby an amounta = Fuzzy Controller OutputGain a : (2)We have performed numerical simulations, bothin C and in Simulink, of Chua’s circuit controlledby the fuzzy logic controller. Figure 3 shows thewhole control system in the form of a block dia-gram, including Chua’s circuit, the fuzzy controller,the peak detector, and the window comparator.Figure 4 gives a sample output of the fuzzy con-troller stabilizing an unstable period-1 orbit by ap-plying a single correction pulse per cycle of oscilla-tion. By changing the control parameters we canstabilize orbits of dierent periods. In Fig. 5 weillustrate more complex higher period orbits stabi-lized by the controller. One can tune the fuzzy con-trol over the circuit to achieve the type of responserequired in a given situation by modifying some orall of the rules in the knowledge base of the system.Of course, in the case of Chua’s circuit the sys-tem equations are available and fuzzy logic is thusnot necessary for control, but this simple examplepermits us to see the possibilities that fuzzy controlprovides, by allowing a nonlinear gain implementedin the form of knowledge based rules.5. ConclusionsWe have introduced the idea of using fuzzy logicfor the control of chaos. Fuzzy logic controllers arecommonly used to control systems whose dynamicsis complex and unknown, but for expositional clar-ity here we have given an example of its use with awell-studied chaotic system. We have shown that itis possible to control chaos in Chua’s circuit usingfuzzy control. Further work is necessary to quantifythe eectiveness of fuzzy control of chaos comparedwith alternative methods, to identify ways in whichto systematically build the knowledge base for fuzzycontrol of a particular chaotic system, and to applythe fuzzy controller to future chaotic systems.ReferencesBezdek, J. C. [1993] \Fuzzy models | What are they,and why?" IEEE Trans. Fuzzy Syst. 1, 1{6.Chen, G. & Dong, X. [1993] \From chaos to order |Perspectives and methodologies in controlling chaoticnonlinear dynamical systems," Int. J. Bifurcation andChaos 3, 1363{1409.Driankov, D., Hellendoorn, H. & Reinfrank, M. [1993]An Introduction to Fuzzy Control (Springer).Hunt, E. R. [1991] \Stabilizing high period orbits in achaotic system," Phys. Rev. Lett. 67, 1953{1955.Johnson, G. A., Tigner, T. E. & Hunt, E. R. [1993]\Controlling chaos in Chua’s circuit," J. Circuits Syst.Comput. 3, 109{117.Kennedy, M. P. [1993] \Three steps to chaos | Part II:A Chua’s circuit primer," IEEE Trans. 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Fuzzy Control of Chaos

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