Fuzzy Control_MModern Control theory.pdf

Fuzzy Control , Modern Control

1. Fuzzy Control
Modern Control Theory
Maha Rauf (ID: 2023220036)
Date: 10th June, 2024
Southwest Jiaotong University

2. Learning Modules
•Robust Control
•Adaptive Control
•System Identification
•Fuzzy Control
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3. Introduction to Fuzzy Control 1/2
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Introduction:Fuzzy control integrates fuzzy logic with traditional
control systems to effectively manage uncertainty and imprecision in
complex environments. Unlike conventional methods, which rely on
precise numerical values and binary logic, fuzzy control uses
linguistic variables (e.g., "hot," "cold," "fast," "slow") and heuristic
rules to approximate human reasoning.
Core Concepts:
•Fuzzy Logic:Deals with degrees of truth rather than absolute
values, making it suitable for handling vague or ambiguous data.
•Linguistic Variables:Represent qualitative terms to interpret
imprecise inputs, enhancing adaptability in real-world scenarios.
•Rule-based Reasoning:Defines how inputs are processed to
generate outputs, mimicking human decision-making processes.
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4. Introduction to Fuzzy Control 2/2
4
Fuzzy control, pioneered by LotfiZadeh in the 1960s, represents a
significant advancement in control systems by extending the
principles of fuzzy logic to model and manage nonlinear and
uncertain systems. This approach has revolutionized various
industries by offering adaptable solutions to complex problems.
Historical Development:
•Origins:Developed as an extension of fuzzy logic, which
challenges traditional binary logic by allowing for degrees of truth,
fuzzy control was initially conceptualized to handle systems with
imprecise inputs and nonlinear relationships.
•Advancements:Over the decades, advancements in computing
power and algorithmic refinement have enabled fuzzy control to
evolve from theoretical concepts to practical applications across
diverse fields.
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5. Key Concepts in Fuzzy Control
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FuzzySets:Unlikecrispsets,fuzzysetsallowelementstobelong
tomultiplesetssimultaneouslywithvaryingdegreesofmembership.
Thisflexibilityenablesfuzzycontroltohandleimpreciseand
uncertaindataeffectively,whichiscrucialinreal-worldapplications
whereprecisecategorizationischallenging.
Rule-BasedSystems:FuzzycontroloperatesonIF-THENrules
basedonfuzzylogic.Theserulesuselinguisticvariables(e.g.,
"high,""low,""medium")tocapturehuman-likedecision-making
processes.Byincorporatingexpertknowledgeintothecontrol
strategy,fuzzyrule-basedsystemscanmakedecisionsandadjust
controlactionsinamannerthatisintuitiveandadaptable.
ControlStrategies:Fuzzycontrolemploysadaptiveandrobust
strategiestoadjustcontrolactionsbasedonreal-timefeedbackand
changingenvironmentalconditions.Thisdynamicapproach
enhancesthesystem'sabilitytomaintainstabilityandperformance
indynamicanduncertainenvironments,wheretraditionalcontrol
methodsmaystruggle.
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6. Case Study: Adaptive Control with Parameterized Sugeno Fuzzy
Approximator 1/2
6
InastudyconductedbyAlata,Su,andDemirli(2001),adaptivecontrol
techniquesemployingafirst-orderparameterizedSugenofuzzy
approximatorwereexplored.Thisinnovativeapproachintegratesfuzzylogic
withadaptivecontrolstrategiestoenhanceperformanceinmanaging
nonlinearsystems.
KeyConcepts:
•ParameterizedSugenoFuzzyApproximator:Thismethodologyinvolves
aspecifictypeoffuzzymodelthatadjustsitsparametersbasedonsystem
variablesandconditions.ByparameterizingtheSugenofuzzymodel,the
controlsystemcanadaptdynamicallytochangesinthesystem'sbehavior,
improvingaccuracyandresponsiveness.
•AdaptiveControlTechniques:Thesetechniquesallowthecontrolsystem
tolearnandoptimizecontrolactionsovertimebycontinuouslyupdatingthe
parametersofthefuzzyapproximator.Thisadaptabilityiscrucialfor
effectivelyhandlingnonlinearitiesanduncertaintiesinherentinreal-world
systems.
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7. Case Study: Adaptive Control with Parameterized Sugeno Fuzzy
Approximator 2/2
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StudyFindings:
✓EnhancedControlPerformance:TheuseofparameterizedSugeno
fuzzyapproximatordemonstratedimprovedcontrolperformance
comparedtotraditionalcontrolmethods.Iteffectivelyaccommodates
variationsinsystemdynamicsandexternaldisturbances,leadingto
smootheroperationandbetterstability.
✓ApplicationInsights:Thestudyhighlightedpracticalapplicationsin
industrieswhereprecisecontrolovernonlinearsystemsiscritical,such
asrobotics,aerospace,andprocessautomation.Byleveragingadaptive
fuzzycontrol,theseindustriescanachievehigherefficiencyandreliability
intheiroperations.
Conclusion:Theresearchunderscorestheeffectivenessofadaptivecontrol
strategiesemployingparameterizedSugenofuzzyapproximatorsin
enhancingcontrolperformancefornonlinearsystems.Thisapproach
representsasignificantadvancementintheapplicationoffuzzylogicto
addresscomplexcontrolchallengesacrossvariousindustrialdomains.
Southwest Jiaotong University

8. Literature Review: Dynamic Fuzzy State Space Models
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◆IntegrationofFuzzyLogicandStateSpace:Cao,Rees,andFeng(1999)introduced
dynamicfuzzystatespacemodelsthatcombinefuzzylogicwithtraditionalstatespace
representations.Thisintegrationallowsfortherepresentationofcomplexsystemswith
uncertaindynamicsmoreeffectivelythanconventionalmethods.
◆EnhancedRobustness:Byincorporatingfuzzylogicintostatespacemodels,the
approachenhancessystemrobustness.Fuzzylogicaccommodatesuncertaintiesand
variationsinsystemparameters,makingthecontroldesignmoreresilientto
unpredictablechangesintheenvironmentoroperatingconditions.
◆AdaptabilitytoUncertainEnvironments:Dynamicfuzzystatespacemodelsadaptto
uncertainenvironmentsbyadjustingtheirparametersandcontrolstrategiesbasedon
real-timefeedback.Thisadaptabilityiscrucialformaintainingoptimalperformancein
complexanddynamicsystems
◆ApplicationinRobustControlDesign:Theresearchdemonstratespractical
applicationsinrobustcontroldesignacrossvariousindustries,includingaerospace,
automotive,andmanufacturing.Thesemodelsofferimprovedstabilityandprecisionin
controllingsystemswithnonlinearitiesanddisturbances.
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9. Literature Review: Dynamic Fuzzy State Space Models
9
PossibilityTheoryandFuzzySets:DuboisandPrade(1996)utilizedpossibilitytheory
andfuzzysetstomodeluncertaintiesandimprecisionsinfaultdiagnosisforsatellite
systems.Inpossibilitytheory,thepossibilitydistributionfunctionΠ\PiΠrepresentsthe
degreetowhichaneventcouldoccur.Fuzzysetsareemployedtohandlevagueor
impreciseinformation,definingmembershipfunctionsμA(x)\mu_A(x)μA​(x)wherexxx
belongstoasetAAA.
ReliabilityEnhancementviaFuzzyLogic:Byintegratingfuzzylogicreasoning,the
methoddetectsfaultsbyevaluatingthepossibilityofsystemstates.Thediagnosticprocess
involves:
Here,II(ObservedSymptoms∣A)denotesthepossibilitythatobservedsymptomsindicate
faultA,andμA​(StateoftheSystem)indicatesthemembershipdegreeofthecurrent
systemstateinthefaultclassA.Thisapproachensuresrobustfaultdiagnosis,criticalfor
maintainingsafetyinsatelliteoperations.
Southwest Jiaotong University

10. Stability Analysis in Fuzzy Control
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Robust Stabilization of Uncertain Nonlinear Systems using Fuzzy Control
1.Lyapunov Functions and Stability:Tanaka, Ikeda, and Wang (1996) employed
Lyapunov functions in conjunction with fuzzy control to achieve robust stabilization of
uncertain nonlinear systems. Lyapunov functions are used to analyze the stability of
systems by evaluating the change in a system's state over time. By integrating these
functions with fuzzy control strategies, they ensured that the systems remain stable despite
uncertainties and variations in operating conditions.
2.Utilization of Linear Matrix Inequalities (LMIs):The researchers utilized Linear Matrix
Inequalities (LMIs) as a mathematical tool to formulate robust control design criteria. LMIs
provide a systematic approach to designing controllers that guarantee stability and
performance specifications across a range of uncertain system parameters. This approach
not only enhances the stability of fuzzy control systems but also improves their reliability
and effectiveness in real-world applications, such as in aerospace, automotive, and
industrial automation.
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11. Mathematical Foundations of Fuzzy Control
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FuzzySets:
MembershipfunctionsμA​(x)definethedegreetowhichanelementxbelongstoafuzzy
setA,facilitatingrepresentationofimpreciseanduncertaindata:
μA​(x):X→[0,1]
whereXistheuniverseofdiscourse.
FuzzyRule-BasedSystems:IF-THENrulesbasedonlinguisticvariablesandfuzzylogic
operationsdeterminecontrolactions,integratinghuman-likereasoningintocontrolsystems:
IFxisATHENyisB
whereAandBarefuzzysets,andxandyarelinguisticvariables.
Thesemathematicalfoundationsenablefuzzycontrolsystemstohandleimpreciseand
uncertaininformationeffectively,enhancingtheirapplicabilityincomplexanddynamic
environments.
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12. Mathematical Model of a Fuzzy Controller
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•Structure:Fuzzycontrollersconsistofthreemainmodules:fuzzification,ruleevaluation,
anddefuzzification.
•Fuzzification:
Inputs�1,�2,…,�??????x1,x2,…,xn​aremappedtolinguisticvariables�1,�2,…,�??????A1,A
2,…,An​usingmembershipfunctions??????��(��)μAi​(xi​):??????��(��),�=1,2,…,??????μAi​(x
i​),i=1,2,…,n
•RuleEvaluation:IF-THENrulesdeterminecontrolactionsbasedonfuzzylogicoperations:
IF�1is�1and�2is�2and…THEN�is�IFx1isA1andx2isA
2and…THENyisB
•Defuzzification:Combinesfuzzyoutputsintoacrispcontrolaction�yusingmethodslike
centroidorweightedaverage:
where��aretheoutputsoftheruleevaluation,�Bisthefuzzysetdefiningtheoutput
variable�,and??????�(��)arethemembershipdegreesof��in�.Thesemodulescollectively
formthemathematicalfoundationofafuzzycontroller,allowingittoeffectivelyprocessand
respondtoimpreciseanduncertaininputsincontrolsystems.
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13. Rule-Based Systems in Fuzzy Control
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✓IF-THENRulesandImpreciseInputs:Infuzzycontrolsystems,IF-THENrulesencode
expertknowledgeandempiricaldatatotranslateimpreciseinputsintoactionablecontrol
decisions.Theserulesareformulatedusinglinguisticvariablesandfuzzylogic
operations,enablingthesystemtohandleuncertainandvariableconditionseffectively.
✓Human-likeReasoningwithFuzzyLogic:Thedecision-makingprocessinfuzzy
controlapproximateshuman-likereasoningbyinterpretingvagueorambiguous
informationusingfuzzysetsandmembershipfunctions.Thisapproachenhancesthe
adaptabilityandrobustnessofcontrolsystems,makingthemsuitableforenvironments
whereprecisenumericaldatamaybelackingorunreliable.
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14. Future Directions in Fuzzy Control Research
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•AdvancedLearningAlgorithms:Integrationwithmachinelearningtechniquesfor
adaptivecontrolanddecision-making.
•Human-MachineInteraction:Enhancingusabilityandinterpretabilityoffuzzycontrol
systemsinreal-timeapplications.
•EmergingTechnologies:Applicationinautonomoussystems,smartcities,and
healthcareforimprovedperformanceandreliability.
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15. Applications of Fuzzy Control
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AutomotiveSystems:Controlofengineoperation,anti-lockbrakingsystems(ABS),and
vehiclestabilitycontrol.
IndustrialAutomation:Robotics,processcontrol,andmanufacturingsystems
optimization.
ConsumerElectronics:Washingmachines,airconditioningunits,andhomeautomation
systemsforenergyefficiencyandcomfort.
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16. Conclusion
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Fuzzycontrolispivotalinaddressinguncertaintyandcomplexityin
moderncontrolsystems,offeringrobustnessandadaptability.By
combiningtheoreticalfoundationswithpracticalapplications,it
contributessignificantlytoadvancementsincontroltheoryand
engineeringpractice.
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17. Recommendations for Implementation
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•SystemDesign:Definesystemdynamicsandidentify
appropriatefuzzylogiccomponents,includingmembership
functionsandrulebases.
•Validation:Validatefuzzycontrolsystemsthroughsimulation
andreal-worldtestingtoensurereliabilityandperformance.
•Integration:Integratefuzzycontrolwithexistingcontrol
strategiesforenhancedsystemperformanceandadaptability.
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18. 18Southwest Jiaotong University
References
1.Alata, M., Su, C.Y., Demirli, K. Adaptive
control of a class of nonlinear systems with a
first-order parameterized Sugeno fuzzy
approximator. IEEE Trans. on Systems, Man and
Cybernetics -Part C, 31(3), August 2001.
2.Cao, S.G., Rees, N.W., Feng, G. Analysis and
design of fuzzy control systems using dynamic
fuzzy state space models. IEEE Trans. Fuzzy
Systems, 7:192–200, 1999.
3.Dubois, D., Prade, H. Handling uncertainty with
possibility theory and fuzzy sets in a satellite
fault diagnosis application. IEEE Trans. on
Fuzzy Systems, 4(3):251–269, Aug 1996.
4.Tanaka, K., Ikeda, T., Wang, H.O. Robust
stabilization of uncertain nonlinear systems via
fuzzy control: Quadratic stability, H1 control
theory and linear matrix inequalities. IEEE
Transactions on Fuzzy Systems, 4(1):1–13, 1996.