APPLICATION OF UNCERTAIN NONLINEAR SYSTEMS PARTIAL STATE VARIABLES CONTROL TO A CLASS OF PENDULUM SYSTEMS

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1 3 st Deceber 0 Vol 46 No JATIT & LLS All rights reserved ISSN: wwwjatitorg E-ISSN: APPLICATION OF UNCERTAIN NONLINEAR SYSTEMS PARTIAL STATE VARIABLES CONTROL TO A CLASS OF PENDULUM SYSTEMS YANJUN LIANG GUOYING LIU School of Coputer Iforatio Egieerig AYag Noral Uiversity Ayag Hea Chia ABSTRACT Because of iaccuracy of odelig paraeters easureet the cotrolled systes are subjected by the iaccuracy ievitably the cotrol of ucertai systes is studied by ay researchers Ucertai oliear systes partial state variables cotrol techique is developed applied to sigle pedulu systes I order to ake a certai state variable asyptotically stable adaptive paraeters idetify the ucertai paraeters usig partial state stability theory adaptive cotrol ethod two cotrollers adaptive regulators are desiged separately for two state variables ucertai paraeters Our cotrollers ca guaratee the uiforly ultiate boudedess of the solutio of the closed-loop syste ake the trackig error arbitrarily sall for ucertai paraeters Nuerical siulatio results illustrate the effectiveess of the proposed cotrollers adaptive regulators Keywords: Noliear Systes Partial State Variables Cotrol Ucertai Systes INTRODUCTION I the actual cotrol systes there are ay oliearity facts Oe ofte usig liear cotrol theory igorig the oliearity of the cotrol systes it is useful for the weak oliearity but for the strog oliearity oe usually ca ot get epected cotrol effect Because of iaccuracy of odelig paraeters easureet the cotrolled systes are geerally subjected by the iaccuracy ievitably Therefore ucertai oliear systes cotrol is take uch attetio recetly There is ot a geeral cotrol ethod for all the ucertai oliear systes cotrol Oe tries to study kids of ucertai oliear systes by ay eas I [] by icorporatig adaptive backsteppig desig techique ito a eural etwork based adaptive cotrol desig fraework adaptive backsteppig desig is developed for a class of oliear systes i strict-feedback for with arbitrary ucertaity I [] the proble of fiite-tie stabilizatio for oliear systes is studied; they prove that global fiite-tie stabilizability of ucertai oliear systes that are doiated by a lower-triagular syste ca be achieved by Hölder cotiuous state feedback A robust adaptive fuzzy cotrol desig approach is developed for a class of ultivariable oliear systes with odelig ucertaities eteral disturbaces [3] I [4] a adaptive cotrol algorith is proposed for output regulatio of ucertai oliear systes i output feedback for uder disturbaces geerated fro oliear eosystes I [5] they ivestigate the robust reliable H cotrol for a class of ucertai oliear syste A uified fraework for adaptive iterative learig cotrol desig for ucertai oliear systes is proposed accordig to the value of a certai paraeter gaa the paraetric adaptatio law ca be a pure tie-doai adaptatio a pure iteratio-doai adaptatio or a cobiatio of both [6] A hybrid cotrol syste itegratig pricipal copesatio cotrollers is developed for ultiple-iput-ultiple-output (MIMO) ucertai oliear systes [7] I [8] a adaptive fuzzy cotrol approach is proposed for a class of MIMO oliear systes with copletely ukow oaffie fuctios the i [9] eteral disturbaces appear i each equatio of each subsyste the disturbace coefficiets are assued to be ukow fuctios rather tha costat oe the uiversal approiatio theore of the fuzzy logic systes is utilized to develop a adaptive cotrol schee for a class of oliear MIMO systes by the backsteppig techique For ucertai systes preceded by ukow dead-zoe oliearity i [0] a ew schee to desig adaptive cotrollers is preseted I [] they prove that the proposed oliear disturbace observer recovers ot oly the steadystate perforace but also the trasiet 77

2 3 st Deceber 0 Vol 46 No JATIT & LLS All rights reserved ISSN: wwwjatitorg E-ISSN: perforace of the oial closed-loop syste uder plat ucertaities iput disturbaces If the ucertaities are bouded while this boud is ot kow i [] a PI-adaptive fuzzy cotrol architecture for a class of ucertai oliear systes is proposed that ais to provide added robustess i the presece of large fast but bouded ucertaities disturbaces Based o the itegratio of slidig ode cotrol adaptive fuzzy cotrol a ovel adaptive fuzzy slidig ode cotrol ethodology is proposed i [3] I the egieerig because soe state variables of the systes are ot available partial state variables or a sigle state variable cotrol is siple effective easy to ipleet I this paper ucertai oliear systes partial state variables cotrol techique is developed for a class of pedulu systes I order to ake a certai state variable asyptotically stable adaptive paraeters idetify the ucertai paraeters usig partial state stability theory adaptive cotrol ethod two cotrollers adaptive regulators are desiged separately for two state variables ucertai paraeters Nuerical siulatio results illustrate the effectiveess of the proposed cotrollers adaptive regulators UNCERTAIN PENDULUM CONTROL SYSTEMS Pedulu is also aed atheatical pedulu it is a iportat odel i physics for its various dyaical behaviors Accordig to Newto laws of otio if cosiderig eviroetal dapig eteral drivig force siple pedulu oveet is a two order oliear differetial equatio Because of iaccuracy of odelig paraeters easureet i the process of odel establishet for pedulu dyaical systes ucertaity is ievitable The we cosider a class of pedulu systes as follows = e + h t a = - q s i () i which are state variables ε η θ are ucertai paraeters produced by the process of syste odelig Figure : The Curves Of Ad We give the curves of the state variables of systes () usig Matlab software i figure i which ε = η = θ = 3 Obviously systes () are ot asyptotically stable The ai purpose of this paper is desigig cotroller u u for cotrol systes () to esure a certai state variable asyptotically stable = e + h ta + u = - q si + u 3 PARTIAL STATE VARIABLE CONTROLLER DESIGN () I order to desig partial state variable cotroller for systes () we itroduce two leas Cosider differetial equatios d f ( t ) = (3) i which f ( t ) C[ I R R ] f ( t 0) 0 = col( y z) = col( ) R + + y = col( ) R p z = col( ) R ( + p = ) z + + / / : = ( i ) y : = ( i ) i= i= / : = ( i ) i= + Lea [4] A ecessary sufficiet coditio of fuctio V ( t ) C( I R R) V ( t 0 ) 0 positive defiite for y is ϕ ( r) K i Ω : = { H} satisfyig 773

3 3 st Deceber 0 Vol 46 No JATIT & LLS All rights reserved ISSN: wwwjatitorg E-ISSN: V ( t ) ϕ( y ) (4) Lea [4] If fuctio V ( t ) C( I R R) V ( t 0 ) 0 satisfies(4) its derivative akes (3) c( y )( c K) trivial solutio of differetial equatios (3) is stable for y 3 Desig Cotroller u Makig Asyptotically Stable I theore we provide cotroller u adaptive cotrol law l ( t) Theore Cotroller u adaptive cotrol law l ( t) i (5) ake state variable of the pedulu cotrol systes () asyptotically stable u = g( t) si - l& = - si i which l ( t) = g( t) - q Proof Take a Lyapuov fuctio (5) V ( t ) = ( + l ( t) ) (6) for the pedulu cotrol systes() Accordig lea it is positive defiite for its derivative alog the systes () is as follows () = + l ( t) l& = (- q si + u ) + l ( t) l& = (- q si + g( t) si - ) - ( g( t) - q) si = - 0 (7) By the lea the cotrol systes () are asyptotically stable for state variable The we eploy Matlab progra to siulate the cotrol effect of the cotroller u adaptive cotrol law l ( t) The results are show i figure - 4 i which the ucertai paraeters are chose as ε = η = θ = 3 Figure : The Curve Of Figure 3: The Curve Of We study asyptotically stability of partial variables i this paper Now our ai purpose is cotrol state variable akes it asyptotically stable Fro figure figure 3 we ca see is asyptotically stable ot for It illustrates that adaptive cotrol paraeter g( t) recogizes ucertai paraeter θ of the pedulu cotrol systes rapidly i figure 4 Figure 4: The Curve Of g ( t ) 3 Desig Cotroller u Makig Asyptotically Stable I the flowig we desig cotroller u adaptive cotrol laws to ake asyptotically stable to recogize ay give ucertai paraeters ε η i theore Theore The cotroller u adaptive cotrol laws described i (8) ake asyptotically stable 774

4 3 st Deceber 0 Vol 46 No JATIT & LLS All rights reserved ISSN: wwwjatitorg E-ISSN: u = (- a( t) + a( t) ( t) - b( t) + b( t) j ( t)) - a( t) - b( t) ta = + ( a( t) - ( t)) j& = ta + ( b( t) - j ( t)) i which ( t) = a( t) - e j ( t) = b( t) - h Proof choose a Lyapuov fuctio (8) V ( t ) = ( + ( t) + j ( t) ) (9) for the pedulu cotrol systes () Apparetly it is positive defiite for its derivative alog the systes () is as follows () = + ( t) + j ( t) j& = ( e + h ta + u ) + ( t)( + ( a( t) - ( t)) ) + j ( t)( ta + ( b( t) - j ( t)) ) = - ( e + h ) 0 (0) Accordig lea the cotrol systes () are asyptotically stable for state variable I order to further check the effectiveess of the cotroller adaptive cotrol laws described i(8) we use Matlab software to siulate the result of the cotrol systes () with the effect of the cotroller adaptive cotrol laws The results are show i figure 5-8 i which the ucertai paraeters are chose as ε = 7 η = 6 θ = Figure 5: The Curve Of Figure 5 figure 6 are the cotrol curves of the state variables fro which we ca see that he state variables has very good asyptotically stability with the effect of the cotroller adaptive cotrol laws Figure 7 figure 8 are the curves of the adaptive cotrol paraeters α( t) β ( t) they show that α( t) β ( t) ca respectively recogize the values of ucertai paraeters ε η have very good real-tie trait strog robustess 4 CONCLUSION Figure 7: The Curve Of α ( t) Figure 8: The Curve Of β ( t) I the practical applicatio the cotrol systes ofte have soe ucertaity the probles of partial variable cotrol are paid attetio to i ay cases I this paper usig Lyapuov stability theory eployig adaptive cotrol ethod we studied the proble of desigig partial state variable cotrollers adaptive laws for a class of pedulu ucertai cotrol systes Two partial state variable cotrollers were desiged respectively adaptive cotrol laws were give for the ucertai paraeters of a class of pedulu ucertai cotrol systes The uerical siulatio were carried out it shows that the cotrollers adaptive cotrol laws are siple easy directly perceived through the seses have strog robustess good cotrol perforace good applicatio Figure 6: The Curve Of 775

5 3 st Deceber 0 Vol 46 No JATIT & LLS All rights reserved ISSN: wwwjatitorg E-ISSN: ACKNOWLEDGEMENTS This work was supported by the Natioal Natural Sciece Foudatios of Chia ( ) Hea Provice Educatio Departet Natural Sciece Foudatios of Chia (A000 A50003) REFERENCES: [] D Wag JHuag Neural etwork-based adaptive dyaic surface cotrol for a class of ucertai oliear systes i strict-feedback for IEEE Trasactios o Neural Networks Vol 6 No 005 pp 95-0 [] X Q Huag W Li B Yag Global fiitetie stabilizatio of a class of ucertai oliear systes Autoatica Vol 4 No pp [3] S S Zhou G Feg C B Feg Robust cotrol for a class of ucertai oliear systes: adaptive fuzzy approach based o backsteppig Fuzzy Sets Systes Vol 5 No 005 pp -0 [4] Z T Dig Output regulatio of ucertai oliear systes with oliear eosystes IEEE Trasactios o Autoatic Cotrol Vol 5 No pp [5] C H Lie KW Yu Y F Li Y J Chug L Y Chug Robust reliable H cotrol for ucertai oliear systes via LMI approach Applied Matheatics Coputatio Vol 98 No 008 pp [6] A Tayebi C J Chie A uified adaptive iterative learig cotrol fraework for ucertai oliear systes IEEE Trasactios o Autoatic Cotrol Vol 5 No pp [7] C M Li L Y Che C H Che RCMAC hybrid cotrol for MIMO ucertai oliear systes usig slidig-ode techology IEEE Trasactios o Neural Networks Vol 8 No pp [8] Y J Liu W Wag Adaptive fuzzy cotrol for a class of ucertai oaffie oliear systes Iforatio Scieces Vol 77 No pp [9] Y J Liu S C Tog W Wag Adaptive fuzzy output trackig cotrol for a class of ucertai oliear systes Fuzzy Sets Systes Vol 60 No pp [0] J Zhou C We Y Zhag Adaptive output cotrol of oliear systes with ucertai dead-zoe oliearity IEEE Trasactios o Autoatic Cotrol Vol 5 No pp [] J Back H Shi Addig robustess to oial output-feedback cotrollers for ucertai oliear systes: A oliear versio of disturbace observer AutoaticaVol 44 No pp [] R Shahazi M- R Akbarzadeh-T PI adaptive fuzzy cotrol with large fast disturbace rejectio for a class of ucertai oliear systes IEEE Trasactios o Fuzzy Systes Vol 6 No 008 pp [3] M Roopaei M Zolghadri S Meshksar Ehaced adaptive fuzzy slidig ode cotrol for ucertai oliear systes Couicatios i Noliear Sciece Nuerical Siulatio Vol 4 No pp [4] X X Liao Matheatical theory applicatio of stability (i Chiese) Wuha Chia: Huazhog Uiversity of Sciece Tec hology Press 00 pp