Journal of Engineering Science and Technology Review 7 (2) (2014) Research Article

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1 Jest Joual of Egeeg Scece ad echology Revew 7 () (4) 75 8 Reseach Atcle JOURNAL OF Egeeg Scece ad echology Revew Fuzzy Backsteppg Sldg Mode Cotol fo Msmatched Uceta System H. Q. Hou,*, Q. Mao, Q. H. Gao ad Aoue, Buam. X a Reseach Ist. of h-ech hogqg ow, X a, 75, Cha Itellget Cotol ad Data Aalyss Cete (IMAC) of BIMD Ic, Los Ageles, CA , Uted States Receved Mach 4; Accepted 5 Jue 4 Abstact Sldg mode cotolles have succeeded may cotol poblems that the covetoal cotol theoes have dffcultes to deal wth; howeve t s pactcally mpossble to acheve hgh-speed swtchg cotol. heefoe, ths pape a adaptve fuzzy backsteppg sldg mode cotol scheme s deved fo msmatched uceta systems. Fstly fuzzy sldg mode cotolle s desged usg backsteppg method based o the Lyapuov fucto appoach, whch s capable of hadlg msmatched poblem. he fuzzy sldg mode cotolle s desged usg -S fuzzy model method, t ca mpove the pefomace of the cotol systems ad the obustess. Fally ths method of cotol s appled to olea system as a case study; smulato esults ae also povded the pefomace of the poposed cotolle. Keywods: Msmatched Uceta System, -S Fuzzy Model, Sldg Mode Cotol, Backsteppg Cotol. Itoducto he poblem of system cotol wth uceta has become a eseach focus. Sldg mode cotol has a wde applcato system wth matchg uceta, foeve, ts obustess s dffcult to guaatee fo msmatched system. A adaptve backsteppg method s a systematc ad ecusve desg methodology fo olea feedback cotol, thee ae uque stuctual desg pocesses ad capable of dealg wth msmatched ucetaty, ad successfully appled acaft, moto ad machey []. H. Bouad desged sldg mode cotol based o backsteppg appoach, ad appled a UAV type-quadoto[].lteatue[3] peseted a backsteppg SMC method to compesate the ucetaty whch occus the slave system, ad appled to mplemet sychozato of factoal-ode Chaotc system. Faz Plta poposed a MIMO fuzzy backsteppg adaptve fuzzy estmato vaable stuctue cotolle desg ad applcato to ege ode to desg hgh pefomace olea cotolle [4]. I lteatue [5], Mohammad Reza Soltapou peseted cotollg of a class of olea systems wth stuctued ad ustuctued ucetates usg fuzzy sldg mode cotol. Abdel Bade Shakawy deved a adaptve fuzzy sldg mode cotol scheme fo obotc systems [6]. Howeve, the backsteppg method eques the ucetaty of the systems ca be paametezed epesetato, ad wth the elatve degee ceasg of cotolled object, thee s a calculated expaso poblem, so that the cotolle s dffcult to acheve. he cocept of Fuzzy cotol s to utlze the laguage ules of system to desg a pactcal cotolle; t s * E-mal addess: 69@63.com ISSN: Kavala Isttute of echology. All ghts eseved. patculaly sutable fo those systems wth uceta, tmevayg ad complex dyamcs [7], [8]. -S fuzzy model s a olea model to descbe the dyamc chaactestcs of complex systems, ad moe effectve way to descbe olea systems [9]. Fo the -S fuzzy model, Lteatue [], [], [] also poved that thee s bette appoxmato pefomace tha the Mamda fuzzy model. I ths pape, a tellget backsteppg sldg mode cotol scheme s peseted fo MIMO olea systems. he scheme s based o the uvesal appoxmato popety of fuzzy systems ad backsteppg sldg mode cotol theoy, ad smulato was pefomed to valdate the aalyss esult.. Poblem Statemet hs pape focuses o a class of MIMO olea uceta systems descbed as # x b x + f (x ) " $# x b u + f (x,x ) + w(t) () f( x ) s olea fucto of the system, x [ x, x, L, x ] s the state vaable of the system, x [ x, x, L, x], (,, L, ). u s the put vecto of the system. ( ) ak( x ) ; w() t Gx ad ae the ucetates ad exteal dstubaces, ad t may ot satsfy the so-called matchg codtos.

2 H. Q. Hou, Q. Mao, Q. H. Gao ad Aoue, Buam./Joual of Egeeg Scece ad echology Revew 7 () (4) 75 8 b jm Assumpto : hee ae costats of satsfy the equalty ad b jm Essetally, -S fuzzy modellg method s appoxmato model, t s descbed by fuzzy IF-HEN ules whch the cosequet pats epeset local lea models.i ode to get Fuzzy -S model of olea system, ad t must obta the lea model of each subsystem. to < b jm < b j < b jm, ( j, ), ad covese matx of matx bj exsts. he am of the tajectoy tackg cotol method s to desg cotol law u(t ), system state s evetually mataed at zeo-fom ay tal state x () 4. Adaptve Backsteppg Sldg Mode Cotol 4. Desg of ackg Cotolle stuatos of ucetaty w (t ). he pape focuses o the fomula () of the msmatched ad uceta system, ths s a secod-ode system; t takes two steps to desg adaptve backsteppg sldg mode cotolle, the fst step s to deteme the mddle tem 3. -S Fuzzy Model wth the stablty -S fuzzy model s a class of olea model, ad t s easy to descbe the dyamc chaactestcs of complex systems. Cosdeg geeal olea uceta system, a compact -S fuzzy model ca be obtaed by a caeful selecto of ule umbe ad paametes [3], [4], [5]. he system s descbed as the followg fuzzy ules, ad the the fuzzy model s composed of the followg ules cotol ad paamete estmato algothm accodg V. he fst step: Fo the posto tackg objectve, defe ew state eo vecto z, z R as f z s F ad z s F L z s F, the z (t ) [ z (t ),z (t ),L,z (t )] pemse vaables. F l z x xd z x τ () y(t) x(t) s x (t ) R s s Fuzzy sets. system, Fuzzy "$ z b x + f ( x ) x d # z b u + f ( x,x ) + w(t) τ %$ ae system vecto ad output vecto. Defe vtual cotol vaable dag(k, k,l, k ), k > (,,L, ). (3) Fom (7) ad (8), we ca obta y (t) (z(t))x(t) z b x + f ( x ) xd M F l ( z (t )) Kz + bz µ l F l ( z (t )), Poof: Assumg the fuzzy system s cotollable, the the fuzzy cotol ules s descbed as Defe the Lyapuov fucto as R : f z s F ad z s F L z s F, V (4) z z + z z, the we ca obta as follows V z z + z z he globally cotol law of state feedback s obtaed as follows z ( Kz + bz ) + z (bu u(t ) α ( z (t ))K x (t ) (9) Poposto : Usg cotol law (8), the closed-loop system s global expoetal stable based o assumpto. mf l s the membeshp fucto of z (t ) fuzzy set F l. the u(t ) K x(t ) (8) K (7) τ as follows τ b- (Kz + f ( x ) xd ) (z(t))[ A x(t) + Bu(t)] x(t) gve by the cotol sgal commads, ad Fom () ad (6), we ca obta as Poduct feece ege, sgleto fuzzfe ad cete aveage defuzzfe ae used to calculate outputs. he fuzzy feed output s wtte as µ xd s τ s a vtual cotol vaable. State u(t ) R m s the put of fuzzy system.,,l, s the umbe of system put. y (t ) R s Fuzzy output of the system. A R ad α ( z (t )) (6) xd ad τ ae the desed state tajectoy of the vaables. B R m by selectg a sutable Lyapuov fucto V ; the secod step s to deteme sldg mode Plat Rules: A x(t) + B u(t) x(t) τ +f ( x,x ) + w (t) τ ) (5) 76 ()

3 H. Q. Hou, Q. Mao, Q. H. Gao ad Aoue, Buam./Joual of Egeeg Scece ad echology Revew 7 () (4) 75 8 Fom assumpto, covese matx of matx bj 4. Stablty Aalyss Defe Lyapuov fucto as follows exsts, assumg uppe boud of ucetaty ad tefeece Geeal tems w (t ) s w, ad the the cotol law s u b [K ( Kz + bz ) + f ( x,x ) V z z + s s + w K 5 () +w τ + b z ] Assumg ucetates w(t ), ts estmated value s w, % ad ts estmated eo s w Fom cotol law () ad (), we obta V z Kz z K z () z ( Kz + bz ) + s (K z + z ) w wˆ K 5 z Kz + z bz + s [K ( Kz + bz ) +b u + f (x,x ) + w (t) τ ] w wˆ K dag(k, k,l, k ), k >. Defe the sldg suface fucto as follows s K z + z (9) 5 z Kz + z b z + s [K ( Kz + bz ) wˆ - K 5 s ) K 5 +bu + f (x,x ) + w τ ] w( (3) V s a postve V z z + s s + w K 5 he secod step: Defe Lyapuov fucto as w* wˆ, K 5 defte dagoal matx. he devatve of Lyapuov fucto s wtte as hough the Lyapuov stablty theoem, thus the system s global expoetal stable. Let K (8) z z + s s, ad the we ca obta ts devatve as follows Fom cotol law fomula (7), adaptve law fomula (6) ad (9), we obta V z z + s s V z Kz + z bz s K 3s K 4 s z ( Kz + bz ) + s (K z + z ) K + K K 3 K K3 K (4) z Kz + z b z + s [K ( Kz Selectg Q +bz ) + bu + f ( x,x ) + w (t) τ ] Fom assumpto, covese matx of matx bj let exsts, assumg uppe boud of ucetaty ad tefeece Geeal tems w(t ) s w, ad the the cotol law s u b [K ( Kz + bz ) + f ( x,x ) +w τ + K 3s + K 4sg(s)] K K 3 + b, K3 ad z z z z, z, the z K + K K 3 K z Qz z z K3 K K K 3 + b z K3 z z K z z b z + z K K 3 K z + z K K 3 z (5) + z K 3 K z + z K 3 z Assumg uppe boud of ucetaty ad tefeece Geeal tems s w(t ), t s ukow ad easy to cause chatteg poblem. I ode to avod usg ts uppe boud, ad we should use adaptve algothm to estmate w(t ), the we ca desg a estmates () () Fom (), we obta V z Kz + z bz s K 3s K 4 s wˆ as follows wˆ K 5 s z Qz K 4 s (6) By choosg the appopate paamete value, we ca Q >, amely Q s a postve defte matx, whch ca make V&.hus, we ca guaatee that cotol obta We obta adaptve cotol law as follows u b [K ( Kz + bz ) + f ( x,x ) +w τ + K 3s + K 4sg(s)] () (7) system s stable. dag(k3, k3,l, k3 ), K 4 dag(k4, k4,l, k4 ), K 3 5. Backsteppg Sldg Mode Cotol Based o -S fuzzy model 5. Fuzzy Backsteppg Sldg Mode cotolle desg Based o the appoxmato popety of -S fuzzy systems, a olea uceta system ca be expessed as the fom of lea paametc model ad a modelg eo tem; the K 5 dag(k5, k5,l, k5 ), k j > (,,L, j 3, 4,5). 77

4 H. Q. Hou, Q. Mao, Q. H. Gao ad Aoue, Buam./Joual of Egeeg Scece ad echology Revew 7 () (4) 75 8 system makes full use of the advatages of fuzzy cotol ad lea cotol. he pape focuses o the fomula () of uceta system, whch s descbed based o -S fuzzy dyamcal model. We obta fuzzy ules, ad the the fuzzy model s composed of the followg ules dag(k, k,l, k ), k > (,,L, ). K Fom (6) ad (7), we obta z b x + f ( x ) xd Plat Rules : f z s F ad z s F z s F, the # x b x + f ( x ) " $# x bu + f ( x,x ) + w (t) F u(t ) R m pemse vaable. vaable. ad b j Let (3) l s Fuzzy sets. x (t ) R s K z + z Defe Lyapuov fucto s State s the put of fuzzy system. ae olea fucto matx. Fom (3), we obta the backsteppg sldg mode cotol Law based o -S fuzzy model as follows (4) f z s F ad z s F z s F, the u b [K ( Kz + b z ) + f ( x,x ) (3) + w τ + K 3s + K 4sg(s)] α ( z (t )). & ˆ s he the adaptve law w Desg step of cotolle Based o Backsteppg Sldg Mode method s wtte as ˆ K5 s w& he fst step: (3) hus, the global cotol law s weghted sum of cotol law of local subsystem; we obta the global cotol law as z, z R, the thee ae the followg fomula u α u (33) (5) 5. Stablty Aalyss Defe the Lyapuov fucto as xd ad τ ae the desed state tajectoy of the system, ad xd s the put sgal of the system. Assumg that τ s vtual cotol vaable, Fom V z z + s s + w%, K5 ad the devatve of Lyapuov fucto s wtte as Equato (4) ad Equato (5), we ca obta V z z + s s + w K 5 (6) z Kz + z b z + s [K ( Kz + b z ) +bu + f ( x,x ) + w τ ] - Selectg vtual cotol vaable τ as follows τ b- (Kz + f ( x ) xd ) (3) +b z ) + bu + f ( x,x ) + w (t) τ ] "$ z b x + f ( x ) x d # %$ z bu + f ( x,x ) + w (t) τ z z + s s, z Kz + z b z + s [K ( Kz fuzzy set F, the the global -S fuzzy model of olea uceta system s wtte as z x xd z x τ z ( Kz + b z ) + s (K z + z ) Assumg that α (t ) s membeshp fucto of z Defe ew eo state vecto as as V V z z + s s ucetates ad exteal dstubaces. α ( z (t )) α ( z (t )) (9) the we ca obta as follows f j w (t ) ae the " x α ( z )(b x + f ( x )) $$ # $ x α ( z )(b u + f ( x,x ) + w(t)) $% K dag(k, k,l, k ), k > (,,L, ), ad defe the sldg suface fucto as follows z (t ) [ z (t ),z (t ),L,z (t )] s Fuzzy (8) Kz + b z (7) wˆ - K 5 s ) w( K5 (34) Fom cotol law fomula (3) ad adaptve law fomula (3) ad (34), we obta 78

5 H. Q. Hou, Q. Mao, Q. H. Gao ad Aoue, Buam./Joual of Egeeg Scece ad echology Revew 7 () (4) 75 8 V z Kz + z b z s K 3s K 4 s he followg -S fuzzy model s used to model of the olea system. (35) Rules: he (35) ca be ewtte as V z Kz + z b z s K 3s K 4 s π, ±π, π,,, π, π ;ad q s π,, π, ±π,, π, π. (Hee oly gve equato ad paamete values at q s π ad q s π ), the If (36) z Qz K 4 s By choosg the appopate paamete value, we ca obta V, ad the we ca guaatee that cotol subsystem s asymptotcal stable. Defe Lyapuov fucto as follows s # x x " $# x f ( x) + b ( x)u V z z + s s + w K 5 +b z ) + bu + f ( x,x ) + w τ ] wˆ - K s ) K ] w( (4) q f, q b ; Othe value of f 7.8 q.9 ad b ae omtted,,l 7. α [ z Kz + z b z + s [K ( Kz 5 q (37) 5 z Qz K 4 s < hus, the global cotol system s asymptotcally stable; the the asymptotcal stablty of system was poved based o Lyapuov theoy. 6. Smulato qd ad qd of qd s(.4π t ) ad q d s(.6π t ) ; state of the system s x [.5,.5,, ]. I ths secto, llustatve umecal smulato example s povded to demostate the effectveess ad obustess of the poposed appoach. he system s epeseted as he poposed cotol law s appled to cotolle desg of system (38), cotol law usg the atcle acheve the cotol of system, ad defg paametes of cotol system ae as follows q f ( x) + b( x)u Assumg Commad sgal (38) sq cos( q )+4cosq +9. 4cos q + cos( q )+4cosq I ths pape, the membeshp fucto of fuzzy logc cotol s agula membeshp fucto, whch completes the fuzzess of put sgal. Fuzzfcato paametes Membeshp fucto I ths pape, to vefy the accuacy ad usefuless of the method, t tasfoms the fomula (38) to state space model, whch t ca be desged based o fuzzy backsteppg sldg mode cotol method. Let s x x [ q, q ], x [q, q ], q ad q.5.5 Fg.. Fuzzy membeshp fucto # x x " $# x f ( x) + b( x)u + w(t) ucetates.6. Fom (38) ca be wtte as follows the.7.3 the state vaable [ x, x ] [q,q, q, q ]. w (t ) ae dstubaces. the tal, K K,., K 4 K 3.. gcos(q + q )(sq ) cos(q )+4cosq +9 f ( x ), gcos(q + q )(cosq + ) cos(q )+4cosq +9 cos( q )+4cosq b( x ) sq cos( q )+4cosq system ae (39) ad At w (t ), the esults of smulato ae show Fgue ad Fgue 3. exteal 79

6 H. Q. Hou, Q. Mao, Q. H. Gao ad Aoue, Buam./Joual of Egeeg Scece ad echology Revew 7 () (4) q Posto tackg Expecato cuve Actual cuve 3 q Posto tackg Expecato cuve Actual cuve.5 q d ad q -.5 q d ad q tme(s) Fg.. Posto tackg cuve of q q d ad q 3 - q Posto tackg Expecato cuve Actual cuve tme(s) Fg. 5. Posto tackg cuve of q at exteal dstubace codto Numecal smulatos demostate the effectveess of the poposed fuzzy backsteppg sldg cotol method, ad thee ae the advatages of stog obustess ad atdstubace capacty. At exteal dstubace codto, the system stll has good cotol capacty ad stog obustess. 7. Coclusos tme(s) Fg. 3. Posto tackg cuve of q At w(t) s q +. q $ # & "# s q +. q %&, ad we do t chage the value of the cotol paametes. he esults of smulato ae show Fgues 4 ad 5, I the Fgue, the dotted le s the expectato cuve ad the sold le s the actual cuve. I ths pape, by combg the Backsteppg method ad the -S fuzzy model method, the fuzzy backsteppg sldg mode cotol was peseted fo a class of msmatched uceta system. I ths ew method, olea system s descbed based o -S fuzzy dyamcal model, ad the olea system s taslated to local lea model by fuzzy method, ot oly make the system s stable, ad avod the chatteg pheomeo of sldg mode cotol, he the global stablty of the cotol law was also acheved by a Lyapuov fucto. Smulato esults dcate that the poposed cotolle s vald ad effectve, ad t has stog obustess to exteal dstubace ad the pefomace of system..5 q Posto tackg Expecato cuve Actual cuve q d ad q tme(s) Fg. 4. Posto tackg cuve of q at exteal dstubace codto Refeeces. Hu Y.A., J Y.-Q, L H.-Y, Robustess of adaptve backsteppg cotol fo olea systems, Publshg House of Electocs Idusty, Bejg,, pp.-8.. Bouad H., Bouchoucha M. ad adje M., Sldg Mode Cotol based o Backsteppg Appoach fo a UAV ype-quadoto, Iteatoal Joual of Mechacal, Idustal Scece ad Egeeg (), 7, pp Wag Z., Sychozato of a Uceta Factoal-Ode Chaotc System va Backsteppg Sldg Mode Cotol, Dscete Dyamcs Natue ad Socety, 3, pp Faz Plta, Sulama N.,Iaj Asad alook ad Fedosal P., Cotol of IC Ege: Desg a Novel MIMO Fuzzy Backsteppg Adaptve Based Fuzzy Estmato Vaable Stuctue Cotol, Iteatoal Joual of Robotcs ad Automato (5),, pp Mohammad Reza Soltapou, Behouz Zolfagha, Fuzzy Sldg Mode Cotol Desg fo a Class of Nolea Systems Wth Stuctued ad Ustuctued Ucetates, Iteatoal Joual of Iovatve Computg, Ifomato ad Cotol 9(7), 3, pp

7 H. Q. Hou, Q. Mao, Q. H. Gao ad Aoue, Buam./Joual of Egeeg Scece ad echology Revew 7 () (4) Abdel Bade Shakawy, Shaaba Al Salma, A Adaptve Fuzzy Sldg Mode Cotol Scheme fo Robotc Systems, Itellget Cotol ad Automato (4),, pp Wag Y.-F., Jag C.-S. Dect adaptve fuzzy sldg mode cotol wth vaable uvese fo ea space vehcle Systems, Egeeg ad Electocs 33(3),, pp Faz Plta, N. Sulama, I. Asad alook ad P. Fedosal, Cotol of IC Ege: Desg a Novel MIMO Fuzzy Backsteppg Adaptve Based Fuzzy Estmato Vaable Stuctue Cotol, Iteatoal Joual of Robotcs ad Automato (5),, pp og S.-C., Wag., Wag Y.-P. ad ag J.-., Desg ad stablty aalyss of fuzzy cotol system, Scece Pess, Bejg 4, pp.-5.. Yg H., Geeal SISO akag-sugeo fuzzy system wth lea ule cosequet Ae Uvesal appoxmatos, IEEE asactos o Fuzzy Systems 6(4), 998, pp Su C.-H., L S.-W. ad Wag Y.-., Relaxed Stablzato codtos fo swtchg -S Fuzzy system wth pactcal costats, Iteatoal Joual of Iovatve Computg, Ifomato ad Cotol 8(6),, pp Mozell L.A., Palhaes R.M., Souza F.O. ad Medes E.M.A.M., Reducg cosevatveess ecet stablty codtos of -S fuzzy systems, Automatca 45(6), 9, pp Chugsh seg. Fuzzy ackg Cotol Desg fo Nolea Dyamc Systems va -S Fuzzy Model, IEEE asactos o Fuzzy Systems 9(3),, pp E. Nechad ad M.N. Hamas, Powe System Stablze Based o Global Fuzzy Sldg Mode Cotol, Balka Joual of electcal ad compute egeeg (), 3, pp La K.-Y., Chag.-S., Chu C.-S., Sythess of Fuzzy Model- Based Desgs to Sychozato ad Secue commucatos fo Chaotc Systems, IEEE asactos o systems, ma ad Cybeetcs-pat B 3(),, pp