ADAPTIVE SLIDING MODE CONROL WITH RADIAL BASIS FUNCTION NEURAL NETWORK FOR TIME DEPENDENT DISTURBANCES AND UNCERTAINTIES

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1 VOL., NO. 6, MARCH 6 ISSN ARPN Jounal of Engneeng an Apple Scences 6-6 Asan Reseach Publshng Netwok (ARPN). All ghts eseve. ADAPIVE SLIDING MODE CONROL WIH RADIAL BASIS FUNCION NEURAL NEWORK FOR IME DEPENDEN DISURBANCES AND UNCERAINIES Mst. Nafsa amanna Shanta an Nosnna Zanul Azlan Depatment of Mechatoncs Engneeng, Intenatonal Islamc Unvesty Malaysa, Jalan Gombak, Kuala Lumpu, Malaysa E-Mal: eeenafsa@gmal.com ABSRAC A aal bass functon neual netwok (RBFNN) base aaptve slng moe contolle s pesente n ths pape to cate fo a -DOF obot manpulato wth tme-epenent uncetantes an stubance. RBF s one of the most popula ntellgent methos to appoxmate uncetantes ue to ts smple stuctue an fast leanng capacty. By choosng a pope Lyapunov functon, the stablty of the contolle can be poven an the upate laws of the RBFN can be eve easly. Smulaton test has been conucte to vefy the effectveness of the contolle. he esult shows that the contolle has successfully compensate the tme-vayng uncetantes an stubances wth eo less than. a. Keywos: aal bass functon netwok, aaptve contol, slng moe contol, tme-vayng uncetantes an stubances, obot manpulato. INRODUCION Robot manpulatos ae beng use n a we vaety of manufactung an techncal applcatons. hey ae use n angeous envonments such as hanlng of hazaous mateals n nuclea plants, chemcal an gas pollute suounngs, an angeous mnng aeas. hey ae also use n many places whee human opeaton s not sutable. In ths applcaton, the obot manpulato s subjecte to constant an tme-vayng uncetantes nclung payloa, fcton an stubances. hese uncetantes eteoate the pefomance of the manpulato. heefoe, t s necessay to esgn an fomulate a contolle whch can cate wth all constant an non-constant uncetantes. In the last few ecaes, many eseaches have fomulate some contolle to ovecome the poblems nclung aaptve contol, Slng Moe Contol (SMC), ntellgent contol an combnaton of aaptve an slng moe contol [-6]. A Raal-Bass Functon Netwok (RBFN) s a banch of neual netwok whch pefoms goo to contol the ynamcs system. RBFN possesses geat mappng ablty an has a smla featue to the fuzzy system. he Raal-Bass Functon Netwok can mpove the contol pefomance even n the pesence of lage uncetanty of the system. hee ae some avantages of RBFN such assmple stuctue, fast an computatonally effcent. o get the bette pefomance, the neuon numbes, the aveage an wth of the aal bass functon (RBF) an the weghts must be selecte caefully. In [7], a obust aaptve tajectoy tackng SMC base on RBFNN popose to contol of two lnks Cleanng an Detectng Robot Manpulatos whch can cate fo mass vaaton, extenal stubance an moellng uncetantes. It s use ue to ts smple netwok stuctue an faste leanng capacty. he popose contol stategy can also euce the chatteng phenomenon an goo obustness n the tajectoy tackng contol. Howeve, ths contolle s esgn base on the toque nput. In [8], a SMC base RBFN s use whee the popose contolle shows satsfe pefomance n the case of -DOF obot manpulatos. Raal bass functon uses cuve fttng moe to obtan the nonlnea mappng. By choosng a sutable Lyapunov theoem, the upate law an slng-moe swtchng gan can be obtane. In [9], the aal bass functon netwok base slng moe contolle s ntouce, whch s a combnaton of aal-bass functon netwok an aaptve slng moe contolle. he stablty s poven by Lyapunov theoy. SMC s use to elmnate the chatteng an contol an SMA actuato. he smulaton esults showe the effectveness of the contolle n thee ffeent sgnals such as- mult-step, snusoal an tangle. Lee an Cho [] consee an Aaptve Neuocontolle wth Raal Bass Functon Netwok fo obot manpulatos. In pevous wok [], a Functon Appoxmaton echnque (FA) base aaptve slng moe contolle s use to compensate tme vayng uncetantes. he contolle use Foue sees to appoxmate the uncetantes. hs technque eques numbe of bass functon an as a esult t consumes hgh computatonal effot. heefoe, n ths pape, a new Raal Bass Functon Neual Netwok (RBFNN) base Aaptve Slng Moe contolle s esgne fo a obot manpulato whch can cay unknown tme-vayng payloa an also unknown stubances an fcton. he RBFN base aaptve SMC uses a low numbe of neuon o bass functon an moe computatonally effcent. he contbuton of ths pape s the voltage-base contol law esgn to compensate fo unknown tme-vayng uncetantes wth the utlzaton of RBFN as the bass functon. Smulaton test esults une thee ffeent cases have been conucte an the contolle has successfully 4

2 VOL., NO. 6, MARCH 6 ISSN ARPN Jounal of Engneeng an Apple Scences 6-6 Asan Reseach Publshng Netwok (ARPN). All ghts eseve. ven the obot manpulato to tack the ese tajectoy espte of the pesence of tme-epenant uncetantes an stubances. hs pape s oganze as follows, the stuctue of RBFN s escbe, next the ynamc moel an the contolle esgn of the popose contolle s pesente n the followng sectons. he esults of the smulaton ae the scusse an fnally concluson s awn. HE SRUCURE OF RBF NEURAL NEWORKS Raal-bass functon netwok (RBFN) s one type neual netwok. hee ae thee layes n RBF netwoks; whch have the nput laye, hen laye an output laye. he unque featue of the RBF netwok s the pocess pefome n the hen laye. he confguaton of RBFN s shown as n Fgue-. [] y M ( x c W exp ),,,... N () ( x c whee, ) Z exp, c ae calle aveage an s calle the wth of the hen laye. W s calle weghts. he estmaton of the uncetantes can be epesente as W ˆ Z ( t ) ; whee, Wˆ s the appoxmaton of W. he appoxmaton eo between the estmate an actual ange can be wtten as ~ W W Wˆ () Fgue-. Stuctue of basc RBF neual netwok Usually the nput vecto s enote by [ x x x n ], output laye as [ y y y n ]. he avantages of ths RBFN s the esgn smplcty, geat speculaton, sol toleance to vaous noses comng wth nputs an eal tme ata pckng up capablty. he chaactestcs of RBF systems make t extemely sutable fo aaptable contol famewoks. In ths stuy, RBFN s use fo contollng the obot manpulato wth -DOF subjecte to tme vayng uncetantes an stubances. heefoe, n ths stuy, the RBFN nput vecto s chosen as [e e ], whee e s the tackng eo an e s the ate of eo vecto. he output y s epesentng aswˆ Z, the achtectue of a RBF neual netwok s gven by [7] y M WZ ˆ () hen, ewte the equaton (), Fgue-. Stuctue of popose RBF neual netwoks. DYNAMIC MODEL OF A -DOF ELECRICAL ROBOIC MANIPULAOR Usng Lagange equaton, the mathematcal moel of an egee of feeom (DOF) obotc manpulato conseng the unknown payloa an stubances, can be epesente X AX ( t) BU ( t) ( t) (4) whee, X (t) s a state vecto an U (t ) s the nput vecto. A s a 9 9 system matx, B s a 9 nput matx an ( t ) s a vecto of the unknown extenal stubances. he measue an ese state vaables can be epesente by vecto X an X espectvely- X (5) X (6) an 44

3 VOL., NO. 6, MARCH 6 ISSN ARPN Jounal of Engneeng an Apple Scences 6-6 Asan Reseach Publshng Netwok (ARPN). All ghts eseve. A A A (7) B B B (8) A an B ae the 9 9 an 9 nomnal matces espectvely, A an B ae the unknown matces of A an B espectvely, whch contans the tme vayng uncetantes, nclung the payloa, m coulomb fcton toque, L F an vscous c fcton toque, V. c he element of A an B contans all the electcal paametes: whee, J m : moment of neta fo th moto (kgm ) m (t) : angula splacement fo th moto (a) Bv kt kv L R : vscous fcton coeffcent fo th moto (Nm/a/s) : toque constant fo th moto (Nm/A) : back emf constant fo th moto (V/a/s) : amatue nuctance fo th moto (H) : amatue esstance fo th moto (Ω) a (t) : amatue cuent fo th moto (A) L (t) : loa toque fo th moto (Nm) V (t) : voltage nput to the th actuato (V) an =, an. he etale expesson fo the element of A an B can be foun n []. CONROLLER DESIGN AND SABILIY PROOF In ths secton, a new Slng Moe Contol- Raal Bass Functon Netwok (RBFN) base aaptve contolle s esgne fo the obot manpulato fo tajectoy tackng. he slng moe contol s chosen ue to ts obustness an RBFN s use fo computatonally effcent. At fst, a stana popotonal ntegal slng suface s chosen n ths stuy, whee, ( t) ( X X ) ( X X ) t (9) (t) s a nvecto, an ae n n agonal postve efnte matx an n s an ntege numbe. he tackng eo X X can also be enote as e, whee e X X. he tme vayng uncetantes can be expesse by the RBFN n Equaton. () an () whee WZ AX BU ( t ) ( t ) () he summaton of the uncetan elements can be wtten as WZ whee ε s the appoxmaton eo n the RBFN expesson. Let s conse the followng contol law u ( t ) u u u u () as: eq PI RBF he RBF neual netwoks upate law can be set ( t) Z ˆ () W an ( t) K ˆ () whee,,,, K ae the eal postve constants. Refeng to Equaton. (4), ueq s the equvalent contol tems s consee fo the appoxmately known nomnal system X AX BU (t ) n the absence of the stubances. Dffeentatng Equaton. (9), ( t ) e e a (4) Utlzng Equaton. (4) by Substtutng e X X, (t) can be wtten as: ( t ) ( X X e ) (5) Now, substtutng X A X B U (t) n Equaton. (5), an equatng (t) to, t) ( X X ) ( AX BU( t) X ) (6) ( whee, X X e (7) Reaangng Equaton. (6) by puttng U ( t ) an assgnng ths tem as u, eq u eq B ( X AX ) (8) 45

4 VOL., NO. 6, MARCH 6 ISSN ARPN Jounal of Engneeng an Apple Scences 6-6 Asan Reseach Publshng Netwok (ARPN). All ghts eseve. he tem u PI s a PI type contolle an t s necessay to enhance the close-loop stablty an mpove the tansent pefomance. V ( AX ( t) BU ( t) AX Bu ( t)) (6) ~ X W Wˆ ~ ˆ u PI B K B K ( e et ) (9) Utlzng Equaton. (), Equaton. (6) can be smplfe as: whee K s a agonal postve efnte constant matx he tem urbf s esponsble fo compensatng the stubances an s efne as base on the estmate stubances u RBF B WZ ˆ () he tem Wˆ Z epesents the uncetantes an unknown stubances of the RBFN as splaye n Fgue-. he tem u a s the tem to elmnate the appoxmaton eo ˆ [sgn( ( ))] () ua B K t whee K s a agonal postve efnte constant matx an ˆ s the estmaton of whch s the uppe boun of (t) whee, s the functon appoxmaton eo Hence, the followng contol law s popose n ths stuy ~ ˆ V ( t) K ( t) W ( Z W ) (7) ~ ˆ ˆ Equaton. (7) can be smplfe as: V ( t) ˆ K( t) W ( Z W ) ( ˆ ) (8) Substtutng Equaton. (), ()- the equaton wth the upate law, Equaton.(8) becomes- V K (9) Snce V an V, ( t ) an e as usng Babalat lemma t he block agam of the popose contolle s shown n Fgue-. u t) B ( X A X K WZ ˆ K ˆ [sgn( )]) () ( o guaantee the stablty of the popose contolle, conse the followng Lyapunov functon: V ( t) ( t) ( t) W W () whee,, ae the eal postve constant an ~ s o ~ (4) ˆ Dffeentatng Equaton. () wth espect to tme, the followng Equaton. can be obtane ~ V ˆ ~ (5) ( e ˆ e) W W Now, utlzng X AX ( t) BU ( t) ( t) an substtutng Equaton. (7) an Equaton. (8) n Equaton. (5), yels Fgue-. Block agam of the popose contolle. SIMULAION RESULS he smulaton esults show the effectveness of the popose contolle. he popose contolle cates fo vaous uncetantes nclung tme-vayng payloa, fcton an extenal stubances. he obot manpulato s eque to follow the ese tajectoy, () t wth =, an, whee, t t () sn ( ) t, () ( ) 46

5 VOL., NO. 6, MARCH 6 ISSN ARPN Jounal of Engneeng an Apple Scences 6-6 Asan Reseach Publshng Netwok (ARPN). All ghts eseve. () s the ntal poston, ( ) s ese fnal estnaton an = ( ) - (), whee =,,. he gaph of ese jont poston tajectoy s shown Fgue-4. In all cases, the contolle paametes ae tune as 5,,, K= 7 an the amount of coulomb fcton toque of Fc, Fc 5 an Fc 6Nman vscous fcton toque Vc 6; Vc 5.5 ; 5 Vc 5. fo jont, an espectvely. he numbe of neuons s chosen as fo all the smulaton cases. (b) (c) Fgue-4. Dese jont poston pofle fo all jonts. Fom the gaph, Fgue-5 (a)-(c) shows the poston esponse of the obot manpulato whee the payloa mass an stubance s consee as ml 5 5.5sn5t an t () 5.5sn5t espectvely. he fgue shows the compason between actual an ese tajectoy poston accong fo jont, an of the obot manpulato. he esult shows that the popose contolle ves the obot accuately espte the snusoally vayng payloa mass an stubance. Fgue-5. ackng pefomance of the popose contolle when the pay loa an stubance s consee as tme-vayng snusoal functon. he contolle s teste wth anom tme-vayng payloa an nose stubance wth aveage=4 an vaance=6. Fom Fgue-6 t can be obseve that, the actual poston tajectoy successfully followe the ese poston tajectoy fo all jonts an the maxmum eo s less than. a. he esult vefes that the popose contolle s obust aganst the unknown fcton an tmevayng uncetantes. (a) () 47

6 VOL., NO. 6, MARCH 6 ISSN ARPN Jounal of Engneeng an Apple Scences 6-6 Asan Reseach Publshng Netwok (ARPN). All ghts eseve. (e) (g) (f) (h) Fgue-6. ackng pefomance of the popose contolle when the pay loa s anom tme vayng an stubance s consee as Gaussan nose. he esult n Fgue-5 an Fgue-6 also show that the RBFN base aaptve slng moe contol eques low numbe of neuons o bass functon to pouce an accuate poston tajectoy tackng, compae to the functon appoxmaton technque popose n []. he aveage an vaance of the nose has been ncease to evaluate the lmt of effectveness of the contolle wth RBFN neuon. Fom the smulaton, t has been the foun that the aaptve slng moe contol wth neuons woks well on noses wth maxmum aveage= 4 an maxmum vaance=6. Fgue-7 shows that the numbe of neuons s suffcent when the aveage an vaance of the nose ae 56 an 7, whch ae geate than the maxmum lmt. () Fgue-7. ackng pefomance of the popose contolle when the pay loa s anom tme vayng an stubance s consee as Gaussan nose whee the aveage an vaance ae geate than the maxmum lmt. CONCLUSIONS In ths pape, a RBF neual netwok base slng-moe contolle s popose to contol a -of obot manpulato. he contolle s obust aganst the tme-vayng uncetantes an stubances whee the tackng eo s less than. a. It gves goo esult thee numbe of neuons. Futue woks nvolves the 48

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