ADAPTIVE FUZZY SLIDING MODE CONTROL FOR X-Z INVERTED PENDULUM

Transcription

1 Journal of heoretcal and Appled Informaton echnology st March 0 Vol 49 No JAI & LLS All rghts reserved ISSN: wwwattorg E-ISSN: ADAPIVE FUZZY SLIDING MODE CONROL FOR X-Z INVERED PENDULUM HENG LIU Department of Mathematcs and Computatonal Scence, Huanan Normal Unversty, Huanan 08, Chna E-mal: luheng@gmalcom ABSRAC X Z nverted pendulum s a new knd of nverted pendulum and t can move wth the combnaton of the vertcal and horzontal forces In ths paper, the control problem of X-Z nverted pendulum wth system uncertantes s addressed, and a par of decoupled adaptve fuzzy sldng mode control method s proposed he fuzzy logc system s employed to approxmate the system uncertantes as well as the complcated ntermedate control functons For updatng the parameter of the fuzzy system, a proportonal-ntegral adaptaton law s proposed Fnally, smulaton studes are carred out to show the stablzaton of the X-Z nverted pendulum under the proposed control method Keywords: Adaptve Fuzzy Control (AFC, Sldng Mode Control (SMC, X-Z Inverted Pendulum INRODUCION he nverted pendulum s nonmnmum phase, nonlnear and underactuated complcated system whch makes t a very dffcult problem that provdes much challengng problem to the controller desgn In the last two decades, there are many lteratures on stablzaton and trackng control for the conventonal nverted pendulum [] Except the wde research on the conventonal nverted pendulum, a lot of researchers pay ther attenton to the other knds of nverted pendulums, such as sphercal nverted pendulum [-4], X Z nverted pendulum [,5-6] Compared wth the conventonal nverted pendulum, the X Z nverted pendulum, n realty, s more lke the real control obect In [5], Maravall establshed a hybrd fuzzy control system that ncorporates a akag Sugeno fuzzy control structure wth PD control for stablzng the X Z nverted pendulum In [6], the PID controllers have been used to the trackng control of the X Z nverted pendulum system And good trackng control performance s obtaned As we know, fuzzy logc control s a modelfree method, and t can handle ll-defned and complex nonlnear systems, even those wth sgnfcant uncertantes and unknown dynamcs Fuzzy rule-based control systems have been extensvely used n many areas, ncludng cluster analyss, the controller desgn, and mage processng he fuzzy control methods have been shown to be effectve for systems wth uncertantes [7-9] he stablty analyss s always an mportant aspect n the completeness of controller desgn Wth tradtonal fuzzy control system, the stablty of the closed-loop system of a Mamdan fuzzy structure s very hard to be proved o guarantee the closed-loop stablty, some certan hybrd control approach s often used [7,0,] Among those schemes, sldng mode control (SMC s a knd of robust stablzng control method by drvng the system states nto a predefned sldng surface he man advantages of sldng mode control are the system robustness wth structured and unstructured uncertantes and satsfactory transent performance can all be preserved [-] Recently, many study results show that corporate the SMC wth the fuzzy control methods not only can allevate the chatterng effects n SMC but can decrease the complexty of fuzzy controller wth reduced number of fuzzy rules [4] In ths paper, the adaptve fuzzy sldng mode control method s used to control X-Z nverted pendulum he fuzzy logc system s employed to approxmate the system uncertantes and the complcated ntermedate control functons PROBLEM DESCRIPION AND PRELIMINARIES he X-Z nverted pendulum on a pvot drven by vertcal and horzontal control forces can be seen n Fg he control nputs of the system are based on the X-Z vertcal and horzontal dsplacements of the pvot 896

2 Journal of heoretcal and Appled Informaton echnology st March 0 Vol 49 No JAI & LLS All rghts reserved ISSN: wwwattorg E-ISSN: and the fuzzy bass functon can be expressed as: n Π µ ( x F p ( x [ Π µ ( x ] F CONROL DESIGN OF HE X-Z INVERED PENDULUM Fg: he Structure Of he X-Z Inverted Pendulum he state equatons of the X-Z nverted pendulum were gven n [] whch can be descrbed by: Mml θ sn θ + ( M + mcos θ Fx mfzsn θcosθ x M( M + m Mml θ cos θ + ( M + msn θ Fz mfxsn θcosθ z g M( M + m Fxcosθ + Fzsn θ θ Ml ( where ( xz,,( xz,,( xz, are the poston, speed and the acceleraton of the pvot respectvely l s the dstance form the mass center of the nverted pendulum to the pvot And g s the acceleraton constant of gravty M, mare the mass of the pvot and the pendulum F z s the vertcal force, and Fx s the horzontal force Descrpton Of he Fuzzy Logc System he fuzzy logc systems that employs sngleton fuzzfcaton, sum-product nference and center-offsets defuzzfcaton can be modeled by θ Π µ ( x F α( x, ( [ Π µ ( x ] where α ( x s the output of the fuzzy system, x s the nput vector, µ ( x s x ' s membershp of F th rule and θ s the centrod of the th consequent set Eq ( can be rewrtten as: α( x ϑψ( x ( wth ϑ [ ϑ,, ϑ ] N, ψ ( x [ p( x, p( x,, p ( ] N x, F Defne the followng transform: x x+ lsn θ, z z+ lcos θ l (4 Based on system (, we have Fxsn θ + Fzsnθcosθ Mml θ snθ x M + m Fxsnθcosθ + Fzcos θ Mml θ cosθ z g M + m Fxcosθ + Fzsnθ θ Ml (5 Let F ( Fxsnθ + Fzcos θ Mml θ / ( M + m, F ( Fxcosθ + Fzsn θ / Ml, we can obtan x Fsnθ z Fcos θ g (6 θ F and Fx ( M + m Fsnθ MlFcos θ, F ( M + m F cosθ + MlF sn θ z (7 If we defne the change of coordnates as tanθ y, u F cosθ, u ( F + θ tan θsec θ, and take the system uncertantes nto consderaton, yelds x uy z u g (8 y u + f( ξ where ξ [ x, x, z, z, y, y ], and f ( ξ s unknown system uncertanty wth unknown bound Let us defne the sldng mode surface as s x + λx, (9 s z + λ z, wth λ > 0,, such that the roots of the polynomal H( s λ + s related to the characterstc equaton of H ( s 0are all n the open-half plane From (8 and (9 we have s u g+ λz, (0 s uy+ λ x 897

3 Journal of heoretcal and Appled Informaton echnology st March 0 Vol 49 No JAI & LLS All rghts reserved ISSN: wwwattorg E-ISSN: hen the transformed control nput u can be constructed as u g λ z ks (, where k > 0 s a desgn parameter If we choose λ and k small enough then we can get u > 0 Let treat y y as an ntermedate control functon, and form the second equaton of (0 we have uy λ x ks (, where k > 0 s controller desgn parameter Note u > 0, the ntermedate control nput y can be descrbed as λx ks y ( u o realze y converges to y, defne e y y and the sldng surface s λ e+ e (4, then we have s λy λy + u + f( ξ y (5 Snce y and y have complcated structure, n ths paper, we employ the fuzzy logc system to approxmate the unknown functon f ( ξ ncorporated wth y and y Let us defne αξ λ ξ (6 (, u y + f( y, then we can approxmate the unknown nonlnear functon αξ (, u, through the fuzzy logc system (, as ˆ( αξ, u, ϑ ϑψ( xu, (7 Let us defne the deal parameter of ϑ as ϑ argmn sup αξ (, u ˆ αξ (, u, ϑ,(8 ϑ and defne the parameter estmaton error and the fuzzy system approxmaton error as ϑ ϑ ϑ, (9 εξ (, u αξ (, u ˆ αξ (, u, ϑ As n lterature [8,9], t s reasonable for us to assume that the fuzzy logc system approxmaton error s bounded, e, there exsts some postve constant ε, such that εξ (, u < ε (0 From above analyss, we can obtan ˆ( αξ, u, ϑ αξ (, u ˆ αξ (, u ˆ ˆ, ϑ αξ (, u, ϑ + αξ (, u, ϑ αξ (, u ( ϑψ ( ξ, u εξ (, u hen the controller u can be chosen as u λy ˆ( αξ, u, ϑ k s k sgn( s, ( 4 where k, k 4 > 0 are desgn parameters he fuzzy system parameter s updated by the followng adaptaton PI law: t ϑ [ σγ s ϑ + γsψ ( ξ, u ] dτ γ δ ( wth 0 δ σ s ϑ sψξ (, u, (4 where σγ,, γ > 0 are desgn parameters `he update law (4 ths paper desgned has a nce performance as the statement of the followng theorem heorem he update law (4 can guarantee that the fuzzy system parameter ϑ L for bounded ntal ϑ (0 Proof Defne the Lyapunov canddate functon as V ( ϑ+ γδ ( ϑ+ γδ (5 γ hen we have ( ϑ+ γδ ( ϑ+ γδ γ ( ϑ γδ ( σs ϑ sψξ u + + (, If δ s chosen as (4, one can obtan σs ϑ + s ϑ ψξ (, u γ δ ( c σ s ϑ ϑ / σ (6 (7 Notng c sup ψξ (, u hen we can conclude that f ϑ > c / σ, V < 0 hus we know that ϑ L hs ends the proof of theorem From above dscusson, now we are ready to gve the followng results heorem Consder system ( or the equvalent system (8 he sldng mode surfaces are defned as (0 and (4, the parameter adaptaton law s gven by ( and (4, the controller s defned by (-( and (, then we have the followng results: I All sgnals n the closed loop system reman bounded II he system states and ther dervatves asymptotcally converge to zero Proof Let defne the followng Lypunov functon: 898

4 Journal of heoretcal and Appled Informaton echnology st March 0 Vol 49 No JAI & LLS All rghts reserved ISSN: wwwattorg E-ISSN: γ ( ( V s + ϑ+ γδ ϑ+ γδ (8 From (0 and (, we have ss ks Accordng to (5 and ( we get s s ϑψξ (, u s + εξ (, u s ks k s 4 (9 Let V ( ϑ+ γδ ( ϑ+ γδ, substtutng γ the fuzzy system adaptaton law ( and (4 nto (5 and by usng the nequalty ϑϑ ϑ + ϑ, one can get ( ϑ+ γδ ( σs ϑ+ sψξ (, u ϑψ ( ξ, u s γ δ (0 σ s ϑ + σ s ϑ hen f we choose that k4 > ε+ 05σϑ, form (0 and ( we know ss + ks whch means that s converges to zero, e e 0 as tme t hen from (0-(, we know s u ( y + e +λ x ( Accordng to ( and e 0, one can obtan that ss ks ( From above dscusson, we can obtan that V k s k s k s ( So, V s always negatve, whch means that the sgnals s,,, and ϑ+ γδ from (, we can easly know ϑϑ, L Integratng (4 yelds: k (0 ( 0 s dt V V < (4 whch mples that s L hen form (0 and (5, we can easly conclude that s L At last by usng Barbalat s lemma [8], we know s 0, and all the sgnals n the closed-loop system s bounded, and the states of the system converge to zero as t hs ends the proof of the theorem 4 SIMULAION RESULS he parameters of the X-Z nverted pendulum are chosen as n able able he parameters of the X-Z nverted pendulum M (kg m (kg l (m g (m/s he parameters of the sldng mode control are chosen as λ λ, λ 05, k k, k 0 he fuzzy logc system uses ξ and u as the nputs For each varable of ξ, we defne three Gaussan membershp functons unformly dstrbuted on the nterval [,] And wth respect to u, we defne fve Gaussan membershp functons unformly dstrbuted on the nterval [ 40, 40] he Gaussan membershp functons of ξ are shown n Fg Fg: Gaussan Membershp Functons Of ξ he ntal values of the system are chosen as x (0 05, x (0 0, z (0 07, z (0 0, θ(0 π / 4, θ (0 0 he ntal values of the fuzzy system are chosen as ϑ (0 0 he system uncertantes n (8 are assumed to be: f( ξ 0θ + sn t (5 he smulaton results are shown n Fg-Fg6 From the smulaton results we can conclude that the stablzaton of the X-Z nverted system s acheved and the system performance s good 899

5 Journal of heoretcal and Appled Informaton echnology st March 0 Vol 49 No JAI & LLS All rghts reserved ISSN: wwwattorg E-ISSN: Fg: Stablzaton Of he X-Z Inverted Pendulum 5 CONCLUSIONS Fg4: he Control Inputs In ths paper, the equvalent transform of the X-Z nverted pendulum s gven and the adaptve fuzzy sldng mode controller s desgned for X-Z nverted pendulum he maor contrbutons of our work can be summarzed as the followng ponts Frstly, we gve the equvalent transform of the nverted pendulum Secondly, the fuzzy system s employed to approxmate the unknown system uncertantes as well as the ntermedate control nput functons he controller we desgned can guarantee the stablzaton of the system and all the sgnals n the closed-loop system keep bounded he smulaton results show that good control performance s acheved REFRENCES: [] JJ Wang, Stablzaton and trackng control of X Z nverted pendulum wth sldng-mode control, ISA ransactons, Vol 5, No 6, 0, pp [] GY Lu, D Nesc, I Mareels, Non-lnear stable nverson-based output trackng control for a sphercal nverted pendulum, Internatonal Journal of Control, Vol 8, No, 008, pp 6- [] GY Lu, I Mareels, D Nesc, Decentralzed control desgn of nterconnected chans of ntegrators: a case study, Automatca, Vol 44, No 8, 008, pp 7-78 [4] VI Utkn, Sldng mode control desgn prncples and applcatons to electrc drves, IEEE ransactons on Industral Electroncs, Vol 40, No, 008, pp -6 [5] D Maravall, C Zhou, Hybrd fuzzy control of the nverted pendulum va vertcal forces, Internatonal Journal of Intellgent systems, Vol 0, No, 005, pp 95- [6] JJ Wang, Smulaton studes of nverted pendulum based on PID controllers, Smulaton Modellng Practce and heory, Vol 9, No, 0, pp [7] YH Chang, CW Chang, CW ao, HW Ln, JS aur,, Fuzzy sldng-mode control for all and beam system wth fuzzy ant colony optmzaton, Expert System wth Applcatons, Vol 9, 0, pp 64-6 [8] A Boulkroune, M adne, M M'Saad, M Farza, Fuzzy adaptve controller for MIMO nonlnear systems wth known and unknown drecton Fuzzy Sets and Systems, Vol 6, No 6, 00, pp [9] H Lu, HJ Yu, W Xang, Adaptve fuzzy nonlnear nverson-based control for uncertan chaotc systems, Chnese physcs B, Vol, No, 005, pp [0] NN Pa, H Yau, CL Kuo, Fuzzy logc combnng controller desgn for chaos control of a rod-type plasma torch system, Expert System wth Applcatons, Vol 7, 00, pp [] A Purs, R Bello, F Herrera, Analyss of the effcacy of a two-stage methodology for ant colony optmzaton: Case of study wth tsp and qap, Expert System wth Applcatons, Vol 7, 00, pp [] ML Chan, CW ao, Lee, Sldng mode controller for lnear systems wth msmatched tme-varyng uncertantes, Journal of the Frankln Insttute, Vol 7, 000, pp 05-5 [] CW ao, ML Chan, Lee, Adaptve fuzzy sldng mode controller for lnear systems wth msmatched tme-varyng uncertantes, IEEE ransactons on Systems, Man, and Cybernetcs, Part B: Cybernetcs, Vol, 00, pp 8-94 [4]N Yagz, Y Hacoglu, Y askn, Fuzzy sldng-mode control of actve suspensons, IEEE ransactons on Industral Electroncs, Vol 55, 008, pp