Hierarchical Sliding Mode Control for Series Double Inverted Pendulums System

Transcription

1 Herarchcal Sldng Mode Conrol for Seres Doble Invered Pendlms Sysem Danwe Qan, Janqang Y, Dongbn Zhao, and Ynxng Hao Laboraory of Complex Sysems and Inellgence Scence Inse of Aomaon, Chnese Academy of Scences 8, Bejng, PR Chna Absrac - Ths paper proposes a herarchcal sldng mode conroller for seres doble nvered pendlms sysem Ths provdes a smple mehod o conrol a class of nder-acaed sysems wh hree sbsysems by sldng mode conrol Frsly, he gven sysem s dvded no hree sbsysems accordng o s srcre characersc Then, he s -level sldng mode srface s defned for every sbsysem and he nd -level sldng mode srface s consed by hem Based on he wo levels srcre, he eqvalen conrol of each sbsysem s dedced and he oal conrol law s derved by he Lyapnov sably heorem The asympocal sably of he enre sldng mode srfaces s proved heorecally Fnally, smlaon resls show he valdy of hs conrol sraegy And he nflence of he conroller parameer changes for he performances s also dscssed Index Terms - herarchy, sldng mode conrol, nvered pendlm, nder-acaed sysem I INTRODUCTION In recen years, here have been growng aenon and ncreasng neres n nder-acaed sysems These sysems wh fewer acaors and ganng fal olerance make for decreasng he nmber of acaors, lghenng he sysems, redcng he cos, e al As a ypcal nder-acaed sysem, seres doble nvered pendlms on a movng car are a sngle-np and ml-op (SIMO) nonlnear me-nvaran sysem wh srcred parameer ncerany Ths sysem s ofen sed as a benchmark for verfyng he effecveness of new conrol approaches Many papers on he conrol of seres doble nvered pendlms sysem have been pblshed Nemann [] presened a robs conroller sng he μ synhess mehod and Weng [] desgned a robs H conroller for sch sysem, respecvely Fjnaka [] presened a self-nng nero-pid conroller and Y [4] proposed a new fzzy conrol based on he sngle np rle modles for seres doble nvered pendlms sysem, respecvely Some of hese conrol algorhms are based on he lnearzed model n he neghborhood of he prgh poson of boh pendlms If he pendlms are far from he prgh poson, hen he lnearzed model s naccrae Thereby, he performances of hose conrollers are no sasfyng The nellgen conrollers can solve he nceran and nknown facors B he conroller srcres and conrol algorhms are nrcae The sldng mode conrol (SMC) s a powerfl nonlnear feedback conrol mehod, whch has been developed and appled o feedback conrol sysems for he las hree decades [5] I provdes a good canddae for he nder-acaed sysems conrol I becomes nsensve o sysem parameer changes or exernal dsrbances, when he sysem sae says n he sldng mode srface For nder-acaed sysems, desgnng a convenonal sldng-mode srface s no approprae, becase he parameers of he sldng mode srface can be obaned drecly accordng o he Hrwz condon Therefore, a smple way o desgn he sldng mode srface for nder-acaed sysems s needed The srcre characersc of a class of nder-acaed sysems, sch as nvered pendlm(s) sysems, s ha hey can be dvded no several sbsysems Based on hs, Lo [6] presened a decopled fzzy sldng-mode conroller for a doble nvered pendlms sysem, whch smplfed he desgn process B ha paper dd no analyse he sably of he sldng mode srfaces Ln [7] presened a herarchcal sldng mode conroller, whose parameers were reglaed by fzzy logc B ha paper only garaneed he second-level sldng mode srface was sable Wang [8] developed a sable herarchcal sldng mode conroller wh he sable wo-levels sldng mode srfaces B ha paper only gave he sably analyss for nder-acaed sysems wh wo sbsysems Y [9] presened a cascade sldng mode conroller for large-scale nder-acaed sysems wh he asympocally sable sldng mode srfaces However, he srcre of ha conroller s more complcaed han ors Ths paper develops he herarchcal sldng mode conrol for a class of nder-acaed sysems wh hree sbsysems In hs approach, seres doble nvered pendlms sysem s dvded no hree sbsysems, ncldng lower pendlm, pper pendlm, and car, respecvely The s -level sldng mode srface s defned for every sbsysem And he nd - level sldng mode srface s defned for he ops of all of he s -level sldng srfaces Based on hs srcre of he sldng mode srfaces, he herarchcal sldng mode conrol law s derved by Lyapnov heorem I drves he sysem s saes o her desred vales and can decrease he chaerng phenomenon The characersc of he proposed conrol sraegy s ha he whole sysem s sldng mode srface and he sbsysem sldng mode srfaces have he asympoc sably Smlaon resls show he conroller s valdy X/6/$ 6 IEEE

2 II SYSTEM DYNAMIC MODEL The srcre of he seres doble nvered pendlms sysem s shown n Fg Consderng he sandard assmpon, e no frcon, e al, he followng sysem dynamc model can be dedced by sng Lagrange s eqaon T A x β α = B+ C () where α and β are he angles of he lower pendlm and he pper pendlm wh respec o he vercal lne, x s he dsan of he car from he ral orgn, s he conrol force added o he car, whch are shown n Fg ; he marxes A, B and C can be represened as follows here a a a a A a a a a a a = a M m m c, B =, C = c c = + + ; ( ) = = a = ml /+ m l cosα ; a ( m l cos α)/; a a ; a = ml /+ m l ; a m l / = [ m ll cos( α β )]/; a = a ; a = a ; x Fg Srcre of seres doble nvered pendlms sysem = ; c = ( ml / + m l ) α sn α + ( m l β sn β) / ; c ml m l g m ll ; = ( / + ) sn α [ β sn( α β)]/ c = ( m l gsn β)/ + [ m ll α sn( α β)]/ where M s he car mass; m s he lower pendlm mass; m s he pper pendlm mass; l s he lengh of lower pendlm; l s he lengh of pper pendlm; g s he gravaonal acceleraon Le x = α, x = α, x = β, x4 = β, x5 = x, x6 = x Accordng o he canoncal form of a class of nder-acaed α β sysems, we can ransform he above mahemac model of he sysem no he followng sae space expresson, brefly x = x x = f( X) + b( X) x = x4 x 4 = f( X) + b( X) x 5 = x6 x 6 = f( X) + b( X) T where X = ( x, x, x, x4, x5, x6) s sae varable vecor; f( X ) and b ( X ) are he nonlnear fncons of he sae varables They are abbrevaed as f and b n he followng descrpon, whch are gven as follows; s he conrol np Le A = a a a a A = a a a a, B = a a a a,, B = aa aa, D = ac ac, D = ac ac, hen we have f = ( BD BD)/( BA BA), f = ( A D A D )/( B A B A ), f = [( aa ab) D+ ( ab aa) D]/[ a( BA BA) ], + c / a b = ( a B a B )/( B A B A ), b = ( a A a A )/( B A B A ), b = [ a ( ab ab ) + a( a A aa )]/[ a( B A B A )] + / a III CONTROL STRATEGY AND STABILITY ANALYSIS Accordng o he srcre characersc of he gven sysem, we consder desgnng he herarchcal sldng mode conroller for hs class of nder-acaed sysems wh hree sbsysems The srcre of he herarchcal sldng mode srfaces s shown n Fg S S S x x x x 4 x 5 x 6 Fg Srcre of herarchcal sldng mode srfaces S ()

3 In he proposed mehod, he seres doble nvered pendlms sysem s dvded no hree sbsysems, ncldng lower pendlm, pper pendlm and car Each sbsysem owns self sldng mode srface They are he s -level sldng mode srfaces And hey make p of he nd -level sldng mode srface From Fg we can fnd ha ( x, x ), ( x, x 4) and ( x5, x 6) are reaed as he sae varables of he lower pendlm sbsysem, he pper pendlm sbsysem and he car sbsysem, respecvely The s -level sldng mode srfaces are consed by he sae varables of each sbsysem Then he nd -level sldng mode srface s consed by he ops of he s -level Accordng o he srcre of he herarchcal wo-levels sldng mode srfaces, he herarchcal sldng mode conroller s desgned as follows A Conroller Desgn For he sae varables of he hree sbsysems, he s - level sldng mode srfaces are defned below s = λ x + x (a) s = λ x + x (b) 4 s = λ x + x (c) 5 6 where λ s posve consan The nd -level sldng mode srface s defned as s = ε s + ε s + ε s (4) where ε s consan; and s sgns can be adjsed accordng o dfferen condons The sbsysem eqvalen conrol can be dedced as = ( λx + f )/ b (5) eq In order o assre each sbsysem o follow s own sldng mode srface, he oal conrol law ms nclde he eqvalen conrol of each sbsysem Ths he oal conrol law can be defned as follows = eq + eq + eq + sm (6) here sm s he correcve conrol of he conroller The correcve conrol law can be derved from Lyapnov heorem The Lyapnov fncon s defned as V = s / (7) Dfferenang V () wh respec o me and from (), (4) and (7), we can oban V = ss = s [ sm ( ε b+ ε b + ε b ) (8) + ( ε b+ εb) eq+ ( εb+ ε b) eq + ( εb+ εb ) eq] Le s = ks η sgn( s), (9) hen we have ε b + ε b ε b + ε b = sm eq eq εb+ ε b+ εb εb+ ε b+ εb ε b + ε b ks η sgn s eq + εb+ ε b+ εb εb+ ε b+ εb Fnally, we can ge he oal conrol law below ε b eq + ε b eq + ε b eq +s = ε b + ε b + ε b () B Sably Analyss Le s check he sably of he s -level and he nd -level sldng mode srfaces Theorem : Consderng a class of nder-acaed sysems (), f he sldng mode srfaces are desgned as () and (4) and he conrol law s adoped as (), hen he nd -level sldng mode srface s asympocally sable Proof: From (8) and (9), we can oban V = ss = η s ks () Inegrang boh sdes of () yelds We can fnd ( η s ks ) dτ V ( ) V () = ( η s + ks ) dτ ( η s ks ) dτ V () = V ( ) + + Frher, we can ge lm ( s + ks ) dτ V () < η () Accordng o Barbala heorem, we can oban ha f hen η s +ks, whch means lm s = Theorem : For a class of nder-acaed sysems (), desgn he sldng mode srfaces as () and (4), and adop he conrol law as () If he followng eqaon s sasfed ( ) ( ) sgn ε s = sgn ε s = sgn( ε s ) f sss () hen s, s and s are sll asympocally sable Proof: Frsly, we prove s, s, s L From (), we have s d τ = ( ε s+ εs+ εs) dτ = ( ε s + εs + εs)dτ + ε εss + εεss + εεss ( )dτ < I s oblvos ha f () s sasfed, hen we can ge s dτ < namely s L (4)

4 Secondly, we prove ha s, s, s L From (), we can oban s L Accordng o (4), f () s sasfed, hen we have (5) s Thrdly, we prove ha s, s, s L From (9), we have s L Dfferenang (c) and (4) wh respec o me, here exs L s = ε s + ε s + ε s, s = λ x + x 5 6 here x 5 and x 6 are velocy and acceleraon, respecvely, whch are conrolled by conrol force; hey belong o L Frher, we can ge s (6a) L L Nex, le s prove s, s by conradcon Case : If arbrary one of s or s doesn belong o L, hen s Ths case conradcs s L L Case : If boh s and s don belong o L, hen we have ) sgnεs = sgnεs, hen here exss s L from s = ε s + ε s + ε s Ths case conradcs s L when, hen we can ge sgn εs sgn εs, Ths conradcs sgn ε s = sgn ε s ) sgn εs sgn εs Ths, we can oban ha s namely sp s = s < (6b) L s L namely sp s = s < (6c) Now, accordng o Barbala heorem, from (4), (5) and (6), lm s =, lm s = and lm s = are sasfed In a word, from heorem and heorem, we can know ha he s -level and he nd -level sldng mode srfaces wll converge asympocally o zero f () s sasfed Hence we choose he sgn of ε s as a benchmark and change he sgns of ε and ε when () s no sasfed condons x =π/6rad, x =rad s -, x =π/6rad, x 4 =rad s -, x 5 =m and x 6 =m s -, hs conrol sraegy can conrol boh he pendlms and he car a he same me n Fg α (rad) β (rad) Fg (a) Angle of lower pendlm Fg (b) Angle of pper pendlm IV SIMULATION RESULTS In hs secon, we shall demonsrae ha hs conrol sraegy s applcable o sablzaon conrol of he seres doble nvered pendlms sysem The conrol objecve of sablzaon conrol s o balance he pendlms prgh and p he car o he ral orgn by movng he car The sysem parameers are seleced as car mass M=kg, lower pendlm mass m =kg, pper pendlm mass m =kg, lower pendlm lengh l =m, pper pendlm lengh l =m, gravaonal acceleraon g=98m s -, whch are as same as [6, 7] The herarchcal sldng mode conroller parameers are seleced as λ =, λ = 585, λ = 7, ε = -, ε =, ε = 8, k = 8, η = afer ral and error For he nal x (m) Fg (c) Car poson In Fg 4, he crves of he s -level sldng mode srfaces can converge asympocally o zero, as we have proved

5 x - (N) Fg 4(a) Phase plane of sldng mode srface s x Fg 5 Conrol force x 4 - Fg 6 s he crves of he angles of boh pendlms and he car poson The nal condons of he sysem s x =π/rad, x =rad s -, x =π/rad, x 4 =rad s -, x 5 =m, and x 6 =m s -, and he desred vales are zero x Fg 4(b) Phase plane of sldng mode srface s α (rad) Fg 6(a) Angle of lower pendlm 5 x x 5 Fg 4(c) Phase Plane of sldng mode srface s Fg 5 s he conrol orqe From, we can fnd hs mehod can decrease chaerng effecvely de o he eqvalen conrol mehod we adop β (rad) Fg 6(b) Angle of pper pendlm

6 From Fg 6, we can fnd ha all of he sae varables can converge o he desred vales as long as he ral and he conrol force aren lmed Ths shows he conroller s very robs V CONCLUSIONS A herarchcal sldng mode conroller has been presened The asympocal sably of he s -level and he nd -level sldng mode srfaces has been proved heorecally In he smlaon, he proposed mehod s appled o a seres doble nvered pendlms sysem The smlaon resls show he valdy of he conrol sraegy for a class of nder-acaed sysems wh hree sbsysems Frher, he nflence of he conroller parameers for he performance has been dscssed Ths provdes a feasble conrol mehod for sch sysems x (m) Fg 6(c) Car poson Now, le s dscss how he parameers η and k affec he performances The η affecs he conrol performances as a convenonal SMC If he vale of η ncreases, hen he chaerng phenomenon becomes large For he nal condons x =π/6rad, x =rad s -, x =π/6rad, x 4 =rad s -, x 5 =m, x 6 =m s - Anoher parameer of he swch conrol η s seleced as drng he smlaon TableⅠ can show he performances wh dfferen k From TableⅠ, we can fnd ha k change almos makes no dfference on he response me of sae varables x and x, b f k decreases, we need he smaller conrol force and he longer ral k TABLE Ⅰ PERFORMANCES WITH DIFFERENT k x max max sx sx sx5 9m 5N 57s 57s 4s 9m 5N 57s 57s 8s 5 5m 8N 5s 5s 68s Here x max s he maxmal car dsance from he orgnal poson; max s he maxmal conrol force; sxj s he response me of sae varable x j (j=,, 5) ACKNOWLEDGMENT Ths work was sppored by NSFC Projecs (No 64, 6444, 6475, and ), MOST Projecs (No CB576 and 4DFB), and he Osandng Overseas Chnese Scholars Fnd of Chnese Academy of Scences (No 5--), Chna REFERENCES [] H Nemann and JK Polsen, Analyss and desgn of conrollers for a doble nvered pendlm, Proceedngs of Amercan Conrol Conference, vol 4, pp 8-88, Jne [] Zh Weng, G Wang, and Y Yao, H conrol of nvered pendlm, Proceedngs of IEEE Conference on Comper, Commncaon, Conrol and Power Engneerng, vol 4, pp , 99 [] T Fjnaka, Y Kshda, M Yoshoka, and S Oma, Sablzaon of doble nvered pendlm wh self-nng nero-pid, Proceedngs of he IEEE-INNS-ENNS Inernaonal Jon Conference on Neral Neworks, vol 4, pp 45-48, Jly [4] J Y, N Ybazak, and K Hroa, Sablzaon Conrol of Seres-Type Doble Invered Pendlm Sysems Usng he SIRMs Dynamcally Conneced Fzzy Inference Model, Arfcal Inellgence n Engneerng, vol 5, pp97-8, [5] O Kaynak, K Erbar and M Ergrl, The fson of compaonally nellgen mehodologes and sldng-mode conrol-a srvey, IEEE Transacon on Indsral Elecroncs vol 48, pp 4-, Feb [6] J Lo, and Y Ko, Decopled fzzy sldng-mode conrol, IEEE Transacons on Fzzy Sysems, vol 6, pp 46-45, Ag 998 [7] Ch Ln, and Y Mon, Decoplng conrol by herarchcal fzzy sldngmode conroller, IEEE Transacons on Conrol Sysems Technology, vol, pp , Jly 5 [8] W Wang, J Y, D Zhao, and D L, Desgn of a sable sldng-mode conroller for a class of second-order nderacaed sysems, IEE Proceedngs of Conrol Theory and Applcaons, vol 5, pp68-69, Nov 4 [9] J Y, W Wang, D Zhao, and X L, Cascade sldng-mode conroller for large-scale nderacaed sysems, Proceedngs of IEEE/RSJ Inernaonal Conference on Inellgen Robos and Sysems, pp -6, Ag 5