Optimal sliding mode control of the pendubot

Transcription

1 International Researh Journal of Coputer Siene and Inforation Systes (IRJCSIS Vol. ( pp. 45-5, April, Available online Copyright International Researh Journals Full Length Researh Paper Optial sliding ode ontrol of the pendubot *Djaila Zehar and Khier Benahaed * Eletroehani Departent, University Mohaed Elbahir Elibrahii of Bordj Bou Arreridj, Algeria LSI laboratory, University Ferhat Abbas of Setif, Algeria Abstrat he ain objetive of this wor it was the ontrol of the pendubot, whih is a seond order underatuated syste, belongs to the seond lass, using an optial inreental sliding ode ontrol, where the syste is deoposed in two sub systes and the hosen sliding surfaes are linearly related. In order to optiize the paraeters of these surfaes we introdue the linear atrix inequality (LMI approah whih has been exploited against the probles in the fore Lyapunov funtion that an be transfored to a for of LMI proble. he siulation results have shown the effiieny of this ontrol, the syste ould be stable and ould follow perfetly the desired trajetory, so we were able to resolve the general and nown proble of swing of the pendubot. Keywords: Pendubot, underatuated syste, sliding od ontrol, optiality, stability, traing. INRODUCION he pendubot is a two lin underatuated roboti ehanis, where the first lin (the shoulder is atuated and the seond lin (the elbow is not atuated, it is a seond order syste and belongs to seond lass of underatuated systes. he underatuated systes are non linear systes that have ontrol inputs less than its degrees of freedo, the interest to synthesizing an underatuated ontroller ours in iproving overall perforanes, and reduing the nuber of the ontrol devies, whih leads to redue the ost and the weight of the systes to be ontrolled, and even the undaratuated ontroller an be used in the ase where one of the atuators of the atuated syste is in failure. he non linear ontrol has nown an iportant expansion beause of the industrial proess ultipliation and roboti appliations, espeially that alost the real systes are ainly non linear. Unfortunately those systes an not be fored in onventional struture, why we annot apply a siple feedba ontrol law, so eah syste is studied separately. Many ontrol strategies for seond order underatuated systes has been proposed (Arai et al., 99, (Astro et al., 996 and (Shiriaev et al., 998, espeially the studies of the pendubot is *Corresponding Author E- ail: Z.dj5@hotail.o proposed, by several tehniques; we find the tehnique of two ontrol algoriths (Blo, 996, the partial feedba linearization (Spong et al., 995 the energy based approah and passivity properties (Fantoni et al.,. All these tehniques are designed to stabilize the pendubot at the upright position and they an solve the swing up proble. Our ain wor is not just stabilization of the pendubot but also the traing of a desired trajetory. he sliding ode ontrol is a non linear ontrol that is largely used in any appliations espeially for perturbed and unertain systes as proven (Utin, 978, beause of its sipliity and robustness property. Several ethods based on sliding ode ontrol are used for seond order underatuated systes; we find the wor (Hao et al., 8 where they proposed the inreental sliding ode ontrol, and the wor of (Riahy et al., 7 whih used the quasi hoogeneous sliding ode ontrol. In the previous wors the hoie of the sliding surfaes oeffiients was done by iteration, our ontribution in this wor onsist of finding a ethodique solution in order to selet the optiu paraeters. At begining we use the proposed inreental sliding ode ontrol (Hao et al., 8, where the syste of the pendubot is deoposed on two sub systes, the variables of the first sub syste are hosen to onstrut the first sliding surfae, the seond surfae is onstruted using the first variable of the seond sub syste and the

2 46 Int. Res. J. Coput. Si. Infor. Syst. θ C C O θ F O Figure. Shee of the pendubot first sliding surfae, the last surfae is oposed of the seond variable of the seond sub syste and the seond sliding surfae. In order to optiize those onstruted surfaes we introdue the linear atrix inequality (LMI approah, where the Lyapunov funtion an be expressed as an LMI for, based on this latter we an find the optial paraeters of the surfaes. Figure In this paper we design the optial inreental sliding ode ontroller, based on LMI tehnique in order to stabilise the pendubot and let it reahing, perfetly a desired trajetory. he perforane of the proposed ontrol law is shown in siulation exaples. Dynais of the pendubot he pendubot is a ehanial syste with two rigid bodies, as depited in figure ( and as it presented by (Aoustin et al.,. he otor atuates the suspension joint O, we assue a diret shaft of the atuated joint, but there is no atuator in the inter lin joint O. Let C and C be the entres of ass of the first and the seond lin respetively, the entre C is loated on line O O. Let the lengths be: OO and = L ; OC l = ; OC = l. are the asses of the first and seond lins. he oent of inertia of the first lin about joint plus the oent of inertia of the rotor of the atuator is denoted I. he oent of inertia of the seond lin about joint is denoted. he generalized oordinates are the angles. he expressions of the ineti energy and the potential energy of the two lin pendulu are well nown: E = a & θ + a os( θ θ & θ & θ + a & θ ( O E = d osθ + d osθ ( p he pendubot is ehanial seond order syste, whih belongs to the seond lass of underatuated systes; we an present it by the following Euleur-Lagrange equation: d L L = F( θ U dt & θ θ L = E E Where: p U : is the ontrol input and F R F ( θ : ( θ = [,] : est un veteur des fores externes. Fro ( and (, the lagrangian yields the following well nown equations of otion: d L L = a && θ + a os( θ θ && θ dt & θ θ (4 a sin( θ θ & θθ& d sin( θ = U d L L = a os( θ θ && θ + a && θ + dt & θ θ a sin( θ θ & θ & θ d sin( θ = Where: a = I + L ; a = l L; a = I d = ( l + L g; d = l g ( g is the gravity aeleration A general for of dynai equations of underatuated systes lass II, an be rewritten, as: M ( θ && θ + C( θ, & θ & θ + G( θ = F( θ U (6 With θ = [ θ, θ ] : is the vetor of onfiguration variables, respetively atuated and non atuated. he inertia atrix is given as: M = Suh that: ( (5

3 Zehar and Benahaed 47 = a; = = a os( θ θ; = a he entrifugal and Coriolis ters C( θ, & θ & θ is given by: a sin( θ θ & θ & θ C( θ, & θ & θ = a sin( θ θ & θ & θ And the gravity ters is G( θ d sin( θ = d sin( θ his lass of underatuated systes an be expressed by the following state representation: = f ( X + b ( X U 4 = f ( X + b ( X U 4 X = [ x, x, x, x ] is the state vetor. 4 f ( X, f ( X, b ( X et b ( X : are non linear funtions sooth and bounded. For the pendubot, sine M is invertible, and using the following hange of oordinate: x = θ, x = & θ x = θ, x4 = & θ we get: = b( g + a ( + sin( x x xx && + g + bu 4 = b ( g + a ( + sin( x x xx && + g + bu 4 Suh that: f = b ( g + a ( + sin( x x + g f = b ( g + a ( + sin( x x + g b =, b = he optial sliding ode ontrol based LMI approah Generally, the oneption of the sliding ode ontrol is divided by three priniple and dependent steps, whih inlude: - he hoie of the sliding surfae. (7 (8 - he establishent of the neessary ondition of existene. - he deterination of the ontrol law. he Inreental sliding ode ontrol (ISMC (Hao et al., 8 used in this setion, onsiste of the onstrution of sliding surfaes, suh that eah surfae is integrated in the next one, and the last surfae regroup all the previous ones. Let be the underatuated syste (8, the objetive is to onstrut the input ontrol U whih drives the states x, x, to tra a desired trajetory. ( x, x and ( x, x are onsidered as the state 4 variables of the two sub systes. he hoie of the sliding surfaes Let be the traing error expressed by: e x (9 i i id Suh that: x id are the desired trajetories. he first surfae is hosen as : s = e + e ( he seond surfae is : s = e + s ( he last sliding surfae is given by: s = e + s ( 4 4 the i ( i =,,, 4 are the optial onstant, we will get the using the LMI approah, whih is given in the next setion. he LMI approah he LMIs onstraints generally are written by the for (Utin, 978: F( x = F + x F < with : i i ( i= n n i Fi = F R F( x is linear funtion in the deision variables x, i {,..., } he solutions set: i {, ( } S R F x < is onvex. here exist three probles of the onvex optiization, but the one we interest is the feasibility proble, whih is written as: x, i,..., suh that F( x < Find { } i Probles of the type: A + BPC + C P B < (P is the unnown, an be

4 48 Int. Res. J. Coput. Si. Infor. Syst. expressed as the for ( For this proble, it is suffiient to deopose the atrix P in a base of syetri atries, so the LMI allows finding the optial atrix to synthesize a ontrol law whih ensures the robustness. Proble forulation For the ase of syste (8 and fro the theore proposed by (Choi et al.,, the prinipal idea for the ontrol and the sliding surfaes hoie by the LMI tehnique an be presented in two steps: he first onsist to hoie the optial oeffiients of the sliding surfaes vetor. he seond onsist to forulate the expression of the ontrol law by introduing the optiu oeffiients, whih allows stabilizing the syste (8 and let it traing perfetly a desired trajetory in finite tie. Now, the proble is to find the atrix K so that the vetor of the sliding surfaes equal to zero. S = zd = Kz, so the equation (4 beoes: z = ( A + BK z & (8 his syste is stable, if for all P P Lyapunov inequality is heed: = >, the following ( A + BK P + P( A + BK < (9 L = KQ et Q = Q = P If we set: We get the following linear atrix inequality, equivalent to (9: QA + AQ + L B + BL < ( Q and L < Q and L are the solutions of the inequality ( suh that, the atrix of the optiu oeffiients is given by : K = LQ ( First step For the syste (8 with the sliding surfaes defined by (, ( and ( we propose the following loal oordinates vetor: z = [ z z z z ] = [ e e& e e& ] 4 he state odel of this vetor is given by: & (4 z = Az + Bz d with: A = ; B = e& z d = e& In order to have a diagonal for of the optiu oeffiients atrix, we have hosen the vetor of the sliding surfaes as follow: ' ' S = s s s s s s s (5 ' s = e fro the theore by (Alaoui et al., 7 and for the syste (8 the vetor of the sliding surfaes (5, an be represented as : S = z Kz (6 d e e& e& S = e e& 4 e& (7 Seond step Now, after deterining the optiu oeffiients we ust forulate the expression of the ontrol fro (, we have : K Whih gives : = = = = 4 4 ( Fro the sliding surfaes defined above, we will dedue three Lyapunov funtions. he first Lyapunov funtion is defined as follow: V s s = ( e + ( e = ( e ( ee + ( e = ( he seond Lyapunov funtion is given by: V = s = (4 4

5 Zehar and Benahaed 49 s = (( e + s = ( e es + s he last Lyapunov funtion is expressed by: V = s (5 s = (( 4 e4 + s = ( 4 e 4 + ( 4 e4s + s We an see that: s < s < s V V V he global synthesized ontrol is deoposed into two ters: the first one allows approahing to the sliding surfaes, the seond ter allows aintaining and desending along those sliding surfaes, it s given by: U = U + U (6 eq Suh as : eq sw U is the equivalent ontrol. U is the swithing ontrol. sw he equivalent ontrol law of the entire syste is onstruted basing on the equivalent ontrols of the sub systes, as follow: We have s& = - e& + ( e& = e& = e& = f + bu eq d ( e& f + d ( ( x d f + d ( ( Ueq = = b b And we have : s& = -e& + s& = - e& e& + ( e& = e& = e& + e& ( ( = ( x + ( x = f + bu ( ( ( ( x d + ( x4 d f + d ( ( Ueq = b 4 d d eq d By the sae way: (7 (8 s& = ( e& + s& = 4 4 ( e& e& e& + ( e& = 4 4 ( f + ( b U ( x eq 4 4d 4 d x + + ( f + ( bu ( = U d eq d eq ( x + x 4 ( 4 f ( f = ( b + ( b 4 ( + ( + ( b + ( b 4 4d d d d 4 (9 We an see that the ontrol U eq ensure that the syste reains on the sliding surfae until all states of s ( e4 and s onverge to zero. he ontrol U eq is degraded to the ontrol U eq whih ensures that the syste reains on the sliding surfae until all states of s ( e and s onverge to zero. he ontrol U eq is degraded to the ontrol U eq, whih leads errors e and e onverging to zero. Fro the equation (5, we an extrat the swithing ontrol expression: We have: V& = ss& = s (( e& + s& 4 4 = s (( f + ( bu ( x 4 4 eq 4 4d 4 + x + + ( f + ( bu ( d d eq d = s (( f x x + ( f ( d d ( d + d + (( 4 b + ( b Ueq + (( 4 b + ( b U sw We hoose: Usw = ( sign ( s s (( 4 b ( b η + + Ueq = (( f ( (( b + ( b d x + x + + ( f ( 4 d d d U = (( f ( (( b + ( b d x + x + + ( f ( 4 d d d + ( ηsign( s + s ( ( ( (

6 5 Int. Res. J. Coput. Si. Infor. Syst x xd. x xd t h e a n g le s x [ r a d ] a n d x d [ ra d ] t h e a n g l e s x [ r a d ] a n d x d [ r a d ] tie [s] Figure. he trajetories of d θ ( t and θ ( t Figure. he trajetories of θ( t and θ d ( t tie [s] t h e s lid in g s u r f a e s s, s a n d s s s s - t h e o n t r o l s i g n a l U [ N ] tie [s] Figure 4. he sliding surfaes s ( t, s ( t and s ( t Figure 5. he ontrol signal U ( t. x xd. x xd.8 t h e a n g le s x [ r a d ] a n d x d [ ra d ] t h e a n g l e s x [ r a d ] a n d x d [ r a d ] tie [s] Figure 6. he trajetories of d tie [s] θ ( t and θ ( t Figure 7. he trajetories of θ( t and θ d ( t t h e s lid in g s u rf a e s s, s a n d s s s s t h e o n t r o l in p u t U [ N ] tie [s] tie [s] Figure 8. he sliding surfaes s ( t, s( t and s ( t Figure 9. he ontrol signal U ( t

7 Zehar and Benahaed 5 & For: V V& = s ( ηsign( s s = η s sign( s s V& = η s s and heore η are positive onstants. (4 CONCLUSION We have presented the optial inreental sliding ode ontrol applied on a seond order underatuated syste alled pendubot, we introdued the LMI approah In order to get the optial paraeters of the sliding surfaes, this robust and optial ontroller allows bringing the states of the syste to desired points and following a desired trajetory, while assuring its stability. Figure Figure 9 we onsider the seond order underatuated syste (8 with the law ontrol defined by (, where the optiu paraeters of the sliding surfaes are given by ( and (, this syste is asyptotially stable and tra perfetly the desired trajetory when all signals are bounded and all errors onverge asyptotially to zero, and the sliding surfaes are asyptotially stable. SIMULAION RESULS We onsider the following paraeters of the pendubot odel: after resolution the feasibility proble, using the the Matlab environent, we have got the following values: Stabilization ase: he initial onditions are: pi pi ( x, x, x, x4 = (,,, With: η = et = And the desired onfiguration is: ( x, x, x, x = (,,, d d d 4d raing ase: he initial onditions are: pi pi ( x, x, x, x4 = (,,, With: η = et = And the desired onfiguration is: pi ( x d, xd, xd, x 4d = (,,, In figures (, (, (6 and (7 we an see that the angles of the pendubot ould reah the objetive, either for the ase of stabilization or the ase of traing in finite tie, with errors of about and under the bounded ontrol signals shown in figures (5 and (9. he sliding surfaes in figures (4 and (8 showed their onvergene to zero and their stabilization. Siulations prove that our ontroller is powerful in both ases stabilization and traing. REFERENCES Alaoui C, Bouhidi J, Mrabti M, (7. Sliding surfae ontroller design based on the LMIs with appliation to a two lins robot, Int. J. Si. and tehniques of autoati ontrol and oputer engineering IJ.SA, speial issue, CSC, pp.-, Deeber 7. Aoustin Y, Foral sii A, Martyneno Y (. Pendubot: obining of energy and intuitive approahes to swing up, stabilization in ereted pose, Published in Multibody Syste Dynais, vol. 5 issue, pp. 65-8,. Arai H, ahi S (99. Position ontrol of a anipulator with passive joint using dynai oupling, IEEE rans. Robot. Autoat., vol. 7, pp , 99 Astro K.J, Furuta K (996. Swinging up a pendulu by energy ontrol, in IFAC 96, Preprints th World Congr. IFAC, vol. E, 996, pp Blo DJ (996. Mehanial Design and ontrol of the pendubot, Master s thesis, Univ. Illinois, Urbana-Chapaign,996. Boyd S, ghaoui L, Feron E, Balarishnan V (996. Linear Matrix Inequalities In Syste and Control heory, SIAM, Philadelphia, 996 Choi HH (. An LMI based swithing surfae design ethod for lass of isathed unertain systes, IEEE ransations on autoati ontrol. 48 ( pp Fantoni I, Lozano R, Spong M.W (. Energy based ontrol of the pendubot, IEEE ransations on Autoati Control, vol. 45, N4, pp , Hao Y, Yi J, Zhao D, Qian D (8. Design of a new inreental sliding ode ontroller, IEEE, 7 th World Congress on Intelligent Control and Autoation, pp. 47-4, 8 Riahy S, Orlov Y, Floquet, Santiestban R, Rihard JP (7. Seond order sliding ode ontrol of underatuated ehanial systes: Loal stabilisation with appliation to inverted pendulu, Int J.l of Robust and Non linear Control, 8, issue , 7. Shiriaev SA, Egeland O, Ludvigsen H (998. Global stabilization of unstable equilibriu point of pendulu, in CDC98, apa, FL, 998. Spong MW, Blo DJ (995. he pendubot: A ehanial syste for ontrol researh and eduation, in 4 th IEEE Conf. Deision and Control,995. Utin VI (978. Sliding ode and their appliation in variable struture systes, Mir, Mosow, 978.