Settling Time Design and Parameter Tuning Methods for Finite-Time P-PI Control

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1 Journal of Conrol Science an Engineering (6) - oi:.765/8-/6.. D DAVID PUBLISHING Seling ime Design an Parameer uning Mehos for Finie-ime P-PI Conrol Keigo Hiruma, Hisaazu Naamura an Yasuyui Saoh. Deparmen of Elecrical Engineering, oyo Universiy of Science, Noa 78-85, Japan. Grauae school of Informaics, Kyoo Universiy, Kyoo 66-85, Japan Absrac: High precision posiion conrol an high spee conrol of he robo manipulaors are funamenal an imporan conrol problems. he effeciveness of finie-ime P-PI conrol was confirme by en-effecor posiion conrol of robo manipulaors. However, parameer uning meho has no been propose o finie-ime P-PI conrol. In his paper, we propose a seling ime esign meho an a parameer uning meho for he finie-ime P-PI conrol. he effeciveness of he propose parameer uning meho is confirme by experimens of en-effecor posiion conrol of a robo manipulaor. Key wors: Nonlinear conrol, finie-ime conrol, P-PI conrol, seling ime esign, robo manipulaor.. Inroucion High precision posiion conrol an high spee conrol of robo manipulaors are funamenal an imporan conrol problems. P-PI cascae conrol is commonly use for robo manipulaor conrol []. In recen years, nonlinear finie-ime conrol [] ha guaranees convergence o a esire sae wihin finie-ime aracs much aenion in nonlinear conrol heory [-9]. In paricular, superior conrol performance of finie-ime P-PI conrol was confirme by en-effecor posiion conrol of robo manipulaors []. However, a parameer uning meho has no been evelope o he finie-ime P-PI conrol. In his paper, we propose a seling ime esign meho for he finie-ime P-PI conrol base on Refs. [9] an []. hen, we confirm he effeciveness of he propose meho for seling ime esign by compuer simulaion. Moreover we exen our propose meho o parameer uning. he effeciveness of he propose parameer uning meho is confirme by experimens of en-effecor posiion conrol of a robo manipulaor. Corresponing auhor: Keigo Hiruma, research fiels: nonlinear conrol an roboics.. Preliminaries In his secion, we summarize efiniions an funamenal properies of nonlinear finie-ime conrol. hroughou he paper,,, Σ / for all, B δ } an.. Sabiliy an Convergence Rae [] In his subsecion, we show efiniions of sabiliy an convergence raes. We consier he following ifferenial equaion: x = f( x), () Where, : is coninuous, an. Sabiliy of he origin of Eq. () an convergence rae are efine as follows: Definiion. (Sabiliy) he origin of Eq. () is sai o be: sable for each ε here exiss δ such ha ;. () globally asympoically sable if he origin is sable an all soluion saisfy he following equaion: lim x ( ) =. ()

2 Seling ime Design an Parameer uning Mehos for Finie-ime P-PI Conrol Definiion. (Convergence rae) he origin of Eq. () is sai o be: raionally sable if he origin is sable an here exiss posiive consan δ,b,b an η such ha b b x ( ) b( + x ) x η,, ; () x B δ exponenially sable if he origin is sable an here exiss posiive consan δ,b,b such ha ( ),, ; (5) b x be x x B δ finie-ime sable if he origin is sable an here exiss a posiive consan δ an a funcion : B such ha lim ( ) =,. (6) ( x ) x x B δ. Naamura s Seling ime Design Meho [9] In his subsecion we consier he following chain inegraor sysem: x () = Ax() + Bu(), (7) c where is a sae, is an inpu, marices an are efine as follows: Ac =,? Bc = (8) For sysem (7), Naamura e al. [9] propose he following nonlinear seling ime esign meho: Proposiion : (Proposiion in Ref. [9]) Consier sysem (7) an saic nonlinear feebac conrol u = q( x) such ha he origin of he close loop sysem is finie-ime sable an efine by c max he following equaion is given for some δ >. = max ( x ). (9) max x Bδ hen, he following inpu finie-ime sabilizes he origin of Eq. (7) an guaranees seling ime for all x B δ : qsx ( ) u =, () n where, an S are efine as follows. = (), max ( ) = n, ( > ) n () S = iag(,,, ). (). Finie-ime P-PI Conrol [] In his subsecion, we inrouce finie-ime P-PI conrol. We consier he following linear conrol sysem: x () = x, x bu θ (5) = +, where, is a sae, is an inpu, b is a nown an θ is an unnown consans. heorem : Consier he following finie-ime P-PI conroller for sysem ()-(5): ˆ θ u = u sgn( u) u sgn( u) τ, (6) b u = x + x sgn( x ), (7) where enoes velociy error,, an are posiive consans as esign parameers, ˆ θ is esimae value of θ. If he origin of he close-loop sysem is asympoically sable, he origin is finie-ime sable. Remar : Noe ha conroller (6)-(7) is ifferen from Eq. (7) in Ref. [] wih respec o he exisence of ˆ θ. However, he proof of Proposiion is sill vali, an heorem hols.

3 Seling ime Design an Parameer uning Mehos for Finie-ime P-PI Conrol. Seling ime Design for Finie-ime P-PI Conrol In his secion, we propose a finie-ime P-PI conroller ha guaranees he esire seling ime. We propose a seling ime esign meho for he finie-ime P-PI conroller for linear conrol sysem ()-(5) an hen we propose a parameer uning meho base on he propose seling ime esign meho for he finie-ime P-PI conroller.. Problem Saemen In his subsecion, we show he problem saemen o be iscusse in his paper. Consier sysem ()-(5) an he finie-ime P-PI conroller (6)-(7). Suppose ha he seling ime ( x ) for a cerain iniial sae x is nown wih given parameers,, an ˆ θ. he objecive of his paper is o esign a conroller o achieve he esire seling ime for he iniial sae x uner he above hypoheses. For his objecive we refine he finie-ime P-PI conroller (6)-(7) base on Proposiion. Noe ha he approach is heavily epenen on he iniial sae x. his implies ha we can esign he seling ime of he finie-ime conrolle sysem in he limie manner.. Finie-ime P-PI Conrol of Linear Conrol Sysem In his subsecion, we propose a finie-ime P-PI conroller ha guaranees he esire seling ime for linear conrol sysem ()-(5) in he following main heorem of he paper. heorem : Consier sysem ()-(5) an finie-ime P-PI conroller (6)-(7). Assume ha parameers, an asympoically sabilize he origin, an he iniial sae x an ˆ θ can be wrien as x = ( x,) an ˆ θ = θ, respecively. Moreover he seling ime ( x ) is assume o be nown. hen he following conroller (8)-(9) finie-ime sabilizes he origin wih he seling ime ( x ) =. ( x) = ux (, x) u sgn( u) ( x ) ˆ θ u sgn( u) τ, (8) b ( x) u = x + x sgn( x ). (9) o prove heorem we prepare he following hree lemmas. Lemma : Assume ha for conroller (8)-(9) he seling ime for he iniial sae x is ( x ). Consier he following coorinaes ransformaion φ : x x : x x, = (), x x = () x = b u x, x sgn u x, x + θ ˆ θ, ( ) ( ( )) () an new ime scale = ( ( x )/ ), where is a consan an is acual ime. hen for an iniial sae x on x -coorinaes he conroller (8)-(9) possesses x( ) =. Proof. We consier sysem ()-(5) an he finie-ime P-PI conroller (6)-(7). he close-loop sysem is obaine as follows: x () = x, = + x b u sgn( u ) x, () = where x is efine as follows: x b u sgn( u ), (5) x = b u sgn( u ) τ θ ˆ + θ. (6) Le a soluion of he ifferenial Eqs. ()-(5) saring a x be ϕ (; x). By using coorinaes ransformaion ()-() an our propose finie-ime conroller (8)-(9), we

4 Seling ime Design an Parameer uning Mehos for Finie-ime P-PI Conrol obain he close-loop sysem as follows: x = x, (7) x b u u x = sgn( ) +, (8) x = b u sgn( u) (9) Recall ha ϕ (; x) is a soluion for Eqs. ()-(5), an he soluion of he ifferenial Eqs. (7)-(9) is ϕ ( ; x). Since ϕ ( = ; x= x ) = in Eqs. ()-(5), ϕ ( = ; x = x ) = is in Eqs. (7)-(9). his complees he proof. Lemma. Suppose ha assumpions in Lemma are saisfie. Assume ha for sysem ()-(5) conroller (8)-(9) finie-ime sabilizes he origin. hen conroller (8)-(9) finie-ime sabilizes he origin in boh ime scale an. Proof: he close-loop sysem wih conroller (8)-(9) is he same as one wih conroller (6)-(7). Noe ha coorinaes ransformaion ϕ is iffeomorphic, an Lemma hols. Lemma. Suppose ha assumpions in Lemmas an are saisfie. hen he iniial sae x on x -coorinaes he seling ime is in ime scale by conroller (8)-(9). Proof. By using he ime scale ransformaion = ( / ), = = =. (9) he conroller (8)-(9) seles he sae a he origin a. Proof of heorem. For he iniial sae x = ( x,) an ˆ θ = θ, an iniial sae of he close-loop sysem (7)-(9) is enoe as x = ( x,). his implies ha in his case he iniial sae is ransforme ino he same sae by coorinaes ransformaion ϕ. Accoring o Lemma, he seling ime of he close-loop sysem (7)-(9) is x ( ) in ime scale. Accoring o Lemma, he seling ime of he close-loop sysem (7)-(9) is x ( ) = in ime scale. By Lemma, conroller (8)-(9) finie-ime sabilizes he origin. his complees he proof. By he same iscussion as he proof of Proposiion in Ref. [9] (Proposiion in his paper) he following corollary of heorem hols. Corollary. Consier sysem ()-(5) an he finie-ime P-PI conroller (6)-(7). Suppose ha max efine by he following equaion is given for some δ > an, an : = max ( x, θ ). () max ( x, θ ) Bδ hen, he following inpu finie-ime sabilizes he origin of ()-(5) an guaranees seling ime for each iniial sae x = ( x, x, θ ) B δ : max u = u sgn( u ) ˆ max θ u sgn( u) τ, () b u ( x, x ) = x + x sgn( x ). () max. Parameer uning Meho In his subsecion, we propose a parameer uning meho by using he propose seling ime esign meho for finie-ime P-PI conrol. We replace he conroller (8)-(9) of / =. hen, we obain he following a new finie-ime conroller ()-(5) for sysem ()-(5). u = u sgn( u) ˆ θ u sgn( u ) τ, () b

5 Seling ime Design an Parameer uning Mehos for Finie-ime P-PI Conrol 5 u = x + x sgn( x ), (5) where >,,, an θ ˆ are esign parameers. Corollary. Assume ha for sysem ()-(5) conroller (8)-(9) finie-ime sabilizes he origin wih given parameers,, an θ ˆ. hen for each > conroller ()-(5) finie-ime sabilizes he origin for he iniial sae x. Noe ha if < he seling ime becomes smaller an if >. Compuer Simulaion one becomes larger. u =.9x +. x sgn( x ), () ˆ θ =.. () Fig. illusraes respecive ime hisories of he sae wih he inpu ()-(). hough he inpu by Eqs. ()-() is much bigger han he original one, we can permi ha he sae is sele a he origin a =.5 s. We confirme ha our propose meho can achieve he esire seling ime. In his paper, we propose he conroller ha guaranees he esire seling ime in heorem. o ensure he basic iea of he propose meho, we consier a seling-ime esign problem for a simple secon-orer linear sysem. We consier he following linear conrol sysem. x (6) = x, x = u+.. (7) For sysem (6)-(7), we consier he following nonlinear finie-ime conroller: u= 5. u sgn( u ) 5. u sgn( u ) τ ˆ θ, (8) u = x +. x sgn( x ), (9) Fig. Original conroller: Sae. ˆ θ =.. () Fig. illusraes ime hisories of he sae for he iniial sae x = [ x, x ] = [ 5.,]. By Fig., we can fin ha he seling ime =.8 s. We se he esire seling ime.5 s for he iniial sae x = [ x, x ] = [ 5.,]. We can obain he following inpu: u=.76 u sgn( u ) ˆ u u τ θ () 55. sgn( ), Fig. Moifie conroller guaraneeing. : Sae.

6 6 Seling ime Design an Parameer uning Mehos for Finie-ime P-PI Conrol 5. Applicaion o Robo Manipulaor In his secion, we apply our propose meho o a posiion conrol of a robo manipulaor. 5. n-lin Robo Manipulaor Sysem [] In his subsecion we consier a serial n-lin robo manipulaor moele by he following equaion: M( φ) φ+ C( φφφ, ) + G( φ) + F( φ) = τ, () where,, consiss of n-join angles, M represens he ineria marix, : enoes he cenripeal-coriolis marix, : represens he graviy effecs, : enoes he fricion effecs, an,, is a orque inpu vecor. his paper consiers nonreunan robo manipulaor; n 6 is suppose an,, enoes he en-effecor posiion in a as-space. q an φ have he following relaion: q= h( φ), (5) where : enoes a forwar inemaics. In his paper, we consier a conrol problem of en-effecor posiion q o consan esire posiion q [,, ] = q qn. In his paper, we assume ha he marix h Jq ( ) = ( ), q φ (6) is non-singular for all q in he omain of ineres. 5. Finie-ime P-PI Conrol for n-lin Robo Manipulaor In his subsecion, we exen our propose meho o en-effecor posiion conrol of he n-lin robo manipulaor. We propose he following inverse Jacobian base on nonlinear conroller for posiion conrol of he robo manipulaor sysem () []. u sgn( u) u ˆ sgn( u) τ θ o τ =, (7) n u n sgn( un) n u ˆ n sgn( un) τ θ n o u = [ u,, u ] n q q sgn( q q) φ J ( φ). (8) = + n qn qn sgn( qn qn) 6. Experimens In his secion we show experimenal environmens an resuls of en-effecor posiion conrol by using he propose meho for a -DOF robo manipulaor. 6. Experimenal Faciliy We implemen he propose conroller on a robo manipulaor PA- prouce by Misubishi Heavy Inusries, L (MHI). We use hree joins of PA- as illusrae in Fig.. he PA- equips an absolue resolver wih.9 5 ra resoluion on each join. he PA- is conrolle by a PC wih ms sampling inerval an limie he maximum orque. he forwar inemaics q= [ x, y, z,] = h( φ) is obaine as follows: x= l cosφ sin φ + ( l + i )cosφ sin( φ + φ ), (9) y= l sinφ sin φ + ( l + l )sinφ sin( φ + φ ), (5) z= l + l cos φ + ( l + l )cos( φ + φ ), (5)

7 Seling ime Design an Parameer uning Mehos for Finie-ime P-PI Conrol 7 Fig. Experimenal resul of propose F-P-PI: Inpu. Fig. Robo Manipulaor PA-. where l =.7 m, l =.5 m an l =.55 m are lin lenghs of PA-. In experimens, he conrol objecive is asympoic sabilizaion of he en effecor posiion q a he esire posiion [ x, y, z ] = [.,.,.6] [m]. In he experimens, we choose [,, ] = [5,5,5],[,, ] = [5,55,5] an [,, ] = [5,,8]. hese parameers are use in experimens of Naamura e al. []. 6. Experimen of Propose Finie-ime P-PI Conrol Fig. 5 Experimenal resul of propose F-P-PI: Sae. In his subsecion, we show experimenal resuls of en-effecor posiion conrol of he robo manipulaor by using our parameer uning meho o he finie-ime P-PI conroller. In he case of =.975, Figs. -6 illusrae ime hisories of inpus, sae variables an posiion conrol error respecively. ables an summarize he performances of he conrollers wih variaions of from 5 s o 55 s. Accoring o ables an, our parameer uning meho can perform.7 μm in mean error an. μm in sanar eviaion. Fig. 6 Experimenal resul of propose F-P-PI: Sae (Error).

8 8 Seling ime Design an Parameer uning Mehos for Finie-ime P-PI Conrol able Resuls of main experimens (sanar eviaion) x[ m].79e-6.e-6.7e-6.7e-6 [ ] zm [ ].e-6 9.9e-7 9.9e-7.99e-6 ym 6.e-7 6.5e-7 6.5e e-7 able Resuls of main experimens (mean error) x[ m] 8.9e-6.6e-7 6.6e-8.855e-7 ym [ ].575e-7.6e-7.756e-7.e-7 zm [ ].6e-7.69e-7.e-8 7.7e-7 6. Experimen of Convenional Finie-ime P-PI Conrol In his subsecion, we show experimenal resuls of en-effecor posiion conrol of he robo manipulaor by convenional finie-ime P-PI conrol. Figs. 7, 8 an 9 illusrae ime hisories of inpus, sae variables an posiion conrol error respecively. able summarizes he performances wih respec o mean error an sanar eviaion from 5 s o 55 s. Accoring o able, convenional meho can perform.77 μm in mean error an. μm in sanar eviaion. 6. Experimen of Convenional P-PI Conrol In his subsecion, we show experimenal resuls of en-effecor posiion conrol of he robo manipulaor by convenional P-PI conrol. In he convenional meho, since i is no possible o use he parameers of he finie-ime conrol, we choose [,, ] = [,,],,, ] [9,9,5], an [ = [,, ] = [,5,]. Figs. - illusrae ime hisories of inpus, sae variables an posiion conrol error respecively. able summarizes he performances wih respec o mean error an sanar eviaion from 5 s o 55 s. Fig. 7 Experimenal resul of convenional F-P-PI: Inpu. Fig. 8 Experimenal resul of convenional F-P-PI: Sae. Fig. 9 Experimenal resul of convenional F-P-PI: Sae (error).

9 Seling ime Design an Parameer uning Mehos for Finie-ime P-PI Conrol 9 able Resuls of convenional finie-ime P-PI experimens. x [m] y [m] z [m] Mean error.855e-7.e-7 7.7e-7 Sanar eviaion.7e-6 8.8e-7.99e-6 Fig. Experimenal resul of convenional P-PI: Sae (Error). Fig. Experimenal resul of convenional P-PI: Inpu. able Resuls of convenional P-PI experimens. x [m] y [m] z [m] Mean error.69e-6.95e-6.689e-6 Sanar eviaion 5.8e-.e- -.9e- Fig. Experimenal resul of convenional P-PI: Sae. Accoring o able, he convenional P-PI conrol can perform.6 μm in mean error an 5.8 μm in sanar eviaion. 6.5 Discussion Fig. illusraes he sanar eviaions. We can confirm ha he sanar eviaion is improve by changing. his confirms effeciveness of our propose parameer uning meho. In paricular, we Fig. Accuracy of sae error in each. can see ha he accuracy of he x-axis is grealy improve when we choose =.975. able 5 illusraes he resul of seling imes. Because s are close o in all cases, he seling imes are no change so much. We can see ha he seling imes of he finie-ime conrol are smaller han he convenional meho, in all cases. On he conrary, mean error has been grealy improve.

10 Seling ime Design an Parameer uning Mehos for Finie-ime P-PI Conrol able 5 Resuls of seling ime ( %). x axis[s] y axis[s] z axis[s] Convenional P-PI Finie-ime P-PI =. Finie-ime P-PI =.975 Finie-ime P-PI =.965 Finie-ime P-PI = Conclusions In his paper, we propose a seling ime esign meho for finie-ime P-PI conrol. he propose meho can esign he esire seling ime an he effeciveness is confirme by compuer simulaion. For en-effecor posiion conrol of he robo manipulaor, our propose meho can improve accuracy of he en-effecor posiion. his paper oes no iscuss racing conrol problem. his remains fuure wor. Acnowlegemen his wor was suppore by JSPS KAKENHI Gran Number 5H. References [] Sciavicco, L., an Siciliano, B.. Moeling an Conrol of Robo Manipulaors. Lonon: Sprinver-Verlag. [] Haimo, V Finie ime Conrollers. SIAM J. Conrol an Opimizaion (): [] Bha, S. P., an Bemsein, D. S Lyapunov Analysis of Finie-ime Differenial Equaions. In Proceeings of he American Conrol Conference, 8-. [] Baccioi, A., an Rosier, L. 5. Liapunov Funcions an Sabiliy in Conrol heory. Lonon: Sprinver-Verlag. [5] Levan, A. 5. Homogeneiy Approach o High-Orer Sliing Moe Design. Auomaica : 8-. [6] Naamura, N., Naamura, H., an Nishiani, H.. Global Inverse Opimal Conrol wih Guaranee Convergence Raes of Inpu Affine Nonlinear Sysems. IEEE ransacions on Auomaic Conrol 56 (): [7] Orlov, Y. V. 9. Disconinuous Sysems. Lonon: Springer Verlag. [8] Polyov, A.. Nonlinear Feebac Design for Fixe-ime Sabilizaion of Linear Conrol Sysems. IEEE ransacions on Auomaic Conrol 57 (8): 6-. [9] Naamura, H., Naamura, N., an Fuji, Y.. Seling ime Design for Nonlinear Finie-ime Conrol Sysems. In Proceeings of IEEE Conference on Decision an Conrol, [] Naamura, H., Nishia, N., an Naamura, N.. High Precision Conrol of Robo Manipulaors via Finie-ime P-PI Conrol. In Proceeings of IEEE Conference on Decision an Conrol, [] Hiruma, K., Masuo, Y., Sao, Y., an Naamura, H.. Seling ime Design for Homogeneous Finie-ime PID Conrol. In Proceeings of SICE Annual Conference, 5-6.