Design of MIMO Controller for a Manipulator Using Tabu Search Algorithm

Transcription

1 Internatonal Conference on Intellgent an Avance Systems 2007 Desgn of MIMO Controller for a Manulator Usng Tabu Search Algorthm I. Hassanzaeh, S. Mobayen, H.Kharrat Research Laboratory of Robotcs, Control Engneerng Deartment Faculty of Electrcal & Comuter Engneerng Unversty of Tabrz, Tabrz, Iran Izaeh@tabrzu.ac.r, smobayen@gmal.com, hame_harrat@yahoo.co.u Abstract - In ths aer, we resent an otmum aroach to esgn a MIMO controller for a manulator usng screte Tabu Search (TS) algorthm. In the frst ste, the TS algorthm s revewe an then we emloy the roose metho n orer to assgn effcently the otmal PID controller arameters. The esgn goal s to mnmze the ntegral absolute error an reuce transent resonse by mnmzng overshoot, settlng tme an rse tme of the system. A 5-bar-lnage s consere as a case stuy. We efne an objectve functon nclung these nexes. Then by mnmzng ths functon usng screte TS algorthm, controller arameters esgn s erforme effcently an qucly. Sueror features of ths algorthm are fast tunng of PID arameters, ra convergence, less comutatonal buren an caablty to avo from local mnma. Smulaton results emonstrate that our roose TS metho comare wth other heurstc metho,.e., the Genetc Algorthm (GA) s more effcent n terms of mrovng the ste resonse of the robot. Keywors: Dscrete Tabu Search (TS) algorthm, Genetc algorthm, ynamc equatons, MIMO controller, robotcs, 5- bar-lnage manulator. Otmzaton [6] an Pattern Recognton [7] have been eveloe, but these methos have some savantages n esgnng of PID controller such as hgh comutatonal tme, low rate of successful otmzaton an hgh usage of memory. In orer to overcome these ffcultes, a new aroach s roose whch emloyng new otmzaton metho calle Tabu Search. In ths aer, we formulate the roblem of esgnng PID controller as an otmzaton roblem. Our goal s to esgn a controller that has a well erformance wth ajustng four erformance nexes, the maxmum overshoot, the settlng tme, the rse tme an the steay state error of ste resonse. Although t has been ale for a robot manulator, t shoul be a romsng way for broa range of MIMO systems. The aer s organze as follows: secton 2 contans a escrton of the system an resents ynamc equatons of the robot. In secton 3 we resent an overvew of TS algorthm an secton 4 resents the roblem formulaton. Secton 5 llustrates mlementaton of both TS an GA algorthms on the lant as well as erformance comarson between them followe by concluson n secton 6. II. DYNAMIC EQUATIONS DESCRIPTION I. INTRODUCTION Proortonal-Integral-Dervatve (PID) controller s the most common tye of control metho for nustral control rocesses. Ths s ue to ts smle structure an owerful erformance n a we range of oeratng contons. The esgn of such a controller requres secfcaton of three arameters; roortonal gan K, ntegral gan K, an ervatve gan K. Almost for MIMO systems whch comose of nonlnear an coule terms, the roblem of tunng these arameters has been accomlshe by a tral an error metho. Ths way taes conserable tme an t s har to etermne otmal or near otmal PID controller arameters for MIMO systems wth hgh comlexty an nonlneartes [1]. To ecrease the comlexty an ffculty of tunng PID arameters usng tratonal methos, several heurstc methos such as GA [2], Partcle Swarm Otmzaton (PSO) [3], Fuzzy Logc [4], Smulate Annealng [5], Ant Colony The 5-bar-lnage manulator robot s frequently use an arorate confguraton n many nustral robotc alcatons such as sray-antng, arc welng an ahesve or sealant alcatons. Fgure 1 shows the tycal one bult n robotcs research lab n our eartment. Also, fgure 2 ects schematc agram of the robot where the lns form a arallelogram [10]. Let q, T an h I be the jont varable, torque an hub nerta of the th motor, resectvely. Also, let I, l, l c an m be the nerta matrx, length, stance to the center of gravty an mass of the th ln, corresonngly. IEEE ~ 1023

2 Internatonal Conference on Intellgent an Avance Systems 2007 Equaton (5) emonstrates that M 12 an M 21 een on both q 1 an q 2, consequently motor torques affect on both angles. In other wors system varables are ntensvely coule. Usng the above equatons, the 5-bar-lnage manulator robot s smulate emloyng Smuln an Matlab acages. Our roose metho s that to aly controller arameters generate by TS algorthm to the bocs n smuln moel of the system, an then calculate the cost functon n the manner that has been resente later n secton 4. III. OVERVIEW OF THE DISCRETE TABU SEARCH ALGORITHM Fg. 1. Planar resentaton of robot Tabu Search (TS) algorthm frst was ntrouce by Glover ( ). The TS metho s a local search aroach wth a flexble memory structure an t can be ale rectly to varous roblems, wthout requrng the lant moels. Ths aroach s a combnaton of otmzaton an search methos base on convergng to the best avalable neghborhoo soluton ont. In the other wors, the rncle ea s to move to the best ont from the vewont of cost functon n neghborhoo, even f t s worse than the current soluton ont. In ths controller esgn metho base on screte TS algorthm, each soluton ont s a bnary strng nclung N bts relate to 3 varables, (K, K, K ), hence the number of bts for each varable s N/3. For nstance, a samle soluton ont wth 12-bts an three varables (K, K, K ), s shown as follows, whch assgne 4 bts for each varable. K K K Fg. 2. Schematc agram of robot The ynamc equatons of ths manulator are [10-11]: M T = ( M + I ) q + M q + q (1) h q 2 + g( m l + m l + m l )cosq 1 c1 3 c M T = ( M + I ) q + M q + q (2) h q1 + g( m l + m l + m l )cosq 2 c c4 2 where g s the gravtatonal constant an M = I + I + ml + m l + m l (3) c1 3 c3 4 1 M = I + I + ml + ml + ml (4) c c4 M = M = ( m l l m l l ) cos( q q ). (5) c2 2 4 c To calculate the cost functon of soluton onts, they must be ecoe at frst. In the followng, ecoe form of the mentone strng s calculate K = 11 0 K = 1 Cost( K, K, K) = Cost(11,1,9) 3 0 K = 9 Neghbors of a soluton ont are obtane by convertng one an only one bt of the strng to nverse state. Consequently, each strng wth N-bts has N neghbors. For examle one of the 12 neghbors of the revous samle that s obtane by convertng 6 th bt from 0 to 1, s as follows: K K K K = K = 5 Cost( K, K, K) = Cost(11,5,9) 3 0 K = ~

3 Internatonal Conference on Intellgent an Avance Systems 2007 TS roceure s utlze wth features that lea to escae from local mnma. Imortant features consst of Short-Term Memory, Long-Term (Frequency Base) Memory an Asraton Crtera. In the Short-term Memory (Tabu Lst) we mantan a lst of soluton onts that have been vste recently an must be avoe. We uate the tabu lst n each teraton base on soluton onts. In the Long-Term Memory we regster the number of occasons that the soluton ont has been vste. The uatng ero of ths memory s very longer than short-term memory. We use ths memory whle all of the non-tabu onts n the neghborhoo lea to ncrease the cost functon an asraton crteron s not satsfe. Asraton Crtera mles that f some tabu moves lea to romsng solutons, these moves are accete. Stages of the algorthm: In ths art we summarze the screte TS algorthm n fve stes. Assume that X s a total search sace an x s a soluton ont samle an f( x) s cost functon: 1- Choose x X to start the rocess. 2- Create a canate lst of non-tabu moves n neghborhoo. ( x, = 1,2,..., N) 3- Decoe all soluton onts that are obtane n ste 2. 4-Fn xwnner N() x such that f( xwnner ) < f( x ), wnner. 5- Chec the stong crteron. If satsfe, ext the algorthm. If not, x= x, uate Tabu Lst an then go to ste 2. wnner Remar: In orer to ext from algorthm, there are several crterons: 1- Determnng a thresholt 0 : If the value of cost functon s less thant 0, sto algorthm. 2- Determne secfc number of teratons. 3- If the value of the cost s remane nvarable or neglgble change for several teratons, ext the algorthm. Flow chart of the algorthm s shown n fgure 3. Ext Algorthm Fn a Startng Soluton Create a Canate Lst of Moves Decoe all non-tabu onts Choose the best non-tabu move Stong Crteron? Uate Tabu Lst Fg. 3. Flow chart of the TS algorthm IV. MIMO CONTROLLER DESIGN AND PROBLEM FORMULATION The PID controller s use to mrove the ynamc resonse an reuce the steay-state error. The transfer functon of a PID controller s escrbe as follows: Gc() s = + + s (6) s where, an are the roortonal, ntegral an ervatve gans, resectvely. A Performance crteron n the tme oman nclues the overshoot M, rse tme t r, settlng tme t s, an steay-state error E ss. We fn the otmal MIMO controller arameters mnmzng the erformance nexes on the loa sturbance an transent resonses. The roose cost functon s: β β f( ) = (1 e )( M + E ) + e ( t t ) (7) ss s r where K s [,, ] an s the weghng factor [3]. If s set to be smaller than 0.7 the rse tme an settlng tme are reuce an f t s set to be larger than 0.7 the overshoot an steay state error are reuce [3]. The screte TS algorthm for searchng otmal PID controller arameters s as follows: Frst of all secfy the lower an uer bouns of controller arameters an fn the ntal soluton ont ranomly. Each ~ 1025

4 Internatonal Conference on Intellgent an Avance Systems 2007 soluton ont s a 24-bts strng an ve to three sectons, each secton relate to a varable,,,. Tabu lst length an neghborhoo sze are selecte 10 an 24, resectvely. Each soluton,.e., K has the neghbor onts. We select the non-tabu neghbors an sen them to blocs n smuln moel of the system. On the other han the values of four erformance crtera n the tme oman namely M, E ss, t r an t s are calculate teratvely n 20 generatons. Then, cost functon s evaluate for each ont accorng to these erformance crtera. Comarng the cost values of all solutons (all avalable neghborhoo onts), the best ont s selecte as the next search ont. At the en of each teraton, the rogram checs the sto crteron. If the number of teratons reaches the maxmum or the stong crteron s satsfe, recors the latest global best soluton an sto the algorthm. GA metho utlzes ts soluton ool as a mechansm for ntroucng versty nto reroucton rocess. TS, on the other han, emloys memory for atonal urose, recsely to revent the search from returnng to a revous exlore soluton onts raly. Also, TS contans a mechansm for controllng the search usng Tabu lst. Ths ensures that some solutons wll be unsatsfactory. However, the constrants comle wth Tabu lst may cause the algorthm get trae at local otma. To comare TS wth GA results, arameters of GA are erve usng tral an error metho. Poulaton sze of chromosomes an number of generaton are 10 an 20, resectvely. Mutaton rate s selecte 0.01 an crossover rate s 0.5. V. SIMULATION RESULTS The lower an uer bouns of the three controller arameters are as shown n Table I. The smuln bloc agram of 5-barlnage manulator robot wth PID controller s shown n Fgure 4 whch shows rastcally nonlnear an coule systems. Fgure 5 llustrates the ste resonse wthout controllers ncatng the system s n the bounary stablty state. TABLE I RANGE OF THREE CONTROLLER PARAMETERS Controller Parameters Lower Bouns Uer Bouns Fg. 4. Bloc agram of two motors wth PID controllers. (motor 1an motor 2 are coule) TABLE II 5-BAR-LINKAGE MANIPULATOR DATA Ln Mass (Kg) Length (m) C of G (m) The manulator secfcaton consstng of mass, length an center of gravty of lns are gven n Table II. Fgures 6 an 7 show the convergence characterstcs of the controller for motor 1 wth two values of usng TS metho comare wth the Genetc Algorthm. Fgures 8,9,10 an 11 show the corresonng ste resonse of motor angles,.e., motors 1 an 2 resectvely. As t can be observe, TS metho can romt convergence an acheve a goo evaluaton value less than 10 teratons for all stuatons. It confrms the ra convergence roerty of TS algorthm. Furthermore, TS-PID controller ntrouces a erfect ste resonse for the robot, ncatng the suerorty of roose controller wth regar to GA-PID controller. The smulaton results of the best soluton for varous values of are summarze n Table III. These results also verfes that TS-PID controller can gve more arorate controller arameters for MIMO system qucly an effcently rather to GA-PID ~

5 Internatonal Conference on Intellgent an Avance Systems 2007 Fg. 5. Ste resonse of the motors wthout PID controller. Fg. 9. Ste resonse of motor2anles wth =0.5 Fg. 6 The evoluton of the cost functon wth =0.5 for motor 1. Fg. 10. Ste resonse of motor1angles wth =1.5 Fg. 7. The evoluton of the cost functon wth =1.5 for motor 1. Fg. 11. Ste resonse of motor2anles wth =1.5 Fg. 8. Ste resonse of motor1angles wth =.5 ~ 1027

6 Internatonal Conference on Intellgent an Avance Systems 2007 TABLE III BEST PID CONTROLLERS WITH DIFFERENT VALUES GAINED BY TS AND GA ALGORITHMS FOR MOTOR 1 AND MOTOR 2 β Iteraton M (%) E t () s t () s ss r s Mn Cost Motor 1 TS GA TS Motor GA VI. CONCLUSION In ths aer we resent mofe Tabu search algorthm for etermnng the MIMO controller arameters of the 5-barlnage manulator robot. Through the smulaton results, the roose controller can erform an effcent search for otmum PID arameters. Ths aer emonstrates that the TS metho can solve searchng an tunng PID controller arameters more effcently an qucly than GA metho. The toc of our future researches s to emloy other heurstc methos n orer to acheve better results of esgnng controller arameters an mrovng ts erformance n real tme an ecreasng elays. Also, mlementaton of mentone algorthm shoul be a challengng tas. VII. REFERENCES [1] J.G. Zegler an N.B. Nchols, Otmum settngs for automatc controllers, ASME Transactons, 64: , [2] R.A. Krohlng an J.P. Rey, Desgn of otmal sturbance rejecton PID controllers usng genetc algorthms, IEEE Trans. Evol. Comuter, vol.5,.78-82, Feb, [3] Zee-Lee Gang, A Partcle Swarm Otmzaton Aroach For Otmum Desgn of PID controller n AVR system, suorte by Natonal Scence Councl of Tawan, R.O.C, gtal object entfer /TEC ( /04$20.00,2004 IEEE) [4] A.Vsol, Tunng of PID controllers wth fuzzy logc, IEE roc. Control theory al., vol, 148, n1, January 2001,.18. [5] G.Zhou an J.D. Brwell, Fuzzy logc-base PID autotuner esgn usng smulate annealng, n roceengs of the IEEE/IFAC jont symosum, comuter ae control system esgn, March 1994, [6] Yng-Tung Hsao, Cheng-Long Chuang, Ant colony otmzaton for esgnng of PID controllers, 2004 IEEE Internatonal Symosum on com. ae control systems esgn, Tae, Tawan, Se [7] T.W. Kraus an T.J. Myron, Self-tunng PID controller uses attern recognton aroach, Control Engneerng, June 1984, [8] Glover, F., Laguna M., Tabu Search n Moern Heurstc Technques for Combnatoral Problems, ages McGraw-Hll, [9] Glover, F. (1990) Tabu Search Part II, ORSA Journal on Comutng, Vol. 2, No. 1, [10] J.P. Hussoon an D. Wang, On the esgn of a 5-barlnage manulator, Robotca (1991) volume 9, [11] D.Wang, J.P.Hussoon an K.Luscott, A teachng robot for emonstratng robot control strateges, manufacturng research cororaton of ontato, February ~