ISSN: Srikanth et al, / International Journal of Computer and Communication System Engineering (IJCCSE), Vol. 2 (4), 2015,

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1 ISSN: Srkanth et al, / Internatonal Journal of Computer and Communcaton System Engneerng (IJCCSE, Vol. (4, 5, Rotary nverted pendulum control and the mpact of tme delay on swtchng between stable and unstable states wth enhanced partcle swarm optmzaton Srkanth Kavrayan Dept.of EEE Gayatr Vdya Parshad College of Engneerng Vsakhapatnam, Inda kavrayansrkanth@gmal.com G V Nagesh Kumar Dept. of EEE GITAM Unversty Vsakhapatnam, Inda drgvnk4@gmal.com Abstract Ths paper presents the analyss of a rotary nverted pendulum wth tme delays. Although, t s a well known fact that tme delays mght cause nstablty, the establshed model that s analyzed as part of ths work was tested under the nfluence of tme delay n real tme for specfc objectve of settlng tme. Ths proposed method gave results whch were better when compared for performance wth and wthout tme delay and whch have shown the mpact of crtcal tme delays. Index Terms Tme delay, Rotary Inverted Pendulum, Partcle swarms Symbol L L l l I I Nomenclature Sgnfcance Pendulum Angle from the vertcal Arm angle from the vertcal Angular Velocty of Pendulum Angular velocty of Arm Length of pendulum Length of pendulum Dstance of the center of mass of pendulum to pendulum s pvot poston The dstance of center of mass of arm to arm s pvot pont Inerta of the pendulum The nerta of arm A typcal rotary nverted pendulum has a rotary head and also a pendulum at the end whch s free to oscllate about ts axes from the unstable equlbrum poston to the stable equlbrum poston. Typcally the arrangement has a motor drven control for such a system usng whch the stablty can be acheved. Researchers have developed smulaton prototypes for the rotary nverted pendulum and also have developed toolboxes usng whch the characterstcs of the plant can be controlled usng mass of the rotary arm, length of the arm and the specfcatons of the motors. Actve research has happened on other versons of the rotary nverted pendulum systems such as the acrobats, pendubots. Ths paper nvestgates the case of ncludng a tme delay n the sgnal transmtted to the motor that control the nput of the rotary nverted pendulum. The analyss s also extended to the case where a partcle swarm optmzaton technque mnmzes the moton of the states of the pendulum and mnmzes ther dsplacement resultng n an enhanced controller desgn. II. PLANT DYNAMICS AND IMPLICATIONS DUE TO TIME DELAY The plant that needs to be controlled can be represented as n fgure wth varous varables as tabulated n appendx A and wth the dynamcs of the rotary nverted pendulum can be gven usng equatons and. I. INTRODUCTION Rotary Inverted Pendulum (RIP s a typcal underactuated non-lnear plant whch has potental applcatons n the feld of defense, aerospace, mechatronc systems and other ndustral applcatons whch use varous levels of manpulators. The plant model has been analyzed for modfcatons n dynamcs and also for extendng the model to sut to ndustry requrements by varous researchers where the emphass was manly on stablzaton[3-6]. RIP as a system can be used for understandng and ads control educaton. Varous types of controllers such as LQR, LQG, H-Infnty, Kalman flter based observer desgn have been appled by researchers. Fg : Rotary Inverted Pendulum Structure 569 P a g e IJCCSE All Rghts Reserved Vol. No.4 June 5

2 ISSN: Srkanth et al, / Internatonal Journal of Computer and Communcaton System Engneerng (IJCCSE, Vol. (4, 5, The dynamcs of the rotary nverted pendulum are gven as [] and the notatons used are explaned n the nomenclature. M M Cos M M 3 MCos M 4 ( M 3 sn ( M sn M sn cos g sn sn cos ( consdered as tme delay. A transport delay was ncorporated nto the system whch would be τ The followng assumpton s addtonally consdered along wth generc assumptons as n [] whch was never nvestgated by researchers earler for a rotary pendulum. a A small delay s consdered n transmttng the nput appled to the pendulum. The plant equatons are consdered as follows under state space representaton as follows: Where n equaton refers to the control nput whch s appled to the shaft of the arm. M (=,,, 3, 4 n equatons and are postve system parameters determned as M I l m ml L (3 M (4 l m M (5 M (6 M 3 lm 4 I l m L m (7 Where (9 ( ( ( The plant equatons consdered under state space representaton are as follows bd E CG E / / qd E bg E / / A u A where a J eq b mlr C mr 4 / 3ML d mgl E ac b The dynamcs of the plant are modfed to accommodate the tme delay nfluence on the system. So, takng the four states that represent as follows a ffth state s added whch wll nfluence the system parameter and the effect of the state s (8 (3 (4 (5 (6 (7 III. POLE PLACEMENT WITH PARTICLE SWARM OPTIMIZATION The objectve of partcle swarm optmzaton (PSO algorthm s to mnmze the ftness functon value where the ftness functon s taken as settlng tme. The partcles are updated based on a ftness functon desgned wth ntally all partcles beng assgned random value. From the partcles best usng the below logcal flow, the partcle best and the global best are found. The partcles of the partcle swarm optmzaton are taken based on the dampng factor and natural frequency n. The objectve of partcle swarm optmzaton (PSO algorthm s to mnmze the ftness functon where the ftness functon s taken as settlng tme. The ftness functon s defned as follows Ftness 4/( n (8 The two partcles are updated based on a ftness functon desgned wth ntally all partcles beng assgned random value and wth the boundary condtons specfed. From the 57 P a g e IJCCSE All Rghts Reserved Vol. No.4 June 5

3 ISSN: Srkanth et al, / Internatonal Journal of Computer and Communcaton System Engneerng (IJCCSE, Vol. (4, 5, partcles best usng, the below logcal flow, the partcle best and the global best are found. The generalzed method s as follows Let D be the dmensonal search space S ε R D The swarm conssts of N partcles and th partcle s n effect n D-dmensonal vector. The poston of the partcle s X ( X, X, X 3... X (9 The velocty of ths partcle s also a D-Dmensonal vector V ( V, V, V3... VD T D S T S. K. C, C are acceleraton coeffcents whch control the dsplacements of a partcle n a sngle teraton. The flow chart s gven as follows ( The best prevous poston encountered by th partcle n S s denoted by Y ( Y, Y, Y3... YD T S ( If Y * be the global best poston among all partcles, and t be the number of teratons nvolved, Durng each teraton, velocty and poston of each partcle/swarm s updated by * V ( t WV Cr ( Y X Cr ( Y X ( The mpact of tme delay s consdered n the update equaton for the partcle poston update as unt tme delay under normal condtons appled to the update equaton Where τ d and s gven below as X( t X( t V ( t * d (3 The mpact of delay due to dead tme or sufferng caused by delay s drectly consdered durng the partcle poston update, ths way the poston updates wth a better accuracy. The delay consdered here s for one second(default Where And, F else Y Y f F X F( Y ( Y X f F X F( Y ( * Y Y, Y,... } such that { Y n (4 (5 (6 Mn( F( Y, F( Y, F( Y, F( Y,... F( * ( Y 3 Y n (7 Where r u(, & r u(, are unform random sequences n range(,. W s the dagonal matrx representng weght of V (t as a contrbuton to V (t+. The constrants nvolved n ths knd of algorthm are such that V s clamped to [V max, V mn ] to prevent PSO from leavng search space V max s chosen to be K* X max wth Fg : Flowchart for the partcle swarm optmzaton The fgure 3 below ndcates the block dagram whch represents the condton where the encoders that are connected to both the pendulum and the rotary arm are added a transport lag and then the pendulum s tred to be swung up and also stablzed. Fg 3: Typcal system tme delay n output IV. RESULTS The tme delay was consdered to be τ=. sec whch gave results whch showed the relatve performance of the controllers wth and wthout tme delay. Fgures 4 and 5 ndcate the varaton of ftness and there relatve settlng. The results are analyzed for the tme delay of. sec n fgure 6 and 7 where the stablzaton s acheved wthout any ssues and a comparatve analyss wth LQR clearly ndcates that the effect of tme delay on the stablzaton s mnmal. The poles placed usng pole placement technque are gven as follows 57 P a g e IJCCSE All Rghts Reserved Vol. No.4 June 5

4 u (V ISSN: Srkanth et al, / Internatonal Journal of Computer and Communcaton System Engneerng (IJCCSE, Vol. (4, 5, based on the expermental values that were obtaned durng ths condton s as follows: p = [ ] (8 The partcle best postons obtaned usng PSO are as follows P_Best =[.7 4] (9 The gan matrx based on partcle swarm and pole placement for the above poles s gven below as K = [ ] (3 Ths clearly depcts that prolonged delays n the control sgnal wll cause nstablty. The analyss was done for two cases as per the followng table. The typcal convergence curve for the objectve functon s gven as below wheren the objectve s clearly acheved wthout any problems durng smulatons Fg 4: Ftness varaton for teratons Ftness Ftness 5 5 Cas e ID Delay For Input:(Encoder (Sec TABLE III: Expermental Condtons Delay Swngup Energy For Tme(Sec Input: (Sec Remarks Only vbrates around downward stable equlbrum poston(swn g up faled Stablzaton acheved Unstable Even after 8 seconds stablzaton could not be acheved Vbratons are very hgh and doesn t have enough energy to move Faled to stablze Table III ndcates the varous test cases that are obtaned under the nfluence of tme delay for whch the performance of the the pendulum was taken for transportaton delay to encoder and whch control the pendulum angle and the rotary arm. (deg (deg Fg 5: Ftness varaton for teratons The tme of computaton for the process s gven n the table II as follow TABLE II: TEST CONDITIONS Test Condton Tme of computaton Populaton Sze =5, Iteratons = 4.36 Seconds Populaton Sze= 5, Iteratons= 7.56 Seconds Fg6: Uprght stablzaton wth tme delay The populaton sze was mantaned constant and the teratons were ncreased and t was seen that the computaton was faster when the number of teratons where ncreased. Also, the values of weghts consdered here n these cases was held constant wthn the lmts of.4 to.9 resultng n a system whose weghts are mostly controlled by a proportonal logc. 57 P a g e IJCCSE All Rghts Reserved Vol. No.4 June 5

5 ISSN: Srkanth et al, / Internatonal Journal of Computer and Communcaton System Engneerng (IJCCSE, Vol. (4, 5, (deg u (V (deg Fg 7: Expermental results wth swng up and stablzaton Fgure 6 ndcates the values of theta, alpha and the nput for the stablzaton about ts uprght equlbrum poston for a small dsturbance. It was found that stablzaton could be acheved slowly. Fgure 7 ndcates the values for swngup and stablzaton, although there were perturbratons t was found that the condton of stablty could be acheved. V. CONCLUSIONS From the analyss, t can be concluded, the mpact of tme delay on the system s very crtcal. Even though stablzaton can be acheved usng partcle swarm optmzaton, t s stll to be consdered that t s only a constrant based optmzaton wheren the maxmum possble tme to whch the tme delay can be ncreased s very much confned as s the case of tme delay becomng 4 seconds whch yelds the unstable condton on one of the states resultng us to conclude the constrants of optmzaton lmt the dynamc system control. The study paves way to analyss of complex dynamc systems where the mpact becomes very crtcal. The analyss can be used as a foresght for the ssues that would arse wth the problems assocated wth such knd of systems. ACKNOWLEDGMENT The author would lke to thank hs mentor for encouragement and support and would lke to thank IIIT, Nuzvd for conduct of real tme experments and Dr.Jagu Srnvas Rao for hs valuable nputs durng the conduct of experments. REFERENCES [] K.Srkanth, Dr. G.V.Nagesh Kumar, Desgn and Control of Pendulum wth Rotary Arm usng Full State Feedback Controller based Partcle Swarm Optmzaton for Stablzaton of Rotary Inverted Pendulum, Proceedngs of 38th Natonal system conference on real tme systems modelng, Analyss and control, JNTU Hyd, Telangana, Inda. pp.59-63, 5-7 Nov. 4 [] V. P. Makkapat, M. Rechhartnger and M. Horn, The output trackng for dual-nerta system wth dead tme based on stable system center approach, 3 IEEE Mult-conference on Systems and Control, Conference on Control Applcatons, Hyderabad, Inda, August 8-3, 3. [3] Qguo Yan, "Output trackng of underactuated rotary nverted pendulum by nonlnear controller," Decson and Control, 3. Proceedngs. 4nd IEEE Conference on, vol.3, no., pp.395,4 Vol.3, 9- Dec. 3 [4] Yuba, K.; Okuhara, K.; Hra, J., "Stablzaton of Rotary Inverted Pendulum by Gan-schedulng of Weght and H Loop Shapng Controller," IEEE Industral Electroncs, IECON 6-3nd Annual Conference on, vol., no., pp.88,93, 6- Nov. [5] Jadlovska, S.; Sarnovsky, J., "A complex overvew of modelng and control of the rotary sngle nverted pendulum system," Power Engneerng & Electrcal Engneerng, vol., no., pp.73-85, 3 [6] M.T. Ravchandran, A.D. Mahndrakar, "Robust stablzaton of a class of underactuated mechancal systems usng tme-scalng and Lyapunov redesgn", IEEE Transactons on Industral Electroncs, Vol. 58, No.9, pp , Sept.. [7] Feng, Pan; Lu, Lu; Dngyu, Xue, "Double nverted pendulum sldng mode varaton structure control based on fractonal order exponental approach law," Control and Decson Conference (CCDC, 3 5th Chnese, vol., no., pp.577,58, 5-7 May 3 [8] Pan Feng; Xue Dngyu; Chen Dal; Ja Xu, "Sldng mode varaton structure network controller desgn based on nverted pendulum," Control and Decson Conference (CCDC, 4th Chnese, vol., no., pp.96,964, 3-5 May [9] Kumar, A.; Kumar, Y.; Mtra, R., "Reduced dmenson fuzzy controller desgn based on fuson functon and applcaton n rotary nverted pendulum," Computng Communcaton & Networkng Technologes (ICCCNT, Thrd Internatonal Conference on, vol., no., pp.,5, 6-8 July [] Sngh, Y.; Kumar, A.; Mtra, R., "Desgn of ANFIS controller based on fuson functon for rotary nverted pendulum," Advances n Power Converson and Energy Technologes (APCET, Internatonal Conference on, vol., no., pp.,5, -4 Aug. [] Kondo, T.; Kobayash, K.; Katayama, M., "A wreless cooperatve moton control system wth mutual use of control sgnals," Industral Technology (ICIT, IEEE Internatonal Conference on, vol., no., pp.33,38, 4-6 March [] Barbosa, D.I.; Castllo, J.S.; Combta, L.F., "Rotary nverted pendulum wth real tme control," Robotcs Symposum, IEEE IX Latn Amercan and IEEE Colomban Conference on Automatc Control and Industry Applcatons (LARC, vol., no., pp.,6, -4 Oct. [3] Alt, B.; Hartung, C.; Svarcek, F., "Robust fuzzy cascade control revsed: Applcaton to the rotary nverted pendulum," Control & Automaton (MED, 9th Medterranean Conference on, vol., no., pp.47,477, -3 June [4] ]Ahangar-Asr, H.; Teshnehlab, M.; Mansour, M.; Pazok, A. R., "A hybrd strategy for the control of Rotary Inverted Pendulum," Electrcal and Control Engneerng (ICECE, Internatonal Conference on, vol., no., pp.5656,5659, 6-8 Sept. [5] Zhang Jan; Zhang Yongpeng, "Optmal lnear modelng and ts applcatons on swng-up and stablzaton control for Rotary 573 P a g e IJCCSE All Rghts Reserved Vol. No.4 June 5

6 ISSN: Srkanth et al, / Internatonal Journal of Computer and Communcaton System Engneerng (IJCCSE, Vol. (4, 5, Inverted Pendulum," Control Conference (CCC, 3th Chnese, vol., no., pp.493,5, -4 July [6] Iles, S.; Matusko, J.; Kolonc, F., "TP transformaton based control of rotary pendulum," MIPRO, Proceedngs of the 34th Internatonal Conventon, vol., no., pp.833,839, 3-7 May [7] unha Hou; Hongwe Zhang; Ka Me, "Vertcal-rotary nverted pendulum system based on fuzzy control," Electrcal and Control Engneerng (ICECE, Internatonal Conference on, vol., no., pp.84,87, 6-8 Sept. [8] Anvar, S.M.M.; Hassanzadeh, I.; Alzadeh, G., "Desgn and mplementaton of sldng mode-state feedback control for stablzaton of Rotary Inverted Pendulum," Control Automaton and Systems (ICCAS, Internatonal Conference on, vol., no., pp.95,957, 7-3 Oct. [9] Feng Pan; Dngyu Xue; Dal Chen; Janjang Cu, "Desgn and mplementaton of rotary nverted pendulum moton control hardware-n-the-loop smulaton platform," Control and Decson Conference (CCDC, Chnese, vol., no., pp.38,333, 6-8 May [] Peng Xue; We We, "An Analyss on the Knetc Model of a Rotary Inverted Pendulum, and Its Intellgent Control," Computatonal and Informaton Scences (ICCIS, Internatonal Conference on, vol., no., pp.978,98, 7-9 Dec. [] Huang Me; Zhang He, "Study on stablty control for sngle lnk rotary nverted pendulum," Mechanc Automaton and Control Engneerng (MACE, Internatonal Conference on, vol., no., pp.67,63, 6-8 June [] Akhtaruzzaman, M.; Shafe, A. A., "Modelng and control of a rotary nverted pendulum usng varous methods, comparatve assessment and result analyss," Mechatroncs and Automaton (ICMA, Internatonal Conference on, vol., no., pp.34,347, 4-7 Aug. [3] Ozbek, N.S.; Efe, M.O., "Swng up and stablzaton control experments for a rotary nverted pendulum An educatonal comparson," Systems Man and Cybernetcs (SMC, IEEE Internatonal Conference on, vol., no., pp.6,3, -3 Oct. [4] Zheng Fang; Naxu Song; Langyong Wang, "Desgn and mplementaton of a novel fuzzy controller wth dsp for rotary nverted pendulum," Control and Decson Conference, 9. CCDC '9. Chnese, vol., no., pp.6,67, 7-9 June 9. [5] Meza, C.; Andrade-Romero, J.A.; Bucher, R.; Balem, S., "Free open source software n control engneerng educaton: A case study n the analyss and control desgn of a rotary nverted pendulum," Emergng Technologes & Factory Automaton, 9. ETFA 9. IEEE Conference on, vol., no., pp.,8, -5 Sept. 9 [6] Kuo, T.C.; Huang, Y.J.; Hong, B. W., "Adaptve PID wth sldng mode control for the rotary nverted pendulum system," Advanced Intellgent Mechatroncs, 9. AIM 9. IEEE/ASME Internatonal Conference on, vol., no., pp.84,89, 4-7 July 9 l r Appendx A: System Parameters of Rotary Pendulum Parameter Descrpton Value M p Mass of Pendulum.7 kg M r Mass of rotary arm.57 Kg R m Motor armature resstance.6 Ohm B p Equvalent Vscous Dampng Coeffcent.4 N.m.s/rad Ƞ g Gear box Effcency.9 Ƞ m Motor electromechancal Effcency.69 g Gravtatonal constant 9.8 m/sec J eq Equvalent moment of nerta w.r.to load. Kg.m J m Motor armature moment of nerta 4.663e-7 Kg.m J r Rotary arm moment of Inerta 9.989e-4 Kg.m L p Pendulum full length.3365m K m Motor back-emf constant.77 V.s/rd K t Motor torque constant.77 N.m/A l p Dstance from Pvot to Centre Of Gravty Rotary arm dstance from pvot to center of mass.556 mt.69 mt VMAX_AMP Amplfer Maxmum Output Voltage 4V IMAX_AMP Amplfer Maxmum Output Current 4A K enc Pendulum encoder resoluton 496 counts/rev τ Tme delay to encoders(mn-max.sec-4sec 574 P a g e IJCCSE All Rghts Reserved Vol. No.4 June 5