ScienceDirect. Adaptive LMS Filter using in Flexible Mechatronics System with Variable Parameter Control


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1 Avilble online t ScienceDirect Procedi Engineering 69 ( 014 ) th DAAA Interntionl Symposium on Intelligent nufcturing nd Automtion, 013 Adptive S Filter using in Flexible echtronics System with Vrible Prmeter Control rtin uhs*, Bohuslv uhsov, Igor Hlenr Institute of AIA FST SUT in Trnv, Hjdoczyho 1, Trnv 917 4, Slovki Abstrct The nlysis of negtive impct of flexible mechtronics system time vrible prmeter nd its reduction or complete elimintion possibility is presented in this contribution. The S dptive filter is proposed to use for reduction of double notch filter insufficient effect, which ws initilly integrted into system for flexible joint prsitic frequencies elimintion. Simultion experiments results  step response nd control qulity nlysis confirmed the correctness of suggestion of S dptive filter using. 014 The Authors. Published by Elsevier td. 014 The Authors. Published by Elsevier td. Open ccess under CC BYNCND license. Selection nd peerreview under responsibility of DAAA Interntionl Vienn. Selection nd peerreview under responsibility of DAAA Interntionl Vienn Keywords: flexible mechtronics system control; vrible flexibility; dptive S filter 1. Introduction This contribution dels with specil cse of flexible mechtronics system DC motor with flexible joint contining intime vrible prmeter ffecting the coupling properties. A DC motor ngulr velocity controller ws designed by stndrd method of controller design  odulus Optimum ethod. A flexible connection is chrcterized by undesirble frequencies occurrence. In cse of time invrible joint prmeters in reltion to this control design method, double notch filter ws used nd enough to eliminte these prsitic resonnt nd ntiresonnt frequencies. Becuse of sttic chrcter of double notch filter, vribility of flexible joint key prmeters cn result in filter efficiency reduction. In cse of joint vrible prmeter occurrence, n dditionl S dptive filter is proposed to use together with simplified model of rigid connection system for generting of desired signl for S lgorithm. * Corresponding uthor. Tel.: Emil ddress: The Authors. Published by Elsevier td. Open ccess under CC BYNCND license. Selection nd peerreview under responsibility of DAAA Interntionl Vienn doi: 11016/j.proeng
2 34 rtin uhs et l. / Procedi Engineering 69 ( 014 ) Flexible mechtronics system model The DC motor (DC) with flexible coupling s nlysed mechtronics system is chosen. A specil type of drive torque motor ws nlysed. This type of drive is chrcterized by high torsion moment t reltively low speed. If the inerti of the trnsmission mechnisms is smll compred to the motor nd lod, the flexible coupling between the motor nd lod cn be treted s twomss motor/lod system [1], [5] described with mthemticl model in form of lgebric differentil equtions system (1) (6) [11]. The electric subsystem [4] is represented by equtions: U di RI U i (1) dt U i cφ ω () T cφ E I (3) The mechnicl subsystem [4] is represented by equtions: dω b1( ω ω ) T1 TE 0 (4) dt dt dt 1 d 1 ω ω 0 (5) dω b1ω ω T1 T 0 (6) dt Tble 1. Anlysed system prmeters. Prmeter Unit Description Vlue R Ω rmture current 0 mh resistnce nd inductnce of rmture winding 100 cφ Nm/A torque constnt 3 kg/m inerti of the motor rotor 10 kg/m inerti of the lod 60 b 1 Nms dmping of the trnsmission 1 d 1 Nm spring constnt of the trnsmission 4 A coupling flexibility is source of the ntiresonnt nd resonnt frequencies, which cn led up to whole system instbility. A double notch filter for this spurious frequencies elimintion is chosen. The filter is represented s subsystem, whose trnsfer function contins complex conjugte poles nd zeros. These conjugte poles nd zeros induce resonnt nd ntiresonnt frequencies in the logrithmic frequency chrcteristics (FCh). The filter coefficients re set up to compenste system ntiresonnt frequency by filter resonnt frequency nd conversely, tht system resonnt frequency ws compensted by filter ntiresonnt frequency []. The numertor nd denomintor of filter trnsfer function re defined s: num_filter ω 1 r s ξ ω r r s 1 (7)
3 rtin uhs et l. / Procedi Engineering 69 ( 014 ) den_filter ω 1 s ξ ω s 1 (8) where: ω r d1 ξ r b 1 d 1 ω d 1 b1 ξ (9) d 1 In (9) re chrcteristic prmeters of twomss system with flexible joint (F): ω r ω ξ r ξ system resonnt frequency system ntiresonnt frequency dmping coefficient of system resonnt frequency dmping coefficient of system ntiresonnt frequency The im of this pper is to focus on the cse where one (or more) prmeter of flexible joint is time vrible. et s ssume tht prmeter d 1, which represents flexibility of trnsition, is vrying in time. Becuse of resonnt nd ntiresonnt frequencies of whole system depends on coefficient d 1 (9), we ssume tht this vrition cn ffect control qulity of this mechtronics system. For best observbility of this vribility influence, the vrition function is set in form: d t π/ 1 1(t). 6 d1 0 0 sin d (10) which mens tht joint flexibility is vrying between 60% with frequency reflecting the whole system dynmics. Simultion model of flexible joint with vrible flexibility [9] ws creted in form s shown in Fig. 1. b Fig. 1. Simultion model of nlysed system: () vrible flexible joint; (b) vrition function for d 1 vrition.
4 36 rtin uhs et l. / Procedi Engineering 69 ( 014 ) An influence of flexibility vribility ws nlysed by tlb lineriztion tool using open loop models of DC with flexible joint, double notch filter nd DC+F with double notch filter together s is shown in Fig. with prmeters specified in Tble 1, bsed on system logrithmic mplitudefrequency chrcteristic (AFCh). Fig.. Simultion model for AFCh nlysis. The nlysis ws performed using tlb script (Fig. 3) in which loop ws used for simultion of joint flexibility vrition during monitored time period. Fig. 3. tlb script for vrition of coupling flexibility nlysis.
5 rtin uhs et l. / Procedi Engineering 69 ( 014 ) Bsed on the results shown in Fig. 5 is evident, tht flexibility vrition cuses reduction of double notch filter positive influence in resonnt nd ntiresonnt frequencies elimintion becuse of moving these frequencies cross frequency spectrum (Fig. 4). Fig. 4. AFCh of mechtronics system nd notch filter independently. Fig. 5. AFCh of system nd notch filter together.
6 38 rtin uhs et l. / Procedi Engineering 69 ( 014 ) Control system design The control system design ws bsed on idelized condition where the infinitely rigid connection ws considered insted of flexible connection between ctutor nd lod. This condition is described s: (11) Oversimplified model of system s trnsfer function of this djusted mechtronics system hs form: G c DCF (s) (1) ( )s ( )Rs c The odulus Optimum ethod for PID controller design ws used from lrge number of existing known methods [3], [8], [7]. For controller of PID type in form: r1 (s) r s G R 0 1 r s (13) hs n opened control loop form: 3r1 s 3r0 s 3r 1 Go(s) (14) 3 7s 14. s 09s Following n ssumption tht idel closed control system trnsfer function hs vlue pproching to one, the eqution involving rel prt of open control loop frequency response hs form: 4 ω ( 4r1 1. r0 ) ω ( 07r0 4r 1 ) Go(s) ω 7ω 0081ω Go(s) 5 (15) The coefficients of PID controller re solved bsed on equtions system in mtrix form solution: r r r1 49 (16) The coefficients of PID controller designed by odulus Optimum ethod re in Tble. Tble. PID (O) controller prmeters. Prmeter Vlue r r
7 rtin uhs et l. / Procedi Engineering 69 ( 014 ) r The simultion ws performed s feedbck control of ngulr velocity by simultion model (Fig. 6) tht consists of: controlled system ctutor electricl prt nd flexible connection of ctutor with lod (with sttic nd vrible flexibility) controller G R (s) double notch filter control qulity mesurement subsystem (Fig. 9) Fig. 6. Simultion model of ngulr velocity feedbck control. Trnsmission prmeter (d 1 ) vribility implies deteriortion of control process qulity becuse of notch filter efficiency reduction, which is evident from the closed loop step response (Fig. 7). Fig. 7. Actutor ngulr velocity in sttic flexibility vs. vrible flexibility system. The proposl for elimintion of flexibility vribility negtive influence is n dptive S filter using. The S filter is represented in tlb s S Filter block using the lest mensqure (S) lgorithm [6], [10]. This lgorithm is defined by the following equtions:
8 330 rtin uhs et l. / Procedi Engineering 69 ( 014 ) y(n) w T (n 1) u(n) e(n) d(n) y(n) w(n) α w(n 1) f( u(n),e(n), μ) (17) The weight updte function for the S dptive filter lgorithm is defined s: * f( u(n),e(n),μ) μe(n) u (n) (18) The vribles re s follows: n the current time index u(n) the vector of buffered input smples t step n u*(n) the complex conjugte of the vector of buffered input smples t step n w(n) the vector of filter weight estimtes t step n y(n) the filtered output t step n e(n) the estimtion error t step n d(n) the desired response t step n μ the dpttion step size As desired signl for S filtering signl from simplified control loop with infinitely rigid connection ws used. The resulting suggested closed loop circuit (Fig. 8) designed for qulity nlysis consists of: control loop with flexible mechtronics system with vrible flexibility, notch filter nd dptive S filter simplified control loop with infinitely rigid connection (for desired signl genertion for S filter) control loop with flexible mechtronics system with sttic flexibility nd notch filter (for simplified control loop using verifiction) control loop with flexible mechtronics system with vrible flexibility nd notch filter (for S filter using efficiency nlysis) Fig. 8. The complex simultion model for qulity nlysis.
9 rtin uhs et l. / Procedi Engineering 69 ( 014 ) Control qulity nlysis The qulity of control process hs been evluted bsed on integrl criteri of qulity for control error spce, cquired through the simultion subsystem shown in Fig. 9 nd consists of: Integrl Squre Error (ISE) Integrl Time Squre Error (ITSE) Integrl Absolute Error (IAE) Integrl Time Absolute Error (ITAE) Fig. 9. The simultion subsystem for integrl criteri mesurement. A set of severl runs of simultion experiments ws relized with different prmeters settings of S filter. A combintion of S filter with nd without notch filter using ws nlysed too. Obtined simultion results confirm ssumption, tht S filter using cn positive ffect control process in cse of flexible joint of mechtronics system flexibility vribility occurrence in both simulted scenrios with or without notch filter using, s is shown in Tble 3 s control qulity comprison nd in Fig. 10 s shpe of motor ngulr velocity chrcteristics. Tble 3. Simultion experiments results. qulity criterion using S without S using notch without notch using notch without notch IAE 5.3E E E E+03 ITAE 7.40E E+04.16E E+05 ISE 1.06E E E E+06 ITSE 8.11E E E E+06
10 33 rtin uhs et l. / Procedi Engineering 69 ( 014 ) b Fig. 1 otor ngulr velocity in vrible flexibility system: () without notch filter; (b) with notch filter. 5. Conclusion In this contribution specil cse of flexible mechtronics system, contining intime vrible prmeter ffecting the coupling flexibility, ws nlysed. This vrible flexibility hs significnt negtive influence to control qulity, while speed controller ws designed by odulus Optimum ethod for PID controller design. In cse of constnt flexibility in reltion to this stndrd method of control design, double notch filter is enough to eliminte prsitic resonnt nd ntiresonnt frequencies. In cse of vrible flexibility, dditionl S dptive filter is proposed to use together with simplified model of rigid connection system for generting of desired signl for S lgorithm. A set of different simultion experiments ws relized with different S filter prmeters settings. A combintion of S filter with nd without notch filter using ws nlysed too. Simultion experiments results  step response nd control qulity nlysis confirmed the correctness of S dptive filter using suggestion, while correctly configured dptive S filter is by IAE, ITAE nd ISE criteri cpble to replce double notch filter s originlly used correcting element. Acknowledgment This contribution ws written with finncil support VEGA gency in the frme of the project 1/0463/13 Study of flexible mechtronics system vrible prmeters influence on its control. References [1] S. Vukosvic,. Stojic, Suppression of Torsionl Oscilltions in HighPerformnce Speed Servo Drive, IEEE Trns. on Industril Electronics, vol. 45, n. 1998, pp [] G. Ellis, Control system design guide, Sn Diego: Elsevier Acdemic Press, 004, pp 464. [3]. Vítečková, A. Víteček, odulus optimum for digitl controllers, Act ontnistic Slovc, vol. 8, no. 4, 003, pp [4]. Klous, C. Krtochvíl nd P. Heribn, Dynmics of Rotry Electromechnicl Drives, ÚT AVČR Centrum mechtroniky, Brno, 007. [5]. Vittek, P. Briš, P. kyš,. Štulrjter nd V. Vvruš, Control of Flexible Drive with PS employing Forced Dynmics, 13th EPE PEC 008, pp IEEE [6] thworks. (013, une). Adptive Filters in Simulink [Online]. Avilble: [7]. uhás, B. uhásová nd P. ydlo, The echtronics System Control Qulity Anlysis Using Simulink nd GUI in tlb, in ecture Notes in Engineering nd Computer Science. WCECS 01. Vol. II. : World Congress on Engineering nd Computer Science 01. Proceedings IAENG & IET. Sn Frncisco, USA, 46 October, 01. Hong Kong : Interntionl Assocition of Engineers, 01, pp [8]. ovnkovič,. Žlmn, echtronické pohybové systémy (), AT&P journl, vol. XIII, no. 3, 006, pp [9]. Foltin, ATAB SIUINK  simulčné prostredie pre tlb (), AT&P ournl. vol. 15, no. 4, 008, pp [10] H.. Butterweck, Stedystte nlysis of the long S dptive filter, Signl Processing, vol. 91, iss. 4, April 011, pp [11] A.A.A.Emhemed, R.B. mt, odelling nd Simultion for Industril DC otor Using Intelligent Control, Procedi Engineering, vol. 41, 01, pp
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