Trajectory Control of a Variable Loaded Servo System by using Fuzzy Iterative Learning PID Control

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1 SCIREA Journa of Eectrics, Counication and Autoatic Contro Deceber 27, 2016 Voue 1, Issue 2, Deceber 2016 Trajectory Contro of a Variabe Loaded Servo Syste by using Fuzzy Iterative Learning PID Contro 1 Oer Aydogdu, 2 Mehet Latif Levent 1,2 Departent of Eectrica and Eectronics Engineering, Secuk University, Aaeddin Keykubat Capus, 42075, Secuku, Konya,Turkey. (e ai: 1 oaydogdu@gai.co, 2 atif@secuk.edu.tr) Abstract: In this study, trajectory contro of the Variabe Loaded Servo (VLS) syste is perfored by using a Fuzzy Logic based Iterative Learning Contro (ILC) ethod. In the study, a Iterative Learning PID (IL-PID) Controer is used as the iterative earning contro structure. Aso, a fuzzy adjustent echanis has been added to the contro syste for specify the initia paraeter of the IL-PID controer. So, with cobining the fuzzy ogic based paraeter adjustent echanis and the IL-PID controer, Fuzzy Iterative Learning PID (Fuzzy IL-PID) controer is designed to iproving the syste perforance. In the designed syste, thanks to the fuzzy adjustent echanis, the IL-PID controer paraeters such as K p, K i, and K d vaues are autoaticay adjusted to the appropriate vaues initiay. To iustrate the effectiveness of the proposed fuzzy IL-PID controer, trajectory contro of the variabe oaded servo syste was perfored by using both Fuzzy PID and Fuzzy IL-PID contro ethods under the sae conditions separatey, and the obtained resuts were copared. It is seen fro the resuts, the proposed Fuzzy IL-PID contro ethod is to better copensate the syste effect as tie varying oads and has reduced the steady-state error ore than other ethod in 85

2 iterations progresses. Keywords: Fuzzy PID contro, Fuzzy IL-PID contro, Trajectory contro, Variabe oaded servo syste 1. Introduction In practice, conventiona PID controers are widey used in any contro systes due to the sipe, robust and effective structure. Athough, PID controer gives very good resuts in the contro of inear, tie-invariant systes, it gives not enough good resuts in the contro of non-inear, tie varying systes. Aso, the effectiveness of the PID controers depends on the precisey odeing of the syste and the settings of the controer paraeters. Conventiona ethods such as Zieger-Nichos used to set the paraeters of PID controers, ony heps to deterine the controer paraeters at the beginning in design stage. Besides, these ethods do not provide the abiity to on-ine setting to the paraeters of the controer in tie progresses [1]. Therefore, the PID controers are insufficient when appied to tie-varying inear or noninear syste and it cannot operate at the desired perforance. On the other hand, fuzzy ogic controer working with intuitive knowedge has been used frequenty in recent years as an aternative to conventiona PID controers. Fuzzy ogic syste is a basic ethod based on huan thinking and knowedge that can be appied successfuy in any areas. Fuzzy ogic is propounded with an artice pubished in 1973 by L. Zadeh [2]. The ost iportant advantage of fuzzy ogic ethod is their intuition and experience of the operator can transit to the controer by eans of fuzzy set theory. As a feature, the advantages of fuzzy contro in the contro syste can be isted as foows [3]; it is robust against noise, it gives better resuts for ode uncertainty and changing syste paraeters, it has very good transient state response, etc. In spite of these advantages, the fuzzy contro ethods are ineffective in the contro of tie varying inear syste and noninear systes, and not enough, especiay to iniize the steady-state error of the syste, as with conventiona PID controer. Therefore, Iterative Learning Contro (ILC) ethod has been proposed to eiinate deficiencies in both the PID and fuzzy contro systes [4]. 86

3 ILC ethod provides very convenient and effective resuts for periodicay repeated in the contro appications such as trajectory contro of a robot ar. Recenty, ILC systes are divided into two categories as traditiona ILC and adaptive ILC ethods in the iterature. Traditiona ILC has a earning echanis that produces the contro inputs based on the error and the previous input vaues. Adaptive ILC ethod uses the advantages of both the ILC and adaptive contro to produce the contro inputs [5]. In the ast decade, any Adaptive ILC appications have been deveoped for unstabe and noninear systes [6]. Proposed Fuzzy IL-PID contro ethod in the study is an adaptive ILC ethod and it has been deveoped to obtain optia PID paraeters of the ILC echanis. In the study, the ain purpose of the cobination of the fuzzy ogic and the ILC ethod to benefit fro the advantages of both fuzzy ogic and ILC ethods and, to iprove the perforance of the ILC syste [6]. In this ethod, obtained initia PID contro paraeters as K p, Ki, and K d by using fuzzy ogic ethod has been used in the ILC syste. 2. Modeing of the Variabe Loaded Servo (VLS) Syste In industria appications, Peranent Magnet DC (PMDC) otors are widey used as an actuator for eectroechanica energy conversion [7]. It directy provides rotary otion or oent and, couped with whees or drus and cabes, can provide transitiona otion or force. In this study, the VLS syste actuated with PMDC otor which has the eectric circuit of the arature and the variabe oaded body diagra of the rotor are shown in Fig R L N 1 Gearbox, i(t) + B, J V(t) eb k M _, _ B N 2 Variabe Load J r Fig. 1 Equivaent circuit of the VLS syste VLS pant and PMDC otor paraeters are given in Tabe 1. We can write the foowing differentia equations based on Newton's aw cobined with Kirchhoff's aw fro the Fig. 1; 87

4 di() t V ( t) Ri( t) L eb ( t), (1) dt 2 d ( t) d ( t) 2 ( t) J B ( t) dt dt, (2) where; it () is the arature current, e b (t) is the back EMF votage, () t is the otor torque, () t is the oad torque, () t is the ange of the arature [8]. Tabe 1 Variabe oaded servo syste and oad paraeters Sybo Definition Vaue V PMDC Motor Votage 6 Vot R Motor arature resistance 2.6 oh L Motor arature inductance 0.18 H k t Motor torque constant N./A k b Motor back-emf constant Vot/(rad/sec) N g Tota gear ratio (N1/N2) 70 η g Gearbox efficiency 0.90 J Motor inertia 4.6 x 10-7 kg. 2 B Motor viscous coefficient 0 (negigibe) J _in Initia oad and gearbox oent of inertia 4.83 x 10-7 kg. 2 J _sub Subsequent oad and gearbox oent of inertia 4.83 x 10-6 kg. 2 B _in Initia oad viscous daping coefficient 4.41 x 10-6 N./(rad/sec) B _sub Subsequent oad viscous daping coefficient 3.41x 10-5 N./(rad/sec) The otor torque () t is reated to the arature current it () by a constant factor k t, and the back EMF e b (t) is reated to the rotationa veocity of the arature () t or d ( t ) / dt by a constant factor k, as given the foowing equations; ( t) k i( t), (3) t e ( t) k ( t) k d () t b. (4) dt In the VLS syste, rotationa ange of the oad () t, transitted by the gear box fro arature ange () t, and equivaent oad torque () t, ay be expressed as; 88

5 1 ( t) ( t), (5) N g ( ) ( ) ( ) ( ) 2 1 d t d t t J B 2 Ng g dt dt, (6) where; N g is tota gear ratio and g is gearbox efficiency. Siuink ode of the VLS syste with variabe oad paraeters ( J, B ) has been obtained using above equations as shown in Fig Trajectory Contro of the VLS Syste In this study, trajectory contro of the variabe oaded servo syste as odeing above has been perfored in Matab-Siuink. First, the servo syste is controed by the Fuzzy PID controer. Next, the sae syste has been controed by Fuzzy IL-PID controer, and the siuation resuts have been copared. Fig. 2 Siuink ode of the VLS syste with variabe oad (J, B ) 3.1 Fuzzy PID Contro In industria appications, PID controers expressed as equation (7) are effective controers widey used in especiay inear tie invariant syste [9]. de() t u( t) K pe( t) Ki e( t) dt Kd (7) dt where; u(t) is the contro signa, e(t) is the error vaue of between desired reference input and actua syste output signa. Aso, as seen fro the equation (7), the design paraeters of the PID controer are proportiona gain (K p ), integra gain (K i ) and derivative gain (K d ) vaues. 89

6 In the iterature, experienta approaches such as Zieger-Nichos ethod is often used in the PID controer design procedure. Many ties however, it needs to be ade further fine tune the paraeters to achieve the optia design. In this respect, deterination of design paraeters takes a ong tie. In addition, various ethods such as fuzzy syste can be used to deterine the PID contro paraeters. In the fuzzy PID ethod, the PID controer paraeters are set using the fuzzy rue base syste. Fuzzy PID ethod is an adaptive ethod, thence, provides better operating of the PID controer in unexpected paraeter changes and aso increases syste perforance [10]. Trajectory contro of the VLS syste by using Fuzzy PID Controer is shown in Fig. 3. As shown in Figure, a Madani type fuzzy bock with two inputs (syste error, et () and derivative of the error, et ()) and three outputs (K p, K i and K d ) has been used in the Fuzzy PID Controer. Aso, PID controer paraeters have been set autoaticay on-ine by the fuzzy syste. Fig. 3 Bock diagra of VLS contro syste by using Fuzzy PID controer Generay, Madani type fuzzy ogic syste consists of four basic coponents naes as fuzzification, rue base, inference engine and defuzzification [11]. In the fuzzifier process, the crisp input vaues ( et (), et ()) are converted into fuzzy vaues [12]. In the study, for fuzzification process, five trianguar ebership function abeed as Negative Big (NB), Negative (N), Zero (Z), Positive (P), Positive Big (PB) are defined for each input vaues. Before the fuzzification process, two input vaues has been noraized to the range of [-2, 2] vaues. Aso, the fuzzy vaues obtained in fuzzy inference echanis ust be converted to crisp output vaues (K p, K i and K d ) by a defuzzifier process. For this purpose, the center-of-gravity ethod has been used and four trianguar ebership functions abeed as 90

7 Zero (Z), Low (L), Mediu (M), High (H) are defined for each output in the defuzzificiation process. Aso, three otput vaues has been denoraized to the range of [0, 2] vaues. In the study, the PID controer paraeters (K p, K i, K d ) have been set to the appropriate vaue with the fuzzy bock. So, fuzzy syste rue base is fored with operator experience, according to the desired syste response. Fuzzy rue base sets defined for K p, K i, and K d have been given in Tabe 2, Tabe 3, and Tabe 4 respectivey. Tabe 2 Rue base for output K p de\ e NB N Z P PB NB H M Z L L N H M Z L H Z H H H H H P H H L M H PB H M L M Z Tabe 3 Rue base for output Ki de\ e NB N Z P PB NB L H H M L N L M H M L Z L L M L Z P L M M M L PB L H H M B Tabe 4 Rue base for output Kd de\ e NB N Z P PB NB Z H H H L N Z H H M M Z L L H L H P L H H M M PB L H H H L Min-Max copositiona rue is used in the deveoped fuzzy ogic syste. In the decision aking operation, iniu inference ethod has been appied in the rue base given in 91

8 above tabes. 3.2 Fuzzy IL-PID Contro Iterative Learning Contro (ILC) iproves the transient response and reduces the error trajectory of the tie varying syste as tie progresses. By eans of iterative earning echanis, contro syste becoe ore stabe and ore robust. Aso, ILC is an effective contro ethod with exceent tracking perforance. Another advantage of the ILC syste, the distorting effects that wi affect the contro syste is abe to copensate. Basic bock diagra of the ILC syste is shown in Fig. 4 [13,14]. Fig. 4 Basic bock diagra of the ILC syste In Fig. 4; u k (t) is the current contro signa, u k+1 (t) is the next contro signa, y k (t) is the output signa and y d (t) is desired output signa. The basic expression of the contro signa for the ILC syste defined by Arioto et a. [15] is given in equation (8). u ( ) ( ) ( ) k 1 t uk t ek t, (8) where; is the earning gain vaue. Equation (8) is adapted to the PID controer paraeters; Iterative Learning PID (IL-PID) controer can adjust their paraeters recursivey created. Accordingy, recursive equations can be written as foows for IL-PID controer [14, 15]. K K ( e( t) e( t) e( t) dt) pk 1 (9) pk K K ( e( t) e( t) e( t) dt) (10) ik 1 ik Kd K ( ( ) ( ) ( ) ) k 1 d e t e t e t dt k (11) where; K p, k i k K and K d are the current paraeters of the PID controer, k p k 1 K, Ki k, and 1 K are the next paraeters of the PID controer, and,, are the earning gain d k 1 vaues for proportiona, derivative, and integra contro respectivey. When equations (9-11) 92

9 are exained, it is seen that PID controer paraeters are continuousy adjusted based on syste error. Under nora conditions, steady-state error can take positive and negative vaues. Therefore, it is very difficut to adjust the PID contro paraeters by IL ethod. In iterature, the absoute vaue of the steady-state error is used to sove this probe. Thus, the PID paraeters are ore easiy oved to the ost appropriate vaue [16]. Aso, the initia vaues of the PID contro paraeters deterined by fuzzy ogic bock and thence syste is achieved in ess tie sef-adjustent. Bock diagra of the Fuzzy IL-PID contro syste proposed in the study is given in Fig. 5. As seen fro Fig. 5, the initia vaues for the IL bock are produced in the Fuzzy Madani bock and PID controer paraeters (K p, K i and K d ) are updated in the IL bock. Trajectory contro of the VLS syste is perfored by PID controer which has continuousy updated paraeters in the IL bock. Fig. 5 Trajectory contro of VLS syste with Fuzzy IL-PID controer 4. Resuts and Discussion In this study, firsty, sinusoida trajectory contro of the VLS syste was perfored with the fuzzy PID controer. Then, the sae trajectory contro syste was carried out with Fuzzy IL-PID controer under the sae conditions. Aso, output responses and trajectory errors of the syste and paraeter variations for each controer have been observed. The syste output responses to both controers are incuded in Fig. 6 for sinusoida reference input. As shown in figure, output response of the Fuzzy PID contro syste has a itte trajectory error for initia oad, but after changing the oad of the servo syste at tie 63 seconds, the trajectory error has increased and so on. 93

10 Fig. 6 Trajectory outputs of the VLS syste On the other hand, it is seen that the trajectory error of the Fuzzy IL-PID contro syste decreases graduay as tie progresses and syste output reaches to the reference trajectory though iterative earning echanis after changing the oad of the syste at tie 63 seconds. Here, it has been understood that Fuzzy IL-PID contro ethod is quicky copensate the effect of oad changes. Zoo area in Fig. 6 is given in Fig. 7 to better observe the syste responses. Trajectory error of the Fuzzy PID and Fuzzy IL-PID contro syste is shown in Fig. 8 and Fig. 9, and aso, paraeter changes of the PID contro for Fuzzy PID and Fuzzy IL-PID contro syste is shown in Fig. 10 and Fig. 11, respectivey. Fig. 7 Zoo area in Fig. 6 94

11 Fig. 8 Trajectory error of the Fuzzy PID contro syste Fig. 9 Trajectory error of the Fuzzy IL-PID contro syste Fig. 10 Paraeter changes of the PID contro for Fuzzy PID contro syste Fig. 11 Paraeter changes of the PID contro for Fuzzy IL-PID contro syste 95

12 5. Concusions In the industria appications, iniization of trajectory error of the DC otor driven servo contro syste is essentia for quaity production process. Especiay, if the syste changes as a variabe oad over tie, the trajectory contro of the servo syste is ore difficut. In this study, Fuzzy IL-PID contro ethod which can be recursivey updated controer paraeters as tie progresses is presented to sove this probe. To iustrate the coparative effectiveness of the proposed ethod, trajectory contro of the tie varying servo syste has been executed for sinusoida input by using both Fuzzy IL-PID and Fuzzy PID controers. Experienta resuts show that the proposed Fuzzy IL-PID contro ethod akes better copensated the effects of tie-varying oad and aso ore reduces the steady state error than other ethod as tie progresses. Finay, Fuzzy IL-PID contro ethod provides better contro by continuousy updating their contro paraeters iterativey. References [1] Baek, S. M., Kuc, T. Y. An adaptif PID earning Contro of DC otors, IEEE Internationa Conference on Coputationa Cybernetics and Siuation, Orando, FL, 1997, vo. 3, pp DOI: /ICSMC [2] Zadeh, L. A. Outine of a new approach to the anaysis of copex systes and decision processes. IEEE trans. on systes, Man and Cybernatics, vo. 3, no. 1, pp.28-44, [3] Eshazy, O., E-Bardini, M. Deveopent of sef-tuning fuzzy iterative earning contro for controing a echatronic syste. Internationa Journa of Inforation and Eectronics Engineering, vo. 2, no. 4, pp , Juy, DOI: /IJIEE.2012.V2.162 [4] Pok, Y. M. Liew, K. H. and Xu, J. X. Fuzzy PD iterative earning contro agorith for iproving tracking accuracy, IEEE Internationa Conference on Systes, Man and Cybernatics, San Diego, CA, 1998, vo. 2, pp DOI: /ICSMC [5] Wei, J. Adaptive iterative earning contro for a cass of noninear tie-varying systes with unknown deays and input dead-zone. IEEE/CAA Journa of Autoatica Sinica, vo. 1, no. 3, pp , Juy, DOI: /JAS

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