A PID Tuning Method for Tracking Control of an Underactuated Gantry Crane

Transcription

1 Univesal Jounal of Engineeing echanics 1 (13), A PID Tuning etho fo Tacing Contol of an Uneactuate Ganty Cane. Nazemizaeh Deatment of echanics, Damavan Banch, Islamic Aza Univesity, Damavan, Ian mn.nazemizaeh@gmail.com Abstact By the cuent eseach, a PID tuning oceue is eveloe to contol of an ovehea ane. To enable the ane to each a esie osition, a contolle esign stategy must be use. This wo esents a simle an efficient enough aoach to contol of the ovehea ane base on esigning a PID contolle. Ziegle-Nichols metho is use to initiate gains setting fo PID contolle an a tuning oceue is eveloe to obtain bette esults. Tuning esign of the PID contolle is consiee fo tacing the esonse of tolley an minimum oscillation of enulum. Finally, some simulations ae efome to illustate the caability of the esente metho fo contol of the ovehea ane. Key Wo an Phases PID Tuning, Ganty Cane, Contol, Tac. 1. Intouction Ganty anes have eceive a lot of inteests in tansotation an inustial alication, ue to thei low cost, easy assembly an less maintenance. They ae comose of a latfom moving in a fixe suot, while a enulum is susene fom a oint on the latfom. Fo the sae of minimum tansotation time, high tacing accuacy, an swing angle, ynamic moeling an motion contol of the ganty ane system become as aealing tass in the fiel of contol science [1-5]. oustafa et al. [6] oose otimal contol of the ovehea anes. They esente a nonlinea ynamic moel of the systems which consiee tavel, tavese, an hoisting/loweing motions of the ane. Rahn et al. [7] use a feebac law to stabilize the hoisting angle of a flexible ovehea ane. Fang et al. [8] use a nonlinea couling law to contol the ovehea obot. oeove, an inut-shaing contol law was oose in [9] to contol the motion of the ane. In this metho, the inut contol ofile is etemine as unwante oscillation uing tavel an esiual enulations ae avoie. Also, a hybi inut-shaing stategy an a PD-tye fuzzy logic contol scheme ae imlemente in [1] to contol a ganty (ovehea) ane system. Howeve, esite of the efficiency of this metho, the inut-shaing metho leas fom being an oen loo contol scheme, an the metho is not obust enough to istubances an aamete uncetainties [11]. ahfouf et al. [1] esigne a fuzzy logic-base contolle to am the sway angle of ovehea ane. oeove, a nonlinea moeling an anti-swing contol metho fo the ovehea anes was esente in [13]. Yu et al. [14] use a etubation technique to seaate the slow an fast ynamics of the ganty ane moel. Then, they use a feebac contol stategy incluing two ineenent PD contolles to tac the e-efine motion ofile an suess ayloa enulations, esectively. Liu et al. [15] oose an aative sliing moe fuzzy contol metho fo the ovehea system. oeove, a neual base metho was use in [16] to contol the motion of the ane. By the esent eseach, a PID tuning oceue is eveloe to contol of an ovehea ane. Hence, Ziegle-Nichols metho is use to stat setting of gains of PID contolle, but a tuning oceue is eveloe to obtain bette esults. The iteion to tune the PID contolle is consiee the tacing esonse of tolley an minimum oscillation of enulum. Finally, some simulation ae efome which iscusse the caability of the esente metho fo contol of the ovehea ane. 45

2 . Nazemizateh. Dynamic oel of the System In this section, ynamic moel of the ganty cane is esente. The set of ynamic equations of the system is eive using Lagange incile. Figue 1 shows an ovehea ane. x f θ l m Fig 1. The ganty ane Paametes of the ovehea system ae esente in Table 1: Table 1. Paametes of the ganty ane Paamete Symbol Cat osition x Cat velocity x& Penulum angula islacement θ Penulum angula velocity & θ Penulum length l ass of the cat system Payloa mass m Gavitational constant of eath g Raius of wheels of cat DC moto voltage of cat e Foce exete to cat f oto amatue esistance R oto toque constant By using the Lagange incile, nonlinea ynamic equations of the system can be esente as [17]: ( + m) && x + ml && θ cosθ ml & θ sinθ f (.1) && x cosθ + l && θ + g sin θ Besies, the linea foce is oiginate fom the toque of tolley moto as [18]: T f T e ω (.) R R x& ω whee ω is the angula velocity of DC moto. This velocity is elate to the velocity of the cat by 46

3 . Nazemizateh Eq.. Futhemoe, by combination of Eq. 1 an Eq., the nonlinea equation of the ovehea ane can be summaize as follows: 1 ( + m) && x + ml && θ cosθ ml & θ sinθ e x& R R (.3) && x cosθ + l && θ + g sinθ T In aition, by efining the state vecto as X [ x x& θ & θ ] [ x x x x ] T an lineaizing the obtaine equations, the esulte equation can be eive in the state sace fom as: & X A X + B u (.4) y CX + Du whee the matices A, B, C an D ae: 1 A C R R l mg ( + m) g l 1 [ 1 ] D B R Rl (.5) 3. PID Contol Fomulation an Simulation Results In this section, the PID contolle is esigne. The fist ty to obtain the gains of the contolle ae efome base on Ziegle-Nichols metho [17}. Then, tuning oceue is continue to obtain esie esults. A lan of contolle is shown in figue : Fig. The contolle lan In Fig., the tansfe function of the PID contolle is shown as G(s), an the tansfe function of the system can be obtaine fom (.5): G 1 X( s) ls + g ( s) C( SI A) B 4 3 E( s) Rls + l s + ( + m) gr s + g s Futhemoe, the tansfe function of contolle is [19]: 1 K I Gc ( s) K 1+ + T s K + + K s Ti s (3.) s whee K, K, K ae the gains of ootional, integal, an eivative contols, esectively. I By using the aamete values of the system as esente in Table, the tansfe function of the system is obtaine as: 1.5 s TF 4 3 (3.3) 3.316s s s s 47 (3.1)

4 . Nazemizateh Table. Paamete values of the ovehea ane Paamete Paamete values Penulum length l. 33m ass of the cat system g Payloa mass m. 3g Gavitational constant of g 9. 81m / s eath Raius of wheels of cat. 6 m oto amatue esistance R. 6Ω oto maximum voltage e max 1V oto toque constant. 767 Vs / a The Ziegle-Nichols metho is eveloe to obtain PID gains of a contolle. In [17] the etail of the metho is esente. oeove, by using the secon metho of Ziegle-Nichols technique fo the ganty ane systems [17], the fist estimation fo contol gains can be obtaine as: K 6.94, P , K.6 K 4. 16, T.5P 6. 98, K K T 7.4, i K T.15P 1.74, K i. 61. Ti It must be notice that in this stuy simulations ae efome using ATLAB Simulin. Emloying fou tuning attemts as it is shown in Table 3, following esults ae achieve via PID contolle: Assuming the unit ste fo inut efeence, the tolley islacement of the ovehea ane obtaine as eicte in Fig. 3: Fig. 3 The islacement of the cat 48

5 . Nazemizateh Accoing to Fig. 3, the fist ty of PID tuning one via Ziegle-Nichols metho has easonable time esonse of the lant while it totally eliminates the eo at the steay state. But the esulte oveshoot an oscillations of the outut ae high. Thus, fo achieving the bette esonse the tuning ocess is continue. As it is obvious fom Fig. 3 afte fou stes of tuning the ocess the esiable esults is obtaine. 4. Conclusions By the cuent ae, a PID tuning oceue has been eveloe to contol of an ovehea ane. Hence, Ziegle-Nichols metho has been use to initiate contolle gains, an then a tuning oceue has been eveloe to obtain bette esults. Finally, numeical simulations have been efome to show the caability of the tuning oceue to contol of the ovehea ane. Refeences 1. Wang Zh., Chen Zh., Zhang J., On PSO base fuzzy neual netwo sliing moe contol fo ovehea ane, Pactical Alications of Intelligent Systems, 4 (1), Lee L. H., Huang P. H., Shih Y. Ch., Ku S. H., Chang Ch., Aative fuzzy sliing moe contol to ovehea ane by CCD senso, Intenational Confeence on Contol Alication, 1(11), Yang J. H., Yang K. Sh., Aative couling contol fo ovehea ane systems, echatonics, 17 (7), Chang Ch. Y., Chiang K. H., Fuzzy ojection contol law an its alication to the ovehea ane, echatonics, 18 (8), Ahma. A., Nasi A. N. K., Raja Ismail R.. T., Ramli. S., Contol Schemes fo Inut Tacing an Anti-sway Contol of a Ganty Cane, Austalian J of Basic an Alie Sciences, 4(1), Al-Gani A. Z., oustafa K. A. F., Javee Nizami S. S., Otimal contol of ovehea anes, Contol Engineeing Pactice, 3 (1995), Rahn C. D., Zhang F., Joshi S., Dawson D.., Asymtotically Stabilizing Angle Feebac fo a Flexible Cable Ganty Cane, J of Dynamics Systems, easuement, an Contol, 11(1999), Fang Y., Dixon W. E., Dawson D.., Zegeoglu E., Nonlinea couling contol laws fo a 3-DOF ovehea ane system, Poceeing of the IEEE Confeence on Decision an Contol., 4(1), Hubbell J. T., Koch B., ccomic D., oen Cane Contol Enhancements, Seattle (199), Ahma. A., Nasi A. N. K., Hambali N., Isha H., Hybi Inut Shaing an PD-tye Fuzzy Logic Contol Scheme of a Ganty Cane System, Int Conf on Contol Alications, Russia (9), Dey R., Sinha N., Chaube P., Ghosh S., Ray G., Active Sway Contol of a Single Penulum Ganty Cane System using Outut-Delaye Feebac Contol Technique, Int. Conf. Contol, Automation, Robotics an Vision, (1), ahfouf. K., Abbo C. H., Linens. F., Fuzzy logic-base anti-sway contol esign fo ovehea anes, Neual Comuting an Alications, 9 (), oustafa K. A. F., Ebei A.., Nonlinea oeling an Contol of an Ovehea Cane Loa Sway, ASE Tans on Dynamic Systems, easuements an Contol, 11 (1988), Yu J., Lewis F. L., Huang T., Nonlinea Feebac Contol of a Ganty Cane. Poc of Ameican Contol Conf., (1995) Liu D., Yi J., Zhao D., Wang W., Aative sliing moe fuzzy contol fo a two-imensional ovehea ane, echatonic, 15 (5), Chiu Ch., Lin Ch. H., Aative outut ecuent neual netwo fo ovehea ane system, Poceeings of SICE Annual Confeence, 1(1), Ghosh S., Ray G., Dey R., Sinha N., Chaubey P., Active Sway Contol of a Single Penulum Ganty Cane System using Outut-Delaye Feebac Contol Technique, Int Conf Contol, Automation, Robotics an Vision, (1) Fielan B., Contol System Design, cgaw-hill, New Yo, Ogata K., oen Contol Engineeing, Aeeizb Publishe, New Jesey,. 49