Discrete Time Sliding Mode Control of Magnetic Levitation System with Enhanced Exponential Reaching Law

Transcription

1 Dscrete Tme Sldng Mode Control of Magnetc Levtaton System wth Enhanced Eponental Reachng Law Dnçer Maden, İrfan Yazıcı Duzce Unversty, Department of Electrcal & Electroncs Engneerng, 86 Konuralp, Düzce, Sakarya Unversty, Department of Electrcal & Electroncs Engneerng, 545 Esentepe, Sakarya, Abstract In ths study, the control of magnetc levtaton system n dscrete tme doman s consdered. Magnetc ball levtaton system model s dentfed, lnearzed near the equlbrum pont and dscretzed n convenent samplng perod. Dscrete tme sldng mode controller wth enhanced eponental reachng law s desgned and compared to tradtonal dscrete tme constant proportonal rate reachng law for the magnetc ball levtaton system whch s subject to many control problems snce t has unstable structure and t has nonlnear dynamcs. The asymptotcally stablty of the system s analyzed by usng Lyapunov stablty condton wth dscrete tme approach. In order to evaluate the performance of the consdered control technque, smulatons are conducted and the results show that dscrete tme enhanced eponental reachng law provdes better performance n terms of both reference trackng and dsturbance nose rejecton as compared to conventonal constant proportonal rate reachng law technques.. Introducton Magnetc levtaton systems provde a non-contact moblty, free of frcton, heatng, nose, and unwanted vbratons from general engneerng problems. Hence, these systems are used n contactless hgh velocty ral systems n Germany and Japan, spacecraft smulators desgned for gravty-free envronments, bomechancal mplant emplacement, and even satellte launchers due to they have specfc advantages []. However, magnetc ball suspenson systems are hghly nonlnear and unstable due to ther electromechancal structure []. The senstvty requrement whch s derved from the system dynamcs s a necessty for desgnng a robust, effcent and practcal controller. Many control algorthms have been proposed n the lterature for the control of magnetc levtaton system. In [3], Prasanta et al. suggested fractonal order SMC (sldng mode control) for magnetc ball levtaton system. They have compared wth tradtonal SMC on the laboratory epermental setup and shown that fractonal order SMC outperform than tradtonal SMC. An observer-based control mechansm amed at movng the non-contact tran structure based on the magnetc suspenson system along wth a gude lne was proposed n [4] and the response of the system to dsturbance effects was eamned. In another study [5], a fuzzy sldng mode controller wth evolutonary programmng based on a magnetc suspenson system was desgned and compared wth other sldng-mode control approaches. System stablty can be ensured when the control methods such as PID or LQR are used. But n such cases the controller's operatng performance s lmted due to the fact that the controller parameters are fed. To remedy ths matter, varable structure control systems are proposed as a soluton for better performance. The sldng-mode control, whch s a varablestructure control technque, was frst ntroduced n the sovet unon n the late 95's and the frst work was done by Emel'yanov n early 96's [6]. Sldng mode control s a hghly robust control technque that can provde the desred dynamc behavor despte the uncertantes n the system, parameter changes and dsturbance effects when approprate condtons are ensured [7]. Presently, SMC s appled n the area of power system control, satellte atttude control, robotc manpulator control, war plane route control etc. The sldng mode control conssts of a set of subsystems supported by approprate swtchng contnuous control functons. It s assumed that ths control technque s subject to dscontnuty on a partcular surface n the system state space [8]. Theoretcally the SMC technque s based on the fact that the error vector of a system s forced nto a desred dynamc and kept n ths dynamc. Lnear and nonlnear systems are drawn on the surface defned n the state space and held on the surface usng the nfnte swtchng feedback control. Ths surface comprses of state varables and s called the sldng surface or sldng manfold. Another advantage of sldng mode control s that t can transform a nthorder control problem nto a frst-order control problem [9]. The sldng mode s realzed n three stages: reachng mode, sldng mode and steady-state mode. Once the system wth controller reaches the sldng surface, the system becomes ndependent of parameter changes and dsturbance effects []. Ths property has known n the lterature as the condton of unformty. In ths study, dscrete tme sldng mode control of the magnetc levtaton system s consdered. In order to obtan satsfactory performance results, dfferent from each other SMC technques whch are dscrete tme constant proportonal rate reachng law (CPRL), and enhanced eponental reachng law (EERL) are desgned. Varous smulatons are conducted for the performance comparson of these control technques.. Magnetc Levtaton System The structure of a magnetc levtaton system s shown n Fg.. The purpose of the magnetc suspenson system s to keep or change the dstance between the electromagnetc col and the steel ball wth mass m. The dstance mentoned here s postoned relatve to the reference value or changes. The ball poston s sensed by a sensor and the electromagnetc force s

2 ELECTROMAGNETİC ACTUATOR changed by ncreasng or decreasng the amount of current suppled from the current source as seen n Fg.. Thus, the am s to determne the poston of the ball. In ths study, a smulaton study was carred out based on the magnetc suspenson system produced by Googol Company. The physcal parameters of the system are gven n Table []. CURRENT SOURCE U Control nput g - f (, ), f (, ) k () m By equatng the dervatve as =, we get g f (, ), (3) where, values can be obtaned. Then k s defned as k mg (4) F(,) Ball Poston () SENSOR (4) provdes the presence of the k coeffcent In order to obtan the transfer functon of the system, the f(,) n the () and () s mplemented by the Tylor seres epanson and hgh order terms are neglected and then Laplace transform s performed. The result s gven as X( s) - (5) I s g s ( ) ( / ) - / F=mg Fg.. General scheme of magnetc levtaton system In the most general case, the nonlnear equaton relatonshp between current, whch est n the electromagnetc col and ball poston, s as follows. m mg - k () In ths equaton, g s the gravtatonal acceleraton and k s the general gan coeffcent for the col parameters. Snce the system model gven n () s n nonlnear form, t s requred to lnearze the system n the vcnty of an equlbrum pont n order to desgn an approprate controller. Table. Realzable parameters of magnetc levtaton system Parameters Values m- Mass of Ball gr g- Gravtatonal Acceleraton 9.8 m/s - Equlbrum Pont of Col Current - Equlbrum Pont of Ball Poston k - The Gan Coeffcent of Col Current k - The Gan Coeffcent of Sensor Current.44 A 35 mm A/V V/m If t s desred to obtan the transfer functon by takng the ponts represented by and as equlbrum ponts, the equaton s obtaned as follows. In the actual transfer functon of the system, the relatonshp between the controlled voltage value of the nput sgnal u and value of the poston sensor output v, can be epressed as follow. Gs () g where b, a system. X () s -( k / k ) b V (6) U ( s) ( / gs ) - / s - a -gk represent the varables of the k As t s seen n (6), the magnetc levtaton system s a second order unstable system snce t has a real root on the rght half s plane. Therefore, t s necessary to combne the closed loop control wth the convenent controller to ensure that the steel ball can be suspended at the desred poston. 3. Desgn and Formulaton of Dscrete Tme Sldng Mode Controller The SMC desgn conssts of two phases. Frst phase s desgnng of sldng surface or sldng manfold and the latter s desgnng of the controller. The sldng surface s desgned n the state space by the root locus method accordng to the desred closed loop system response []. The epresson of the sldng surface accordng to the error value and the dervatve of the error s gven as follows S e e (7) In the second stage, the control nput sgnal s obtaned by equvalent controller, reachng law technques and Lyapunov

3 stablty theory. The Lyapunov sldng condton s forces the system to reach the surface and keep t on the surface. Then the formulaton s obtaned as d S dt - S (8) where s a strctly postve constant coeffcent. When the condton of (8) s satsfed t keeps the system trajectores remanng on the sldng surface []. Lyapunov based sldng mode reachng condton can be epressed as below SS (9) Up to ths pont, contnuous epressons about sldng mode control are gven. But n today's applcatons, nstead of analog elements, dgtal sgnal processors, mcrocontrollers are wdely used due to ther fleblty and ther ablty to perform comple control algorthms. The controllers and systems must be epressed n dscrete tme form for a sutable samplng perod to be able to perform sldng mode control wth these equpment s. In ths regard, Lyapunov stablty condton n (9) should be eamned n dscrete tme doman. Dote and Hoft frst nvestgated dscrete-tme SMC systems n lterature []. They have derved dscrete-tme reachng and stablty condton from contnuous-tme approach n (). In ths study, ths formulaton has been used whle Lyapunov stablty condton s analyzed. s( k ) - s( k) s( k) () The state varables of the system modeled n secton are determned and transformed nto the state space matr form as follows u() t b a () When a dscrete-tme sldng-mode controller s devsed, the system must be transformed nto a dscrete form as n () where we have ( k ) G( k) Hu( k) () Then the swtchng functon S() s determned as n the sldng surface. That s, S()= and S(k) s obtaned at an approprate samplng perod. The last stage s obtanng the control nput sgnal U() n dscrete-tme form as U(k). The dsplacement of the system at the reference nput sgnal and ts output are represented by r and, respectvely. The steel ball poston error e(t) n the system and the dervatve e(t) of ths error are defned as follows z e( t) r( t) - z e( t) r( t) - r( t) - (3) In ths study, EERL, whch s the compound of the reachng laws n the dfferent concepts n [3] and [4], was proposed n order to facltate the nterventon of the reachng and sldng phases. The property of the EERL method provdes faster reachng speed to the sldng surface than the constant rate reachng law method, whch has smlar or the same K value. The EERL s newer n the lterature than the other reachng laws []. The dscrete-tme model of the magnetc levtaton system requred for dscrete-tme controller desgn s derved from the contnuous-tme model gven n Eq. (6) and (). In order to dscretze the system, the A and B matrces are transformed nto G and H matrces usng the ZOH (Zero Order Hold) method n T S =ms. samplng perod. Then, dscrete tme form of the swtchng surface s obtaned by usng contnuous tme form n the same samplng perod for desgnng the dscrete tme constant proportonal rate reachng law [5]. S( k ) (- KT ) S( k) - sgn( S( k)), K s (4) The followng result s obtaned when stablty condton gven n () s nvestgated. ( ) - ( ) ( ) - ( ) - ( ) s s k s k s k KT S k S s (5) So, the system can meet the condton n (). The followng equaton could be wrtten f the estence of a lnear sldng surface s consdered. S k ( ) C z( k) (6) when the state varables reach the sldng surface, From the Eq. (5) S( k ) S( k) (7) C k KT S k sgn S k ( ) (- ) ( ) - ( ( )) (8) Usng the Constant Proportonal-rate reachng law, the dscretetme control nput sgnal s obtaned as follows u( k) where C - - C H.6 H=.98, C G. ( k) (- KT ) S( K) sgn( S( k)) [ c ],.3. G= (9) and

4 Snce the dscrete-tme controller desgn s consdered wth the EERL approach proposed n ths study the swtchng functon s dscretzed by the same samplng perod. Dscrete tme swtchng functon has the form of S (k ) (- T ) S (k ) - KT S (k) sgn( S ( k )) () where (- )e - S (k ) () For analyzng the Lyapunov condton n (), the result s obtaned as follow - Ts S (k ) - KT S (k) S (k ) () From whch we observe that ths system can also meet the condton. Fg.. Poston trackng curves As t s seen from Fg., both controllers have good trackng performance and no overshoot s observed n both reachng laws. But t s obvous that the enhanced eponental reachng law method converges more quckly to the reference nput sgnal. The samplng perod depended sldng manfolds of constant-proportonal rate reachng law and enhanced eponental reachng law are shown n Fg. 3. From (6), (7) and (), one can get C (k ) (- Ts ) S (k ) KTs S (k) sgn( S (k )) (3) Fnally, usng the Enhanced eponental reachng law, the dscrete-tme control nput sgnal s obtaned as follows -C G.(k ) (- T ) S (k )... - u (k ) C H - KT S (k ) sgn( S (k )) (4) 4. The Smulaton Study In ths paper, the dscrete-tme EERL sldng mode control and CPRL sldng mode control have been adapted on the magnetc ball levtaton system model. System and two dfferent sldng mode controllers are dscretzed n same samplng tme, then closed loop controller scheme has formed. The model of magnetc ball levtaton system wth sldng mode controllers s bult n MATLAB/SIMULINK envronment. The physcal parameters of ths system are presented n Table. Gan coeffcents of the controller system n CPRL are chosen as follow, K=5, ε.5 and c=5 respectvely. For EERL, the coeffcents are determned as α=., β=.9 and γ=.. When the system s smulated, a square wave sgnal wth ampltude of. unt and a perod of 4 sec. s used as a nput sgnal. The smulaton results of the system are shown n Fg.. Fg. 3. Sldng surfaces of CPRL& EERL. A systematc dsturbance effect was appled to the system output n order to eamne the dsturbance nose rejecton behavor of the system. As can be seen n Fg. 4 enhanced eponental reachng law method provdes better results than constant proportonal rate reachng law technque. Fg.4. Dsturbance nose rejecton behavor of controllers. The control nput sgnals of both controllers n the dscrete tme form are shown n Fg.5

5 Fg.5. Control nput sgnal of both controllers 5. Conclusons [] S. M. Mozayan, M. Saad, H. Vahed, H. Fortn-Blanchette, and M. Soltan, Sldng Mode Control of PMSG Wnd Turbne Based on Enhanced Eponental Reachng Law, IEEE Trans. Ind. Electron., vol. 63, no., pp , 6. [3] C. J. Fallaha, M. Saad, H. Y. Kanaan, and K. Al-Haddad, Sldng-mode robot control wth eponental reachng law, IEEE Trans. Ind. Electron., vol. 58, no., pp. 6 6,. [4] W. Gao and J. C. Hung, Varable Structure Control of Nonlnear Systems: A New Approach, IEEE Trans. Ind. Electron., vol. 4, no., pp , 993. [5] J. J. W. L. Slotne, APPLIED NONLINEAR CONTROL. NEW JERSEY,USA: ENGLEWOODS CLIFFS, 99.. In ths paper, the control of magnetc levtaton ball system s consdered n dscrete tme. In order to obtan an effectve control performance, enhanced eponental reachng law control technque (EERL) s mplemented and ts performance s compared wth the conventonal technque that s constant proportonal rate reachng law (CPRL). The results show that EERL shows superor performance n terms of reference pont trackng and dsturbance nose rejecton behavor. 6. References [] N. Effect, N. Dma, C. Fang, Y. Jng, Y. Zong, and Z. Ln, Modelng and control wth neural networks for a magnetc levtaton system, Int. J. Adhes. Adhes., vol. 7, no. February 6, pp. 3, 6. [] X. Shao, F. Meng, Z. Chen, and Q. He, The Eponental Reachng Law Sldng Mode Control of Magnetc Levtaton System, no. 35, pp , 6. [3] P. Roy, S. Sarkar, B. K. Roy, and N. Sngh, A Comparatve Study between Fractonal Order SMC and SMC Appled to Magnetc Levtaton System, no. 33, 7. [4] E. M. Junad, E. Sadaqat, and U. Rehman, Observer Based Controller for Magnetc Levtaton System, vol. 4, no., pp. 9 33, 5. [5] T. H. S. L, C. L. Kuo, and N. R. Guo, Desgn of an EPbased fuzzy sldng-mode control for a magnetc ball suspenson system, Chaos, Soltons and Fractals, vol. 33, no. 5, pp , 7. [6] V. Utkn, Varable structure systems wth sldng modes, IEEE Trans. Automat. Contr., vol., no., pp., 977. [7] K. D. Young, V. I. Utkn, and Ü. Özgüner, A control engneer s gude to sldng mode control, IEEE Trans. Control Syst. Technol., vol. 7, no. 3, pp , 999. [8] V. I. Utkn and H.-C. Chang, Sldng mode control on electro-mechancal systems, Mathematcal Problems n Engneerng, vol. 8, no pp ,. [9] M. F. Uslu, SLIDING MODE SPEED CONTROL OF INDUCTION MOTORS, Kırıkkale Unvercty,Insttue of Scence, 5. [] W. Gao, Y. Wang, and A. Homafa, Dscrete-tme varable structure control systems, Ind. Electron. IEEE Trans., vol. 4, no., pp. 7, 995. [] Googol Technology Lmted Chnese, New Magnetc Levtaton System User Manuel, Vol. GMLA, 7