Stabilization of a Magnetic Levitation Control System via State-PI Feedback

Transcription

1 Stabilizatio of a Mageti Levitatio Cotrol System via State-PI Feedbak Withupog Wiboojaroe* ad Sarawut Sujitjor Abstrat State feedbak tehique through a gai matrix has bee a well-kow method for pole assigmet of a liear system. The tehique ould eouter a diffiulty i elimiatig the steadystate errors remaied i some states. Itroduig a itegral elemet to work with the gai a effetively elimiate the errors. This paper presets desig ad implemets the state-pi feedbak otroller for otrollig the mageti levitatio system. First, a liear model that represets the oliear dyamis of the mageti levitatio system is derived by the feedbak liearizatio tehique. The, the state-pi feedbak otrol developed from the liear model is proposed. Results are ompared betwee the ovetioal state feedbak tehique ad the proposed method. I additio, we pratially implemeted the otroller i a experimetal mageti levitatio system ad ivestigated its regulatig performae. The experimetal results show the effetiveess of the proposed method for disturbae dampeig ad stabilizig the system. Keywords state feedbak, observer, state-pi feedbak, mageti levitatio, pole plaemet, stability ad stabilizatio I. INTRODUCTION ageti levitatio tehology elimiates mehaial Motat betwee movig ad statioary parts. This implies that this tehique also elimiates the fritio problem. Therefore, they are widely used i various fields, suh as high-speed trais, mageti bearigs, vibratio isolatio systems ad so o. Mageti levitatio systems are iheretly ustable ad uertai oliear dyamial systems. Therefore, it is always a hallegig task to ostrut a high performae feedbak otroller to fix the positio of the mageti levitatio system rapidly ad exatly. I reet years, may proposals have bee preseted i literatures based o liear ad oliear system models for otrollig this system [-]. The stadard liear tehiques are usually based upo a approximatio liear model by whih a liear otrol law a be ostruted to meet the desig speifiatio. A wide variety of otrol methods are proposed ragig from PID ad lassial state feedbak otrols to omplex oliear ad adaptive otrols. Several advaed otrol algorithms are applied for otrollig mageti levitatio system, suh as model referee otrol [4], robust otrol [5], slidig mode otrol [6], feedbak liearizatio method [7] et. Reetly, state-pi feedbak [8,9] has bee proposed for regulatio problem of a LTI system. The oept is exteded to stabilizatio otrol of a mageti levitatio system as reported by this paper is show o Fig.. I this paper we osider stabilizatio otrol of a mageti levitatio system. First, the state-pi feedbak otrol is applied to ahieve stabilizatio ad disturbae rejetio via poleplaemet. Seod, a liear model represetig the oliear dyamis of the mageti levitatio system is derived by the feedbak liearizatio. The ahieved results are ompared with those obtaied from the ovetioal state feedbak approah. Setio 2 presets the desigig of state-pi feedbak otroller. Setio gives a brief o model represetatio of a mageti levitatio system. Experimetal results for stabilizatio of the mageti levitatio system follow i Setio 4. Setio 5 provides the olusio. Fig. Mageti levitatio system II. POLE PLACEMENT BY STATE-PI FEEDBACK Let s osider a delay-free ompletely otrollable LTI system desribed by x Ax B x x () = u, ( t) = where x R is the state vetor, ad u R is the otrol iput. A ( ) ad B ( ) are the system matrix ad the otrol gai vetor, respetively. From A, the harateristi polyomial a be writte as Mausript reeived Marh 5, 2. This work was supported by Rathamagkala Uiversity of Tehology Isar Thailad. W. Wiboojaroe is with the Shool of Eletroi Egieerig, Rathamagala Uiversity of Tehology Isar, Muag Distrit, Nakho Rathasima, Thailad. *orrespodee: (vihupog@gmail.om). S. Sujitjor is with the Syhrotro Light Researh Istitute (Publ. Org.), Muag Distrit, Nakho Rathasima, Thailad. det( si A) = as a s as a = (2) Issue 7, Volume 7, 2 77

2 Where a = [ a a a ], a = det( A) = ( ) det( A) ad a = trae( A ). The otrol u of the state-pi feedbak is u = Kx p KI x ( τ) dτ, () where Kp, KI R are the desiged gai matries to ahieve a desired losed-loop harateristi polyomial. The losedloop system a be represeted by Eq. (4). t x' = ( A BKp) x BK x ( τ) dτ (4) Eq. (5) represets the losed-loop harateristi equatio, while Eq. (6) represets the presribed harateristi polyomial. I BK det[ si ( A BK ) I p ] = (5) s () s = α α s α s α s α s (6) d It is otied that the -order of the ope-loop system is ireased by due to the itegral term. A. 5BFrobeius Caoial Form The pole plaemet problem herei osiders the Frobeius aoial form of a delay-free LTI system. Eq. (7) represets the state trasformatio - ξ = Tx, x = T ξ, (7) where ξ(t) ( ) is the trasformed state variable vetor, ad T ( ) is the trasformatio matrix. The matries A( ) ad B( ) are the trasformed system matrix ad the otrol gai vetor, respetively. Both matries a be alulated as follows: where A - = TAT, B = TB, (8) T= q qa qa. (9) The vetor q ( ) i (9) is T - - T q =e w, () i whih w is the otrollability matrix of the system () 2 - w = [ B AB A B A B], () ad the uit vetor e = [ ] T. The Frobeius aoial form a be expressed as ξ = Aξ Bu (2) B. 6BPole Plaemet For State-PI Feedbak The sigle-iput LTI system () is assumed to be ompletely otrollable, ad B is of full olum rak. State feedbak through a PI otroller a be ahieved via the gai matries K P ad K I respetively. Note that due to the itegral elemet, oe additioal losed-loop pole is eeded. This imposes a oditio for derivatio of the gai matries, ad results i a irease i the order of the system by oe. The system () with its Frobeius form of (2) is subjet to the otrol iput u = Kx p KI x ( τ) dτ or u = Kξ F t Kξ ( τ) dτ i whih [ K, K ] = [ K, KF,] T. There exist F p I F the followig gai matries to ahieve a desired harateristi polyomial () s = α α s α s α s α s d Kp KI [ a a a2 a α] T [ ] = = α α α α T( ) 2 See [8], Propositio 2., for proof of Eq. (). The desig proedures are as follows:.calulate the trasformatio matrix for a -order LTI plat - usig = T T - T q qa qa where q =ew, e = [ 2 - ]T ad w = [ B AB A B A B ]. 2. Calulate the matries A ad B usig - A = TAT ad B = TB for the Frobeius form of (2).. Assig the losed-loop pole loatios of a -order for state-pi feedbak, add oe egative real pole havig a fast time-ostat (i.e. a egative real pole with a large magitude) 4. Determie the presribed harateristi polyomial d () s havig the order of or orrespodig to step. 5. Calulate the gai matries for state-pi feedbak use (). Cosider the followig sigle-iput otrollable systems: Example. A 2 =, = B The system i example is origially ustable with its poles at ad -2. It is desirable to have the losed-loop poles at Issue 7, Volume 7, 2 78

3 -4.2±.8486j. As a result of trasformatio, the aoial form of the system model is = 2 ξ ξ u To ahieve the presribed pole loatios for step, a additioal pole at -2 is osidered. The desired harateristi polyomial is ( s ) = d s 28.2 s s The obtaied gai matries are Example 2. K P = [ 26.2 ] = [ ] K. I INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES A = , B = 2 The system is origially ustable with its poles at ±.5 ad -. It is desirable to have the losed-loop poles at -±j ad -2. As a result of trasformatio, the aoial form of the system model is ξ = ξ u To ahieve the presribed pole loatios for step, a additioal pole at - is osidered. The desired harateristi polyomial is ( s ) = d 2 4 s 26 s 9 s s The obtaied gai matries are K P = [ ] K I = [ ] 2 4 The results show i Figs. 2 ad have the iitial states of ( t ) =. T.5 T. x [ ] ad [ ] (a) Resposes of states: x, x 2. (b) Cotrol sigal u Fig. 2 Time resposes ad otrol sigal of the umerial example with state-pi feedbak from the proposed method Issue 7, Volume 7, 2 79

4 (a) Resposes of states: x, x2 ad x (b) Cotrol sigal u Fig. Time resposes ad otrol sigal of the umerial example 2 with state-pi feedbak from the proposed method III. MAGNETIC LEVITATION SYSTEM The mageti levitatio system is a mageti ball suspesio system whih is used to levitate a steel ball o air by the eletromageti fore geerated by a eletromaget. Cosider a steel ball of mass M plaed uder a eletromaget at distae y as show i Fig. 4. The objetive of the otrol system is to keep the steel ball i a dyami balae aroud its equilibrium poit. The desig of the suspesio system preseted here uses the eletromageti attratio fore. The ball positio i the mehaial system a be otrolled by adjustig the urret through the eletromaget where the urret through the eletromaget i the eletrial system a be otrolled by applyig otrolled voltage aross the eletromaget termials, thus the ball will levitate i a equilibrium state. But it is a oliear, ope loop, ustable system that demads a good dyami model ad a stabilized otroller. Eletromageti fore produed by urret is give by the Kirhoff s voltage law. The voltage equatio of the eletromageti oil is give by R L Eletromaget i v where v = Ri L( y) i (4) v : iput voltage, i : widig urret, R : widig resistae ad L : widig idutae. y ki y The total idutae L is a futio of the distae ad give by M Steelball Mg Fig. 4 Ball suspesio system. The mageti ball suspesio system a be ategorized ito two systems: a mehaial system ad a eletrial system. Ly ( ) Ly y = L (5) Where L is the idutae of the eletromageti (oil) i the absee of the levitated objet, L is the additioal idutae otributed by its presee, ad y is the equilibrium positio. Issue 7, Volume 7, 2 72

5 Assumig the suspeded objet remais lose to its equilibrium positio, y=y, ad therefore Ly ( ) = L L (6) Also assumig that L>>L, Eq. (4) a be simplified as v = Ri Li (7) The priipal equatio for the suspeded objet omes by applyig Newto s seod law of motio. For this oe degree of freedom system, a fore balae take at the etre of gravity of the objet yields where ki My = Mg (8) y M : ball mass, y : ball positio, g : gravitatioal ostat ad k : mageti fore ostat. The state variables are defied as x = yx, 2 = y ad x = i. The state equatios of the system are x = x, 2 k x x = g, (9) 2 Mx R v x = x L L Let us liearize the system about the equilibrium poit y = x = ostat, whih results i state vetor as T x = [ x x2 x]. At equilibrium, time rate derivative of x must be equal to zero i.e. x 2 = x = ad y =. The equilibrium poit of the system is at / kue u e x = (2) gmr R [ ] Thus we a write the liearized model i state spae form as uder; / g ( gmr) gr, A = / B = ( kue ) ue R L L T (2) The umerial values of the experimetal system parameters are show i Table. Table. Parameters of the mageti levitatio system Parameters Desriptio Values y ball positio at operatig 5-2 poit (m) M mass of steel ball (kg) 4. - R oil resistae ( Ω ).7 L oil idutae (H) 5. - i oil urret at operatig.5 poit (A) K ostat (kgm 5 /s 2 /A). -6 u e oil applied voltage at.79 operatig poit (V) G gravitatioal ostat (m/s 2 ) 9.8 BIV. REAL TIME IMPLEMENTATION The mageti levitatio system is preset with fousig o stabilizatio ad disturbae rejetio issues. Results are ompared with those desiged by the pervious method iludig Akerma s formula []. A. 7BState-PI Feedbak Cotroller The state-pi feedbak otroller is applied to the stabilizatio ad disturbae rejetio problems of the mageti levitatio system. The blok diagram i Fig.5 represets a mageti levitatio system with state-pi feedbak. For ompariso purposes, the method based o Akerma s formula is also used. u B K D K I Fig. 5 Blok diagram represetatio of a mageti levitatio system with state-pi feedbak. The mageti system is desribed by the followig statevariable models: x x = x u (22) The system is iheretly ustable sie it has ope-loop poles at ± 8.52 ad To stabilize this system, the system poles are to be plaed at - ad 5 ± 5j. As a result of trasformatio, the aoial form of the system model is s A K P s x C y Issue 7, Volume 7, 2 72

6 ξ = ξ u (2) To ahieve the presribed pole loatios, a additioal pole at - ad -2 are osidered for 2 ases. Usig the proposed method, the followig gai matries are obtaied: - (for addig poles at -) K p = [ ] K i = [ ] - (for addig poles at -2) Fig. 6 Diagram of the mageti levitatio system. K p = [ ] K i = [ ] For a ompariso, usig the Akerma s formula oe a obtai the gai matrix K P = [ ]. B. 8BExperimetal Setup Cosider the mageti levitatio system show i Fig. 6, i whih a eletromaget exerts attrative fore to levitate a steel ball. We pratially implemet the proposed state-pi feedbak otroller i a experimetal setup. A image of the experimetal apparatus system a be see i Fig. 7a. The oi ell batteries atig as disturbae to the ball a be otrolled as show i Fig. 7b. These experimets poit out that the proposed otroller is robust. I order to test the state-pi feedbak otroller o a real plat, the regulator was desiged i Simulik of Matlab (Fig. 8). It was the implemeted with the RTW of Matlab via a digital board o the real system. The digital board is a Rapo, 2-bit iput/output ard [], used with a Itel ore TM 2 duo omputer. The aalog iput ad output bloks i the simulik sheme of Fig. 7 are iput/output bloks ompatible with the Rapo digital board with samplig time.s. The mageti levitatio uit is omposed of a eletromaget, of a steel ball, ad of a liear hall effet sesor set that measures the positio of the ball. (a) without disturbae (b) with disturbae Fig. 7 Proposed mageti levitatio system C. 9BExperimetal Results The experimetal results show below were very satisfatory ad demostrated the robustess ad the effetiveess of the state-pi otroller. Fig. 9 shows the resposes ad the otrol iput aordig to the proposed method, ad the states are disturbed by hages i the mass at the time t=2.5s. It a be observed that usig the proposed method the states possess very good resposes, the disturbaes are ompletely dampeed out, ad the otrol iput is reasoable. With the ovetioal pole plaemet method, some states otai a large amout of steady-state errors due to disturbae as depited i Fig.. Fig. 8 Implemetatio of the state-pi feedbak otroller o Simulik. Issue 7, Volume 7, 2 722

7 (a) Respose of system state (x ). (b) Respose of system state (x ). () Cotrol sigal. Fig. 9 Respose of system states for 6% variatio of the mass with the proposed state-pi feedbak. (a) Respose of system state (x ) Issue 7, Volume 7, 2 72

8 (b) Respose of system state (x ). () Cotrol sigal. Fig. Respose of system states for 6% variatio of the mass with the ovetioal state feedbak. I Fig., the large effet of a high step disturbae o the equilibrium positio exeeds the liear rage of the sesor, deterioratig the system performae. However, this effet does ot our with the state-pi feedbak, as illustrated i Fig., whih idiates that this otroller produed a appropriate atio fast eough to avoid large deviatios o the steel ball positio. The state feedbak otroller ould ot stabilize the plat for large variatios o the mass. From Figs. -2, oe sees that the robust otrollers ahieve better disturbae rejetio tha the ovetioal state feedbak otroller ad that the robust otrollers perform very well i brigig the ball bak to the adopted operatig positio eve whe the system is subjeted to hage i the mass. Further results are illustrated i Fig. to show the effets of the additioal real pole due to the desig step o the dyami resposes. It is foud that a additioal fast real pole results i better trasiet resposes i a exhage of high gais. Moreover, the system is more robustess to exteral disturbaes. (a) Respose of system state (x ) Issue 7, Volume 7, 2 724

9 (b) Cotrol sigal. Fig. Respose of system states for 2% variatio of the mass with the ovetioal state feedbak. (a) Respose of system state (x ) (b) Respose of system state (x 2 ) () Respose of system state (x ) Issue 7, Volume 7, 2 725

10 (d) Cotrol sigal. Fig. 2 Respose of system states for 2% variatio of the mass with the proposed state-pi feedbak. (a) Respose of system state (x ) (b) Respose of system state (x 2 ) (b) Respose of system state (x ) Issue 7, Volume 7, 2 726

11 (d) Cotrol sigal. Fig. Respose of system states for 2% variatio of the mass with the proposed state-pi feedbak (added pole at -2). V. CONCLUSION We have demostrated that the proposed state-pi feedbak otrol is effiiet whe used i motio otrol i whih the displaemet, veloity ad urret are usually eeded as feedbak sigals. By ompariso with the ovetioal state feedbak otrol, its simple struture meas less effort to be made i the implemetatio of the otroller. This is very attrative for a pratial desig of a feedbak otrol system. The mageti levitatio system has bee used i this paper to pratially demostrate the effetiveess of the proposed otrol sheme. Experimetal results idiate the state-pi feedbak otrol sheme a result i a losed-loop system with good regulatig performae as well as good robust property agaist high step disturbaes. Also, the effets of the positio of oe additioal pole required aordig to the itegral term are ivestigated. It is reommeded that a fast real pole be added to ahieve more robustess to exteral disturbaes bearig i mid o the irease i the feedbak gais. REFERENCES [] Z.J. Yag, K. Miyazaki, S. Kaae, ad K. Wada, Robust positio otrol of a mageti levitatio system via dyami surfae otrol tehique, IEEE Trasatio o Idustrial Eletroi, Vol.5, No., 24, pp [2] I. Ahmad, M.A. Javaid, Noliear model & Cotroller Desig for mageti levitatio system, i Proeedig of the 9 th WSEAS Iteratioal Coferee o Sigal Proessig, Robotis ad Automatio (ISPRA ), 2, pp [] F. Gazdos, P. Dostal, ad J. Marholt, Robust otrol of ustable systems: algebrai approah usig sesitivity futios, Iteratioal Joural of Mathematial Models ad Methods i Applied Siees, Issue 7, Vol.5, 2, pp [4] D.S. Liu, J. Li ad W.S. Chag, Iteral model otrol for mageti suspesio systems, i Proeedigs of the 4 th Iteratioal Coferee o Mahie Learig ad Cyberetis, 25, pp [5] Yag ad Tateishi, Adaptive robust oliear otrol of a mageti levitatio system, Automatia, Vol.7, 2, pp.25-. [6] N.F. Al-muthairi ad M. Zribi, Slidig mode otrol of a mageti levitatio system, Mathematial Problems i Egieerig, Vol.2, 24, pp [7] A.E. Hajjaji ad M. Ouladsie, Modelig ad oliear otrol of mageti levitatio systems, IEEE Trasatio o Idustrial Eletroi, Vol.48, No.4, 2, pp [8] W. Wiboojaroe ad S. Sujitjor, Stabilizatio of a Iverted Pedulum System via State-PI Feedbak, Iteratioal Joural of Mathematial Models ad Methods i Applied Siees, Issue4, Vol.5, 2, pp [9] S. Sujitjor ad W. Wiboojaroe, State-PID feedbak for pole plaemet of LTI system, Mathematial Problems i Egieerig, ID.9294, Vol.2, 2 pages. [] K.Ogata, Moder Cotrol Egieerig, Third Editio, Pretie Hall, 997. [] M. S. Kim, Y. S. Byu, Y. H. Lee ad K. S. Lee, Gai Shedulig Cotrol of Levitatio System i Eletromageti Suspesio Vehile, WSEAS Trasatios o Ciruits ad Systems, Vol.5, 26. pp [2] Fratisek Gazdos, Petr Dostal, Polyomial approah to robust otrol of ustable proesses with appliatio to a mageti system, i Proeedigs of the th WSEAS iteratioal oferee o Automati Cotrol, Modellig & Simulatio (ACMOS ), 2, pp [] Real-time Rapid Cotrol Platform. [Olie]. Available: Withupog Wiboojaroe was bor i Nakho Rathasima, Thailad, i 979. He reeived the B.S ad M.S. degrees i otrol system egieerig from Kig Mogkut s Istitue of Tehology Ladkrabag (KMITL), Bagkok, Thailad, i 22 ad 24, respetively. He is assistat professor of eletroi egieerig at the Departmet of Eletroi Egieerig, Rajamagala Uiversity of Tehology, Thailad. His researh iterests ilude applied AI ad otrol systems. Professor Sarawut Sujitjor reeived his BS (EE) i 984 from the Royal Thai Air Fore Aademy, ad PhD (EE) i 987 from the Uiversity of Birmigham, UK. He is urretly the Diretor of Syhrotro Light Researh Istitute (Publi Orgaizatio), Thailad. His researh iterests ilude otrol ad itelliget systems, metaheuristis, power overters ad drives. Issue 7, Volume 7, 2 727