Robust Control Design for Maglev Train with Parametric Uncertainties Using µ - Synthesis

Transcription

1 Proceeing of the Worl Congre on Engineering 007 Vol I WCE 007, July - 4, 007, Lonon, UK Robut Control Deign for Maglev Train with Paraetric Uncertaintie Uing - Synthei Mohaa Ali Sarnia an Atiyeh Haji Jafari Abtract : The agnetic uenion yte that they are bai of aglev train ivie in two clae: electroynaic uenion (EDS an electroagnetic uenion (EMS EDS i bae on reulive force acting on a agnet an i inherently table yte an even ha well robutne in any cae with oen loo control But EMS i bae on attractive force acting on a agnet, i inherently untable yte that without feebac control ha a oor erforance So, we ut ue feebac an we nee to an exact atheatical oel of lant to ynthei the feebac control yte Thi oel houl contain ifferent uncertaintie to ae it ore iilar to 1 INTRODUCTION Magnetic levitation (aglev i an innovative tranortation technology that via relaceent of echanical coonent by electronic overcoe the technical retriction of wheel on rail technology Coare with traitional railway aglev yte have high ee, high afety, le ollution, low energy conution an high caacity Since the agnetic uenion yte ue in thee train i untable, we ut ue feebac In aition we ut ay attention to robutne of reone in control eign Thi ean that the yte houl have robut tability an erforance uner oel uncertaintie In thi tuy we conier the erturbation of the oel araeter a the ource of uncertainty Since we can rereent thee erturbation in tate ROBUST CONTROL Robut erforance roble i ecribe with general fraewor hown in fig1 Firt an Secon Author are with Shahroo Univerity of Technology, Shahroo/Iran (Phone: ; fax: ; e-ail: aarnia@yahooco actual oel Therefore, control yte houl have robut tability an erforance uner oel uncertaintie Above eire will be acceible with a controller in fraewor In thi aer, we aue that the uenion yte i EMS an erturbation of the oel araeter are coniere a the ource of uncertainty Since we can rereent thee erturbation in tate ace araeter (A, B, C, D,uncertainty will be tructure Key-Wor: robut control, technique, aglev train ace araeter, uncertainty will be tructure At each art of yte that exit uncertainty it can be coniere uncertainty a a bloc about certain araeter In thi anner each of uch bloc ha one inut an one outut Putting in orer all bloc will for uncertainty et In thi way all bloc are coniere out of lant an obtaine uncertainty et ha ecial tructure: it i blociagonal an bloc on iagonal are all that in fact ulle out fro interior of lant Thu, if we have n bloc, then = iag,, We conier ( 1 n uncertaintie in noralize for in foralization of tanar conition at robut eign ie 1 j P i noinal oel an rereent yte interconnection an K i controller y i eaureent outut, u i control inut, i external inut an iturbance, e i error ignal an z an w are the inut an outut ae uncertaintie

2 Proceeing of the Worl Congre on Engineering 007 Vol I WCE 007, July - 4, 007, Lonon, UK (4 Structure ingular value that rovie uitable tet for robut tability an erforance i efine a: Fig1 Fraewor Thi fraewor can be hown a cobination of analyi an ynthei roble by linearfractional tranforation (LFT efinition ( M 0 if et( I M 0 B = 1 {inf σ ( ( j : et( I M } (5 B We can conier yte erforance by external iturbance inut For thi uroe we augent fictitiou erforance uncertainty bloc to analyi tructure Then extene uncertainty et i efine a * = iag(, (6 P F L F U (b Fig Cobination of analyi an ynthei roble In robut analyi tranfer function [ w ] to [ e] F L fro z i hown by lower LFT: M = F (1 L K = P11 + P1K( I P P1 (To efine Pij, we have e = P11 + P1u y = P1 + Pu An in robut ynthei hown by uer LFT: K (a F U for to e i F = F ( U ( M, = M + M 1( I M 11 M 1 (To efine M ij, we have z = M11 + M1 e = M 1 + M It will be clear that F ( L[ FU, K ] = FU [ FL K, ] Deign ecification i to obtain tabilizing controller K uch that (for all B,that B = { : σ ( 1} cloe loo yte be table an atifie F U [ F K, ] 1 L Fig Rereentation of yte with augente uncertainty Definition of the noinal an robut tability an erforance are exree a: Noinal Stability (NS: M i (interior table Noinal Perforance (NP: ( M < 1 P Robut Stability (RS: Robut Perforance (RP: ( M 11 < 1 ( < * M 1 The uroe of ynthei i to iniize ea of value ( of cloe loo tranfer * function F L K uon tabilizing controller K: in ax (, ( (7 * P K jw K tabilizing There in t any otiu olution for thi iniizing roble, but D-K iteration roceure that cooun analyi an H ynthei often ha goo reult Thi roceure attet to olve in ax D( F K( jw D (8 K tabilizing that D { D D = D} = { } F L D( S D; table,in hae Thi roble can be olve reeately an by a conecutive olution uon one of the L

3 Proceeing of the Worl Congre on Engineering 007 Vol I WCE 007, July - 4, 007, Lonon, UK variable D an K an fixing another variable It worthy of ention that atrix D i aue equal to I at firt iteration becaue of we on t acce to D MAGLEV TRAIN MODEL In thi aer a oel i ue with two egree of freeo with one car boy an one agnet Degree of freeo are tranlational ilaceent at the center of a of the car boy an bogie To balance the car we houl aue the car ha two bogie but a art of thi yte i only coniere ue to avoiance colexity an high aount of equation becaue of rotational ilaceent equation at car boy will be ae an alo the nuber of equation relate with bogie an agnet will be two tie a uch The yte contain two uenion yte: agnet uenion an uenion caue by ring an ahot of bogie Moveent equation atify Newton low; f i agnetic force an all of the araeter introuce in table 1 v + c ( v v + ( v v (9 v + c ( v v + ( v v = f In agnetic uenion oel the attractive force between the ole urface an the ferroagnetic late (fig4 i: 1 F = B (A 0 (10 That i obtaine with regar to relationhi in [1] a F n ab 4h o = ( N I N i c c + t Fig4 Electroagnetic an ferroagnetic late configuration (11 The relationhi between voltage an tri current i erive fro Kirchhoff voltage low: ( Bab u = Rci Nt t (1 That with relaceent of B fro [1] we have: 0abNt i abnt ( NcIc + Nti h u = R (1 ci + h t h t The agnetic force equation i linearize at noinal air ga where h = h0, i = i 0 M1 M1 F N (14 cic NcIc( h h0 + M1Nti h0 An the ynaic agnetic force i exree a M1 f N (15 cic ( h h0 + M1Nti h0 The voltage law i alo linearize at the noinal air ga, where h = h0, i = i 0, h = h0, i = i0 i M u M i + M araeter Priary uenion (bogieagnet a Seconary uenion (car boy a Vertical ilaceent of bogie Vertical ilaceent of the car boy at the center of gravity Seconary aing Seconary tiffne Pereability of air Flux enity acro the air ga Face area of each agnet ole Nuber of agnet in each oule Refer to fig Refer to fig Magnetic air ga Noinal air ga Nuber of turn of coil in each agnet roviing a contant force Contant current Nuber of turn of coil in each agnet roviing tri 4 h ybo l v v c 0 B A a b h n h 0 N c I c N t (16

4 Proceeing of the Worl Congre on Engineering 007 Vol I WCE 007, July - 4, 007, Lonon, UK current Tri current Voltage Reitance of tri coil Wave nuber Roughne Gauian white noie with zero ean Vehicle velocity Bea vertical ilaceent Funaental frequency i u R c r f h w V v g Table 1 Paraeter of Maglev Moel It i aue that the roughne caue by ier elevation ifference an cree eforation will be a non-white tationary rano roce, which i oele a the reone of a firt orer filter to a tationary white excitation: 1 h (17 ( x + h ( x = w ( x rf x That 1 it i in tie oain a: h ( t + h ( t w ( t = (18 r V f The guie way oel i exree in thi fro: v (19 g + vg That v i the bening of the guie way uner g the train an i eene on bea bening rigiity, a of bea an itance of two an of bea In thi anner eviation of noinal air ga i obtaine with thi equation: h h = h + v v (0 0 g 4 CONTROLLER DESIGN The araeter, c,,, are not contant ue to variou factor uch a train loa change, tolerance of reitance of agnetic uenion yte, etc an contain uncertaintie Range of uncertaintie that be coniere for each araeter i in thi for: = 05δ = δ (1 ( ( ( = 04δ c (4 = c 0δ c M ( (5 = M 1+ 05δ M r (6 f = r f 0005δ r f = 045δ (7 or = 075δ (8 V = V 05δ (9 V All δ are in the interval [-1, 1] We for uncertaintie a the atrice by uing efinition of LFT for exale for we have: 04 (0 = (1 + 04δ M = 1 0 Fig5 Uncertain a LFT Thu the linear yte interconnection will be hown in fig6 Fig6 Interconnection tructure of linear yte We for thi yte by "yic" rogra in -tool toolbox an then engage in eign of controller Three tye of controller LQG, H an are eigne for thi yte To evaluate the robutne of reulting cloe loo yte, frequency analyi for any of the will be hown in earate curvature

5 Proceeing of the Worl Congre on Engineering 007 Vol I WCE 007, July - 4, 007, Lonon, UK 5 SIMULATION RESULTS The aglev train yte ha two eaureent outut; car boy vertical acceleration an air ga between bogie an guie way an two inut; voltage an roughne inut The value of the araeter are given in table By atrice lie hown in fig4 weight of uncertaintie are coniere in yte interior tructure an a a reult the uncertaintie will be coniere out of lant in noralize for (uch a Fig1 erforance, robut tability an robut erforance tet in equence are hown in figure 7 to 10 It i clear that the yte ha only uitable noinal erforance an oe not atify robut tability an erforance characteritic ybo l value unit 500 g 500 g Fig7 boe iagra of LQG controller c 4 10 N/ 0 A π N/ weber/a a b n h 0 N t Fig8 noinal erforance tet of cloe loo yte with LQG controller R c oh r f V / ra/ Table value of oel araeter At firt, we engage in eign of LQG controller We for cloe loo yte by uing of lower LFT an then by cloe loo yte an uncertaintie a uer LFT we o ynthei For each of thee efinition: noinal an robut tability an erforance (in ection we raw relate curvature In each cae that the ea value of curvature i aller than 1 can be reulte that we reach to eire characteritic For LQG control, boe iagra of controller an noinal Fig9 robut tability tet of cloe loo yte with LQG controller

6 Proceeing of the Worl Congre on Engineering 007 Vol I WCE 007, July - 4, 007, Lonon, UK Fig10 robut erforance tet of cloe loo yte with LQG controller At next tage we engage in eign of H controller The reult are hown in figure 11 to 14 in equence lie LQG controller In thi cae the yte ha uitable noinal erforance, too but becaue of the ea value of M an ( i P ( * M equal to 1, we on t attain robut tability an erforance Fig11 boe iagra of H controller Fig1 robut tability tet of cloe loo yte with H controller Fig14 robut erforance tet of cloe loo yte with H controller At the en, we eign controller for thi yte By uing of D-K iteration roceure we will reach boe iagra of controller, noinal erforance, robut tability an robut erforance tet hown in figure 15 to 18 at thir iteration With regar to the ea value of lat three curvature are aller than 1, thi tye of controller rovie uitable noinal erforance, robut tability an robut erforance conition Fig1 noinal erforance tet of cloe loo yte with H controller Fig15 boe iagra of controller

7 Proceeing of the Worl Congre on Engineering 007 Vol I WCE 007, July - 4, 007, Lonon, UK REFERENCES: Fig16 noinal erforance tet of cloe loo yte with controller Fig17 robut tability tet of cloe loo yte with controller [1] Huiguang Dai, Dynaic Behavior of Maglev Vehicle/Guieway Syte with Control, ubitte in artial fulfillent of the requireent for the egree of Doctor of Philoohy, 005 [] Fujita M an et al, Synthei of an Electroagnetic Suenion Syte, IEEE Tran Autoatic Control, vol 40, 1995, H Control of [] Zhang F an Suyaa K, Magnetic Suenion Syte Proceeing of the t Conference on Deciion an Control, 1994, [4] Sinha PK, Electroagnetic Suenion Dynaic an Control, IEE Control Engineering Serie, 1987 [5] Muller PC, Deign of Otial State Oberve an It Alication to Maglev Vehicle Suenion Control, IFAC, Multivariable Technological Syte, 1997, [6] Gottzein E an Graer W Critical Evaluation of Multivariable Control Technique Bae on Maglev Vehicle Deign, IFAC, Multivariable Technological Syte, 1997, [7] Jong L an Boon Ch, Analyi an -bae Controller Deign for an Elecroagnetic Suenion Syte, IEEE Tran on Eucate, vol 41, no, 1998, [8] Sinha PK, Analytical an Deign Aect of Paenger Carrying Vehicle Uing Controlle DC Electroagnetic Suenion, IFAC, Multivariable Technological Syte, 1997, [9] Kitio I an Pienie T, Structure-Secifie Robut-Multivariable-Controller Deign for Practical Alication Uing Genetic Algorith, IEEE Proc Control Theory Al vol 150, no,00, 17- [10] Bala GJ an et al, Analyi an Synthei Toolbox for Ue with MATLAB, The MathWor, Inc Natic, 001 Fig18 robut erforance tet of cloe loo yte with controller 6 Concluion For aglev train yte that entione in thi aer with available ynaic equation an with tructure uncertainty a uncertainty in oel araeter, LQG an H controller on t atify the robutne ecification for tability an erforance, but controller attain uitable robutne in tability an erforance at thir iteration of D-K iteration roceure