Robust linear and non-linear control of magnetically levitated systems

R o b u st linear and n on-lin ear co n tro l o f

m a g n etica lly le v ita te d sy ste m s

T h esis S u b m itted to T h e U niversity o f W ales

for T h e D egree o f D octor o f P h ilosop h y

Alexandre Nikolov Pechev

Institute of Magnetics, Electronics and Electrical Systems

School of Engineering

Cardiff University

UMI Number: U584675

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A c k n o w le d g e m e n ts

I would like* to express my g ra titu d e to Professor P.K. Sinha, E m eritu s Professor a t the U niversity of H eading, for his financial and academ ic su p p o rt d u rin g th e course of this work. No d o u b t t h a t w ith o u t his su p p o rt this project would n ot have been possible.

I would like also to th a n k Professor H. Bolton for giving me th e o p p o rtu n ity to be a p art of C ard iff U niversity an d to bo th him and Dr. A. H addad for th e ir constructive’ com m ents and su g g estio n s on th e thesis.

I am g re a tly in d e b te d to E lena for her love, patience and u n d erstan d in g .

Last but n o t least I would like to express my g ra titu d e to my p a re n ts in B ulgaria for their love.

To m y so n D e n n is

,

m y f a m i ly a n d m y p a re n ts

A b stra ct

T he two m ost advanced applications of contactless m agnetic lev itatio n are high-speed m ag­ netic hearings and m agnetically levitated vehicles (Maglev) for ground tra n sp o rta tio n using superconducting m agnets and controlled d.c. electrom agnets. T h e repulsion force from su­ perconducting m agnets provide stable levitation w ith low d am p in g , while th e suspension force generated by electrom agnets is inherently unstable. T h is instab ility , due to the in­ verse force-distance relationship, requires the ad d itio n of feedback controllers to sustain stable suspension.

T he problem of controlling m agnetically levitated system s using d.c. electrom agnets under different o p eratin g conditions has been studied in th is thesis w ith a design process prim arily driven by experim ental results from a rep resentative single-m agnet test rig and a m ulti-m agnet vehicle. T he controller-design stages are presented in d e tail an d close rela­ tionships have been constructed between selection of perform ance c rite ria for th e derivation process and desired suspension characteristics. B oth linear an d no n lin ear stabilising com ­ pensators have been developed. Sim ulation and experim ental resu lts have been studied in parallel to assess o perational stab ility and the m ain em phasis has been given to assessing perform ance under different o p eratio n al conditions. For th e ex p erim en tal work, a new dig­ ital signal processor-based hardw are platform has been designed, b u ilt w ith interface to M atlab/Sim ulink.

T he controller design m ethods and algorithm ic work presented in th is thesis can be divided into: non-adaptive, adaptive, o p tim al linear and nonlinear. A d ap tiv e algorithm s based on model reference control have been developed to im prove th e perform ance of the suspension system in th e presence of considerable variations in ex tern al payload and force disturbances. New design m ethods for M aglev suspension have been developed using ro­ bust control theory ('Hoc, and fi—synthesis). Single- and m ulti-m agnet control problem s have been tre ated using the sam e fram ew ork. A solution to the /H 00 co n tro ller-o p tim isatio n prob­ lem has been derived and applied to M aglev control. T he sensitivity to robustness has been discussed and tools for assessing th e robustness of the closed-loop system in term s of sus­ taining stab ility and perform ance in th e presence of uncertainties in th e suspension m odel have been presented. M ultivariable controllers based on 'H00 and / i —synthesis have been developed for a lab o rato ry scale experim ental vehicle weighing 88 kg w ith four suspension magnets, and experim ental results have been derived to show su p erio rity of th e proposed design m ethods in term s of ability to deal w ith external d isturbances. T h e concept of Hoo

control has been extended to the nonlinear settin g using the concepts of energy and dissipa- tivity, and nonlinear state-feedback and outp u t-feed back controllers for M aglev have been developed and reported. Sim ulation and experim ental results have been presented to show the improved perform ance of these controllers to a tte n u a te guidew ay-induced d isturbances while m aintaining acceptable suspension qualities and larger o p eratio n al b andw idth.

C ontents

1 In tro d u c tio n 1 1.1 B a c k g r o u n d ... 1 1.2 C ontrol work m o t i v a t i o n ... 2 1.3 C ontrol work b a c k g ro u n d ... 4 1.4 Scope of this t h e s i s ... 9 1.5 Overview of th e t h e s i s ... 10

2 E lectro m a g n etic S u sp en sio n S y stem : m o d e llin g , sim u la tio n an d tra n sp u ter-based co n tro l 16 2.1 Electrom agnetic suspension m o d e l... 16

2.2 E xperim ental s y s t e m ... 18

2.3 State-feedback control for M a g l e v ... 20

2.4 T ransputer-based platform for c o n t r o l ... 23

2.5 Software environm ent for tra n sp u ters-b ased c o n t r o l ... 26

2.6 Host i n t e r f a c e ... 28

2.7 T ransputer im plem entation of th e s ta te feedback c o n t r o l l e r ... 29

3 A d a p tiv e p o le p la c em en t an d m o d e l referen ce c o n tro l o f M a g le v s y s te m s 31 3.1 On-line identification of M aglev m o d e l ... 31

3.1.1 Problem form ulation and background i n f o r m a t i o n ... 31

3.1.2 Im plem entation an d results ... 34

3.2 A daptive pole-placem ent c o n t r o l ... 36

3.2.1 O utline of im plem entation issues ... 39

3.3 Model reference ad ap tiv e control of a M aglev system ... 41

3.3.1 Im plem entation of th e ad ap tiv e a l g o r i t h m ... 43

3.3.2 R e s u lts ... 44

4 D S P en v iro n m en t for M a g lev co n tro l 48 4.1 D igital signal p r o c e s s o r s ... 48

4.2 DSP based h a r d w a r e ... 49

4.3 D SP software for Maglev c o n tr o l... 53

4.4 C ontrol algorithm s p o rted to D SP-based control fra m e w o rk ... 53

4.4.1 State-feedback c o n t r o l ... 54

4.4.3 Model reference c o n tr o l... 56

4.5 Fuzzy logic c o n t r o l ... 59

4.6 Fuzzy logic control for M a g le v ... 63

4.6.1 Fuzzy control with co nstant a c c e l e r a t i o n ... 63

4.6.2 Fuzzy controller for Maglev with fuzzy a c c e l e r a t i o n ... 69

4.6.3 Fuzzy controller using three state* v a ria b le 's ... 70

5 D esig n o f D S P hardw are for M a g lev co n tro l 7 4 5.1 Design prelim inaries ... 74

5.2 Com m ercial D SP hardw are*... 75

5.3 Hardware* d e s c r ip tio n ... 79

5.4 Software description ... 84

5.5 Software fram ew ork for control applications ... 85

5.6 C o n c lu s io n s ... 87

6 Tioo co n tro llers for M a g le v s y s te m s 88 6.1 The 'H.qq control p r o b l e m ... 88

6.2 D erivations of tra n sfe r f u n c t io n s ... 90

6.2.1 T ransfer function from w to q (Tqw) ... 90

6.2.2 S ensitivity m i n i m i s a t i o n ... 90

6.2.3 We»ighte*d sen sitiv ity m in im is a tio n ... 91

6.2.4 Mixed sen sitiv ity m in im is a tio n ... 92

6.3 A lgorithm s for co m p u tin g th e 00 norm of a system ... 94

6.3.1 S tate-sp ace m odel of P ( s )... 95

6.3.2 A lgorithm for co m p u tin g th e 00 n o r m ... 96

6.4 D eriving state-feed b ack 'H00 controllers for M a g le v ... . 99

6.4.1 C losed-loop re q u ire m e n ts ... 101

6.4.2 Selection of W ( s ) an d W c ... 102

6.4.3 S tate-feed b ack design for M a g l e v ... 104

6.5 D eriving o u tp u t-feed b ack 1~L00 controllers for M a g le v ... 109

6.5.1 C losed-loop r e q u ire m e n ts ... I l l 6.5.2 M a tla b design e x a m p l e ... I l l 6.5.3 A ssessm ent of th e l i00 d e s i g n ... 119 6 .6 E x p e rim en tal s t u d i e s ... 121 6.7 C oncluding co m m en ts ... 128 7 R o b u st a n a ly sis a n d c o n tr o l for M a g le v sy s te m s 130 7.1 P roblem definition ... 130

7.2 M aglev m odel w ith u n certain ty , G ( s )... 132

7.2.1 Sources of u n c e rta in ty in th e Maglev model ... 132

7.2.2 P erform ance of th e closed-loop M aglev system in th e presence of u n certain tie s in th e p a r a m e t e r s ... 133

7.2.3 M odelling p aram etric u n c e r t a i n t i e s ... 134

7.3 R obustness of closed-loop Maglev s y s t e m s ... 137

7.3.1 Maglev model w ith additive uncertainty ... 138

7.3.2 Nom inal Maglev model with m ultiplicative u n c e r t a i n t y ... 142

7.4 R obustness analysis using the stru c tu red singular value*-//... 145

7.4.1 R obust s t a b i l i t y ... 145

7.4.2 R obust perform ance ... 148

7.5 R obust controller design for Maglev s y s t e m s ... 152

7.5.1 Design of //-optim al controllers for M aglev system s ... 153

7.5.2 Sim ulation r e s u l t s ... 155

7.5.3 E xperim ental r e s u l t s ... 159

7.6 C o n c lu s io n s ... 161

8 M u l t i v a r i a b l e M a g le v c o n t r o l 165 8.1 M ultivariable Maglev c o n tr o l... 165

8.2 M ultivariable model of the v e h ic le ... 168

8.2.1 M odelling the m otion of th e v e h i c l e ... 168

8.2.2 S tate-space model of the 3D O F Maglev: M odel C ... 172

8.3 V alidation of the electro-m echanical p aram eters in T able 8 . 1 ... 175

8.4 R obust m ultivariable control for M aglev v e h ic le s ... 177

8.4.1 M ultivariable vehicle controller design using //-s y n th e s is ... 179

8.4.2 Selection of perform ance weights ... 179

8.4.3 Selection of th e u n certain ty bound ... 185

8.4.4 //—s y n t h e s i s ... 186

8.5 Sim ulation and experim ental r e s u l t s ... 187

8.6 M ultivariable Maglev control w ith guidance s u p p o r t ... 205

8.6.1 S tate-sp ace m odel of th e 6D O F m o d e l ... 208

8.6.2 C ontroller design an d sim ulation results ... 211

8.7 C oncluding com m ents ... 213

9 N o n l i n e a r Woo c o n t r o l f o r M a g le v 2 1 9 9.1 T he Hoo gain of nonlinear s y s te m s ... 219

9.1.1 N onlinear m odel of th e electrom agnetic s y s te m ... 219

9.1.2 D issipative dynam ical s y s t e m s ... 221

9.2 N on-linear Woo s ta te - f e e d b a c k ... 223

9.2.1 D erivation of th e c o n tro lle r... 223

9.2.2 Solution to th e H am ilton-.]acobi-Isaacs i n e q u a l i t y ... 224

9.2.3 A lgorithm for deriving non-linear state-feedback Woo controllers . . 227

9.2.4 A pplication to Maglev model ... 228

9.2.5 Sim ulation r e s u l t s ... 231

9.3 N onlinear output-feedback Woo c o n tr o lle r ... 232

9.3.1 D erivation of th e c o n tro lle r... 232

9.3.2 A lgorithm for deriving non-linear output-feedback Hoo controllers . 237

9.3.3 A pplication to Maglev s y s t e m ... 238

9.3.4 Sim ulation r e s u l t s ... 239

9.4 E xperim ental r e s u l t s ... 240

9.4.1 D SP im plem entation of the nonlinear c o n tr o lle r s ... 240

9.4.2 E xperim ental r e s u l t s ... 242

9.5 C o n c lu s io n s ... 248

10 C o n clu sio n s and futu re work 252 10.1 C o n c lu s io n s ... 252

10.2 Future research r e c o m m e n d a tio n s ... 258

A C V I en viron m en t for M a g lev co n tro l 260 B D S P -b a sed co n tro l hardw are 262 C S o lu tio n s to Hoo co n tro l p ro b lem s 279 C .l State-feedback H00 o p tim isation p r o b l e m ... 279

C.1.1 P ro o f of r e s u l t s ... 284

C.1.2 A lgorithm for deriving state-feedback Hoo control l a w s ... 287

C.2 O utput-feedback Hoo op tim isatio n p r o b le m ... 287

C.2.1 E stim ation problem s using th e Hoo design c rite ria ... 288

C .2.2 Value of th e cost function in Eqn. C.34 when x — x in Eqn. C.47 . . 293

C .2.3 A lgorithm for deriving Hoo s t a t e - e s t i m a t o r s ... 295

C .2.4 Design example: deriving Hoo s ta te e s t i m a t o r ... 296

C .2.5 O utput-feedback Hoo d e s i g n ... 298

C .2.6 A lgorithm for deriving output-feedback Hoo c o n tr o lle r s :... 300

D C o m p u tin g th e str u c tu r ed sin g u la r valu e /i 303 E R o b u stn e ss a n a ly sis 306 F M u ltiv a ria b le co n tro ller d e sig n 310 F .l Local control of M aglev vehicle using s ta te - f e e d b a c k ... 310

F.2 Hoc design for m ulti-m agnet s y s t e m s ... 310

F.3 N um erical form of the m ultivariable //- c o n tr o lle r ... 316

G M a tla b p rogram s for d e riv in g n o n lin ea r H00 sta te -fee d b a c k an d o u tp u

t-feedb ack co n tro llers 320

C hapter 1

Introduction

1.1

Background

Defying gravity by suspending a body freely in space and controlling it in all six degrees of freedom to perform some useful functions has been a focus of a tte n tio n and research for m any years [1,2]. W ith o u t a d o u b t, this technology has a considerable in d u strial p o ten tial since it offers absence of friction, wear and o th e r dynam ic effects in m oving bodies. T he two m ost advanced applications of contactless m agnetic lev itatio n are high-speed m agnetic bearings and m agnetically levitated vehicles (M aglev) for ground tra n s p o rta tio n using su­ perconducting m agnets and controlled d.c. electrom agnets [1, 2, 3]. T h e repulsion force from superconducting m agnets provide stab le lev itatio n w ith low d am p in g , while the sus­ pension force generated by electrom agnets is inherently u n stab le. T h is instability, due to th e inverse force-distance relationship, requires the ad d itio n of feedback control system s to sustain stab le suspension.

Suspension based on electrom agnetic m ethods for passenger carry in g vehicles has been known and researched since th e early 1900. These technologies, known as m agnetically levitated vehicles or Maglev, are proposed as an altern ativ e to air, au to m o tiv e and rail tra n sp o rtatio n . Various experim ental an d research work has been done, w ith m ain p ar­ ticipants G erm any, Jap a n and UK [1, 2, 3, 4, 5, 6]. High-speed M aglev system s have been design and tested w ith speeds of up to 581 k m /h (Y am anashi P refecture, Jap an , 2004), which is alm ost twice the m axim um speed of com m ercial rails cu rren tly in oper­ ation [5, 7]. T he first com m ercial tra n sp o rta tio n system based on magnetic* suspension was im plem ented for th e B irm ingham In te rn atio n al A irp o rt and has o p erated w ith high reliability from 1985 to 1995 [6]. Test tracks w ith a to ta l length of 100 miles are under active research in G erm any and Ja p a n . Recently C h in a in co llab o ratio n w ith G erm any has built the first high-speed com m ercial Maglev system to connect Shanghai w ith P udong In­

te rn a tio n a l A irp o rt [8, 9, 4, 10, 7] and to roach operatio n al speeds beyond 450 k m /h . This high-speed link, now in full op eratio n , has shown th a t Maglev can be a practical tra n s p o rta ­ tion system which can be* superior to the com m ercial tra n sp o rta tio n system s in term s of o p eratio n al cost and reliability, safety, aesthetics and en v iro n m en tal accep tab ility [11, 12].

1.2

C ontrol work m otivation

E a rn sh aw ’s theorem (1842) sta te s th a t a pole placed in a s ta tic field of force does not have a position of stab le equilibrium [2]. T h e repulsion forces betw een m ag n ets of fixed stren g th are therefore unstable and im practical for suspending a body in space. T h is inevitably requires some form of an active control to m an ip u late one or b o th fields of force* to achieve* a stab le suspension.

A full review on th e known principles of electrom agnetic m eth o d s for su p p o rtin g masses known a t present tim e can be found in [1, 2]. C u rren t research on advanced ground tra n s­ p o rta tio n is m ostly based on the following two forms: (1) electro d y n am ic (superconducting) repulsion system s and (2) electrom agnetic suspension system s w ith controlled DC electro­ m agnets. Each has proved to have p o ten tial for in d u strial ap p licatio n s [4, 5, 10, 7].

E lectrodynam ic suspension uses su p erconductors to produce* m ag n ets w ith very high flux densities. M otion between such m agnets and conducing sheets produces repulsive forces leading to suspension effects. T h is m ethod is inherently sta b le above som e critical speed (« 8 0 k m /h ) and th u s does not require active control system s for lev itatio n and guidance. T he first full-scale system w ith superconducting m agnets was b u ilt in J a p a n to su p p o rt work for applicatio n s for ground tra n sp o rta tio n . Test vehicles based on su p erconducting m agnets are under co n stan t research and developm ent w ith te st lines near Tokyo [5]. T he first com m ercial M aglev link in Ja p a n is planned to connect O saka and Tokyo and to reduce the 500 km ro u te to a one-hour-journey (cu rren tly the Shinkansen tra in takes 2h and 30 m inutes).

Suspension w ith controlled DC electro m ag n ets is by far th e m ost advanced in term s of research and developm ent for altern a tiv e ground tra n sp o rta tio n and contactless m agnetic bearings [1, 2, 13, 4, 3]. T he advantage of th is principle, in c o n tra st to electrodynam ic suspension, is its ability to provide suspension a t zero-speed. T h e system , however, is inherently u n stab le and requires a feedback control system to m a n ip u la te the forces of a t ­ tra ctio n and hence the suspension airgap (gap between the m agnet an d th e track, typically less th a n 10 m illim etres). T he nonlinear n a tu re of th e dynam ics due to the square

rela-tionship between the excitation current and the magnetic force, the open-loop instability and the necessary operational bandwidths, require a considerable analytical effort in the derivation of the control algorithms and the design of the digital hardware and supporting circuits. A quick glance of the control problem is given below.

linear motor

tracks

position sensor magnet

accelerometer

Figure 1.1: Block diagram of a magnetically levitated vehicle (suspension only). Four magnets, one in each corner, provide four suspension forces f \ to / 4, which give three degree in freedom in the chassis: pitch 0, roll <t> and heave z. A linear motor provides the fourth degree of freedom, propulsion along x.

A typical Maglev suspension vehicle system is assumed to behave as a rigid body in free space, Fig. 1.1, which is capable of linear and rotational motions along three orthogonal axes: (A", Y, Z). The main requirement from the suspension control system is to decouple the body from the guideway by suspending it freely in space and following the track- profile. Additionally, external disturbance forces, track and load irregularities have to be accommodated. To achieve this, the suspension forces have to be actively controlled. The linear motion along the X axis is typically controlled by the propulsion system and therefore five modes are controlled by the vehicle’s control system. Although the minimal requirements to control these five modes require five independent magnets, for practical reasons four electromagnets are used to provide suspension forces along Z and rotational modes along A, K, while other four electromagnets (not shown in the figure) provide guidance support.

Taking only one corner, a simplified schematically representation of each magnet is given in Fig. 1.2. Excitation current ik(t) flowing through the m agnet’s winding produces magnetic flux and thus electromagnetic suspension force f k ( ^ z , t ) which suspends the the magnet and the body toward the guideway. Under certain assumptions, the suspension

track airgap position J r sensor I <*(*)! control current accelerometer Zk(t) Zk{t)

Figure 1.2: Electromagnet suspension system: /*(z, z, t) is the suspension force for the A;-th magnet, /«* is the disturbance force acting on the vehicle, Zk(t) is the suspension airgap and

Zk(t) is the acceleration of the magnet.

force /*, k = 1..4 can be described by [1]

**(<) k = 1 ..7i

A*o 4

where Dk(t) is the magnetic flux density in the airgap, A is the magnetic pole-face area, /i0 is the vacuum permeability, N is the number of coil turns, and Zk(t) is the k-th airgap length. By controlling the excitation current, a balance can be reached where the suspension force will be equal to the gravitational and disturbance forces /<*(£) and then the body will be suspended in air in equilibrium. The suspension force, however, is proportional to the square of the the control current and inversely proportional to the square of the airgap and thus the system is highly nonlinear. In addition, the m ulti-m agnet configuration and the cross-coupling effects in the rigid body (Maglev vehicle in Fig. 1.1) would require a special control design to account for the multivariable nature of the system and to reduce undesirable dynamic effects from disturbances in the suspended rigid body.

1.3

C o n tro l w ork b ack grou n d

The first registered attem pt to control DC-electromagnets dates back in the early 1900’s, when Graeminger patented a feedback system for stable suspension [1]. The first known prototype, which wras able to levitate a mass of 156 kg at 15 mm airgap, was demonstrated in the late 1930s. With advances in solid-state devices in the mid-1960s and the use of the linear motors, actively developed and promoted by Professor Eric Laithwaite, the potential of magnetic suspension for transportation became a reality. In the past few decades, a considerable research effort has been undertaken worldwide to prove th a t magnetic levita­

tion based on controlled electrom agnets has a valuable p o ten tial for in d u stria l applications. W ith th e co n stan t research in th e area of dynam ic control and w ith th e em ergence of faster processors, th is area continues to be very a ttra c tiv e and to offer a basis for b o th applica­ tions of control-system s design an d real-tim e im plem entation. T est-rigs have been b u ilt in different universities for b o th single- and m ulti-m agnet ap p licatio n s in order to reduce the tim e delay between theoretical advances in th e control th eo ry an d p ractical engineering applications.

Some of the earlier work on M aglev vehicle design using d.c. electro m ag n ets was pi­ oneered in th e UK by th e U niversity of Sussex an d several im p o rta n t results had been rep o rted [14, 13, 15, 16]. A 1-to n research vehicle was b u ilt to stu d y th e dynam ics of coupled m agnets and control of th e m ultivariable system [15]. A d etailed analysis of the analytical and engineering asp ects of of th e design of controllers for m u lti-m ag n et vehi­ cle suspension system s has been presented in [1, 13]. It has been estab lish ed th a t the due to th e m ultiple-degree of freedom in th e vehicle, in te ra c tio n o f th e various control loops through th e dynam ics of th e vehicle is an im p o rta n t facto r in th e form ulation of the overall control strategies [13]. Two fu n d am en tal control designs were developed: (a) inte­ g rated control which uses coo rd in ate tran sfo rm atio n s an d in d ep en d en t controllers for the heave, pitch and roll com ponents an d (b) local control th a t stabilises each m agnet inde­ pendently. R esults showed th a t in teg rate d control poses less overshoot in th e response and faster tra n sien t responses. T h e analysis of th e ride an d track -clearan ce characteristics of an electrom agnetic suspension system travelling along a guidew ay w ith ran d o m roughness has been analysed in [1, 16] to show th a t low-speed m agnetically suspended vehicles (70 k m /h ) can travel w ith o u t th e aid of a secondary suspension. T h e tra ck roughness and dis­ tu rbances, however, have a considerable influence on th e ride an d s ta b ility ch aracteristics of a ttra c tio n -ty p e of suspension system s due to th e track-clearance requirem ents. Special atte n tio n has to be th u s paid on th e control-design asp ects to a tte n u a te these undesired effects on th e suspended body.

T h e problem of designing m agnetic suspension system s for higher speeds (500 k m /h ) and th e m odelling of th e ir dynam ics is discussed in [17]. T ogether w ith ex tern al d istu r­ bances such as wind force and track tolerances, early work a t M BB paid som e a tte n tio n to u ncertainties in th e m odel (referred as system d istu rb an ces) such as in stru m en tatio n errors, accelerom eter biases and gap and cu rren t m easurem ents errors. An e stim a to r for th e track -d istu rb an ce is included in th e control system design b u t th e u n certain ties in the

model are not accounted for. T he Linear O ptim al Control synthesis technique has been used to design a m ultivariable com pensator for the vehicle. Some prelim inary results from digital im plem entation of the controllers have been presented. T his work was extended in [18] and a new model using finite-element m ethods has been designed to account for elas­ ticity in the chassis, which is difficult to deal w ith using rigid-body m odelling techniques. These results showed th a t several resonance frequencies are presented due to th e elasticity in the body. T he controllers have observer-based stru c tu re s and use in p u ts from either current and position or acceleration and position to generate a feedback. It has been noted th a t the observers are, however, sensitive to uncertainty in th e m odel and th u s a careful account for the selection of their param eters has to be taken to avoid in stab ility in the system . An im p o rtan t conclusion has been draw n th a t eddy c u rren t losses are produced due to the high velocity in th e vehicle and this degrades th e perform ance [1]. T h e digital control work in [18] has been carried out on a Honeywell d ig ital co m p u ter w ith a m axi­ mum control-sam pling frequency of 100Hz. T he sam pling frequency has been increased to 18 KHz in [19] to im plem ent a digital state-feedback control system for a single-m agnet electrom agnetic suspension system . C om parison is m ade w ith an analogue controller to show the benefits from a flexibility and quality prospective to em u late th e real-tim e per­ form ance of the classical op-am p based controller. D igital controllers provide a basis for im plem enting different controllers in software w ithout a h ardw are rework.

A one-dim ensional Maglev model which considers only th e heave response has been presented in [20]. A secondary suspension is included and m odelled as a spring-dam per system . A set of inequalities are solved to get a com pensator w ith two in p u ts (position and acceleration) and one o u tp u t (control voltage) to m aintain th e position error, the acceleration and the drive o u tp u t w ithin pre-defined lim its. Sim ulation resu lts are presented to show airgap response due to guideway irregularities.

A sm all test vehicle (88 kg) running along a 5 m track using a linear m o to r was designed in [21]. T he vehicle chassis is equipped w ith four m agnets for suspension, while th e guidance is provided from the inherent lateral stiffness. T ransputer-based hardw are was designed w ith ap p ro p riate in p u ts for m agnet transducers. Four sw itched-m ode cu rren t controllers were im plem ented to drive the m agnets providing up-to 10 A of cu rren t. Each m agnet was equipped w ith a position and an acceleration sensors. Using th is configuration, a new force control configuration was proposed and designed in [21] which employed a detailed electrom agnetic m odel to provide a linear force actu atio n . In [21] th e control strateg y was

firstly im plem ented on a single-m agnet test-rig and then grouped to th ree independent controllers for the three m odes (heave, pitch and roll) to control th e suspension forces of th e vehicle. T h e linear force actu atio n of this m ethod is considered as an advantage, but low accuracy in the air-gap m easurem ent a t small airgaps was observed [21]. T h e control work was im plem ented in the digital dom ain using the on-board floating-point tra n sp u te rs an d sam pling frequencies of 500Hz. T he m echanical chassis, sensors an d power am plifiers from [21] are used in this thesis.

/ / —Synthesis of a single-m agnet electrom agnetic suspension system has been reported in [22]. A pool of uncertainty where the system is assum ed to reside has been m odelled by a complex uncertainty m odel. T he controller synthesised using a m echanism which iterates between Hoo design and //-synthesis (D K -iteration) provides ro b u st stab ility and perform ance for the selected range of m odel p e rtu rb atio n s. T h is work has been extended in [23] by adding robust analysis in term s of linearision errors, p a ra m e tric u n certain ties and neglected dynam ics. A m ixed sensitivity problem is solved com bining real and com plex uncertainties. T he system perform ance is analysed using ex p erim ental results from a single­ m agnet system and step responses in the force disturbance.

A frequency-shaping linear q u a d ratic (LQ) controller for a single-m agnet suspension system has been rep o rted in [24]. Stochastic and determ in istic d istu rb a n c e in p u ts are targ eted w ith th e aim of frequency-dom ain weightings used in th e perform ance criterion integral for th e LQ o p tim isation. R esults from a single-m agnet suspension system are presented to show th a t th e frequency-shaping LQ m ethod is b e tte r th a n th e classical LQ m eth o d for determ in istic in p u ts. T h e design co n strain ts are chosen to accom m odate good ride qualities by lim iting acceleration and position variation levels.

Linearising control for a single-m agnet electrom agnetic suspension system using observer- based s tru c tu re was rep o rted in [25]. Using a su itab le coordinate tran sfo rm atio n , th e non­ linear dynam ics is linearised to get a system w ith th ree sta te variables: position, velocity and acceleration. Two weakly coupled R iccati equations are solved to derive th e feed­ back and th e observation gains. T he sta b ility is analysed using Lyapunov techniques. T he sim ulation results presented in [25] show th a t th e linearising controller has a superior perfor­ m ance in term s of coping w ith p aram eter variations com pared to th e linear observer-based controller. Sim ilar work on th e design of a nonlinear feedback linearising controller for a single-m agnet suspension system is presented in [26]. E xperim ental results are presented from a te st rig to show th a t th e proposed nonlinear controller is ro b u st ag ain st p e rtu rb

tion in th e suspended m ass and external force disturbance com pared to the classical linear state-feedback controller. T he control work in [26] was im plem ented on a TM S320C31 DSP w ith 3 kHz sam pling rate. Feedback signals are the m ag n et’s cu rren t, the position and the* acceleration levels. Experim ental results from a single-m agnet system using a linearising controller are also reported in [27]. Special atten tio n is given to th e fact th a t although excellent results are obtained from the design process, a very a ccu rate m odel is required to im plem ent th e coordinate transform ation and hence the controller. R obust properties in the design of linearising nonlinear controllers for a single-m agnet system was also presented in [28]. P aram etric uncertainties are incorporated into th e design to g u aran tee robustness. Experim ental results are included to show th a t the nonlinear ro b u st controller m anages to track b e tte r large variations in the desired airgap com pared to a nom inal linear controller. This work has been extended in [29] by adding ad ap tiv e features to th e linearising controller to cope with p aram etric u ncertainty in the design. A single-m agnet system is used for the experim ental analysis.

An Hoc loop-shaping design for a single m agnet system w ith a secondary suspension is presented in [30]. A mixed sensitivity problem is solved num erically using an evolution­ ary algorithm . Sim ulation responses were presented to show th e air-gap response and the acceleration levels for a chosen reference in p u t. A detailed account was given of the nu­ m erical solution and a com parison is m ade w ith other genetic algorithm s. T he advantage in th e developed approach was considered to be the ability to use a m ix tu re of discrete and continuous p aram eters in the problem form ulation. T he design of an H00 controller for a m agnetically suspended vehicle w ith four m agnets is also rep o rted in [31]. T he au th o rs had chosen to neglect th e cross-coupling in th e rigid body and to design four Hoo com pensators stabilising each corner independently. E xperim ental results from a 100kg vehicle were in­ cluded to show th a t th e Hoo controllers give b e tte r dynam ic responses to disturbances applied to one of th e corners of th e vehicle. S im ulation results from a sw itched-m ode H0o controller-design w ith gain scheduling is presented in [32]. A single-m agnet system is con­ sidered where the a u th o rs assum e a scenario w ith a very large variation in the o p erating airgap. To cope w ith this, a gain-scheduling m echanism was used in com bination w ith an

Hqo design. Two com pensators are alte rn a te d w ith a threshold m onitor point on the o p erat­

ing airgap. T he sim ulation results were com pared w ith the linearising controller developed in [26] to show th a t the gain-scheduling controller achieved b e tte r tracking perform ance in th e presence of p aram eter p ertu rb atio n .

Sine recently more exotic applications of m agnetic levitation have been rep o rted such as th e M agnetic Launch Assist, which is a m agnetically levitated vehicle w ith propulsion to provide an initial velocity by using electrical power from ground source for launching space sh u ttle s into space [33, 34]. Maglev is proposed as a su itab le concept in reducing costs for fu tu re space explorations [33].

1.4

Scope o f this thesis

A lthough several results for the control of single-m agnet suspension system s have been reported, little account has been given to th e developm ent of a d etailed design framework for the control of Maglev vehicles since the results presented in [17, 18, 15, 1, 21]. In this respect, the research work presented in th is thesis has aim ed to reduce th e gap between recent advances in control theory and th eir engineering ap p licatio n s to M aglev. T he prob­ lem of controlling m agnetically levitated system s using DC electro m ag n ets under different o perating conditions has been studied w ith a design process p rim arily driven by experim en­ tal results. In previous research work, independent suspension controllers for each m ode of the vehicle (heave, pitch and roll) have been designed using force control algorithm s em ploying a detailed model of th e electrom agnet in conjunction w ith cu rren t and airgap feedbacks [21]. It was noted in [21] th a t th e quality of th e proposed control stra te g y de­ pends on th e accuracy of th e m odel an d th e airgap m easurem ents a t sm all air-gaps. T he research work u n d ertak en in th is thesis has aim ed to continue th e developm ents in [21] by em ploying o p tim al control theory to the design of Maglev controllers. New tools for the vehicle control have been developed to provide a basis for deriving feedback com pensators by m inim ising cost functions using o p tim isatio n . W ith th is approach, single- and m u lti­ m agnet control problem s have been tre a te d using th e sam e framework. T he design process has aim ed to provide convenient tools for assessing th e robustness of th e closed-loop sys­ tem in term s of su stain in g stab ility and perform ance in the presence of u n certain ty in the suspension m odel. T h e research work has been p rim arily driven from experim ental results from a single m agnet test rig and a sm all m ulti-m ag n et vehicle (88 kg) equipped with position and acceleration sensors an d power am plifiers [21]. T he controller-design stages are presented in detail and close relationships have been constructed between selection of perform ance crite ria for th e derivation process and desired suspension characteristics. Both linear and nonlinear com pensators have been developed. Sim ulation and experim ental re­ sults have been studied in parallel to assess operatio n al stab ility an d th e m ain em phasis

has been given to assessing perform ance under different o p eratio n al conditions. To su p p o rt the ex p erim ental work, a new digital signal processor-based hardw are platform has been developed to m eet constraints from the com putational and o p eratio n al b andw idths. Some of th e m ain areas of work undertaken during this research p roject arc* listed below.

• A daptive algorithm s based on model reference control have* been developed to improve* the perform ance of the suspension system in the presence of considerable variations in external payload and force disturbance's [35].

• Design of custom -built DSP hardw are and corresponding softw are libraries have been developed to control a three-degrees-of-freedom M aglev vehicle (8 8 kg w ith four m ag­ nets) and logging in real-tim e and delivering experim ental d a ta to a host com puter for analysis (M atlab /S im u lin k ). A software control fram ew ork for D SP has been de­ veloped which is fully custom isable to provide the su p p o rtin g ex p erim en tal results presented in this thesis.

• Recent advances in robust control theory (H<x, and //—synthesis) have been applied to b o th single- an d m ulti-m agnet suspension system s. An a lte rn a tiv e solution to the Hoo controller-optim isation problem has been derived and applied to M aglev control. Sensitivity to robustness has been discussed. M ultivariable controllers based on Hoo and / i —synthesis have been developed for an 88 kg vehicle an d experim ental results are derived to show superiority in term s of ride qualities, acceleration levels and ab ility to deal w ith guidance- and track-induced disturbances.

• T h e concept of Hoo has been extended to th e nonlinear settin g using th e concepts of energy and dissip ativ ity an d nonlinear state-feedback and output-feedback controllers for M aglev have been developed for a first tim e [36]. Sim ulation and experim ental results have been presented to show th e superior perform ance of these controllers to a tte n u a te guidew ay-induced d istu rb an ces while m aintaining acceptable ride qualities and larger operatio n al ban d w id th .

1.5

O verview o f th e th esis

C h a p ter 2: E lectro m a g n etic S u sp en sion System : m o d ellin g , sim u lation and tra n sp u ter-b a sed con trol sy stem

In this chapter, the electrom agnetic suspension dynam ic model of a single-m agnet system is developed. T he nonlinear relationship between the force of a ttra c tio n , the control current and th e distance, is linearised around a nom inal operating point to derive a state-space m odel of the suspension system. Its param eters are m atched w ith those of a single-m agnet experim ental test-rig. To modify the force-current relationship, a state-feedback controller is derived. The suspension characteristics are analysed experim entally using a network of three transputers (two 32-bit floating point units and one 16-bit integer unit) which was previously built in [21]. Experim ental results in changing the desired reference airgap are com pared with sim ulation results. Some im plem entation issues of th e real-tim e control im plem entation are discussed.

C hapter 3: A d a p tiv e p o le-p la cem en t and m od el reference con trol o f M aglev sy stem s

For Maglev system s which are under th e influence of external force and mass disturbances, a m ethodology of checking the stability properties of the closed-loop Maglev system is developed using real-tim e identification based on recurrent least-squares algorithm s. An identification loop running in parallel w ith the m ain suspension controller is developed to m onitor the location of the closed-loop poles of the experim ental system and hence the force-airgap relationship. Analysis has shown th a t external force and mass disturbances can be m odelled as a variation in the dynam ic characteristics of the suspension system. To provide a framework for coping w ith external disturbances and uncertainties, two adaptive algorithm s for Maglev control are developed: (a) adaptive-pole placem ent control and (b) adaptive m odel reference control. The adaptive pole-placem ent controller is derived using a recurrent least-square error-m inim isation algorithm . T he Diophantine equation, which defines an error m easure between the desired and current location of the closed-loop poles, is m inim ised to ad ap t in real-tim e the gains of the state-feedback controller. The adaptive model-reference controller is derived by m inim ising analytically a cost function constructed from the error between the theoretically derived (using the model from C h ap ter 2) and the experim entally obtained state-variables (position, velocity and acceleration) and their error rate. T he result is a mechanism for m odifying the controller’s gains in real-tim e to m aintain the cost-function defined above m inim al. E xperim ental responses in coping w ith external mass and force disturbances are presented to show the benefits of this controller com pared

to the ordinary state-feedback. Some constraints from the real-tim e im plem entation of the ad ap tiv e pole-placem ent controller on the tra n sp u ter hardw are are discussed.

C h a p ter 4: D S P environ m en t for M aglev control

T he event-driven nature of the tra n sp u te r architecture does not allow running real-tim e applications with a fixed sam pling tim e and some jitte r in the clock was observed during the controller im plem entation. A dditionally, advanced control algorithm s for Maglev put additional constraint on the signal-processing bandw idth. Because in the m iddle 1995’s transputers were discontinued from m anufacture, a new processor for th e control work was needed. A suitable choice for real-tim e Maglev control was found to be the Analog Devices SHARC family of D SPs offering 40 M IPS and 80 M FLO PS sustained processing power and an in terru p t driven architecture capable of delivering fixed sam pling tim e. To explore com patibility of these D SPs for real-tim e Maglev control, this chapter develops a single­ m agnet D SP control hardw are using a commercial EZ-K IT Light D SP hardw are and a custom -build A D C /D A C interface for m agnets. T he state-feedback controller developed in C h ap ter 2 was firstly p orted to th e new hardw are, followed by th e adaptive pole-placem ent controller and the adaptive m odel-reference controller from C h ap ter 3. The sam pling of the com putationally intensive adaptive pole-placem ent controller was considerably reduced from 950 /xs on th e tra n sp u ters to 200 /xs on th e DSP hardw are and successful experim ental results are obtained in a tten u atin g 120N force disturbance. T he adaptive model-reference controller was also p orted to the D SP hardw are and new experim ental responses w ith

200 /xs sam pling tim e were analysed in atte n u a tin g external force and mass disturbances. An account is also given of selecting tu n in g param eters for the ad ap tatio n rate versus tra n sien t response in the disturbance atten u atio n . A pplications of Fuzzy-Logic control for electrom agnetic suspension system s are also discussed and three different Fuzzy controllers are derived using position, acceleration and velocity feedback from the suspended m agnet. Experim ental results in dealing w ith reference dem and change are analysed and some of the key aspects of im plem enting Fuzzy-Logic controllers using DSP are discussed.

C h ap ter 5: D esig n o f D S P hardw are for M a g lev control

The prelim inary control work carried out in C h ap ter 4 using th e SHARC fam ily of DSPs has shown th a t th is processor provides enough processing bandw idth to im plem ent com pu­ tatio n ally intensive control tasks. T he m ultiprocessing capabilities, Super H arvard Archi­ tectu re w ith em bedded program and d a ta m em ory on-the-chip and 80 M FLO PS sustained

processing ban d w id th m ade the SHARC DSP a suitable candidate for the m ulti-m agnet control work. Also (mid 1999) no commercial hardw are was capable of fulfilling the re­ quirem ents for the m ulti-m agnet control work. A new DSP board hence was built to offer: a direct interface to four m agnets (inputs from eight transducer and four o u tp u ts to current controllers), a fast and reliable E th ern et interface, a custom isable digital interface, up-to 32 M bit SRAM memory on-board and facilities for m ultiprocessing. H ardw are design aspects and design of software libraries, such as kernel, T C P /IP interface, interfaces to M atlab and Simulink, are addressed in this chapter. This hardw are is used for all experim ental work described in the rem aining chapters of this thesis.

C hapter 6: Hoo con trollers for M aglev sy stem s

In this chapter a detailed account of the derivation and im plem entation of Hoo state- feedback and output-feedback controllers for Maglev system s is given. T he m ixed-sensitivity optim isation settin g is introduced and details of the selection of perform ance weights for single-m agnet electrom agnetic suspension system s to satisfy pre-defined ride and perfor­ m ance qualities are given. A lthough algorithm s for deriving Hoo controllers are readily available, in this chapter an alternative analytical solution is developed using Lagrange m ultiplier m ethods and differential game theories. The m otivation for this work is based on its analogy w ith the classical LQG solution which is well understood. Despite the fundam ental difference in the derivation steps, th e analytical solution for the Hoo control problem produces identical results as reported in [37]. A full account of the derivation of

Hoo controllers for Maglev is given and the correspondence between perform ance weights and desired suspension characteristics. Sim ulation and experim ental results are presented to highlight th e fact th a t the suspension stiffness and the dam ping are well controlled with the new Hoo controllers.

C h ap ter 7: R o b u st an alysis and con trol for M a g lev sy stem s

U ncertainties in the linearised Maglev m odel used for the controller design arise from changes in op eratin g conditions due to the nonlinear force-current relationship and external force and payload disturbances. Up to some degree, the electrical param eters of the m agnet are also considered as uncertain. For a given bounded variation in the M aglev’s param eters, it has been established th a t the Hoo controller developed in C h ap ter 6 for the nominal model fails to provide robust stability and perform ance for variations in the param eters above 10% from the nom inal values. A m easure based on singular values has been derived

to tost Maglev controllers against uncertainty in the model. The test for robustness has been extended by employing the definition of the stru ctu red singular value //. It has been established analytically th a t the robustness (both stab ility and perform ance) is more sensitive to variations in the operation condition and external payload and less sensitive to v ariations in the electrical param eters. To provide a framework for robust Maglev design, th e concepts of //—synthesis based on D K -iteration are used to derive robust controllers using the performance requirem ents derived in C h ap ter 6. A selection of sim ulation and experim ental results are presented to show the robust properties of the new controller in term s of sustaining perform ance in the presence of large variations in the model.

C hapter 8: M u ltivariab le M a g lev con trol

This chapter develops a design framework for the m ulti-m agnet vehicle controller using the developments of W 00 and //—synthesis from C hapters 6 and 7. A three degree of freedom (3- D OF) state-space m odel of a representative suspension vehicle (four m agnets) is developed, which is com patible for W 00 design. Experim ental results from a 88 kg test rig are used to validate th e model. A robust m ultivariable controller w ith seven inputs (four airgap m easurem ents from each m agnet and reference pitch, roll and heave) and four o u tp u ts (control signals to m agnet amplifiers) is derived using the //—synthesis algorithm s from C h ap ter 7. An account of the selection of perform ance weights for achieving predefined ride qualities is given and a large selection of sim ulation and experim ental results are given. These results show th a t the new m ultivariable controller is capable of sustaining good suspension qualities and, contrary to ordinary state-feedback controllers stabilising each corner independently, is also capable of m aintaining robust stability in atten u atin g guidance induced disturbances. T he design framework is extended to a 6-D O F model of a vehicle w ith facilities for active guidance control. Sim ulation results are used in the 6-D O F analysis of the suspension and guidance qualities.

C h ap ter 9: N on lin ear W ^ con trol for M aglev

The concept of local dissipativity and supply power are used in this chapter to develop a design framework for the derivation of nonlinear Woo controllers. Using the values supplied and stored energy as criteria, th e controller is designed to keep th e energy of a penalty vector bounded and sm aller th an the energy of the disturbance input. T he controller derivation requires finding a solution to H am iltonian-Jacobi-Isaacs inequalities. W hile the analytical solution to these inequalities is not readily available, an approxim ate solution is

found by deriving sequential term s in its Taylor series expansion. T hus instead of deriving one single nonlinear controller, the algorithm developed in this chapter derives a whole class of Maglev controllers by collecting the appropriate term s in the series expansion. T h is m ethodology is used in both the derivation of a nonlinear state-feedback controller and a nonlinear estim ator in conjunction w ith the nonlinear state-feedback controller. The derivations of the controller are perform ed analytically. T he perform ance of the suspension system w ith the nonlinear Hoo controller is analysed by studying th e atten u atio n properties of the closed-loop system in atte n u atin g track disturbances. The nonlinear output-feedback controller has been observed to provide a significant im provem ent in term s of dealing with track irregularities over the linear state-feedback controller.

C hapter 10: C on clud in g co m m en ts

Concluding com m ents w ith future research recom m endations are included in this chapter.

C h a p te r 2

E le c tr o m a g n e tic S u sp e n sio n S y stem :

m o d ellin g , sim u la tio n an d

tr a n sp u te r -b a se d co n tro l

2.1

E le c tr o m a g n e tic su sp e n s io n m o d e l

Electromagnets with d.c excitation have the ability to a ttra c t ferromagnetic m aterials with the force of attraction being controllable. As a device, an electromagnet consists of two poles and a magnetisation winding. Excitation current i(t) flowing through the m agnet’s winding produces magnetic flux (</>m(£)) and thus electromagnetic suspension force F(t)

which suspends the the magnet and the body toward the guideway as shown in Fig. 2.1. There also a leakage flux (f>i, (t) which flows from one magnet pole to other and entirely depends on the m agnet’s shape, m aterial and design. Usually it is desirable to minimise this by suitable m agnet design [1].

F ( i , z , t ) f z(t) position Jr sensor i\ z(t) control current accelerometer fd(t)

Figure 2.1: Electromagnet configuration [1]

W ith the assumption th at for the iron track /jlt = oo, then the following equation

describes the force of attraction [1]

B 2A fi0N 2A i(t)

z(t) (2.1)

Ho 4

where: B is the airgap flux density, A is the pole face area, Mo is the perm eability of free space (4 e l0“ 7 h /m ), N is the num ber of the coil turns, i(t) is the coil current, and z(t)

is the airgap. This force of a ttra c tio n is a non-linear function of the current i(t) and the airgap z(t). If R is the to tal resistance of the circuit, for an instantaneous voltage v(t)

across the m agnet winding

v(t) = R i( t) + — [L(i,z)\

where L(i, z) is the inductance of the m agnet winding, given by [1]

L ( i , z ) = Mo N 2A »(*)

z(t)_

(2.2)

(2.3)

For vertical force balance of the system in Fig. 2.1

d?z(t)

m-dt2 = —F(i, z) + f d(t) + m g (2.4)

where fd is a disturbance force input. For the equilibrium point («o, Zq)

m g = F0(i0, z 0) = Mo N 2A i_o_

LZ0 (2.5)

S u b stitu tin g Eqn.2.3 into 2.2 gives the following equations for the electrical and mechanical ch aracteristics for suspension dynam ics

di(t) dt i(t) m z(t) d2z(t) dt2 dz(t) dt _

r

" 2 i(t) + f i ( t ) + mg (2.6) (2.7)

where T = v°N^ A is a characteristic feature of the suspension m agnet. Solving num erically Eqns. 2.6 and 2.7, th e exact dynam ics of the system m ay be obtained.

By linearising th e non-linear equations (2.6 and 2.7) around a nom inal operating point (Mb zq) th e linear model for the suspension dynam ics is given by

d Ai (t ) dt i_o I Z 0 1 m d?Az(t) _ dF(i , z) dt dz d ? A z ( t) d A z ( t ) dt (*o,zo) ^ z { t ) R Lq\ Ai{ t) + d F( i, z) 1 1 A f(^) di LT0J (io,zo) T Afd(t) m dt kzA z ( t ) - ki Ai(t) + A f d{t) 17 (2.8) (2.9) (2.10)

where

if

k z = T — ; ki = T Lq = — ; kf = kzL 0 and mg = — ^

L 2 0 J a t («o, Zq)

Zq z q 1 2

By co n stru ctin g a vector using Az(£), A i(£), and A i(t) and using Eqns. 2.9 and 2.10, the state-space model of the suspension system can be derived

A z(t) A2(£) = A i(t) 0 hz. m 0 1 0 0 A* _ i ki Lq A z(t) A z(t) A?(£) + 0 0 i L Lq 0 J_ m 0 A v(t) fd{t) (2.1 1)

Choosing the voltage signal Au(£) as the inp u t and the air-gap between the m agnet and the track z(t) as the o u tp u t, the block diagram of the system in tim e-dom ain is as shown in Fig. 2.2. T he corresponding transfer function becomes

ki_

(2.12) G(a) = A z(s) _{c.3 | R r.2} mLp _{kz R}

* ^ L0 * m Lq

Eqn. 2.12 is the linearised single-input-single-output model of the non-linear m agnetic suspension system in Fig. 2.1. Since ki and k z depend on the linearisation of the system around the nom inal operating point, the dynam ics of the system are dependent on the choice of this point. If a considerable variation in the choice of (z0, z0) is expected, the controller has to com pensate for this. This is analysed in subsequent chapters. Eqn. 2.12 is used for all designs in the following chapters. In C h ap ter 8, this m odel is extended to describe force and torque relationships in m ulti-m agnet vehicles.

Due to th e lack of dam ping in the m echanical dynam ics (Eqn. 2.10), the closed-loop system has one positive pole and hence the system is inherently unstable in open loop. A feedback com pensator is therefore required to stabilise the m agnet underneath the track at a specified airgap. In this respect, the m ain em phasis in this thesis is the development of control algorithm s for electrom agnetic suspension system s working under different o perat­ ing conditions. T he experim ental work is carried out on single m agnet representative rig and a m ulti-m agnet vehicle. The single-m agnet rig is described in the following section. D etails of th e m ulti-m agnet vehicle are given in In C hapter 8, page 165.

2.2

Experim ental system

The experim ental system consists of an electrom agnet, an accelerom eter, and an airgap sensor (Fig. 2.3). T he cantilever m agnet and the track are supported by a base. The m echanical com ponents are designed to enable the m agnet to move freely in the vertical

z(t)

z(t) z(t)

m

Figure 2.2: Block diagram based on the linearised sta te equations.

direction (closer to the track and away from the track). The m echanical dimensions have been worked out such th a t a t the nom inal operating point, th e m agnet is parallel to the track. TRACK M AGNET POSITION SENSOR ACCELEROMETER

Figure 2.3: E xperim ental single-m agnet system . Photographic image of the test rig is shown in Fig. 2.4.

T h e m agnet used in this experim ental system is an E-shaped m agnet w ith lam inated core w ith dim ensions shown in Fig. 2.5 [1]. It is designed to work around th e equilibrium point (iq=2 A, 2:0=2.5 m m ), and can be used w ith current up to 10 Amps. The constants of th e m agnet a t th e nom inal o perating point are:

R = 1.1ft

Lq = 3.0 mH

ki = 12.61 N /A

k z = 6305.3 N /m

m = 1.8 kg

T he experim ental system is designed to work w ith airgap distances between 0.5 and 9 mm. T he transducer used to m easure the airgap (Az(t)) is an inductive non-contacting transducer m anufactured by Pepperl & Fuchs model IA8-M1K-I3. T he o u tp u t is a current signal linearly proportional to the distance (0-20mA, 3% error). T he m easurem ent accuracy

Figure 2.4: Photographic image of the magnet-track configuration.

of this sensor is given to be less than 30mm in the full +10° to 4-40° C tem perature range. To construct the full state-space model, the acceleration of the magnet is required. For this purpose an accelerometer is used with voltage output proportional to accelerations in the range of ± 5 g (± 5 0 ra /s2) (ICSensors, type 3110-005). The full-scale voltage output is ± 2 volts, with a 2.5 volt offset. The accelerometer generates internally a 2.5 volts reference for the conversion circuits. The accuracy is ±1% over the full —10° to 4-40° C tem perature range.

Figure 2.5: The E-core magnet dimensions [mm], left: side view; right: front view [1]

2.3

S ta te -fe e d b a c k co n tro l for M a g lev

Substituting the param eters for the experimental m agnet shown in Fig. 2.3, the numerical form of Eqn. 2.12 becomes

Az(s) _ 2335.3

^ ~ Av(s) ~ s34- 366.67s2 - 1.284 x 106 20

As discussed earlier, the system has three real poles (p i = -356.56, p2 = - 65.28 and p3 = 55.17) w ith one located in the right-hand-side of the s-planc. To keep the magnet, stable and close to a pro-specified nominal point beneath the track, a linear state-feedbaek is derived below.

To stabilise the Maglev system, all right-hand s-plane poles have to be relocated on tin* left-half of the s-plane, the precise' locations being related to the given set of elosed- loop requirem ents. Tin* linear sta te feedback control law is aim ed at deriving the feedback gains of the three state variables such th a t the closed-loop poles are moved to a set of pre-specified locations. The open-loop state-space model of the system is given by

x(i) = A x (i) + Bu(i)

y(t) = Cx(<)

whore for the Maglev system , x G R Sxl, A € R 3x3, B G R 3 x l, C G R lx3, « is the in p u t to the system and y is the o u tp u t . The state variable x here has three elements corresponding to the airgap, velocity and acceleration (current being related to force or acceleration). The dynam ics of the system in sta te space form is fully determ ined by the eigenvalues of the m atrix A, or the roots of the characteristic equation [38]

dot (si - A) = 0 (2.14)

Introducing a control signal in the form:

u(t) = v(t) — K x(t) (2-15)

where v = Azp(t) is the desired o u tp u t (position) from the system , and K is a vector containing the feedback gains, the closed-loop state space representation becomes

x(f) = Ax(t) + B(v(t) - Kx(f))

= (A - B K )x(f) + Bv(f) ^ j

By choosing different values for K , the eigenvalues of this closed-loop system may be explicitly determ ined to make the system stable and satisfy given closed-loop perform ance requirem ents. Com bining the state-space model in Eqn. 2.12 of the experim ental system w ith the sta te feedback controller (for the linear model in Fig. 2.2):

Az R(t) - Az{t)

Av(t) = [kp k v k A\ Az(t)

A i(t)

= k p A z p ( t ) — k p A z ( t ) + k y A z ( t ) + k AAi ( t )

th e closed-loop suspension system is transform ed to

(2.17) A z ( t ) ' 0 1 0 ‘ Az(t) ' ' 0 ' A z(t) = k z_m 0 _ bi. rn A z(t) + 0 Ai(t) _ ( k P | k Ak z \ L V I'O ^ m L 0 ) (k y 1 kz_\ \ L 0 ^ k j ( R , k t k A \ \ L 0 m L 0 ) . _ Ai{t) _ k p L L0 J A zp(t) (2.18) 21

T he input to the system is the desired position value (distance between the m agnet and the track v = zT,.j = Ac/e(/,)) with the sta te variable's being the position, the velocity and the acceleration signals. The block diagram of the closed-loop system (Eqn. 2.17 added to Fig. 2.2) is shown in Fig. 2.6. T he characteristic polynomial of the closed-loop is [1]

k.

Xx)A J

% kt

+ - A+

Figure 2.6: Linearised model with linear sta te feedback controller.

d ct(sl - (A - B K )) = s3 + ( ^ - + k’kr \ L q TTlLo

* , ki kv , 1

s s _|_ _{(kikp - k z R) (2.19)}

m L o rnLo

Using the m a g n e t’s param eters from Section 2.2, and feedback controller’s gains speci­ fied by experim entally derived values: k p =20833.0, k y —250.0, and ka = 4.0, the closed-loop system is successfully stabilised. A sim ulation of the m agnet system w ith a state feedback controller was im plem ented in MATLAB. Two step inputs in changing the desired airgap Azref(t) were applied at tim es shown as arrows in Fig. 2.7. The responses of the sim ulated magnet indicate a rise tim e in the order of 40ms.

A key feature in the design is the choice of the nominal operating point such th a t the param eters (kt, k z and L 0) rem ain unchanged for the whole of the operating ranges for the current and the airgap. An analysis of robustness and the effects of variations in the operating conditions is analysed in C h ap ter 7. If (_\ 7_'■'() ^- + _{TTIIjQ /} > > 1, the closed-loop system can be modelled as a seco