Control of an Electromagnetic Vehicle Suspension

University of Reading

Control of an Electromagnetic

Vehicle Suspension

A thesis submitted for the degree of Doctor of Philosophy

by

Neil Stuart McLagan

Department of Engineering

June 1992

Control of an Electromagnetic

Vehicle Suspension

i

Control of an electromagnetic vehicle suspension

This dissertation describes the analysis of an electromagnetic vehicle

suspension and the proposal and synthesis of a sophisticated suspension

control system. The difficulties associated with the control of

electromagnetic suspension providing both primary and secondary

suspension functionality are first discussed in the light of existing

research and development progress. The strengths and weaknesses of

existing control techniques are then identified, and a structured control

strategy is proposed. This in turn involves a new, nonlinear

electromagnet force control algorithm which employs air gap and

current feedback, and a sophisticated suspension control algorithm

which consists of an absolute position controller, with a position

reference supplied by a guideway following algorithm. Three feedback

states are measured for each electromagnet, namely air gap, current,

and acceleration. The suspension control algorithm is applied to the

vehicle heave, pitch, roll and torsion motions independently. The

resultant electromagnet force demands are fed into force controllers

which provide dominantly linear and independent electromagnet force

actuation. An experimental control system is developed using

transputers, and the control algorithms are implemented using the occam

parallel programming language. The proposed control theory is

validated by presenting both simulation results, and responses from the

experimental system. The results clearly show the efficacy of the

proposed control method.

ii

Acknowledgements

I am very grateful to my supervisor, Pradip Sinha, for igniting my interest in the

problems of controlling electromagnetic suspensions, and also for giving me the

opportunity to carry out the research described in this dissertation.

I would like to thank my friends and family for their encouragement and support. I am

particularly indebted to Greg Pye, Nigel Kneebone and Jeremy Hinton for the many

hours spent discussing various issues and reviewing this dissertation. Last, but not

least, I would like to thank Gail Tucker for her patience, tolerance and help.

The research and equipment described in this dissertation was funded by the Science

and Engineering Research Council.

iii

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Wheels and bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Controlled d.c. electromagnetic suspension . . . . . . . . . . . . . . . . . . . 2

1.3 Vehicle suspension configuration . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Review of electromagnetically suspended vehicles . . . . . . . . . . . . . 4

1.5 Anatomy of an electromagnetic vehicle suspension . . . . . . . . . . . . . 7

1.5.1 Suspension force actuation . . . . . . . . . . . . . . . . . . . . . . . 8

1.5.2 Decoupling the electromagnet motions . . . . . . . . . . . . . . . 9

1.5.3 Control of the vehicle mode motions . . . . . . . . . . . . . . . 10

1.6 Proposed vehicle suspension control strategy . . . . . . . . . . . . . . . . 12

1.7 Proposed implementation strategy . . . . . . . . . . . . . . . . . . . . . . . . 14

1.8 Direction and scope of this research . . . . . . . . . . . . . . . . . . . . . . 15

2 Electromagnet analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Electromagnet geometry and specification . . . . . . . . . . . . . . . . . . 18

2.3 Steady-state analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.1 Magnetic force characteristic . . . . . . . . . . . . . . . . . . . . . 20

2.3.2 Lift and lateral force components . . . . . . . . . . . . . . . . . . 22

2.3.3 Air gap reluctance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.4 Iron path reluctance . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.5 Leakage path reluctance . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3.6 Steady-state model equations . . . . . . . . . . . . . . . . . . . . . 26

2.3.7 Accuracy of the steady-state model . . . . . . . . . . . . . . . . 29

2.4 Dynamic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4.1 Magnetic force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4.2 Magnetic flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4.3 Coil current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4.4 Eddy currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4.5 Magnetic hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4.6 Accuracy of the dynamic model . . . . . . . . . . . . . . . . . . . 38

2.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

iv

3 Electromagnet force control . . . . . . . . . . . . . . . . . . . . . . . . 423.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2 Operational envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3 Electromagnet transfer function . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Force control strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4.1 Force feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4.2 Flux feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4.3 Acceleration feedback . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4.4 Air gap and current feedback . . . . . . . . . . . . . . . . . . . . . 52

3.4.5 Proposed force control strategy . . . . . . . . . . . . . . . . . . . 54

3.5 Force controller design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.6 Force controller performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4 Suspension mode control . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2 Functional requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3 Suspension control strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.4 Synthesis of the suspension control system . . . . . . . . . . . . . . . . . 68

4.4.1 Position control transfer function . . . . . . . . . . . . . . . . . . 69

4.4.2 Position, velocity and acceleration feedback gain . . . . . . 71

4.4.3 Force actuation bandwidth . . . . . . . . . . . . . . . . . . . . . . . 72

4.4.4 Position error integral time constant . . . . . . . . . . . . . . . . 73

4.4.5 Guideway following algorithm . . . . . . . . . . . . . . . . . . . . 73

4.4.6 State integration filters . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.4.7 Suspension controller design specification . . . . . . . . . . . . 76

4.5 Lateral guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.6 Performance of the experimental mode suspension . . . . . . . . . . . . 79

4.6.1 Position controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.6.2 Full suspension system . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

v

5 Vehicle suspension control . . . . . . . . . . . . . . . . . . . . . . . . . 885.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2 The experimental research vehicle and guideway . . . . . . . . . . . . . 88

5.3 Control strategy for the vehicle suspension . . . . . . . . . . . . . . . . . 90

5.4 Synthesis of the vehicle control system . . . . . . . . . . . . . . . . . . . . 92

5.4.1 Decoupling the electromagnet motions . . . . . . . . . . . . . . 93

5.4.2 Normalising the vehicle mode motions . . . . . . . . . . . . . . 94

5.4.3 Control of the vehicle mode motions . . . . . . . . . . . . . . . 95

5.4.4 Configuration of the vehicle mode suspension controllers . 95

5.5 Lateral vehicle guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.6 Performance of the experimental vehicle suspension . . . . . . . . . . . 99

5.6.1 Performance and stability of the mode position

controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.6.2 Decoupling of the vehicle modes and load sharing . . . . . . 104

5.6.3 Ride quality of the vehicle suspension . . . . . . . . . . . . . . 107

5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6 Control system implementation . . . . . . . . . . . . . . . . . . . . . 1116.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.2 System requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.2.1 Bandwidths and sampling rates . . . . . . . . . . . . . . . . . . . 112

6.2.2 Range, resolution and accuracy . . . . . . . . . . . . . . . . . . . 115

6.3 Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.3.1 Accelerometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.3.2 Air gap sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.3.3 Electromagnet current controller . . . . . . . . . . . . . . . . . . . 120

6.4 Signal processing, conversion and communication . . . . . . . . . . . . . 123

6.4.1 Transputers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.4.2 Control system hardware structure . . . . . . . . . . . . . . . . . 125

6.4.3 Analogue signal conditioning and conversion . . . . . . . . . 126

6.4.4 Digital signal processing and communication . . . . . . . . . . 128

6.5 Software design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.5.1 Occam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.5.2 Control system software structure . . . . . . . . . . . . . . . . . . 130

6.5.3 Discrete time domain integration . . . . . . . . . . . . . . . . . . 134

6.5.4 Numerical accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.5.5 Process configuration . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

vi

Appendices

A Electromagnet analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 151

B Electromagnet force control . . . . . . . . . . . . . . . . . . . . . . . 168

C Suspension mode control . . . . . . . . . . . . . . . . . . . . . . . . . 173

D Control system hardware . . . . . . . . . . . . . . . . . . . . . . . . . 178

E Control system software . . . . . . . . . . . . . . . . . . . . . . . . . . 220

F Operating instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

Introduction 1

1

Introduction

1.1 Wheels and bearings

The wheel is without doubt one of man’s most impressive early inventions. The

important difference between a wheeled cart and its predecessor, a sledge, lies in the

arrangement and quality of the bearing surfaces. The Concise Oxford English

Dictionary defines a bearing as: ‘a carrier or support for moving parts of any machine;

any part of the machine that bears the friction’. Even a crude wheeled cart has

relatively smooth and small bearing surfaces whereas a sledge has much larger bearings,

one of which (the ground) can be quite rough. The linear motion bearing between the

sledge and the ground, was thus transformed through the use of wheels, to a rotary

bearing between two controlled surfaces.

For the majority of applications the advantages of the mechanical transformation from

linear to rotary bearings have stood the test of time well. However, by harnessing

magnetic forces to support a moving body, a bearing with no physical contact between

its surfaces is possible. The lack of physical contact offers superior performance over

mechanical bearings in terms of friction and wear. For certain wheel-on-rail transport

applications, the benefits can be even greater, since magnetic linear bearings can be

used to replace both the wheel and its rotary bearing. For such applications, the wheel

may in future become as unusual as a horse-drawn carriage is today.

There are a number of different electromagnetic methods for supporting moving or

rotating masses1 (see Table 1.1). The attraction schemes are conventionally referred

to as electromagnetic suspension (EMS), whilst repulsion schemes are referred to as

electrodynamic levitation (EDL). A comprehensive review of the various EDL and

EMS schemes and their development potential can be found in reviews by Jayawant,2

Sinha3 and Weh.4 This dissertation describes the development of new, improved

techniques for controlling d.c. electromagnets for vehicle suspension applications.

Introduction 2

Table 1.1 Electromagnetic methods of supporting moving or rotating masses

Levitation using:

• forces of repulsion between permanent magnets.

• forces of repulsion between diamagnetic materials.

• superconducting magnets.

• forces of repulsion due to eddy currents induced in a conducting surface or body.

• force which acts on a current-carrying conductor in a magnetic field.

• mixed µ system, where µ is the permeability of the material.

Suspension using:

• a tuned LCR circuit and the magnetic force of attraction between an electromagnet

and a ferromagnetic body.

• controlled d.c. electromagnets and the force of attraction between magnetised bodies.

1.2 Controlled d.c. electromagnetic suspension

The force of attraction between two magnetised bodies is proportional to the inverse

square power of their separation, thus there is no point of equilibrium between two

magnetised bodies (Earnshaw’s theorem5). The force between an electromagnet and

its reaction rail is therefore open-loop unstable and closed-loop feedback control of the

electromagnet is necessary to stabilise the force and provide a satisfactory suspension

response. The essential elements of an EMS system, therefore, consist of an

electromagnet and its ferromagnetic reaction rail (see Figure 1.1), feedback sensor(s)

and associated control algorithm processing, and finally a current controller for the

electromagnet.

Graeminger appears to have been the first to propose a controlled electromagnetic

attraction system6 in 1912. He proposed a U-shaped electromagnet suspended beneath

an iron rail to carry letters. A measure of the air gap between the electromagnet and

the track was coupled mechanically to a rheostat which varied the electromagnet coil

current.

Twenty-five years later, Kemper built the first prototype EMS7 which supported 210 kg

at an air gap of 15 mm with a power consumption of 270 W. A capacitive

displacement sensor was used to measure the air gap. Thermionic valves were used to

amplify the air gap signal and a velocity signal, and also to drive the electromagnet

coil.

Introduction 3

However, the weight of the thermionic valve power controllers used to implement

Figure 1.1 Configuration of electromagnet and reaction

rail

Kemper’s EMS precluded their use in transport applications. It was after 1970, with

the advent of transistor technology capable of handling suitably high power levels, that

research into the use of EMS for transport applications flourished. Before reviewing

the more significant developments in vehicular EMS systems, it is instructive to

consider the configuration of the functional components of a conventional train

suspension.

1.3 Vehicle suspension configuration

Conventional trains are supported by a primary suspension which is coupled via a

secondary suspension to the vehicle chassis. The function of the primary suspension

(typically stiffly sprung wheels), is to maintain contact with the track, and hence avoid

derailment. The function of the secondary suspension is to provide a low bandwidth

coupling between the primary suspension and the vehicle, thus decoupling the vehicle

from high frequency track irregularities. The secondary suspension employs suitable

stiffness and damping components so that an acceptable passenger ride quality is

maintained for a given train speed and track profile. The secondary suspension also

reduces the wear and tear on the track and the primary and secondary suspensions.

This is because the decoupling of the vehicle mass from the wheels reduces the

dynamic forces generated by track irregularities.

Introduction 4

The two essential elements of a conventional train suspension are therefore the primary

suspension, which can be viewed as a high bandwidth track follower, and the secondary

suspension, which acts as a low pass filter with a low bandwidth.

Kemper’s approach of using air gap clearance and velocity feedback provides the

electrical equivalent of mechanical stiffness and damping respectively. The controlled

electromagnet, therefore, behaves in a similar manner to a conventional suspension with

the stiffness determined by the sum of the air gap feedback gain and the negative

electromagnet stiffness, and the damping is determined by the velocity feedback gain.

In principle, a conventional secondary suspension design could be utilised, but with an

electromagnet replacing the wheel. The feedback gains of the electromagnet controller

are designed to give the appropriate primary suspension stiffness and damping.

Alternatively, the electromagnet could be used to replace the secondary suspension by

setting the feedback gains to give the stiffness and damping required for the secondary

suspension. A primary suspension is not required in this case due to the lack of

physical contact between the secondary suspension and the rail. This arrangement

eliminates all moving parts and associated maintenance requirements from the vehicle

suspension, plus all of the size and weight associated with wheels, axles, bogies, springs

and dampers. Whether this trade of a mechanical for an electromagnetic systems pays

off obviously depends on the cost, weight, size and maintenance requirements of the

electromagnet and its associated control circuitry. In addition, Hrovat8 has shown that

reducing a vehicle’s unsprung weight (that of the primary suspension) enables the ride

quality to be improved. Since an EMS incorporating both primary and secondary

suspension would have no unsprung weight, ride comfort quality above that of

mechanical suspensions is theoretically possible.

The two basic suspension configurations for EMS systems are therefore electromagnetic

primary with conventional secondary suspension, or electromagnetic primary plus

secondary suspension, the latter scheme offering a suspension with no moving parts.

1.4 Review of electromagnetically suspended vehicles

Table 1.2 lists some of the milestones in the development of electromagnetic vehicle

suspensions. The first vehicles developed incorporated both primary and secondary

Introduction 5

suspension in the EMS. Subsequently however, the high speed systems and some low

speed systems have reverted to using conventional secondary suspensions.

The Magnetmobil9 was the first full-scale system capable of carrying passengers. It

Table 1.2 Electromagnetically suspended vehicles

Organisation Vehicle DateWeight

/t

Speed

/kphCSS Propulsion

MBB, FRG Magnetmobil 1971 7 90 x DLIM

Krauss-Maffei, FRG Transrapid-02 1972 11 165 x LIM

Rohr Industries, USA Romag IMPS 1972 1 low x LIM *

University of Sussex, UK Sussex 1t 1974 1 low x LIM

Japan Airlines, Japan HSST-01 1975 1 300 x LIM

British Rail, UK BR 3t 1976 3 low x LIM

Transrapid Consort.,FRG Transrapid-06 1981 122 400 LSM *

PMG Consortium, UK PMG 8t 1984 8 48 x LIM

Timisoara Poly., ROM Magnibus-01 1988 4 72 LIM *

HSST Consortium, Japan HSST-05 1989 - 200 LIM

Key: CSS - Conventional Secondary Suspension

DLIM - Double-sided Linear Induction Motor

LIM - Single-sided Linear Induction Motor

LSM - Linear Synchronous Motor

* - Suspension is combined with propulsion system

used a pair of orthogonally orientated electromagnets at each corner to provide

independent lift and lateral suspension. Propulsion was provided by a double sided,

short stator linear induction motor mounted on the vehicle, with an aluminium reaction

rail on the guideway. The Transrapid-0210 vehicle, used two in-line electromagnets

at each corner which were slightly offset either side of the rail. Both electromagnets

contributed lift force, but a lateral force component was also generated by adjusting the

relative drive levels between the electromagnet pair. The propulsion configuration was

the same as the Magnetmobil except for the use of a single-sided linear induction

motor. The Romag system11 took a different approach, using one linear induction

motor at each corner to provide combined lift and propulsion, but there was no active

control of lateral motion. The University of Sussex12 vehicle, the BR13 and PMG14

vehicles and the HSST-0115 used essentially the same configuration as the

Introduction 6

Transrapid-02, incorporating the Krauss-Maffei offset electromagnet arrangement for

lateral guidance.

For practical reasons, the nominal operational air gap of EMS systems is limited to

about 15-20 mm, above which the weight and power consumption of the electromagnets

become excessive.16 For high speed systems, meeting ride comfort requirements with

such a small allowable air gap deviation would require excessively expensive track

construction and alignment maintenance. As a consequence, practical high speed EMS

systems need to use an electromagnetic primary suspension with a conventional

secondary suspension capable of a much larger suspension deflection.17

The Transrapid-0618,19 system uses a continuous array of ‘magnetic wheels’ down the

full length of both sides of each coach. Each magnetic wheel is coupled to the coach

via an air-spring secondary suspension and operates autonomously offering a highly

modular system with significant redundancy and hence robustness to individual module

failure. The lift and propulsion are both provided by a long stator (active track) linear

synchronous motor.20 The lift force is controlled by varying the effective resistance

of coils on the lift/propulsion ‘rotor’. Lateral guidance is provided independently of

lift/propulsion in each magnetic wheel by controlled d.c. electromagnets mounted

orthogonally with respect to the lift units. The problem of non-contacting power

collection for propulsion at high speed was overcome through the use of an active track.

The power required for the onboard controllers and general vehicle services is provided

by linear synchronous generators which are incorporated into each lift/propulsion unit.

All other vehicles use power rails with sliding shoes for power collection. The

Magnibus-0121, a low speed system, appears to be functionally equivalent to the

Romag system, but with the addition of conventional secondary suspension. The HSST-

0522,23 system is the latest development of the HSST-01 and now employs a

conventional mechanical secondary suspension.

In addition to passenger transport applications, EMS based materials transportation

systems for use in automated factory production lines have been researched in

Japan.24,25 These systems typically use a small vehicle weighing about 10 kg to carry

a load of just over 10 kg. Propulsion is provided by a number of short stator linear

induction motors distributed suitably along the guideway with an aluminium reaction

plate mounted on the vehicle. Such systems use onboard battery power with special

recharging stations to reduce power collection problems. To maximise the operational

time between recharges, hybrid magnets are used consisting of permanent magnets plus

control coils. The suspension controllers are designed to operate the hybrid magnets

Introduction 7

at an air gap which requires zero average current rather than at some fixed nominal air

gap.

Maximising the benefits of the non-contacting nature of electromagnetic suspension

requires the other vehicle systems to use a non-contacting technique. Whilst linear

induction or synchronous motors provide an appropriate propulsion technique, power

collection and route switching remain practical problems. Power collection from an

active track as employed by the Transrapid system is likely to prove too expensive for

low speed applications. Route switching is also a problem because a gap must be

introduced into a suspension rail at a junction to allow an electromagnet plus its support

structure to cross the rail. The suspension force is therefore lost as the electromagnet

traverses the rail gap. This problem can be overcome by using a duplicate set of

electromagnets with a suitable arrangement of duplicate rails at the junction. However,

such a solution is undesirable due to the resulting poor suspension utilisation and the

increase in weight. A novel solution was proposed by Jayawant and Wheeler26 where

two sets of magnet pole faces were connected to a single electromagnet, but this still

incurred a significant weight penalty over conventional electromagnets. An acceptable

configuration for low cost, non-contacting power collection, and route switching without

track movement or contact has yet to be established.

Having reviewed the general configuration of some representative electromagnetically

suspended vehicles, the detailed structure of an electromagnetic vehicle suspension

system is examined next.

1.5 Anatomy of an electromagnetic vehicle suspension

The anatomy of an electromagnetic vehicle suspension is largely determined by the

functional requirements of the suspension system. The main functional requirement is

to decouple the passengers from guideway irregularities whilst following the general

guideway profile. In addition, external disturbance forces such as wind gusts must be

resisted, and passenger load variations must be accommodated. The forces generated

by the electromagnets suspending an EMS vehicle must therefore be controlled to meet

these requirements. The suspension parameters depend on factors such as guideway

profile, guideway stiffness and natural frequency, operational air gap range, disturbance

forces, passenger load variations and the required level of passenger comfort.

Introduction 8

An electromagnetic vehicle suspension consists of three important components. Firstly,

a set of electromagnets is required to provide force actuation to the vehicle body.

Secondly, a technique for decoupling the electromagnet motions is required, and finally,

suspension control algorithms are required for each of the decoupled motions. Each of

these components will now be considered.

1.5.1 Suspension force actuation

The number and configuration of the electromagnets required to suspend and guide a

vehicle depends mainly on the number of degrees of freedom to be controlled. Practical

factors such as vehicle shape and the required electromagnet redundancy (needed to

provide system availability under partial failure conditions) also contribute to the

vehicle configuration.

A vehicle assumed to behave like a perfectly rigid body in free space is capable of

linear motion and rotation with respect to three orthogonal axes. Convenient horizontal

reference axes for a tracked vehicle are the longitudinal and lateral axes of the

guideway, with the third orthogonal axis being vertical. The linear motion of the

vehicle along the guideway is controlled by the propulsion system, thus leaving five

degrees of freedom to be controlled by the vehicle suspension system. The vehicle

mode motions, which are conventionally referred to as heave and sway (vertical and

lateral motions), and pitch, roll and yaw (lateral, longitudinal and vertical axis

rotations), are illustrated in Figure 1.2.

Figure 1.2 Vehicle mode motions

Introduction 9

In reality, vehicle bodies are not perfectly rigid, and so additional degrees of freedom

exist. These correspond to the various linear and torsional bending motions which can

occur due to vehicle flexibility. Complete and independent control of the motion of a

vehicle body thus requires five independent electromagnet actuators for the rigid body

motions, with additional actuators needed if control of vehicle bending is required.

For practical reasons, most EMS vehicles have used eight electromagnets, with two

located at each corner of the vehicle to provide lift and lateral forces. Due to

redundancy in the configuration, the four lateral forces produce only two independent

vehicle mode forces/torques, namely sway and yaw. The four lift forces produce four

independent vehicle mode forces/torques, namely heave, pitch, roll and torsion (vehicle

body twist axial to its length).

1.5.2 Decoupling the electromagnet motions

The simplest vehicle suspension control strategy would be to use independent

suspension controllers, with identical parameters for each electromagnet. With this lift

control configuration, the resulting heave, pitch, roll and torsion motions of the vehicle

would all experience the same controller parameters. Unfortunately, for a vehicle with

electromagnets mounted directly on the chassis, the resultant stiff coupling between the

electromagnets results in the independent control configuration being generally

unacceptable. This is because the high controller stiffness and zero steady-state air gap

error required for the vehicle heave mode also applies to the other vehicle modes.

When zero torsion error cannot be achieved, for example due to normal track/vehicle

misalignment, the vehicle would be largely supported by a diagonal pair of

electromagnets, with the other electromagnet pair sitting virtually idle. This poor load

force distribution would cause two of the electromagnets to be overloaded. Independent

lateral electromagnet control would also produce equal suspension parameters for the

yaw and sway motions. This may be acceptable since the lateral motions are largely

decoupled. For general ride comfort considerations, it may also be desirable to have

different settings for the heave, pitch and roll mode suspension controllers.

If a conventional secondary suspension is used to couple the electromagnets to the

vehicle, then the low stiffness of the secondary suspension largely decouples the high

stiffness primary suspension electromagnets from the vehicle and hence from each

other. In this case, the electromagnets can be controlled independently, and autonomous

‘magnetic wheel’ modules can be employed as exemplified by the Transrapid-06

Introduction 10

vehicle. The ride comfort characteristics of the vehicle’s heave, pitch, roll, sway and

yaw motions are then determined by the conventional secondary suspension.

For vehicles employing electromagnets for secondary suspension, the direct attachment

of the electromagnets to the vehicle chassis results in a tightly coupled, multivariable

system. In this case, the independent electromagnet control scheme is unacceptable for

the reasons given earlier. A multivariable controller is therefore required which must

decouple the electromagnet motions (eg. by transforming them to independent vehicle

motions) and apply independent suspension controllers to each decoupled mode.

Multivariable control of the vehicle system is complicated due to the nonlinear and

unstable nature of the electromagnet force characteristic. As a consequence, all of the

vehicles listed in Table 1.2 that employed electromagnets for secondary suspension used

independent electromagnet force stabilisation controllers. The controllers significantly

reduced the force instability by using feedback of the derivative of the air gap flux for

each electromagnet. The multivariable control schemes used linear decoupling to

transform between the electromagnet motions and the vehicle mode motions.

Independent suspension controllers were then applied to each vehicle mode motion.

Results from Sinha and Jayawant27 showed that the multivariable control scheme could

achieve a superior control performance relative to an independent electromagnet

suspension control scheme. However, the nonlinear electromagnet force characteristic

resulted in these schemes suffering from significant cross-coupling between the heave,

pitch, roll and torsion modes which impaired the dynamic response of the vehicle

suspension. In addition, the elimination of steady-state pitch and roll offsets through

the use of error-integral feedback action could not be achieved on the Sussex and PMG

vehicles. This was due to low frequency cross-coupling problems between the vehicle

heave, pitch and roll modes.

1.5.3 Control of the vehicle mode motions

Having examined the configuration of the vehicle suspension control system, the

independent vehicle mode controllers that are applied to the decoupled motions can now

be considered. It is these controllers that must achieve the main functional requirements

of the vehicle suspension system outlined earlier.

The early EMS vehicles used air gap feedback to provide stiffness relative to the rail

and absolute velocity feedback to provide damping. The stiffness had to be high in

Introduction 11

order to counter load variations and disturbance forces within the small available

operational air gap range. The high stiffness resulted in a suspension bandwidth of

around 6 Hz, which was too high to meet ride comfort specifications for a cost effective

guideway. The PMG vehicle overcame this problem by using air gap stiffness at low

frequencies (below about 1.5 Hz), with vehicle position stiffness used for higher

frequencies. This was achieved through the use of a complementary pair of low and

high pass filters on the air gap and position feedback signals respectively. The

suspension thus provided a low frequency coupling to the guideway with a high

absolute stiffness to load variations and disturbance forces. Damping was provided by

applying phase-lead compensation to the complementary stiffness signal. The absolute

velocity and position signals were obtained by integration and double integration

respectively of the output from an accelerometer mounted near each electromagnet. The

integrators were given a low frequency cutoff to prevent drift problems. Acceleration

feedback was also employed in an attempt to reduce the nonlinearity of the

multivariable system.

For a guideway without gradients, a simple two pole filter design was satisfactory for

the complementary filters. However, the gradient entry and exit characteristic required

for the PMG guideway caused unacceptable air gap deviations when using two pole

filters. The final design used two and three pole filters to provide a compromise

between ride comfort and air gap deviation at guideway gradients.

The parameters of the vehicle guideway can have a critical influence on the steady-state

and dynamic behaviour of the vehicle suspension due to the coupling between the

vehicle and the guideway. If the static deflection of the guideway due to the weight

of the vehicle is to be accommodated without disturbing the passengers, then the

guideway deflection must be less than the operational range of the secondary

suspension. For vehicles with an electromagnetic secondary suspension, the small

operational air gap range requires a much stiffer guideway than that required for

vehicles employing a conventional secondary suspension. The high guideway stiffness

generally simplifies suspension design by enabling the track to be assumed to be rigid.

However, to avoid resonant oscillations, an adequate margin between the natural

frequencies of the various system components must be ensured. For example, the stiff

track required on the elevated concrete guideway for the PMG vehicle was required to

have a natural frequency above 10 Hz, to give adequate separation from the

suspension-guideway coupling bandwidth of about 1.5 Hz and the force rejection

bandwidth of about 6 Hz.

Introduction 12

1.6 Proposed vehicle suspension control strategy

The objective of the research described in this dissertation is to improve the

performance of electromagnetic secondary suspension for vehicles through the use of

improved control techniques.

The design of the electromagnetic suspension scheme for the PMG vehicle is the most

sophisticated of those employed for electromagnetic secondary suspension. However,

it has two areas of weakness. The first weakness is due to the nonlinear electromagnet

force actuation which causes cross-coupling between the independent vehicle mode

motions. This impairs the dynamic response of the suspension and prevents the use of

self-levelling roll and pitch mode controllers. The inaccurate nominal air gaps which

result from the lack of a self-levelling response reduce the allowable air gap deviation,

and hence give poor electromagnet utilisation.

To overcome this problem, a novel force control algorithm is proposed which is capable

of providing a sufficiently linear and stable force actuation. First, a detailed nonlinear

model of the electromagnet force characteristic is developed. The proposed control

algorithm then employs the model to determine the appropriate electromagnet excitation

for any required operating point.

The second weakness of the PMG vehicle is structural and stems from the use of a

single control block for the vehicle mode controllers. This enables the disturbance force

rejection characteristic to be freely chosen, but the guideway coupling for flat

guideways and guideways with gradients are both determined by the air gap feedback

filter. The implementation of the guideway interaction functions is thus tightly coupled

and the resultant performance of guideway following is therefore compromised. For

example, it may be possible to improve the vehicle guidance at the entry and exit of

curves by employing a matched filter technique. The matched filter could use the

functions defining the guideway curves to actively identify curves and guide the vehicle

appropriately.

The proposed solution to this structural problem is to partition the vehicle mode control

algorithm into two independent blocks. The first block is fed with the guideway

position which it processes using a suitable guideway following algorithm to produce

a vehicle position demand. This is then fed as the reference into a vehicle position

control algorithm which employs position error integral feedback to eliminate

steady-state position errors, and high stiffness in order to resist load variations and

Introduction 13

disturbance forces. The force demands from the vehicle position controller are then

sent to the electromagnet force controllers. Figure 1.3 outlines the structure of the

proposed vehicle control scheme.

If active guideway damping is also required,28 then assuming linear superposition, the

proposed system could be augmented with another independent control block. This

would receive the guideway velocity and, using a suitable algorithm, determine the

required damping force demand to be added to the force demands from the vehicle

position controller. Such a configuration would require a detailed analysis of the

coupling between the various control blocks since linear superposition of the vehicle

position control action and the track damping action is likely to be impaired by

nonlinearities within the vehicle suspension and guideway.

Having outlined the structure of the proposed vehicle control strategy, the design must

Figure 1.3 Structure of the proposed vehicle control

system

be developed and validated. For the vehicle suspension system, the features which are

difficult to model accurately are critical to the performance of the overall system. The

modelling difficulties are attributed to effects such as the nonlinearity and higher order

characteristics of the electromagnets, the vehicle chassis and the track. Proof of concept

using simulation as a validation tool is therefore considered to be inappropriate for the

Introduction 14

vehicle suspension system. The development of an experimental system (using

simulation as a design tool) is therefore considered to be necessary to facilitate

validation of the proposed vehicle control strategy.

1.7 Proposed implementation strategy

Three principle options are available for the hardware implementation of the suspension

controller for the experimental vehicle. These options are, analogue electronic circuitry,

programmable digital processors, or a combination of both analogue and digital

hardware. In order to determine the best implementation strategy, each of the control

system components must be considered.

For the electromagnet force control algorithm, analogue electronic circuitry is

impractical due to the complexity of the nonlinear electromagnet model which is

embodied within the algorithm. The use of a programmable digital processor is

therefore required for the force controller. The vehicle mode decoupling algorithms and

mode position controllers are linear and can thus be readily implemented using either

analogue or digital techniques. For the final control system component, the guideway

following algorithm, a programmable processor implementation is required to ensure

maximum flexibility in the choice of algorithm. In addition to the signal processing

requirements of the control system, executive control of the vehicle in terms of startup,

shutdown, fault detection and data monitoring is needed, and this is more readily and

flexibly achieved through the use of a programmable controller. The only system

functions that could feasibly use analogue signal processing are the vehicle mode

decoupling and the vehicle mode position control. Since these functions are logically

located between components that need digital signal processing, additional

analogue/digital conversions would be required if analogue signal processing were

employed. Therefore, to simplify the system hardware configuration, eliminate the

problems of drift and offsets, and maximise the flexibility of the implementation, the

use of digital processing for all system functions is proposed.

Since the control algorithms for the electromagnet forces and the vehicle mode motions

are independent of each other, they can be readily implemented using a coarsely grained

parallel processing29 approach. The prime benefit of this approach is that the vehicle

signal processing can be performed by a number of low cost microprocessors.

Additional benefits include easily scalable electromagnet configurations, and the

possibility of achieving fault tolerance at low cost through the use of spare processors.

Introduction 15

The main disadvantage of a parallel processing approach is due to the overhead

associated with the provision and use of the required inter-processor communication.

Overall, the benefits of parallel processing are considered to outweigh the disadvantages

for this particular application. The proposed implementation strategy can use one

processor per electromagnet force controller and vehicle mode motion controller. One

processor can conveniently run both of these functions because only one is active at any

instant in time.

The microprocessor family selected to implement the parallel processor control system

was the Inmos transputer.30 This was chosen because it provides a range of processor

powers, parallel language support using occam, a parallel processor development

system, a low cost/performance ratio, and also because Transputers have inter-processor

communication interfaces included on-chip.

1.8 Direction and scope of this research

This dissertation describes the theoretical and practical research work involved in the

analysis, design, implementation and validation of the proposed new electromagnetic

vehicle suspension control strategy.

In order to validate the proposed control strategy, a small experimental vehicle chassis

(capable of carrying one person) was constructed and equipped with an implementation

of the proposed suspension control system. Since the lateral motions of the vehicle do

not suffer from problematic cross-coupling, electromagnets to provide lateral force

actuation were not employed. This results in only four electromagnets rather than eight

being required, which significantly reduced the cost and complexity of the experimental

vehicle. The electromagnets, track and linear induction motor from an earlier research

project were provided for use with the new experimental vehicle. A single

electromagnet experimental rig was also constructed to facilitate algorithm testing on

an independent suspension configuration.

This account of the research work is partitioned into seven chapters as listed in

Table 1.3. In each chapter the theoretical basis is described and then results from the

experimental systems are discussed and conclusions are drawn.

Introduction 16

In Chapter 2 the steady-state and dynamic behaviour of the electromagnets used on the

Table 1.3 Summary of the research work

1 General literature review and proposed research strategy.

2 Analysis of the electromagnet force characteristic.

3 Synthesis of the electromagnet force control algorithm.

4 Analysis of an independent mode suspension and synthesis of the control algorithm.

5 Analysis of the vehicle motions and synthesis of the vehicle control system.

6 Selection/development of the hardware and software for the control system.

7 Overall conclusions and identification of areas for further research.

experimental research vehicle is analysed. Equations are developed which model the

steady-state lift and lateral forces in terms of core dimensions, air gap, coil current and

lateral displacement. The models include the effects of air gap flux fringing, leakage

flux between the electromagnet poles pieces, and the variability of core permeability

due to saturation effects. Dynamic model equations are developed for the flux lag time

constant due to the electromagnet coil circuit and the eddy current circuits within the

electromagnet and rail cores. The model for the flux lag time constant is a function of

the core dimensions, air gap, number of coil turns, coil resistance, core construction and

core material resistivity.

Chapter 3 describes the analysis of the performance of various feedback control

strategies in terms of their capability to reduce the instability and nonlinearity of the

electromagnet force characteristic. The strategies considered include feedback of force,

flux, rate of change of flux, acceleration, air gap and current. A novel feedback control

algorithm is then proposed using only air gap and current feedback, and embodying the

detailed electromagnet model developed in Chapter 2.

In Chapter 4 the requirements for an independent suspension are analysed. A

suspension control scheme is then proposed which consists of two separate functions.

Firstly, a track coupling algorithm uses the measured track position to determine a

required suspension position. Suspension positioning with disturbance force rejection

is then performed by a state feedback mode position controller which incorporates

acceleration, velocity, position error and position error integral feedback. All feedback

signals for the proposed scheme are derived from measured values of acceleration and

air gap. The validity of the proposed strategy is then tested using simulations and an

experimental single electromagnet suspension rig. Since the experimental vehicle track

Introduction 17

has no significant gradients, a second-order, low pass filter is used for the track to

suspension coupling algorithm to obtain experimental results.

In Chapter 5 the suspension requirements for the complete multivariable vehicle

suspension are analysed. The transformations required to decouple the electromagnet

motions are identified so that independent vehicle mode control loops can be realised.

Extensive experimental suspension tests covering stability, mode decoupling, ride

comfort over simulated rail steps, disturbance force rejection and linear motor force

coupling are then presented and evaluated.

Chapter 6 describes the implementation of the experimental vehicle control system.

First, the development of the electronic hardware for the control system and the

selection criteria for the feedback sensors are described. Hardware development

includes closed-loop electromagnet current controllers, a transputer module motherboard,

transputer based analogue to digital and digital to analogue converter cards, and fibre

optic interface cards for the transputer communication links. The structure of the

concurrent software which implements the vehicle suspension control algorithms is

described next. Topics include selection of sampling rate, discrete digital

implementation of the continuous time design, required numerical accuracy of the

digital processors and finally it proposes a scalable multi-processor configuration

strategy which can efficiently utilise one processor per electromagnet. Practical

implementation features such as real-time data monitoring and logging, control of

vehicle suspension startup and shutdown, and system fault detection are also included

in the experimental system design.

The last chapter draws overall conclusions about the success and limitations of the

results of this research and suggests some areas for further work.

Electromagnet analysis 18

2

Electromagnet analysis

2.1 Introduction

The characteristic behaviour of the electromagnets used to suspend the experimental

research vehicle must be analysed and modelled before an electromagnet control law

can be synthesised. The behaviour of suspension electromagnets is complicated by their

nonlinear and unstable nature, and the dynamic geometry changes associated with the

electromagnet moving along an uneven track.

Kortüm and Utzt31 have shown that a linearised model of a suspension electromagnet

is inadequate for effective simulation of the full operational envelope of a suspension

electromagnet. This chapter therefore presents a detailed nonlinear analysis and a set

of model equations which give good accuracy over the full operational envelope of the

experimental electromagnets.

The geometry and specification of the experimental electromagnet are outlined first.

Then the steady-state behaviour of the electromagnet is modelled. This is followed by

modelling of the dynamic behaviour of the electromagnet flux. The chapter is

concluded with a summary of the electromagnet model equations and their accuracy.

2.2 Electromagnet geometry and specification

Figure 2.1 shows the physical arrangement of the experimental U-shaped electromagnet

and the inverted U-shaped track. Magnetic flux passes through the air gap between the

poles of the electromagnet and track and this generates a force of attraction which

suspends the electromagnet beneath the track.


The electromagnets used for this research were previously used on a research vehicle

Figure 2.1 Physical arrangement of the electromagnet

and track (cross-section perpendicular to track axis)

at the University of Warwick.32,33 They consist of insulated copper windings wound

on steel cores and were designed to lift a maximum load of 50 kg at a nominal

operating air gap of 3-4 mm. Table 2.1 lists the electromagnet dimension indices along

with the relevant values for the experimental electromagnet.

2.3 Steady-state analysis

The role of an electromagnet in a vehicle suspension application is that of a controlled

force actuator. Excitation of the electromagnet coils generates a magneto-motive force

which causes a magnetic flux to flow through the electromagnet, air gaps and track.

The interaction of the air gap flux and field strength generates a force of attraction

between the electromagnet and the reaction rail. The steady-state force characteristic

of the experimental electromagnet will now be analysed and model equations developed.

The analysis is performed by considering the following:

• fundamental magnetic force characteristic.

• lift and lateral force components.

• air gap reluctance.

• iron path reluctance.

• leakage path reluctance.


2.3.1 Magnetic force characteristic

Table 2.1 Experimental electromagnet dimension indices and values

Index Value Dimension

l 200 mm Length of the electromagnet

w 33 mm Width between the electromagnet pole pieces

h 63 mm Height of the pole pieces above the yoke

p 9.5 mm Width of the pole pieces

t 30 mm Width between the track pole pieces

g 0-7 mm Suspension air gap length

N 274

turns

Total number of coils

Rcoils 0.8 Ω Resistance of the coils

ρ 100 nΩm Resistivity of the steel cores (estimated value)

m 7.3 kg Mass of the electromagnet

By assuming that the poles of the electromagnet and the track have equal magnetic

potential over their working faces, the force of attraction across each air gap, Fairgap,

between the electromagnet and track is given by:34

where Hairgap is the magneto-motive force gradient across each air gap and Φairgap is the

2.1Fairgap

1

2H

airgapΦ

airgapnewtons

air gap magnetic flux. As force is generated across two air gaps, the total

electromagnet lift force, Fmagnet, is given by:

Equation 2.2 can be more conveniently expressed by considering it in terms of the

2.2Fmagnet

2 Fairgap

Hairgap

Φairgap

newtons

magneto-motive force across each air gap, Mairgap, and the reluctance of each air gap,

Rairgap.


The magneto-motive force gradient and air gap flux can be expressed as:

where g is the length of each air gap. Substituting Equations 2.3 and 2.4 into Equation

2.3Hairgap

Mairgap

gamperes/metre

2.4Φairgap

Mairgap

Rairgap

webers

2.2 gives:

2.5Fmagnet

M2

airgap

g Rairgap

newtons

A first order approximation for the electromagnet lift force may be made by assuming

the iron paths to have zero reluctance (i.e. infinite permeability), and that the air gap

flux density between the poles, is uniformly distributed over an area equal to the pole

face area. This gives the first order approximations for the air gap magneto-motive

force, M airgap, and reluctance, R airgap, as:

where N is the total number of coil turns, I is the current flowing through the coils, µo

2.6Mairgap

NI

2amperes

2.7Rairgap

g

µo

Apole

amperes/weber

is the permeability of free space and Apole is the pole face area. These two expressions

may be substituted into Equation 2.5 to give a first order approximation for the lift

force, F magnet, as:

Equation 2.8 shows that the magnetic force is a nonlinear function of both current and

2.8Fmagnet

µo(NI)2 A

pole

4g2newtons

air gap length. Also, it shows that for a constant current the force decreases with

increasing air gap, hence it has a negative stiffness coefficient. There is therefore no

point of equilibrium between two magnetised bodies,35 and so the open-loop force-air

gap characteristic of an electromagnet is unstable. Figure 2.2 shows a graph of the


electromagnet force characteristic predicted by the first order approximation given by

Equation 2.8.

The assumption of uniform air gap flux distribution is valid only for air gaps that are

Figure 2.2 First-order model of electromagnet lift force (Equation 2.8)

much smaller than the pole width. At larger air gaps, flux fringing increases the

effective air gap flux area and hence decreases the air gap reluctance. Since the first

order approximation neglects flux fringing and cannot determine the effects of lateral

displacement of the electromagnet relative to the track, a more detailed analysis is

required.

2.3.2 Lift and lateral force components

The experimental suspension electromagnet is relatively long and thin. Therefore,

end-effects can be neglected and a 2-dimensional analysis can be performed by

considering the electromagnet cross-section. This assumption is not strictly true with

regard to eddy-currents when the electromagnet is moving along its rail, so they are

analysed independently in Section 2.4.4. By assuming the pole surfaces to be magnetic

equipotentials, the electromagnet force can be determined using conformal mapping


techniques, but an exact analysis considering all four corners is difficult because the

expressions involve complex elliptic integrals and require the solution of implicit

equations containing elliptic functions.36 However, by assuming that an interval of

uniform magnetic field exists in the air gap, the problem may be reduced to the sum

of 2 two-corner problems, which produce the following simpler formulas for lift and

lateral forces:37

where F magnet is the first order approximation for electromagnet force (Equation 2.8),

2.9Flift

Fmagnet

12g

πp

1y

gtan 1 y

gnewtons

2.10Flateral

Fmagnet

2g

πptan 1 y

gnewtons

p is the pole width and y is the lateral offset between the electromagnet and track poles.

The uniform magnetic field assumption limits the useful range of these expressions to

a maximum air gap and lateral offset of about two-thirds of the pole width.

Figure 2.3 shows a graph of the electromagnet lift force characteristic predicted by

Figure 2.3 Electromagnet lift force model with air gap fringe flux (Equation 2.9)

Equation 2.9. This graph indicates a maximum increase in force relative to the first


order approximation (Equation 2.8) of 40% at an air gap of 7 mm, falling to a decrease

of 7½% at an air gap of 1 mm. The decrease in force at 1 mm is due to the slight

difference between the track pole separation and the electromagnet pole separation.

2.3.3 Air gap reluctance

The flux fringing correction factor in Equation 2.9 cannot be applied directly to the air

gap reluctance (Equation 2.7) because of the inclusion of the orthogonal force

components. However, by considering the case of zero lateral offset, the effects of flux

fringing can be modelled simply and to a good degree of accuracy. This does not

prejudice the application of the full accuracy model for orthogonal forces later on. The

air gap reluctance incorporating lateral fringe flux is given by:

Equations 2.9, 2.10 and 2.11 model the lift and lateral force components and the air gap

2.11R

airgap

g

µo

l

p2g

π

amperes/weber

reluctance, but the magneto-motive force across the air gap (Equation 2.6) still neglects

the magneto-motive force needed to overcome the reluctance of the iron paths due to

their finite permeability.

2.3.4 Iron path reluctance

The reluctance of the iron paths is a function of their geometry and the relative

permeability (µr) of the core material. Since the permeability is a non-linear function

of past and present flux density, incorporating the effects of hysteresis and saturation,

it is very difficult to quantify exactly. It is presumably for this reason that most

researchers in this field choose to neglect its effect by assuming infinite permeability.

The experimental suspension electromagnets are assumed to be made of mild steel for

which the typical maximum permeability is about 2000-3000,38 and the variation of

permeability with flux density39 is outlined in Table 2.2. To accommodate the fact

that the cores of the experimental electromagnet and track have been machined and

welded without subsequent heat treatment, a maximum value for µr of 2000 has been

assumed. With air gaps ranging from about 1% to 7% of the iron path length, the iron


path reluctance is significant, causing a reduction in force at small air gaps and/or high

Table 2.2 Typical variation of µr for mild steel

Flux density µr / maximum µr

0 T 10 %

0.15 T 50 %

0.4 - 1.0 T 100 %

1.3 T 50 %

1.5 T 10 %

2.1 T µr = 1 (saturation)

flux densities.

The reluctance of the track, Rtrack, and electromagnet core, Rmagnet, are given by:

where t is the width between the track poles, w is the width between the electromagnet

2.12Rtrack

(t 4p)

µT

µo

lpamperes/weber

2.13Rmagnet

(2h w 2p)

µM

µo

lpamperes/weber

poles, p is the pole width (and rail pole height), h is the pole height above the yoke, l

is the length of the electromagnet and µT, µM are the relative permeabilities of the track

and electromagnet respectively.

Consideration of the iron path reluctance is required to determine the operational limits

of the electromagnet (due to the onset of saturation) and also to enable it to be

controlled over its full operational envelope. To evaluate the reluctance of the iron

paths, the permeability must be known, and this in turn requires knowledge of the flux

density. Therefore, the leakage flux between the electromagnet poles must be

investigated since the electromagnet core carries both useful suspension flux and the

parasitic leakage flux.


2.3.5 Leakage path reluctance

The magneto-motive force between the poles of the electromagnet causes a parasitic

leakage flux to flow between them in addition to the useful suspension flux which flows

through the air gaps and the track (see Figure 2.4). Finite element analysis has shown

that the leakage flux can exceed the suspension flux at large operational air gaps.40

The leakage flux path reluctance must therefore be modelled. By assuming the

magneto-motive force to be generated linearly over the length of the vertical pole

pieces, the effective height of the poles is halved. A first order approximation to

leakage flux reluctance can now be made by neglecting fringe flux. This gives:

where w is the width between the poles, l is the length of the electromagnet and h is

2.14Rleakage

w

µo

l

h

2

amperes/weber

the height of the poles. Using the expression obtained earlier for the air gap path

reluctance (Equation 2.11), the above expression for the leakage reluctance can be

modified to include leakage fringe flux at the ends and on top of the electromagnet.

This gives:

The incorporation of flux fringing above the top of the electromagnet poles is bound

2.15R

leakage

w

µo

l2w

π

h

2

w

π

amperes/weber

to overestimate the leakage flux at small air gaps due to the close proximity of the track

poles and yoke. However, since the effect of leakage flux on the steady-state force is

significant only at larger air gaps, this fact is not a problem.

Flux leakage around the outer faces of the electromagnet poles has not been modelled

because observation of flux plots obtained using finite element techniques41 shows that

such leakage is not significant for the U-shaped electromagnets.

2.3.6 Steady-state model equations

The model equations for the reluctance of the air gap, leakage and core flux paths are

now combined to build model equations for the electromagnet lift and lateral forces.


The magnetic flux paths of the electromagnet and rail are modelled by the equivalent

circuit shown in Figure 2.4. The coil windings generate a magneto-motive force, Mcoils,

given by:

This drives flux through the electromagnet (reluctance = Rmagnet), where it splits between

2.16Mcoils

NI amperes

the useful path carrying suspension flux (reluctance = 2Rairgap + Rtrack), and a parasitic

path carrying leakage flux (reluctance = Rleakage).

The expression for electromagnet force given in Equation 2.5 is repeated here for

Figure 2.4 Magnet flux paths and flux model circuit diagram

convenience:

The above equation requires an expression for Mairgap in terms of Mcoils. Network

2.17Fmagnet

M2

airgap

g Rairgap

newtons

analysis of the flux circuit diagram is detailed in Appendix A and produces the

following result:

2.18Mairgap

Rairgap

Rleakage

(Rtrack

2Rairgap

)(Rleakage

Rmagnet

) Rmagnet

Rleakage

Mcoils

Substituting for Mairgap (Equation 2.18) into Equation 2.17 gives:


where Rairgap, Rleakage, Rtrack and Rmagnet are defined by Equations 2.11, 2.15, 2.12 and 2.13

2.19F

magnet

Rairgap

R2

leakage

g (Rtrack

2Rairgap

)(Rleakage

Rmagnet

) Rmagnet

Rleakage

2M

2

coils newtons

respectively. The lift and lateral forces can now be expressed by using the factors in

Equations 2.9 and 2.10 and compensating these for the fact that the air gap reluctance

already incorporates flux fringing at zero lateral offset, this gives:

These equations are needed to model the difference between the pole spacings of the

2.20Flift

Fmagnet

1y/g tan 1 y/g

1 πp/2gnewtons

2.21Flateral

Fmagnet

tan 1 y/g

1 πp/2gnewtons

electromagnet and track in addition to modelling lateral offsets between the track and

the electromagnet.

An expression for the air gap flux can now be obtained by substituting for Mairgap

(Equation 2.18) into Equation 2.4 giving:

2.22Φ

airgap

Rleakage

(Rtrack

2Rairgap

)(Rleakage

Rmagnet

) Rmagnet

Rleakage

Mcoils

webers

An expression for the leakage flux is obtained similarly giving:

The flux which flows through the yoke of the electromagnet is the sum of the air gap

2.23Φ

leakage

Rtrack

2Rairgap

(Rtrack

2Rairgap

)(Rleakage

Rmagnet

) Rmagnet

Rleakage

Mcoils

webers

and leakage fluxes above, giving:

2.24Φ

magnet

Rtrack

2Rairgap

Rleakage

(Rtrack

2Rairgap

)(Rleakage

Rmagnet

) Rmagnet

Rleakage

Mcoils

webers

Figure 2.5 shows a graph of the electromagnet lift force characteristic given by

Equation 2.20 with a fixed value for the permeability of the iron paths. This graph

predicts a decrease in lift force relative to the previous graph (Figure 2.3) of


Equation 2.9 due to the effects of the iron path reluctance, and leakage flux. The lift

Figure 2.5 Electromagnet lift force model with fixed µiron (Equation 2.20)

force is reduced by 4% at an air gap of 1 mm and by 11% at 7 mm.

Figure 2.6 shows a graph of the full model of the electromagnet lift force predicted by

Equation 2.20 with the iron path permeability dependent on the iron flux density

according to the values given in Table 2.2. This graph illustrates the effect of the fall

in the permeability of the electromagnet core as it approaches saturation. It also shows

clearly the effect of increased leakage flux at larger air gaps, which causes reduced

maximum lift force due to the electromagnet core saturation.

2.3.7 Accuracy of the steady-state model

The steady-state characteristics of the electromagnet have been analysed and model

equations have been developed. The lateral force of the electromagnet will not be

controlled by the suspension control system and therefore the quality of the lateral force

model is not investigated in this work. However, to verify the accuracy of the lift force

model equations, a simple electromagnet controller was implemented to permit accurate

experimental measurements. Experimental data has been obtained by measuring the


electromagnet current whilst suspending a range of different loads at various suspension

Figure 2.6 Electromagnet lift force model with variable µiron (Equation 2.20)

air gaps. Figure 2.7 shows a graph of the predicted lift force (from Equation 2.20)

along with the measured data from the experimental electromagnet which are plotted

as diamonds. The uncertainty of each experimental data point due to measurement error

is represented approximately by the area of the diamond markers.

Magnetic hysteresis within the electromagnet and track cores causes the experimental

current measurements to vary by approximately ±5%, ±1½% and ±1% at air gaps of

1 mm, 3 mm and 5 mm respectively. To isolate this effect from the underlying

steady-state force characteristic, the experimental values plotted in Figure 2.7 are the

mean of measurements taken with both a rising and a falling flux level.

The performance of the air gap fringe flux model is evident at low forces, before the

onset of core flux saturation. The model accuracy is very good for air gaps of 1-4 mm,

and then steadily deteriorates as the air gap to pole width ratio exceeds about ½. At

higher flux densities, when the core reluctance becomes significant even at large air

gaps, the overall model accuracy remains good. The force model error ranges from -

2½% to +7% for air gaps up to 5 mm, and then increases to a maximum of about +9%

and +13% at air gaps of 6 mm and 7 mm respectively.


Figure 2.7 Steady state lift force model (Equation 2.20) and measured values.

2.4 Dynamic analysis

The models developed so far have dealt with the steady-state characteristics of the

electromagnet. Analysis of the dynamic characteristics is also required because the

open-loop instability of the electromagnet necessitates the use of closed-loop feedback

control. The dynamic relationship between force and flux is considered first. This is

followed by models of the flux and coil current dynamics. Eddy current effects are

then modelled and finally, the effect of magnetic hysteresis is discussed.

2.4.1 Magnetic force

The instantaneous force exerted between the electromagnet and its track can be

approximated as a function of air gap flux Φairgap, by using Equations 2.4, 2.5 and 2.7

to give:

2.25Fmagnet

Φ2

airgap

µoA

pole

newtons


The predicted leakage flux between the electromagnet poles varies from being less than

the air gap flux at small suspension air gaps to being larger than the air gap flux at the

maximum air gap. This effect is significant because even for a constant force, the

electromagnet core flux must double as the air gap varies from minimum to maximum.

Since a change of core flux is required both for changes in force and changes in

suspension air gap length, the dynamics of the flux circuit must now be considered.

2.4.2 Magnetic flux

The relationship between electromagnet core flux Φmagnet, applied coil terminal voltage

V and coil current I, is given by:

where N is the number of coil turns and Rcoils is the total coil resistance.

2.26V NdΦ

magnet

dtI R

coilsvolts

Equation 2.26 is useful for dimensioning the voltage requirement of the power

controller for the electromagnet. The maximum supply voltage needed is the sum of

the parasitic voltage needed for the maximum steady state current requirement plus the

voltage needed to provide the maximum required flux slew rate.

The dynamic behaviour of the electromagnet current must now be considered to

determine the time constant associated with changes in applied coil terminal voltage,

current, flux and air gap.

2.4.3 Coil current

A first order approximation to the electromagnet flux, Φ magnet, can be obtained by

substituting Equations 2.6 and 2.7 into Equation 2.4, giving:

where N is the total number of coil turns, Apole is the pole face area and g is the

2.27Φmagnet

µo

Apole

NI

2 gwebers

suspension air gap length. Substituting Equation 2.27 into Equation 2.26 gives:


where L coils is an approximation to the self inductance of the coils which is given by:

2.28Vd

dtILcoils IR

coilsvolts

The response of Equation 2.28 to a step change in applied coil terminal voltage

2.29Lcoils

µo

N 2 Apole

2 ghenrys

(assuming constant air gap and hence coil inductance), is given by:

where the electrical time constant Tcoils is given by:

2.30IV

Rcoils

1 et /T

coils amperes

2.31Tcoils

Lcoils

Rcoils

seconds

Equations 2.30 and 2.31 show that the electromagnet coil current experiences a first

order lag characteristic relative to the applied terminal voltage. The current lag time

constant given by Equation 2.31 also approximates the flux lag time constant due to the

coil for changes in applied coil terminal voltage or air gap (see Equation 2.27).

A more accurate model of the coil inductance can be obtained by using Equation 2.24

for Φmagnet instead of Equation 2.27 which gives:

This expression varies with air gap and core flux density. Table 2.3 lists the predicted

Lcoils

N2R

track2R

airgapR

leakage

(Rtrack

2Rairgap

)(Rleakage

Rmagnet

) Rmagnet

Rleakage

henrys 2.32

inductance (Equation 2.32) and the consequent lag time constant (Equation 2.31) for a

range of air gaps, but for a constant iron path permeability.

Table 2.3 illustrates how the leakage flux increases the electromagnet flux

(Equation 2.27) at larger air gaps, preventing the inductance from falling as the inverse

of air gap (Equation 2.29). The finite inductance at zero air gap is due to the finite

permeability of the iron paths. This inductance is an overestimate because the

mechanical joints in the iron circuit have been neglected. The table clearly shows that


as the air gap is reduced, the time constant associated with coil current and

electromagnet flux is increased.

Table 2.3 Predicted coil inductance and time constant

Air gap Model Lcoils Model Tcoils

0 mm 1440 mH 1800 ms

1 mm 108 mH 135 ms

2 mm 69 mH 86 ms

3 mm 56 mH 70 ms

4 mm 50 mH 62 ms

5 mm 47 mH 59 ms

6 mm 45 mH 56 ms

7 mm 44 mH 55 ms

2.4.4 Eddy currents

The inductance of the electromagnet coils has been modelled and it has been shown that

there is a phase lag characteristic between a change of terminal voltage or air gap and

the consequent change of flux. Changes in flux level also generate electro-motive

forces in the electromagnet and track cores which cause eddy currents to circulate

within the cores. The eddy currents generate a magneto-motive force which opposes

the flux change. The resistivity of the core material ensures that the eddy currents

decay, so the flux is subject to a lag characteristic. Analysis of the eddy currents is

complicated by the distributed nature of the eddy current circuits, each of which

encloses only a limited portion of the total core flux.

Yamamura and Ito42 performed a sophisticated frequency domain analysis of the

temporal and spatial effects of eddy currents on core flux. Their analysis produced a

model which predicted a first order flux lag with a time constant which increases from

the periphery of the core towards its centre. This spatial flux distribution is due to the

presence of more effective ‘enclosing turns’ around the centre of the core compared to


the periphery of the core. However, the model required to assist in the synthesis of a

control algorithm for the electromagnet does not require a spatial flux distribution

model. A novel time domain analysis is therefore proposed which produces a temporal

model without a spatial model.

The time domain analysis is performed by considering elemental core circuits, which

are then translated to equivalent external coil circuits, ie. coil circuits located outside

the core. The external circuits are equivalent to the internal circuits in terms of their

total core flux coupling. The contributions of each equivalent external circuit are then

combined to give the total flux lag characteristic for a given core geometry. This

analysis, for long, thin pole cross sections, is described in Appendix A and results in

a first order flux lag model with a time constant given by:

where p is the electromagnet pole width, lc is the length of the iron flux circuit, g is the

Teddy

µo

lcp2

24ρgseconds 2.33

suspension air gap and ρ is the resistivity of the core material. To simplify the

analysis, this model neglects the core reluctance, air gap flux fringing and leakage flux.

The temporal component of the model proposed by Yamamura and Ito makes the same

simplifying assumptions and gives a very similar time constant. The only difference

lies in the denominator coefficient which for the frequency domain analysis is 2π2

versus 24 for the time domain analysis. The frequency domain approach therefore

predicts a time constant which is 22% larger than that predicted by the time domain

analysis.

Flux lag due to eddy currents appears in two distinct guises. The first is a reaction to

variation of the flux level in the electromagnet and track due to changing airgap or

changing coil magneto-motive force. For this case, the iron circuit includes both the

electromagnet and the track. To improve the accuracy of the model, a correction factor

is applied to model the effects of air gap flux fringing (see Equation 2.11), and leakage

flux (see Equation 2.15). The effect of leakage flux is modelled by adding it to the

suspension flux passing through the track. The time constants predicted by the model

are shown in Table 2.4 for a range of air gaps.

The second cause of eddy currents is due to the motion of an electromagnet along its

track. Even though the magneto-motive force and air gap are constant, the motion of

the electromagnet is continuously magnetising fresh track at the front of the

electromagnet. A similar effect occurs at the trailing end of the electromagnet as the


track is demagnetised. The flux lag due to the eddy currents reduces the suspension

Table 2.4 Predicted eddy current lag time constant for the electromagnet and track

Air gap Eqn 2.33 Teddy Leakage & fringe

flux correction

Model Teddy

1 mm 11.7 ms 1.36 15.8 ms

2 mm 5.9 ms 1.73 10.2 ms

3 mm 3.9 ms 2.09 8.1 ms

4 mm 2.9 ms 2.46 7.1 ms

5 mm 2.4 ms 2.82 6.8 ms

6 mm 2.0 ms 3.18 6.4 ms

7 mm 1.7 ms 3.54 6.0 ms

force and generates a drag force which retards the motion of the electromagnet.43,44

The iron circuit length for the model (Equation 2.33) is in this case just the track flux

path length. To improve the accuracy of the model, a correction factor is again applied

to model the effects of air gap flux fringing (see Equation 2.11). The time constants

predicted by the model are given in Table 2.5.

Due to the motion of an electromagnetically suspended vehicle along its guideway, the

track flux lag characteristic appears spatially along the length of the electromagnet. The

speed at which the electromagnet length corresponds to 1 flux lag time constant

represents a breakpoint above which a significant loss of suspension force occurs. The

predicted eddy current lag time constant for the track is 3.5 ms at an air gap of 1 mm

(see Table 2.5). This gives a speed break point of 57 m/s for the experimental

electromagnet which is 0.2 m long. Since the experimental track for the research

vehicle is only 5 metres long, the consequent low maximum vehicle speed prevents this

effect from being observed.

2.4.5 Magnetic hysteresis

The magnetisation characteristic for the electromagnet and track steel cores includes

significant magnetic hysteresis.45 Therefore, in addition to the iron permeability being

a nonlinear function of flux density, it is also a nonlinear function of flux history.


Since hysteresis affects the core permeability and hence the core reluctance, it is an

Table 2.5 Predicted flux lag time constant due to eddy currents in the track core

Air gap Eqn 2.33 track

Teddy

Fringe flux

correction

Model track

Teddy

1 mm 3.31 ms 1.07 3.53 ms

2 mm 1.65 ms 1.13 1.87 ms

3 mm 1.10 ms 1.20 1.32 ms

4 mm 0.83 ms 1.28 1.06 ms

5 mm 0.66 ms 1.34 0.88 ms

6 mm 0.55 ms 1.40 0.77 ms

7 mm 0.47 ms 1.47 0.69 ms

effect which increases in significance at smaller air gaps. There are very few references

to hysteresis in the published literature on electromagnetic suspension control, but

Limbert et al46 referred to doubling the damping feedback gain of their suspension

controller to overcome problems which they attributed to hysteresis. Magnetic

hysteresis is very difficult to model analytically with even moderate accuracy because

of its nonlinear dependence on the history of the core flux.

The steady-state model error caused by the iron core hysteresis increases for smaller air

gaps as described in Section 2.3.7. However, due to the small size of the model error

for air gaps greater than 2 mm, and the difficulty of modelling hysteresis, the design

of the suspension control system is to proceed without the aid of an analytical model

for magnetic hysteresis.

Finally, annealing47 the electromagnet core to relieve the stresses built up during

manufacturing increases the iron core permeability and significantly reduces the

hysteresis envelope for minimal cost. This reduces the detrimental effects of both

hysteresis and iron core reluctance prior to saturation for the experimental

electromagnets.


2.4.6 Accuracy of the dynamic model

The flux lags associated with changes in both air gap and applied coil terminal voltage

have been modelled. The flux lag time constant is a function of both the coil

resistance, which is temperature dependent, and most significantly the inductance, which

is a function of air gap. Eddy currents generated within the cores also cause a flux lag

characteristic. Both the coil and the eddy current circuits are magnetically coupled to

the core with little leakage inductance. The combined effect is therefore modelled by

a first-order flux lag (see Appendix A) with a time constant equal to the sum of the coil

and eddy current time constants.

The quality of the flux lag model is now tested by comparing the model predictions

with experimental measurements from the electromagnet and track. All of the measured

flux responses exhibited a dominantly first order lag characteristic.

The electromagnet coil time constant was determined by applying voltage steps to the

coil and measuring the flux rise time constant using a Hall plate flux probe and a

storage oscilloscope. The measured flux time constant includes a lag contribution from

both the coil and the eddy current circuits. The eddy current time constant is therefore

subtracted from the measured value to give the coil time constant. The accuracy of the

coil time constant is insensitive to errors in the measured eddy current time constant

because of the large relative size difference. Table 2.6 lists the model predictions and

experimentally measured values for the electromagnet time constant for a range of air

gaps.

The error between the predicted and measured electromagnet coil time constants ranges

Table 2.6 Predicted and experimental electromagnet coil time constants

Air gap Model Tcoil Experimental Tcoil Error factor

1 mm 135 ms 111 ms 1.21

4 mm 62 ms 76 ms 0.82

7 mm 55 ms 68 ms 0.81

from around +20% to -20%. With a measurement accuracy of about ±5% the model


accuracy is seen to be quite satisfactory. This model relies heavily on the static model

equations thus giving additional confidence in their accuracy. The error at 1 mm is in

part due to an overestimate of leakage flux which is inherent in the static force model

equations (see Section 2.3.5). The discrepancies that exist suggest that the iron

permeability may be lower than assumed and that the leakage flux may be slightly

larger than modelled for large air gaps.

In order to measure the eddy current time constant, the effective coil circuit time

constant was significantly reduced. This was achieved through the use of a closed-loop

controller incorporating high gain (x 100) current feedback. Analysis of this

closed-loop configuration (see Appendix B) shows that current feedback reduces the

effective coil time constant, whilst leaving the eddy current time constant unaltered.

Step changes in coil current reference were input to the controller and the flux change

time constant was again measured using a Hall plate flux probe and a storage

oscilloscope. The measured flux time constant now includes the eddy current lag time

constant plus 1% of the coil circuit time constant. The contribution of the coil circuit

is therefore subtracted from the measured time constant to give the eddy current time

constant. The relative size of the two components gives an eddy current time constant

error approximately equal to the measurement error plus one third of the coil time

constant error. Table 2.7 lists the predicted and measured values for the eddy current

time constant for the electromagnet and track cores over a range of air gaps.

The predicted values are all about 3 to 3.5 times the experimental values. The rather

Table 2.7 Predicted and experimental eddy current lag time constants for the

electromagnet and track

Air gap Model Teddy Experimental Teddy Error factor

1 mm 15.8 ms 4.6 ms 3.4 x

4 mm 7.1 ms 2.4 ms 3.0 x

7 mm 6.0 ms 1.8 ms 3.3 x

large discrepancy between the model and the experimental system is attributed mostly

to the model assumption of uniform core flux distribution and zero core reluctance. In


reality, the flux lag is composed of spatially distributed components, with time constants

which are small near the perimeter of the core, but which increase towards the centre

of the core. The interaction between distributed flux components is further complicated

by the effects of the core permeability.

In addition to the above factors, the model parameters represent an additional error

source. All of the core dimensions were measured with high accuracy, however, the

core resistivity could not be measured with available equipment. Manufacturing records

for the cores do not exist, but they appear to be made of mild steel. If, however, the

cores are made of magnetic steel with even as little as 0.5% Silicon content, then the

resistivity would be approximately double that of mild steel.48 In view of the

uncertainty associated with the core resistivity, and the otherwise good fit of the first

order lag model, further experimental work with a known core resistivity would be

required to determine the exact accuracy of the eddy current model time constant.

For the purposes of developing an electromagnet control algorithm, the quality of the

first order lag model is considered to be sufficiently accurate when combined with the

measured time constants.

It is assumed that if the track cores are made of the same material as the electromagnet

cores, then the predicted eddy current time constants for the track alone (see Table 2.5)

are also approximately three times too large.

2.5 Concluding remarks

Models have been developed for the significant steady-state and dynamic characteristics

of the experimental suspension electromagnet. The force between the electromagnet

and its reaction rail is dominantly a function of the core dimensions, the square power

of the coil current-turns and the inverse square power of the length of the suspension

air gap. The resulting negative stiffness makes the electromagnet open-loop unstable.

The operational envelope for a given electromagnet geometry depends on the maximum

achievable air gap flux. This is determined by the flux saturation level of the cores,

and the air gap which determines the reduction of air gap flux due to the significant

flux leakage between the electromagnet pole pieces. Finite core permeability at low

flux levels is also significant for small air gaps. The electromagnet steady-state model

equations therefore incorporate variable core permeability and leakage flux.


The dynamic characteristics of the electromagnet force are dominated by the dynamic

behaviour of the electromagnet coil current and the eddy currents within the cores.

These are a function of the coil inductance and resistance, the core dimensions and

resistivity, and the suspension air gap and core permeability. In addition to the

electromagnet core flux varying with force, the flux also varies with air gap (at constant

force) due to leakage flux varying with air gap. The dynamic behaviour of the

electromagnet flux is modelled by a first order lag with a time constant given by the

sum of the coil and eddy current time constants.

The accuracy of the steady-state model equations is very good. The characteristic

behaviour of the dynamic model equations is also good, but the predicted time constant

for the eddy current lag is about 3 times that of the measured time constant. This

discrepancy may be largely due to an erroneous estimate for the core resistivity.

However, further experimentation with a material of known resistivity is required to

validate the model time constant more fully.

Electromagnet force control 42

3

Electromagnet force control

3.1 Introduction

The multivariable vehicle suspension control system proposed in Chapter 1 requires

electromagnetic force actuators which are independent, and suitably linear and stable,

in order to decouple and control the vehicle mode motions successfully. This chapter

describes the steps involved in the synthesis and validation of a novel electromagnet

force control scheme which meets the above requirements.

The open-loop force/voltage transfer function of the electromagnet is first examined to

identify the dynamic structure and the parameters which characterise the experimental

suspension electromagnet. Existing force control strategies, which use linear algorithms,

are then examined and found to be unsatisfactory for providing independent and linear

force actuation. A novel control scheme, employing a detailed nonlinear model of the

electromagnet, is proposed to meet the force actuation requirements. After describing

the design of the proposed force controller, the results of some experimental

performance tests are presented and discussed. Finally, some conclusions are drawn on

the work described in this chapter.

Before examining the electromagnet transfer function, the operational envelope for the

experimental suspension electromagnet is defined, and some operational parameters are

identified and discussed.

3.2 Operational envelope

The experimental electromagnet has a mass of 7.3 kg and was designed to support a

maximum load of about 50 kg, giving a lift/weight ratio of about 7. A 50 kg load calls


for a maximum lift force capability of about 550 N to allow for some acceleration of

the suspended load. The design of the experimental electromagnet results in a large

leakage flux which limits the maximum air gap to about 5 mm for a lift force of 550 N

(see Figure 2.7). This allows a nominal operating air gap of 3 mm, with a deflection

of ±2 mm, to be realised. Mechanical limit stops are positioned to restrict the air gap

range to 0.5-5.5 mm. The current range required to suspend 50 kg is 3.5-20 A for air

gaps of 1-5.5 mm respectively. This gives a power/lift ratio of about 1.6 W/kg at the

nominal operating air gap.

The minimum suspended mass for the experimental suspensions is 15 kg per

electromagnet. Using Equation 2.22, the predicted air gap flux density varies from

about 0.35-0.65 Tesla over the full operational envelope. The corresponding predicted

electromagnet core flux density using Equation 2.24 is 0.45-1.50 Tesla. The operational

envelope for the electromagnet force actuator is summarised in Table 3.1. The

power/lift and lift/weight ratios for the experimental electromagnet are very similar to

those achieved by electromagnets with a nominal suspension load of 250-1000 kg.49

Table 3.1 Operational envelope for the electromagnet

Parameter Operating range

Lift force 0-550 N

Suspended mass 15-50 kg

Air gap 3 mm ±2 mm

Current 0-20 A

3.3 Electromagnet transfer function

A transfer function for the electromagnet is now derived to characterise its dynamic

behaviour and to analyse the nature of the force instability. The approximate model of

the electromagnet developed in Section 2.3.1 is used as the starting point. This model

is linearised around a nominal operating point to permit modelling of the electromagnet

in the frequency domain using the Laplace operator50 s.


Since the electromagnet is in practice excited by a variable voltage source, the

electromagnet force/voltage transfer function is derived and the location of the

open-loop poles and zeros is identified. Figure 3.1 illustrates the configuration of the

electromagnet and track and the variable nomenclature used in this chapter. To simplify

the analysis, the natural frequency of the electromagnet reaction rail is assumed to be

sufficiently high to be neglected.

Four equations which characterise the electromagnet motion for small perturbations

Figure 3.1 Electromagnet and track configuration

around a nominal operating point (io, co) are identified. These are then combined to

form the force/voltage transfer function. All equations are expressed in the frequency

domain using the Laplace operator s.

The air gap flux is the key element in the operation of the electromagnet, and it is

approximately proportional to the coil current divided by the air gap (see

Equation 2.27). This is linearised for small perturbations by:

and where ∆C, ∆Φ, and ∆I are the perturbations of air gap, its flux, and current

3.1∆Φ(s) kφi∆I(s) kφc

∆C(s) where kφi

∂φ(io,c

o)

∂i, kφc

∂φ(io,c

o)

∂c

respectively, and the partial derivative coefficients, kφi and kφc are both positive.


The electromagnet force, which is approximately proportional to the square power of

the air gap flux (see Equation 2.25), is linearised by:

and where ∆F is the force perturbation, and the partial derivative coefficient kfφ is

3.2∆F(s) kfφ∆Φ(s) where k

fφ

∂f(io,c

o)

∂φ

positive.

The acceleration of the suspended load due to the electromagnet force is now given by:

where m is the suspended mass.

3.3s 2 ∆C(s)∆F(s)

m

The final equation determines the relationship between electromagnet coil voltage, air

gap flux, and coil current (see Equation 2.26). This relationship is expressed by:

where ∆V is the coil voltage perturbation, N is the number of coil turns, and Rcoils is the

3.4∆V(s) sN ∆Φ(s) Rcoils

∆I(s)

coil resistance.

The force/voltage transfer function for the electromagnet is now obtained by substituting

current from Equation 3.1, flux from Equation 3.2, and force from Equation 3.3, into

Equation 3.4. The substitutions are detailed in Appendix B and result in:

This is more conveniently expressed as:

3.5∆F(s)

∆V(s)

kfφkφi

Rcoils

s 2

s 3 Nkφi/R

coilss 2 k

fφkφc/m

where:

3.6∆F(s)

∆V(s)

ki

Rcoils

s 2

s 3 Tflux

s 2 kc/m


The partial derivative coefficient ki represents the force/current gain factor, whilst kc

3.7k

ik

fφkφi

∂f

∂φ∂φ∂i

∂f

∂i, k

ck

fφkφc

∂f

∂φ∂φ∂c

∂f

∂c,

Tflux

Teddy

Tcoil

, Tcoil

N kφi/R

coils

represents the magnitude of the electromagnet’s negative stiffness. Tflux is the total flux

lag time constant (see Section 2.4.6), which is comprised of Teddy, the eddy current time

constant, and Tcoil, the coil time constant (see Appendix B).

Equation 3.6 shows that the force/voltage transfer function has two zeros at the origin

of the s-plane and three poles. The negative electromagnet stiffness generates a positive

denominator root, which places one of the poles in the right-hand side of the s-plane,51

thus producing an open-loop unstable force. Figure 3.2 illustrates the open-loop

electromagnet force/voltage transfer function, with the scope of the nonlinear elements

denoted by the two dashed boxes. The configuration of the eddy current flux lag loops

is identical to that of the coil current flux lag loop, but with a larger resistance to reflect

the smaller eddy current time constant. The eddy current loops are omitted from

Figure 3.2 for the sake of clarity. The open-loop transfer function is augmented in

Figure 3.2 with a second-order model of a flexible track which shows how the track

motion couples with that of the electromagnet. The track model has a natural

undamped frequency, ω, and damping ratio, ζ.

Figure 3.2 Block diagram of the electromagnet system


Table 3.2 shows how the measured parameters of the experimental electromagnet vary

as a function of air gap and suspension force. The partial derivative coefficients

defined in Equation 3.7, namely ki and kc, are calculated from the measured data used

to produce the graph shown in Figure 2.7. The coil and eddy current time constants are

taken from the measured data summarised in Tables 2.6 and 2.7 respectively.

Due to the nonlinear nature of the electromagnet, the location of the poles of the

Table 3.2 Linearised electromagnet model parameters

Air gap1 mm 3 mm 5 mm

Max

/minParameter

ki 140-400 N/A 50-90 N/A 25-50 N/A 16

kc 260-950 N/mm 95-350 N/mm 60-150 N/mm 16

Tcoil 111 ms 85 ms 72 ms 1.5

Teddy 4.6 ms 3.0 ms 2.1 ms 2.2

Notes: The coefficients ki and kc were measured over the force range of 150-550 N

transfer function varies with the operating point. The unstable pole reaches a maximum

value of s = 77, when kc=950 N/mm, m=15 kg, and Tflux=0.1 s, whilst the two remaining

poles form a complex conjugate pair at s = -44 ±70j. The worst case instability time

constant is therefore 13 ms.

In order to be able to neglect the effects of track flexibility and hence vibration, the

track poles must be sufficiently separated from the electromagnet force poles. For the

open-loop voltage controlled electromagnet, the track poles should ideally have

frequencies of at least 16 Hz. The acceptable natural frequency for any closed-loop

electromagnet configuration requires adequate separation of the track poles from the

dominant poles of closed-loop response.


3.4 Force control strategies

The force/voltage transfer function of the electromagnet has been shown to be

open-loop unstable due to the electromagnet’s negative stiffness. A control strategy is

now needed to provide electromagnetic force actuation with sufficient stability,

accuracy, linearity, and bandwidth.

The force actuation stability is most conveniently expressed in terms of the residual

electromagnet stiffness that is superimposed upon the controlled electromagnet force.

The acceptable level for residual stiffness clearly depends on the requirements of the

system using the electromagnet force actuator. For this application, the suspension

control system uses a suspension stiffness gain of 250 N/mm and is designed to

accommodate total transducer errors of up to ±25% (see Section 4.4.2). Therefore, a

force accuracy of ±20% with a corresponding residual stiffness of up to ±50 N/mm will

not significantly impair the suspension controller response. The linearity of the force

actuators will affect the success of the vehicle mode decoupling (see Chapter 5) and so

a force nonlinearity of no more than 10% is acceptable.

The suspension controller requires a force actuation bandwidth of about 50 Hz or more

(see Section 4.4.3). The ideal force actuation transfer function therefore consists of a

single pole low-pass filter with a time constant ≤ 3.2 ms. For small force perturbations,

a linear approximation between flux and force can be assumed. The dynamic behaviour

of the electromagnet force and flux is therefore considered to be equivalent for the

purpose of developing a force control strategy.

The most direct method to reduce the electromagnet force instability, to improve its

linearity, and to increase its force actuation bandwidth, is to apply force feedback, or

some functionally equivalent feedback such as acceleration or air gap flux. An

alternative approach is to target the cause of the instability, and use air gap feedback

to reduce the instability. Since the largest flux time constant for the force/voltage

transfer function is about 100 ms, an air gap feedback strategy must be augmented by

an additional technique to increase the force actuation bandwidth. Before discussing

these force control strategies, it is useful to consider the constraints imposed by the

electromagnet power control hardware.

In order to achieve an acceptable weight and cost for the electromagnet power control

hardware, the power conversion efficiency must be high, and the use of a switch-mode

power controller rather than a linear one is therefore required.52 This imposes an


upper limit on the voltage actuation bandwidth which is determined by the required

voltage and current ratings, and the type of semiconductor device used to switch the

power.53 An additional constraint on the power controller performance is due to the

provision of a finite voltage source, with the attendant potential problems of voltage and

hence current slew-rate saturation.

The steady-state electromagnet current ripple which arises due to the use of a

switch-mode power controller,54 causes the magneto-motive force of the electromagnet

to experience a similar superimposed oscillation. This continuous cycling of the

magneto-motive force increases the effective differential permeability of the core

material for small current perturbations. The use of a switch-mode controller therefore

ameliorates the control problems caused by the magnetic hysteresis of the electromagnet

and track cores.

3.4.1 Force feedback

The open-loop force/voltage transfer function (see Equation 3.6) has two zeros at the

origin and three poles, one of which is located in the right-hand side of the s-plane, thus

making the open-loop system unstable. However, the two zeros at the origin suggest

that if sufficient force feedback is applied, the unstable pole can be drawn in close to

the origin, thus reducing the force instability. Figure 3.3 illustrates the closed-loop root

locus of the electromagnet for a force feedback gain varying from 0 to 120. The

maximum gain is determined by limiting the time constant of the fast ‘flux lag’ pole

to 1 ms to accommodate an acceptable minimum response time for the power controller.

Attempts to use excessive gain would result in the additional pole due to the power

controller causing the left branch of the root locus to split away from the real axis. The

resultant pair of high frequency complex conjugate poles would produce an undesirable

oscillatory force response.

Figure 3.3 shows that the use of force feedback can improve the worst case time

constant of the unstable pole from its open-loop value of 13 ms to 44 ms. Since the

closed-loop stiffness is inversely proportional to the square power of the unstable pole

time constant,55 the effective negative stiffness is reduced by a factor of 11.

The principle problem with applying force feedback is that of measuring the force. In

the case of a single electromagnet suspending a load many times its own weight, a force

transducer could be located between the electromagnet and its load. However, for the


experimental vehicle, the measured force would suffer disturbances from the other

Figure 3.3 Closed-loop root locus for force feedback

(kc = 950 N/mm, m = 15 kg, Tflux = 116 ms, Gain=0-120)

electromagnets which are directly coupled to the vehicle chassis. Since independent

electromagnet stabilisation is required, it is unacceptable to transform the electromagnet

instability into a multi-variable instability problem. Therefore, direct force measurement

and feedback is not a feasible option. Flux and acceleration are two possible substitutes

for force feedback, so these are considered next.

3.4.2 Flux feedback

Since the electromagnet lift force is proportional to the square power of the air gap flux

(see Equation 2.25), the r.m.s. level of the flux over the entire pole face area represents

the lift force. This factor complicates matters because the pole face flux distribution

is not always uniform. This is due to three dominant factors. Firstly, when changes

in flux are demanded, eddy currents in the electromagnet and track cores cause flux to

be concentrated near the pole edges during the flux transient. Secondly, the motion of

an electromagnet along its track produces a similar effect, with track-borne eddy

currents causing a flux lag which is distributed spatially along the length of the

electromagnet pole face and across its width. Finally, gaps between track joints can

cause local flux disturbances.


Two different transducers have been used to measure flux in electromagnetic suspension

systems, namely Hall plate devices and search coils. Hall plate sensors56 must be

mounted near the electromagnet pole face to minimise the measurement of leakage flux.

This exposed location is environmentally harsh and the electromagnet core experiences

both low and very high temperatures which can cause problems with the delicate Hall

plate devices. Positioning the sensor on the pole face is an additional problem due to

the non-uniform distribution of flux across the pole face area. Use of laminated

electromagnet and track cores can significantly reduce the flux distribution problem, but

it is an expensive option. Alternatively, a number of sensors can be used, distributed

over the pole face to provide sufficient measurement accuracy and also redundancy to

cope with device failures.

Hall plate sensors can enable excellent flux control, both dynamically and in the

steady-state, although a nonlinear controller is needed for linear force actuation.

However, due to their questionable robustness, they have not advanced from laboratory

prototypes.

The prime benefit of flux control can be achieved more readily and robustly through

the use of a search coil wound around the electromagnet poles, close to the pole faces.

The coil voltage gives a measure of the rate of change of flux which must be integrated

to provide a measure of the flux. To avoid problems due to integrator drift, a low

frequency roll-off is required. The resultant lack of a d.c. response does not impair the

stabilisation function since the flux dynamics are the problem, not the steady-state flux.

However, linear control of the lift force is impossible since there is no absolute flux

level measurement.

The merits of the search coil sensor are that it senses the average flux level over the

whole pole face area, and it provides a cheap and robust method of reducing the force

instability. However its major disadvantage for this application is that it cannot produce

a linear force actuator.

3.4.3 Acceleration feedback

The final alternative to direct force measurement is to measure electromagnet

acceleration to derive the lift force. An acceleration signal is required by the

suspension controller, and is obtained by attaching an accelerometer to the

electromagnet. If a conventional secondary suspension is used, this technique is


available directly. However, the rigid coupling of the electromagnets and

accelerometers to the chassis of the experimental vehicle leads to a tightly coupled

multi-variable instability problem. Acceleration feedback, like force feedback, is thus

incapable of providing independent electromagnet stabilisation.

3.4.4 Air gap and current feedback

The electromagnet is open-loop unstable due to its negative stiffness coefficient. The

instability of the electromagnet force can therefore be reduced by using air gap feedback

with the controller stiffness set as close as possible to the magnitude of the open-loop

electromagnet stiffness. The force actuation time constant can then be significantly

reduced by using high gain current feedback. A current loop gain of 100 is sufficient

to reduce the coil current lag time constant from a worst case value of 110 ms down

to approximately 1 ms. The transfer function of the open-loop current controlled

electromagnet is given by Equation 3.6, but with Tflux now given by:

where kamp is the loop gain of the current controller (see Appendix B). The worst case

3.8Tflux

Teddy

Tcoil

kamp

time constant of the unstable pole for the current controlled electromagnet is 5.6 ms (for

kamp = 100). The higher force actuation bandwidth incurs the penalty of a faster

instability time constant relative to a voltage controlled electromagnet.

Air gap and current feedback can theoretically provide a stable and linear force actuator,

but only for a single operating point in terms of air gap and force. The problem with

this technique is that the negative stiffness coefficient (see Table 3.2) varies by a factor

of about 16 over the full operational envelope of the electromagnet. The closed-loop

root locus for the current controlled electromagnet with air gap feedback (see

Figure 3.4) shows that the closed-loop force becomes more unstable as the air gap

feedback gain error increases. If the feedback gain is too high, a pair of unstable

complex conjugate poles replaces the single unstable pole which is present if the gain

is too low. If the feedback gain is approximately equal to the electromagnet stiffness,

the two poles near the origin can be considered to cancel with the two zeros at the

origin. The resultant force transfer function is then dominated by the flux lag pole.


In order to match the variation in the open-loop stiffness over the full operational

Figure 3.4 Closed-loop root locus for air gap and current feedback

(kc=950 N/mm, m=15 kg, Tflux=5.7ms, Gain=0-2kc)

envelope, a nonlinear control algorithm is required. A disadvantage of this technique

is that in order to achieve accurate gain scheduling, the required control algorithm must

embody a complex model of the electromagnet (see Chapter 2). If sufficient accuracy

can be achieved in the model and signal measurements, a reasonably stable and linear

force actuator is theoretically possible. A further disadvantage associated with air gap

feedback stabilisation, is that unlike force or flux feedback, air gap feedback does not

encompass the magnetic hysteresis of the electromagnet and track cores.

Accurate measurement of the r.m.s. air gap over the full length of the electromagnet is

ideally required. In practice, a large area sensor and/or a number of smaller sensors

could be used to measure the average air gap. For example, using triple mode

redundancy, three sensors could provide average air gap measurement, with a graceful

degradation of accuracy if one of the sensors failed. Various industrial sensors using

inductive, capacitive and eddy-current techniques57 are available for non-contacting

displacement measurement.


3.4.5 Proposed force control strategy

The major problem associated with producing a well controlled suspension force from

an electromagnet stems from the variation of the parameters of the open-loop transfer

function rather than the fact that it is unstable. However, in reality it is only the

parameters of the linearised model which vary. The parameters of the nonlinear models

developed in Chapter 2 are mostly constant. The coefficient parameters of the static

force characteristic are determined by the physical dimensions of the electromagnet and

track, and the permeability of air, both of which are essentially constant. The dynamic

behaviour of the electromagnet is only slightly less ideal since temperature affects the

coil resistance and core resistivity which in turn affects the coil current and

eddy-current time constants respectively. With such detailed models of the nonlinear

behaviour of the electromagnet available, it would be unwise to discard most of the

information in order to use only classical linear control algorithms.

The only force feedback scheme which is suitably robust is flux derivative feedback

using a search coil. This can stabilise the electromagnet force but it cannot provide

linear force actuation. For electromagnetically suspended vehicles which use

conventional secondary suspensions, the electromagnets are effectively decoupled by the

secondary suspensions and they can therefore be considered to be independent. For

such systems, flux derivative feedback and/or acceleration feedback is employed58,59

to stabilise and linearise the force actuation.

Alternatively, since air gap feedback is required by the suspension controller, air gap

feedback stabilisation can be used with no additional transducer requirements.

However, to achieve satisfactory force stability and linearity, the air gap feedback gain

must be changed dynamically as a function of the operating point. The proposed

central element of the electromagnet force controller is therefore, air gap feedback into

a nonlinear model of the electromagnet static force characteristic.

The force controller must increase the open-loop force actuation bandwidth to the 50 Hz

required by the suspension controller. This corresponds to a force actuation time

constant of about 3.2 ms. The force time constant for the electromagnet is

approximated by the flux time constant which is composed of the coil and eddy current

lag time constants (see Equation 3.7). Since the worst-case open-loop coil and eddy

current time constants (see Table 3.2) are 111 ms and 4.6 ms respectively, both of these

time constants must be reduced. The massive reduction required for the coil time

constant can be most robustly achieved through the use of high gain current feedback,


so that is the proposed method. The current controller also enables coil resistance

changes due to varying temperature to be neglected. The additional hardware

complexity of an electromagnet current controller over a voltage controller is not great

and it can readily provide safe current limiting under fault conditions.

Since the reduction ratio required for the eddy current time constant is not large,

pole-zero cancellation using a phase-lead compensator60 is proposed to achieve the

necessary reduction. The increase in core resistivity due to elevated temperatures

reduces the eddy current time constant and is therefore neglected. The phase-lead

compensation should be applied to the electromagnet core flux demand, which

incorporates the leakage flux as well as the air gap flux. In a practical system, it would

be better to design the electromagnets to have an acceptable eddy current time constant

by using either narrower pole pieces, a higher resistivity core material, or laminated

cores (see Appendix A). This would reduce the power controller voltage requirement

and applies irrespective of the stabilisation strategy used.

Figure 3.5 illustrates the configuration of the proposed force control strategy. The eddy

Figure 3.5 Proposed force control configuration

current flux lag loop has been omitted for the sake of clarity. The detailed design of


the force control algorithm is described next. The implementation of all control system

components is described in Chapter 6.

3.5 Force controller design

The stabilisation and linearisation element of the proposed force control strategy

consists of feeding the measured air gap and the demanded force into a model of the

electromagnet, which then specifies the requisite current demand. The full nonlinear

model of the electromagnet is given by Equation 2.20 and is illustrated by Figure 2.7.

This equation cannot be used directly because the magnetic reluctance of the

electromagnet core depends on the core flux density. Therefore the core flux density

must be evaluated en route to the current demand. The full force control algorithm is

listed in Figure 3.6, where c is the air gap between the electromagnet and the track, y

is the lateral offset between the electromagnet and track poles (which is assumed

constant), p is the pole face width, and the air gap, track, leakage and electromagnet

reluctances are defined in Chapter 2.

The lateral offset in the control algorithm refers to the fixed pole offset due to the

difference in separation between the electromagnet poles and the reaction rail poles.

The electromagnet lateral offset is not measured in order to reduce the cost and

complexity of the experimental vehicle. Therefore, lateral displacement of the

electromagnet relative to the rail, results in a reduction of the suspension lift force. The

force reduction does not affect the linearity of the force actuation and only slightly

impairs the electromagnet stabilisation function. The same argument applies to the loss

of suspension force due to reaction rail eddy currents induced by the electromagnet

motion along its guideway.

To increase the open-loop force bandwidth to that required by the suspension controller,

the proposed force control strategy employs current feedback and phase-lead

compensation of the electromagnet core flux. The required force actuation time

constant is about 3.2 ms, whilst the worst-case open-loop coil and eddy current time

constants (see Table 3.2) are 111 ms and 4.6 ms respectively. To meet this

requirement, a current controller feedback gain of 100 is appropriate to reduce the

maximum coil current time constant to a maximum value of about 1.1 ms, giving a total

flux lag time constant of 5.7 ms. The total flux lag time constant can be further

reduced to a maximum value of 2.5 ms by employing pole-zero cancellation. This is

implemented in the controller by applying phase-lead compensation to the electromagnet


core flux demand using the compensator whose transfer function is given by:

Figure 3.6 Electromagnet force control algorithm (sequential program)

Fper_pole

Fdemand

2

1y/c tan 1(y/c)

1 πp/2c

gross force per pole

Φairgap

Fper_pole

2c

Rairgap

(c)air gap flux

Mairgaps track

Φairgap

2Rairgap

(c) Rtrack

air gap mmfs track mmf

Φleakage

Mairgaps track

/Rleakage

leakage flux

Φmagnet

Φairgap

Φleakage

magnet core flux

Φcomp

Phase_Leadeddy

(Φmagnet

) eddy current compensation

Mmagnet

Φcomp

Rmagnet

(Φmagnet

) magnet core mmf

Icoils

Mairgaps track

Mmagnet

/Ncoil_turns

current demand

The maximum value for the compensated flux time constant of 2.5 ms gives a minimum

3.9Phase_Lead

eddy(s)

1 s Tflux

1 s Tflux

/nwhere T

flux5.7 ms, and n 2.3

predicted force actuation bandwidth of about 64 Hz. The actual values used in the

experimental system are Tflux = 4.6 ms and n = 3, thus giving a slightly higher minimum

force bandwidth.

3.6 Force controller performance

The performance of the electromagnet force control algorithm cannot be conveniently

tested in isolation due to the residual closed-loop stiffness that always exists due to

imperfections in the control algorithm and its implementation. Therefore, a closed-loop

air gap controller was used to assist in testing the accuracy, linearity, and stability of

a practical implementation of the force control algorithm. The suspension control


system test results presented in Chapter 4 are used to validate the force actuation

bandwidth of the experimental force controller.

To impose a realistic test environment on the force controller, the suspension controller

developed in Chapter 4 is used to obtain test results for the experimental force

controller. The physical arrangement of the experimental single electromagnet

suspension rig used for the performance tests is illustrated in Figure 3.7. All measured

signals are provided by the control system from the sampled input data.

Two tests are used to determine the effectiveness of the force control strategy and

Figure 3.7 Single electromagnet suspension experimental rig

implementation. Both tests are performed over the operational range of air gaps and

suspension forces. The first test measures the static force accuracy and linearity by

plotting measured suspension forces versus reference force demands. The second test

determines the residual force actuation stiffness, and hence the residual force instability.

Figure 3.8 illustrates experimental static force measurements for suspended loads of

140, 220, 360 and 510 N. The graph plots the suspension load forces versus the

reference force demands required to generate those suspension forces. The

measurements were taken over the operational air gap range of 1-5 mm. All of the

force measurements are within the anticipated force error tolerance of ±15% except for

the 1 mm and 5 mm points with the 510 N load. These are out by just under 19%.

The larger error at the 1 mm air gap is attributed to slight bending of the electromagnet

support beam which causes air gap measurement errors. The higher error for the

maximum air gap is due to the larger inaccuracies involved in modelling the core

permeability at the very high core flux level at that operating point.


Figure 3.8 clearly shows that the proposed force controller is successful in producing

Figure 3.8 Experimental controller reference force demands and actual suspension

forces

a dominantly linear static force actuation over the full operational envelope. The force

nonlinearity, determined by measuring the maximum deviation from a straight line fitted

through all of the experimental data, is typically about ±5%, with a worst case value

of ±7% at an air gap of 1 mm.

The residual stiffness exhibited by the force controller is investigated by experimenting

with the closed-loop stiffness of the suspension controller by means of the air gap

feedback gain. The error integral action is first removed and then the controller air gap

feedback gain is reduced to determine the minimum value for which the system remains

stable. The worst case electromagnet negative stiffness predicted by Table 3.2 is

950 N/mm, which occurs at maximum load and minimum air gap. For the experimental

system, the minimum stable air gap feedback gain for the operating point (1 mm,

510 N) was about 50 N/mm. This represents a residual stiffness of 5% of the predicted

electromagnet stiffness. For smaller loads and larger air gaps, the minimum stable air

gap feedback gain was approximately 25 N/mm. This result shows that the proposed

force control strategy achieves a residual stiffness of 10-20% of the suspension

controller stiffness gain of 250 N/mm (see Section 4.4.2).


Table 3.3 summarises the measured performance of the experimental electromagnet

force controller in terms of its general accuracy, linearity and residual stiffness. As

expected, the performance of the experimental system was found to be highly dependent

on the accuracy of the air gap measurement. However, calibration of the air gap sensor

offset at the most critical operating point (at a gap of 1 mm), ensured a satisfactory

performance from an inexpensive industrial air gap sensor. The impact of magnetic

hysteresis is just noticeable at the minimum operational air gap, but it did not

significantly impair the behaviour of the force controller.

Table 3.3 Performance of the electromagnet force controller

Parameter Typical Worst case

Accuracy ± 15 % - 19 %

Linearity ± 5 % ± 7 %

Residual stiffness ≤ 25 N/mm ≤ 50 N/mm

3.7 Conclusions

The force/voltage transfer function of the experimental suspension electromagnet has

been analysed. Force control schemes employing linear feedback techniques have been

shown to be unsuitable for providing independent linear force actuation in an

environment where a number of electromagnets are rigidly coupled. Therefore, a new

force control algorithm has been proposed, which employs a detailed nonlinear

electromagnet model, in conjunction with air gap feedback, to provide an independent

force actuator for rigidly coupled electromagnets. The bandwidth of the electromagnet

force is increased through the use of closed-loop current feedback and series

compensation of the electromagnet core flux.

The proposed electromagnet force control scheme has been shown to possess significant

advantages compared with existing stabilisation techniques using flux derivative

feedback due to its dominantly linear force actuation. It does however, suffer from a

slight disadvantage due to its increased reliance on an accurate air gap measurement at

small air gaps. This drawback could be overcome if desired, by developing a hybrid


control approach combining flux derivative feedback, for stability, with the proposed

scheme, for force linearity. However, the test results from the experimental

implementation of the proposed force control strategy show that an acceptable

performance has been achieved. Therefore, this force control strategy will now be used

by the suspension control systems described in Chapters 4 and 5.

Suspension mode control 62

4

Suspension mode control

4.1 Introduction

The previous chapter describes an electromagnet control scheme which is capable of

producing an electromagnetic force actuation that is dominantly linear and stable. This

chapter now develops the sophisticated suspension mode control strategy proposed in

Chapter 1 and employs the aforementioned force actuator to provide a non-contacting

suspension system.

The suspension mode controller is a component of the multi-electromagnet vehicle

control system described in Chapter 5. However, it is first developed and tested using

a single electromagnet suspension configuration in order to simplify the design and

verification stages.

The description of the development of the suspension mode control system is partitioned

into six sections. First, the functional requirements of the suspension are identified.

A sophisticated suspension control strategy is then proposed, and the strategy is

developed by analysing the proposed closed-loop suspension system and synthesising

each system component. The characteristic behaviour of the passive lateral

electromagnet guidance motion is then briefly analysed. Finally, some simulated and

experimental responses are presented and discussed, and conclusions are then drawn on

the performance of the proposed system.

4.2 Functional requirements

The primary functional requirement for an electromagnetic suspension system is to

follow the general guideway profile whilst providing a quality of ride consistent with


passenger comfort.61 This requirement calls for a control system with two conflicting

aims. Guideway following requires the suspension to be effectively coupled to the

track, whilst ride comfort considerations require the suspension to be isolated from track

irregularities.

Passenger ride comfort is a subjective measure which depends on the individual in

Figure 4.1 ISO reduced comfort boundary (for a 1 hour exposure)

question. However, it has been recorded that irrespective of the individual in question,

the human body is particularly sensitive to vibration over a frequency range of about

1-20 Hz. In 1974 the International Standards Organisation (ISO) produced a guide62

which incorporated a specification for vertical acceleration versus frequency

corresponding to the threshold between human comfort and discomfort. For urban

transport applications, the ISO’s one hour exposure characteristic is appropriate (see

Figure 4.1) for which the peak acceleration limit is 0.4 m/s2 over the frequency range

4-8 Hz.

For an electromagnetic suspension with no conventional secondary suspension (ie. no

springs and dampers), suspension travel is limited to the operating air gap range of the

electromagnet. This suspension configuration therefore requires a stiff track, with a

deflection restricted to about 50% of the normal operating air gap deviation.


The ride comfort analysis for vehicle suspensions is generally performed by considering

the track roughness.63 However, since the guideway for the proposed low-speed

system consists of stiff, smooth track sections, which are joined together, a ride quality

specification in terms of a worst case track joint misalignment is more convenient. A

suspension designed to accommodate significant track steps is desirable since it permits

cost savings in track production and maintenance.64 Therefore, after allowing for

operational air gap deviations, it is desirable for the suspension to accommodate a

worst-case track step size of 50% of the nominal operating air gap deviation. The

suspension acceleration when negotiating such a track step must be within the ISO

comfort specification (see Figure 4.1). To improve upon existing light rail systems,

which have a typical peak vertical acceleration65 level of 0.4 m/s2, a design target of

0.2 m/s2 is considered desirable for the proposed electromagnetic suspension system.

In addition to the track oriented functional requirements discussed so far, the suspension

system must reject disturbance forces due to variations in the weight of the payload,

wind gusts, and vertical forces generated by the linear inductance motor66 which

propels the vehicle along its guideway. Accommodation of disturbance forces of

30-50% of the maximum vehicle weight is typically required.67 Therefore, the design

target for the proposed system is for a transient air gap deflection of 30-50% of the

operating gap deviation, for a disturbance force equal to 30-50% of the maximum

suspended weight. In addition, there must be no steady-state air gap deflection due to

disturbance forces in order to maximise the available air gap deviation.

Having identified the functional requirements of the electromagnetic suspension, the

suspension system structure proposed in Chapter 1 is now examined and developed.

4.3 Suspension control strategy

The vehicle suspension control system structure proposed in Chapter 1 is applied to a

single electromagnet suspension as illustrated in Figure 4.2. The suspension strategy

involves the calculation of the absolute track position, which is then suitably processed

to provide an absolute position reference demand for the high stiffness electromagnet

position controller. The force actuation demand from the position controller is then fed

to the electromagnet force controller which is described in Chapter 3. Strategies for

designing the electromagnet position controller and the guideway following algorithm

are outlined next.


The electromagnet position controller is the heart of the suspension system. It must

Figure 4.2 Suspension control system hierarchy

control the absolute position of the suspension with zero steady-state error, and it must

also resist disturbance forces and accommodate load variations. The minimum

properties required of the position controller therefore include stiffness, damping, and

position error integral action.

Implementing this control system structure is complicated by the lack of a physical link

between the electromagnet and an absolute datum. Measurement of the absolute

position of the electromagnet is therefore not readily achievable, but absolute

acceleration can be cost-effectively measured using a standard industrial accelerometer.

The acceleration measurement must be integrated to give the absolute velocity, which

must in turn be integrated to give the absolute position. Pure integrators would suffer

from drift problems due to erroneous offsets, so a high-pass filter must augment each

integrator. Since acceleration is measured, the position controller can incorporate

acceleration feedback to assist in the control task. The proposed state-vector used for

feedback thus consists of position, and integral of position error, together with output

velocity and acceleration.

The absolute position of the track can be calculated using the absolute electromagnet

position and the electromagnet air gap measured using an industrial non-contacting


displacement sensor. The function of the guideway following algorithm is to receive

the track position and to transform it to a suspension position reference signal which

avoids contact with the track and which provides a comfortable ride quality. The

design of the guideway following algorithm depends on several factors. The most

important ones are the operational air gap deviation, the vehicle speed, the size of track

discontinuities, and finally, the parameters of track gradient entries and exits.

Since acceleration is the second derivative of displacement, a guideway following

algorithm consisting of a second-order low-pass filter can be used to limit acceleration

for a given size of track discontinuity on an otherwise flat guideway. However, Pollard

and Williams68 showed that the use of solely linear control algorithms presents

considerable difficulties and operational limitations when designing for gradient entries

and exits. It is therefore envisaged that the use of techniques such as

matched-filtering69 could provide better performance than traditional linear filtering.

For example, a matched-filter could be programmed with the functions used to define

the gradient entries and exits. These may consist of straight track segments, smoothly

curved sections, or a combination of both. Having identified the underlying guideway

profile curvature, linear filtering may then be suitable for rejecting discontinuities

superimposed on the curved profile.

If a guideway profile is encountered which cannot be comfortably negotiated, the

guideway following algorithm must attempt to limit the air gap deviation in order to

prevent the mechanical air gap limit stops from being reached. Similar techniques may

also be required for lateral guidance of the suspension.

The design of suitable guideway following algorithms, for both vertical and lateral

motion, represents a large research project in itself and is peripheral to the main

objective of this research work. Therefore, for the experimental research system, the

use of a second-order linear filter is proposed for the vertical motion guideway

following algorithm. In order to reduce the cost and complexity of experimental system

hardware, the lateral motion is not actively controlled.

The logical arrangement of the suspension system is outlined in Figure 4.3, and the

configuration of the proposed suspension control system is illustrated in Figure 4.4.

Table 4.1 lists the control system variables and parameters.


Inspection of Figure 4.4 shows that the position error signal consists of a component

Figure 4.3 Logical arrangement of the suspension system

Figure 4.4 Suspension control system configuration

of both the absolute position signal and the position signal relative to the track. The

proportion and frequency spectrum of each component is determined by the guideway

following algorithm. The coupling of the electromagnet suspension to the track is

therefore dominated by the guideway following algorithm. In order to be able to isolate

the suspension design procedure from the guideway dynamics, a sufficient margin must

be provided between the natural frequency of the track and the suspension to track

coupling frequency.

The analysis and synthesis of the position controller, the guideway following filter, and

the state integration filters are described next.


Table 4.1 Suspension control system nomenclature

Identifier Variable / Parameter

Z(s) Electromagnet absolute position (ˆ - calculated position)

P(s) Track absolute position (ˆ - calculated position)

C(s) Electromagnet to track air gap

Zref(s) Position controller reference signal

Fdemand(s) Position controller force demand

Fdisturb(s) Disturbance force input

kacc Acceleration feedback gain (virtual mass)

kvel Velocity feedback gain (damping)

kpos Position feedback gain (stiffness)

Terr Position error integral action time constant

ωint State integration filter corner frequency

Tforce Electromagnet force actuation time constant

m Suspended mass

4.4 Synthesis of the suspension control system

The proposed suspension control system is developed by analysing the closed-loop

system and then synthesising each system component. The position, velocity and

acceleration feedback gains are designed first, then an acceptable force actuation

bandwidth is determined. Next, a suitable error integral action time constant is chosen.

Finally, the guideway following algorithm is designed, and an acceptable cut-off

frequency for the state integration filters is determined. The development is concluded

by identifying a simplified transfer function which characterises the suspension system

response.


4.4.1 Position control transfer function

In order to be able to design the position controller and the guideway following

algorithm on a largely independent basis, two restrictions must be imposed on the

guideway algorithm. Firstly, the guideway algorithm must pass the track position signal

for frequencies below ωint so that the bandwidth of the position error signal extends

down to dc. Also, the guideway following algorithm must have a gain not exceeding

unity at all frequencies so that the position error feedback gain is determined solely by

kpos. In practice, these are fundamental requirements of all guideway following

algorithms and therefore they do not impose any operational restrictions.

The Laplace transform of the full position control law illustrated in Figure 4.4 is given

by:

The acceleration of the suspended load due to the control force demand and the

4.1

Fdemand

(s)

kpos

kpos

sTerr

s 2 Z(s)

s ωint

2Z

ref(s) k

vel

s 2 Z(s)

s ωint

kacc

s 2 Z(s)

disturbance force (see Figure 4.4) is given by:

The Laplace transfer function for the full closed-loop position controller is obtained by

4.2s 2 Z(s)F

demand(s)

m 1 sTforce

Fdisturb

(s)

m

substituting Equation 4.1 into Equation 4.2. After rearranging and collecting terms (see

Appendix C), the reference position transfer function is given by:

and the disturbance force transfer function is given by:

4.3

Z(s)

Zref

(s)

kpos

s 1/Terr

s ωint

2

s 6 mTforce

s 5 kacc

m 2ωint

mTforce

s 4 kvel

2ωint

(kacc

m) ω2

int mTforce

s 3 kpos

ω2

int(kaccm) ω

intk

vels 2 k

pos/T

err

The reference position transfer function has 6 poles and 3 zeros so the closed-loop

4.4

Z(s)

Fdisturb

(s)

1 sTforce

s ωint

2

s 5 mTforce

s 4 kacc

m 2ωint

mTforce

s 3 kvel

2ωint

(kacc

m) ω2

int mTforce

s 2 kpos

ω2

int (kaccm) ω

intk

vels k

pos/T

err

position control system is a sixth-order system. However, since the guideway following

algorithm passes frequencies below ωint, the position feedback signal can be assumed


to be dc coupled. This results in the transfer function zeros due to the state integration

filters moving to the origin of the s-plane, where they cancel with the poles at the

origin. The closed-loop response is therefore effectively that of a fourth-order system,

with two poles due to the sprung mass, one pole due to the position error integral

action, and another pole is contributed by the force actuator. Furthermore, the high

stiffness required of the position controller permits the use of a slow error integration

time constant. The position controller can therefore be designed as a dominantly

third-order system with a fourth pole located close to the origin due to the error integral

action. If the force actuation time constant is sufficiently small, then the system

reduces to a dominantly second-order system. However, since the electromagnet force

actuation bandwidth is limited by cost and complexity considerations, the force

actuation pole cannot be neglected.

Synthesis of the position control algorithm can be considerably simplified if the

dominant response of the controller is that of a second-order system.70 To permit this

design approach, the position control system is reduced to a second-order system by

assuming infinite force bandwidth, by neglecting error integral action, and by assuming

that the suspension velocity and position are measured directly. After designing the

state feedback gains for the reduced-order system, the constraints for the neglected

terms are calculated to ensure an acceptable full closed-loop transfer function.

Since the guideway following algorithm determines the passenger ride characteristic,

the detailed characteristic behaviour of the position controller transfer function is not

critical. The primary requirements are that the position controller bandwidth is

sufficiently high, and that the response is adequately damped. In practice, the high

stiffness required for the force disturbance rejection ensures sufficient bandwidth, and

the damping is an independent design factor.

The reduced-order position transfer function is obtained by substituting values of

Tforce = 0, Terr = ∞ and ωint = 0 into Equations 4.3 and 4.4. This substitution gives:

The characteristic behaviour of this second-order transfer function can be conveniently

4.5Z(s)

kpos

s 2 (kacc

m) skvel

kpos

Zref

(s)1

s 2 (kacc

m) skvel

kpos

Fdisturb

(s)

expressed in terms of its undamped natural frequency, ωn, and damping ratio, ζ. The

standard form for a second-order transfer function and the location of its poles is given

by:


The undamped natural frequency and damping ratio of the closed-loop position control

4.61

s 2/ω2

n 2ζ s /ωn

1with poles at: s ω

nζ ±ω

nζ2 1

transfer function (Equation 4.5) are thus given by:

4.7ωn

kpos

kacc

mζ

kvel

2 (kacc

m)kpos

4.4.2 Position, velocity and acceleration feedback gains

The most exacting requirement for the position controller is the rejection of disturbance

forces. The controller must limit the transient deflection due to a force disturbance of

50% of the full load force, to half of the maximum suspension deviation (see

Section 4.2).

The maximum total suspended load for the experimental electromagnet is 50 kg and its

nominal operating air gap is 3 mm with an operational deflection of up to ±2 mm (see

Table 3.1). Therefore, accommodation of a 50% load change, within half the

operational air gap deflection, requires a suspension stiffness, kpos, given by:

kpos = 50% × 50 kg × 9.81 Nkg-1 ÷ 1 mm ≈ 250 N/mm

If kacc is initially assumed to be zero, kpos = 250 N/mm and m = 15 kg, then the

undamped natural frequency of the position controller (see Equation 4.7) is 20 Hz,

while for m = 50 kg, the undamped natural frequency falls to 11 Hz. The use of

acceleration feedback can reduce this variation in natural frequency due to a load

change by providing ‘virtual mass’. Acceleration feedback is functionally equivalent

to force feedback, so it effectively increases the force bandwidth and enhances linearity

and stability. However, due to the constrained achievable force bandwidth (see

Chapter 3), kacc must be kept to a sensible minimum. The breakpoint for the

effectiveness of acceleration feedback in ameliorating the impact of load changes occurs

when kacc = minimum mass, so this is proposed as an acceptable compromise. With

kacc = 15 kg, the undamped natural frequency is reduced to 15 Hz and 10 Hz at no load


and full load respectively. The acceleration feedback gain is effectively equivalent to

a force feedback gain of 1 at the minimum load.

Since the underlying electromagnetic force actuator is open-loop unstable, a well

damped response is considered prudent in order to produce a robust suspension which

is insensitive to force actuation errors by the electromagnet force controller. In

addition, since the operational air gap range is limited, overshoot of the position

response is undesirable. Therefore, a worst case damping ratio given by the critical

damping ratio71 of 0.707 is required. To allow a margin for feedback gain deviations

due to transducer errors of up to ±25%, a minimum nominal value for ζ of 0.8 is

required. This requires a velocity feedback gain (see Equation 4.7) of

kvel = 6500 N/(m/s), which gives a nominal damping ratio of 0.8 and 1.2 at full and

minimum load respectively. In practice, the minimum damping ratio will be higher if

the suspended load is a passenger. This is because the passenger’s mass will not be

rigidly coupled to the vehicle at the natural frequency of the position controller.

4.4.3 Force actuation bandwidth

Having designed the response of the reduced-order position controller, the required

force actuation bandwidth is now determined. The transfer function for the position

control system, neglecting only the low frequency components due to the state

integration filters and position error integral action, is obtained by substituting values

of Terr = ∞ and ωint = 0 into Equation 4.3. This substitution yields:

Table 4.2 lists the location of the closed-loop poles for minimum and maximum

4.8Z(s)

Zref

(s)

kpos

s 3 mTforce

s 2 (m kacc

) skvel

kpos

suspended masses over a range of force actuation bandwidths. The location of the poles

is strongly influenced by both the force actuation bandwidth and the suspended mass,

and the damping ratio relates to the complex conjugate pole pair or the dominant pair

of real poles. As expected, the table shows that the damping ratio falls sharply once

the force actuation bandwidth approaches that of the reduced-order position control

system. To ensure an adequately damped response, and to allow a margin for

transducer errors, a force actuation bandwidth of about 50 Hz is required, which is

equivalent to a force time constant, Tforce = 3.2 ms.


Table 4.2 Variation of closed-loop poles with force bandwidth

Force

bandwidth

Closed-loop poles

(m=15kg)

Closed-loop poles

(m=50kg)

Hz rad/s Location ζ Location ζ

∞ ∞ -∞, -167, -50 1.19 -∞, -50 ±j37 0.80

1000 6283 -12347, -170, -50 1.19 -8067, -50 ±j37 0.80

100 628 -993, -214, -49 1.28 -708, -55 ±j38 0.82

50 314 -290 ±j155, -48 0.88 -284, -62 ±j41 0.83

40 251 -227 ±j188, -48 0.77 -190, -69 ±j44 0.84

30 188 -165 ±j197, -47 0.64 -88, -78 ±j67 0.75

20 125 -102 ±j186, -46 0.48 -63, -50 ±j86 0.50

4.4.4 Position error integral time constant

The final feedback gain factor to be determined is the time constant for the position

error integral action. Since the stiffness of the controller is high, the functional

requirement for the integral action, that of eliminating steady-state position errors, can

be performed quite slowly. This enables the closed-loop pole due to the integral action

to be placed close to the origin of the s-plane where it has negligible impact on the

other system poles. The slowest closed-loop pole is located at s = -48 (see the

highlighted row in Table 4.2), and has a time constant of 21 ms. Therefore, a value for

the integral action time constant, Terr, of 1 second will cause negligible disturbance of

the other closed-loop poles. Position error integral action is thus introduced without

impairing the dynamic response or damping of the position controller.

4.4.5 Guideway following algorithm

The guideway following algorithm is required to limit the acceleration due to a ±1 mm

step change in track height to about ±0.2 m/s2 (see Section 4.2). Since acceleration is

the second derivative of position, the proposed second-order low-pass filter can be


designed to limit the acceleration to any desired level for a given track step size. The

Laplace transform of such a filter, with a damping ratio of 1, is given by:

where ωfollow is the guideway following filter corner frequency. For a step input of

4.9Guideway_filter(s)ω

follow

s ωfollow

2

amplitude ∆pos, the acceleration of the filter response is given by:

The corresponding acceleration response in the time domain is given by:

4.10Acc(s) s 2 ∆pos

s

ωfollow

s ωfollow

2∆pos

s ωfollow

s ωfollow

2

Inspection of this time domain response reveals that the peak acceleration occurs at time

4.11acc(t) ∆pos ω2

follow eω

followt

1 ωfollow

t

t=0. The guideway filter time constant required to limit the acceleration to accmax is

thus obtained by setting t=0 and acc(t)=accmax. After rearranging, this gives:

Therefore, to accommodate a track step size of ±1 mm, with a peak acceleration of the

4.12ωfollow

accmax

∆pos

filter output of ±0.2 m/s2, a low pass filter, with two poles at s = 10,10 is required.

However, the poles of the closed-loop position controller transfer function limit the

initial acceleration of the suspension to zero. Since an exact numerical solution for the

maximum acceleration of the full suspension control system is unduly complicated, the

two pole guideway following filter is simulated along with a representative pole from

the position controller transfer function. Examination of Table 4.2 shows that the

position controller can be effectively modelled for this purpose by a single pole at

s = 48. The simulation results show a peak acceleration for three poles at s = 10,10,48

of 0.05 m/s2, which is much lower than that required. The guideway following filter

poles are therefore moved so that s = 25,25,48 where they produce a peak acceleration

of 0.21 m/s2. The bandwidth of the guideway following filter, ωfollow is thus set to 25

rad/s, and the overall track/suspension position response is approximated by three poles

at s = 25,25,48.


To prevent the suspension system from exciting track vibrations, the natural frequency

of the track must be above the track following frequency of 4 Hz.

4.4.6 State integration filters

The position controller synthesis has so far assumed dc coupled feedback signals for

acceleration, velocity and position. However, since the state integration is augmented

with high-pass filtering, it is necessary to determine the constraints that must be

imposed upon the filter corner frequency in order to limit the dislocation of the

closed-loop poles by the filters.

Neglecting the high frequency component due to the force actuation time constant, the

transfer function of the closed-loop position control system is obtained by substituting

the value Tforce = 0 into Equation 4.3. This substitution gives:

The integration filter corner frequency, ωint, is present in the characteristic polynomial

4.13

Z(s)

Zref

(s)

kpos

s 1/Terr

s ωint

2

s 5 kacc

m s 4 kvel

2ωint

(kacc

m) s 3 kpos

ω2

int(kaccm) ω

intk

vels 2 k

pos/T

err

as a factor in the s4 and s3 terms. These terms are therefore used to determine

constraints on ωint such that the characteristic polynomial is not materially altered by

the integration filters. The first term produces the constraint given by:

The second constraint is simplified using the first constraint and is given by:

4.142ωint

(kacc

m) kvel

∴ ωint

kvel

2(kacc

m)

For the controller feedback gains determined earlier (see Table 4.3), the filter corner

4.15ωint

ωint

(kacc

m) kvel

kpos

∴ ωint

kpos

kvel

frequency, ωint, must be much lower than 38 rad/s.

An additional constraining factor is due to the bandwidth of the guideway following

algorithm. Since this provides a relative position signal at frequencies below 25 rad/s

(see previous section), the state integration filters must pass frequencies above 25 rad/s.


Therefore, ωint must be much less than 25 rad/s in order to be able to assume that the

position feedback is dc coupled, and hence assume that the state integration filter zeros

cancel the poles at the origin of the closed-loop transfer function (see Equation 4.13).

The low frequency limit that can be achieved in practice is determined by the system

implementation (see Chapter 6). The limiting factors include the hysteresis,

non-linearity, and resolution of the accelerometer and its analogue-to-digital converter,

along with the numerical techniques and precision used to implement the filters. Since

achieving a very low frequency response is costly, an acceptable performance and cost

trade-off is required.

Experimentation with a sensor which met the other system requirements demonstrated

that a low frequency cut-off of 0.6 rad/s is achievable. With ωint = 0.6 rad/s, the two

state integration filters reduce the position signal amplitude by a total of 5-7.5%

at 15-25 rad/s. Due to offsets present within the analogue parts of the experimental

system, two additional high-pass filters are applied to the measured acceleration signal

before it can be used. For frequencies of 15-25 rad/s, this gives rise to a total loss of

position signal amplitude of 10-15%. An overshoot of approximately this size is

therefore expected on the closed-loop suspension position response to a track step. The

size of the overshoot is increased further by the velocity feedback signal. If desired,

it may be possible to compensate for most of the position signal gain loss by increasing

the gain of the air gap signal to match the low frequency gain loss of the position

signal.

4.4.7 Suspension controller design specification

By adhering to the constraints identified in the preceding section for the state

integration filter bandwidth, the closed-loop transfer functions for the position controller

(see Equations 4.3 and 4.4) can be expressed more simply by:

and

4.16Z(s)

Zref

(s)

kpos

(s 1/Terr

)

s 4 (mTforce

) s 3 (kacc

m) s 2 kvel

skpos

kpos

/Terr


The closed-loop suspension/track position transfer function is obtained by augmenting

4.17Z(s)

Fdisturb

(s)

s (1 sTforce

)

s 4 (mTforce

) s 3 (kacc

m) s 2 kvel

skpos

kpos

/Terr

Equation 4.16 with Equation 4.9. This gives:

This is dominated by the guideway following filter and can therefore be approximated

4.18Z(s)

P(s)

ω2

follow kpos

(s 1/Terr

)

s ωfollow

2 s 4 (mTforce

) s 3 (kacc

m) s 2 kvel

skpos

kpos

/Terr

by Equation 4.9 alone, giving:

The suspension control system parameters designed earlier are summarised in Table 4.3,

4.19Z(s)

P(s)≈

ω2

follow

s ωfollow

2

and the closed-loop poles and zeros that these produce (see Equation 4.18) are listed

in Table 4.4.

Table 4.3 Suspension control system parameters

Parameter Name Value

Stiffness kpos 250 N/mm

Damping kvel 6.5 N/(mm/s)

Virtual mass kacc 15 kg

Error integral time constant Terr 1 s

Force actuation time constant Tforce 3.2 ms

State integration filter frequency ωint 0.6 rad/s

Guideway following frequency ωfollow 25 rad/s


4.5 Lateral guidance

Table 4.4 Closed-loop suspension system zeros and poles

MassPosition Controller Guideway Filter

Zero Poles ζ complex.poles Poles

15 kg -1 -290 ±j157, -48, 0 0.88 -25, -25

50 kg -1 -281, -63 ±j40, 0 0.84 -25, -25

In order to reduce the complexity of the implementation of the experimental vehicle and

single electromagnet rig, provision has not been made for lateral force control. The

problem of controlling the lateral force is minor compared with that of controlling the

lift force, since the lateral behaviour of the electromagnetic suspension is open-loop

stable.

Without active control, the suspension experiences lateral stiffness due to the geometry

of the shear flux between the electromagnet and track poles. However there is

negligible lateral damping so a brief analysis to ascertain the lateral stiffness and natural

frequency of the lateral motion is considered prudent.

For lateral offsets of up to 2/3 of the electromagnet pole width, the lateral force (see

Equation 2.10) is approximated by:

where Fmagnet is the gross electromagnet force, c is the air gap, y is the lateral offset, and

4.20Flateral

Fmagnet

2c

πptan 1

y

c

p is the pole width. The lateral stiffness can therefore be approximated by:

where mt is the total suspended mass and ag is the acceleration due to gravity.

4.21klateral

Flateral

ym

ta

g

2c

πpytan 1

y

c

Although this is a nonlinear function, it can be linearised for small lateral offsets by:


4.22klateral

≈2m

ta

g

πpfor y<c

The laterally sprung mass is therefore assumed to have a dominantly second-order

response, with the natural undamped angular frequency (see Section 4.4.1) of the lateral

motion approximately given by:

where mc is the effective, rigidly coupled suspended mass.

4.23ωn

≈2m

ta

g

mcπp

For any suspension load which is rigidly coupled to the electromagnet, the model

predicts a natural undamped frequency of 4 Hz. However, with a human passenger

load, the natural frequency of the human body lateral coupling is likely to be below

4 Hz, and so the natural undamped frequency of the lateral motion could be as high as

7 Hz. In practice, it is likely to lie somewhere in the range of 4 Hz to 7 Hz. The

natural lateral suspension frequency is thus similar to that designed for the vertical track

following algorithm. A beneficial side effect of human body to vehicle coupling for

the experimental systems is that it provides a degree of lateral damping.

The presence of the air gap in the lateral force model (see Equation 4.20) presents the

interesting possibility of providing lateral damping by controlling the lateral force

through modulation of the air gap. Clearly, such a technique would impose a

disturbance on the vertical motion, but it would provide lateral damping without the

need for additional electromagnets. However, since no problems have been experienced

with regard to lateral motion oscillations, this technique has not been investigated.

4.6 Performance of the experimental mode suspension

In order to validate the theoretical basis of the design of the proposed suspension

control system, the system was simulated and various step responses were obtained and

examined. An experimental single electromagnet suspension system was then

developed (see Chapter 6) and the experimental system was tested by comparing a

number of simulated and experimental responses.


The control system was simulated using the Advanced Continuous Simulation Language

(ACSL)72 which provides high level simulation constructs and has powerful signal

monitoring capabilities. The model of the suspension control system is listed in

Appendix C and incorporates a discrete-time suspension controller with analogue-digital

conversion quantisation, and continuous-time models of the electromagnet and control

system transducers.

Figure 4.5 (Figure 3.7 repeated) shows the physical arrangement of the experimental

Figure 4.5 Single electromagnet suspension experimental rig

single electromagnet test rig in which the long pivoted beam allows the electromagnet

and sensors to move with negligible stiffness and damping in the vertical direction. The

control system was designed using the suspension configuration shown in Figure 4.4

with the parameters defined in Table 4.3.

Step changes in track height are simulated by injecting an offset into the track profile

calculation. The full bandwidth response of the position controller is tested by

bypassing the guideway following algorithm, thus effectively setting the guideway

following filter bandwidth to infinity. This results in the position error signal consisting

solely of the air gap signal with no contribution from the absolute position signal. After

testing the position controller response, the system is reconfigured with the correct

guideway following filter to verify that the suspension meets the required ride comfort

specification.

4.6.1 Position controller

Figure 4.6 and Figure 4.7 show the simulated and experimental position controller

responses respectively, for a 1 mm step size, and a suspended mass of 15 kg.

Figure 4.8 and Figure 4.9 show the respective responses for a suspended mass of 45 kg.


The air gap responses are well damped, with a small, very low frequency overshoot.

Figure 4.6 Simulated suspension response to a 1 mm air gap reference step

(ωfollow = ∞ rad/s, m = 15 kg)

Figure 4.7 Experimental suspension response to a 1 mm air gap reference step


The 7% overshoot on the simulated responses is due solely to the high-pass filtering of

the velocity feedback signal. This is augmented on the experimental responses by an

additional 5% overshoot due to the force actuation error as the electromagnet operating

point changes with the air gap. The experimental response for the 45 kg suspended

mass has a further 5% overshoot (bringing its total to 17%) which can be attributed to

the damping ratio being slightly below the critical damping ratio.

The sharp peaks for the acceleration responses are caused by current slew rate limiting

in the electromagnet current controller. This occurs because the force slew rate is


designed for comfortable passenger acceleration levels rather than the high levels

Figure 4.8 Simulated suspension response to a 1 mm air gap reference step


Figure 4.9 Experimental suspension response to a 1 mm air gap reference step


experienced during these tests. The time to the acceleration peak is approximately 9 ms

for both the experimental and simulated responses. The peak acceleration amplitudes

of the experimental responses are 10-20% lower than the simulated responses which

suggests that the bandwidth or force slew rate of the experimental system is slightly

lower than the design target.

The low amplitude oscillation superimposed on the transient experimental acceleration

responses (up to time = 150 ms) is partly due to the current slew rate limiting, but is

mostly due to the force pulse causing the experimental rig to vibrate. The natural


frequency of the rig is about 60 Hz and the oscillation decays rapidly. However, the

natural frequency can be reduced down to about 10 Hz by using flexible rubber

mountings. As expected, at rig natural frequencies close to that of the position

controller (approximately 10-15 Hz), the rig could be excited with a step position

reference to produce continuous steady-state oscillation. The rig oscillation presented

no electromagnet stability problems, but it is obviously unacceptable. Setting the rig

natural frequency to double that of the air gap controller is sufficient to cause any

oscillation to decay rapidly.

The acceleration ripple which is apparent in the steady state on both the experimental

and simulated responses is due to limit cycling73 of the air gap. The period and

magnitude of the limit cycle is primarily a function of the air gap measurement

resolution and hysteresis, and conversion quantisation, but it is also affected by the

controller sampling period and the electromagnet operating point. The amplitude of the

limit cycles for the experimental responses is approximately one third that of the

simulated responses. This is attributed to measurement noise in the experimental

system increasing the effective analogue to digital conversion resolution.74

The results discussed above give confidence in the linearity and bandwidth of the

electromagnet force actuator developed in Chapter 3. However, additional responses

were obtained to test the performance of the suspension over its full operational

envelope. Figure 4.10 presents the results of a test in which three step responses were

obtained, each stepping up and then down by 1 mm, from initial air gaps of 1.5, 2.5 and

3.5 mm. The consistent responses demonstrate the fact that the electromagnet force

actuator provides an acceptably linear response over the full operational air gap range.

The disturbance force rejection requirement of the suspension is tested by rapidly

applying and removing a 15 kg load. This produces a peak air gap deflection of

±0.73 mm, whilst the expected theoretical deflection is ±0.6 mm. The 20% discrepancy

is attributed to sensor and force actuation errors. The air gap deflection recovers by

90% within 2 seconds of the application of the force disturbance, and is subsequently

eliminated.

The overall correspondence between the steady-state and dynamic characteristics of the

simulated and experimental responses is good, and the air gap response is well damped.

Further position controller test results, including disturbance force rejection and the

frequency domain response are presented in Chapter 5 for the multi-electromagnet

vehicle.


Figure 4.10 Experimental suspension responses to three 1 mm air gap reference steps


4.6.2 Full suspension system

The ability of the suspension control system to meet the required ride comfort level is

now examined. Since the reaction rail of the experimental electromagnet rig cannot be

moved, track steps are simulated by injecting a step disturbance into the track position

calculation before the guideway following algorithm. Figure 4.11 and Figure 4.12

illustrate the simulated and experimental responses respectively, for a (simulated) 1 mm

step in track height, with a suspended mass of 15 kg. Figure 4.13 and Figure 4.14

show the respective responses for a suspended mass of 45 kg.

The peak acceleration of the simulated responses is 0.21 m/s2 which is very close to the

design value of 0.21 m/s2. The peak accelerations of the experimental responses are

0.28 and 0.30 m/s2 for the 15 and 45 kg suspended masses respectively, which are 27%

and 36% above the design target. Since the peak acceleration is quite sensitive to the

location of the position controller poles, the experimental acceleration levels are

attributed to errors in the assumed position controller poles. These errors arise due to

inaccuracies in the sensor measurements in general, and the electromagnet force actuator

in particular. The response for the 45 kg mass shows evidence of a low amplitude

oscillation of the experimental rig at a frequency of about 5 Hz. Both experimental

acceleration responses are however, well within the ISO ride comfort specification of

0.4 m/s2, and a consistent response is achieved even when the suspended mass is tripled.

The sensitivity of the acceleration response to the location of the position controller

poles can be ameliorated through the use of a more complex guideway following


algorithm, for example, by using a third-order filter.

Figure 4.11 Simulated suspension response to a 1 mm simulated track step

(m = 15 kg)

Figure 4.12 Experimental suspension response to a 1 mm simulated track step

(m = 15 kg)

The low frequency overshoot on the air gap step responses for both the simulated and

the experimental results is around 23%. The overshoot is dominantly due to the state

integration filters which reduce the amplitude of the position and velocity feedback

signals at low frequencies. The 23% overshoot is close to that anticipated with a

10-15% contribution from the position signal (see Section 4.4.6) plus a 7% contribution

from the velocity signal which was observed in the position controller test. For the

experimental system, this overshoot is considered acceptable.


The track step response test described above is in fact rather harsh, since a step change

Figure 4.13 Simulated suspension response to a 1 mm simulated track step

(m = 45 kg)

Figure 4.14 Experimental suspension response to a 1 mm simulated track step

(m = 45 kg)

in track height for a moving electromagnet is transformed to a ramp change as the

electromagnet passes by the step. Since full-scale suspension magnets are quite long75

(typically 1 m, for speeds of 0-25 m/s) this transformation adds a further filter to the

system, which can be approximated by a first order lag with a pole frequency of

0-25 rad/s (speed divided by electromagnet length). A real track step would therefore

produce a lower peak acceleration than the simulated track steps.


4.7 Conclusions

A novel suspension control scheme has been proposed, developed, and tested, which

permits the conflicting requirements of disturbance force rejection and guideway

following to be designed independently. Simulated and experimental test results show

that the system meets the functional requirements, and that the response is stable and

well damped.

For a full-scale system, it may be desirable to reduce the air gap overshoot associated

with step changes in the track height. Further research investigating higher performance

accelerometers, more complex state integration filters, and air gap gain compensation

should permit a reduction in the air gap overshoot.

The development of a sophisticated guideway following algorithm suitable for use on

guideways with gradients is an area that requires further research. The use of

matched-filter techniques is considered to present potential improvements over low-pass

filtering. This research could be performed very largely through simulation studies.

Having established the validity of the proposed suspension control system, it is now

ready to be applied to the suspension control of the experimental multi-electromagnet

vehicle.

Vehicle suspension control 88

5

Vehicle suspension control

5.1 Introduction

This chapter describes the development and validation of the multi-electromagnet

vehicle suspension control strategy outlined at the end of Chapter 1. The suspension

controller developed in Chapter 4 is used to control independent vehicle modes, and the

force controller developed in Chapter 3 is used to provide independent electromagnet

force actuation.

The development of the vehicle control system is divided into six parts. First, the

characteristics of the vehicle chassis and guideway are identified. The decoupling and

control requirements are then considered and a vehicle suspension control strategy is

developed. Next, the control system is synthesised using the independent force and

suspension controllers developed in Chapters 3 and 4 respectively. The lateral motion

of the vehicle is then briefly analysed. Experimental test results are presented next, to

verify that the proposed vehicle suspension system meets the operational requirements.

Finally, some conclusions are drawn on the merits and limitations of the proposed

system.

5.2 The experimental research vehicle and guideway

The chassis of the experimental research vehicle is designed to be very stiff in order to

provide a rigid coupling between the electromagnets and this exacerbates the problems

of controlling the vehicle.76 Figure 5.1 shows a photograph of the experimental

vehicle and its guideway. The vehicle is suspended by four electromagnets, one at each

corner, and is propelled and braked by a linear induction motor mounted centrally


underneath the chassis. The vehicle is equipped with all of the signal processing and

power control equipment necessary to implement the proposed control schemes.

The chassis is constructed from welded steel tubing, with an aluminium alloy used for

Figure 5.1 Vehicle chassis

the electromagnet hangers to prevent conduction of the electromagnet flux. The

torsional stiffness of the chassis, measured axially about the length of the vehicle is

about 112 kNm/rad. The flexibility of the electromagnet support hangers gives rise to

a coupling stiffness of about 1.5 kN/mm between each electromagnet pole face and the

vehicle chassis. The chassis is assumed to be rigid apart from the torsional and

electromagnet hanger flexibility.

The experimental electromagnets and reaction rail are too narrow to permit direct

measurement of the electromagnet air gap. Therefore, the gap sensors measure the

distance from the chassis down to the top of the track, and the electromagnet air gaps

are calculated using an appropriate formula. This arrangement is not ideal due to the

flexibility of the electromagnet hangers which introduces small errors in the air gap

measurement. In order to limit the magnitude of such errors, the air gap sensors are

calibrated with a preloaded mass of 35 kg per electromagnet. A suspended mass of

20-50 kg per electromagnet thus gives rise to a maximum measurement error of

±0.1 mm due to steady-state hanger deflection. The accelerometers are rigidly mounted


to the underside of the electromagnets and are therefore assumed to be perfectly

coupled.

The guideway is constructed from stiff steel girders which carry ferromagnetic reaction

rails for the electromagnets and a steel-backed, aluminium alloy reaction rail for the

linear motor. When excited with a force impulse, the guideway exhibits lightly damped

oscillation modes at natural frequencies of around 20 Hz and 40 Hz.

The mass of the fully equipped vehicle is 88 kg which permits a maximum passenger

load of about 110 kg. Table 5.1 lists the mass of each of the major vehicle components

and the dimensions between the centres of the electromagnets. The linear motor which

propels the vehicle produces a maximum thrust of about 50 N and an associated

repulsion force of around 200 N.

Table 5.1 Component masses and dimensions of the experimental vehicle

Vehicle parameter Index Size

Chassis Mass mchassis 27 kg

Total Electromagnet Mass mmagnets 29 kg

Linear Induction Motor Mass mLIM 14 kg

Control System Equipment Mass mcontrol 18 kg

Total Vehicle Mass m 88 kg

Chassis length (between electromagnet centres) L 0.8 m

Chassis width (between electromagnet centres) W 0.4 m

5.3 Control strategy for the vehicle suspension

The free-body motion of the vehicle has six degrees of freedom which can be

considered in terms of three cartesian modes of linear motion, namely, heave, sway and

track progress, and the three corresponding cartesian rotation modes, pitch, roll and

yaw. These modes are illustrated in Figure 5.2, along with the electromagnet indices.


The experimental vehicle has no provision for active lateral force actuation, so the sway

Figure 5.2 Free-body motion of the vehicle

and yaw modes are not actively controlled. Lateral guidance is, however, provided by

the inherent lateral stiffness which exists between the suspension electromagnets and

the reaction rails, whilst the progress of the vehicle along the guideway is controlled

by the linear induction motor. The four lift suspension electromagnets on the

experimental vehicle therefore control three free-body motions, namely, heave, pitch and

roll. The redundancy associated with four electromagnets controlling only three

free-body motions, results in the control of a fourth degree of freedom, namely,

torsional distortion of the vehicle chassis.

The suspension requirements for the independent vehicle modes are functionally

equivalent to those identified in Chapter 4, but different suspension parameters are

required for the different vehicle modes. For example, in order to maintain the correct

nominal air gaps, and hence maximise the available air gap deviation range, the heave,

pitch and roll modes require position error integral action. However, an important

operational requirement for an electromagnetically suspended vehicle is for the

suspension forces to be evenly distributed among the electromagnets. This is required

because suspension electromagnets are designed with only a moderate overload

capability due to the weight penalty that it incurs (see Appendix A). Since a cost

effective vehicle and guideway configuration will always have some finite torsional

displacement error, the application of torsional position error integral action would

cause a severe load imbalance between diagonal pairs of electromagnets. This would

clearly be unsatisfactory from an electromagnet utilisation viewpoint. The application

of position error integral action to the vehicle torsion motion is thus precluded. For

full-scale vehicles, it may also be desirable to have different settings for the heave,


pitch and roll mode suspensions,77,78 as is found on conventional trains and road

vehicles.

Since the required suspension characteristics for the vehicle modes differ, independent

vehicle mode suspension controllers are required.79,80 The proposed vehicle control

strategy therefore employs the suspension control algorithm developed in Chapter 4 to

control independently each of the vehicle modes, heave, pitch, roll and torsion.

Figure 5.3 shows the configuration of the proposed vehicle suspension control strategy,

where C, Z and F represent air gaps, accelerations and forces/torques respectively. The

electromagnet feedback signals for air gap and acceleration are first transformed to

vehicle mode coordinates and then fed into independent mode suspension controllers.

The force and torque demands from the suspension controllers are then transformed to

electromagnet force demands. The electromagnet force controller proposed and

developed in Chapter 3 is then used to achieve independent electromagnet force

actuation. The synthesis of the proposed vehicle control strategy is described next.

Figure 5.3 Configuration of the vehicle suspension control system

5.4 Synthesis of the vehicle control system

The synthesis of the vehicle suspension control system is performed in three stages.

First, the transformations required to convert the electromagnet coordinate signals to and

from vehicle mode coordinates are identified. Then, for convenience of design, the

vehicle mode angular measurements are normalised so that they are equivalent to the

linear motions which they generate at the electromagnets. This is achieved by

reformulation of the decoupling transformations, and conversion of the vehicle mode

inertias. Finally, the parameters of the suspension control algorithm developed in the

previous chapter are configured for each vehicle mode.


5.4.1 Decoupling the electromagnet motions

The maximum angular displacement of any vehicle motion is less than 1°. Therefore,

the vehicle mode motions (see Figure 5.2) can be accurately approximated by

Equation 5.1 where h, θ, φ and ψ are the positions of the vehicle modes, heave, pitch,

roll and torsion respectively, zm1-zm4 are the positions of the electromagnets, and L and

W are the vehicle length and width measured between the electromagnet centres (see

Table 5.1).

5.1

h

θφψ

≈ 1

4

1 1 1 1

2/L 2/L 2/L 2/L

2/W 2/W 2/W 2/W

2/W 2/W 2/W 2/W

zm1

zm2

zm3

zm4

The corresponding transformation between vehicle mode forces and torques, and the

electromagnet forces is given by Equation 5.2. This transformation matrix reflects the

additive nature of force and torque translations, compared with the averaging nature of

displacement translations.

5.2

Fm1

Fm2

Fm3

Fm4

≈ 1

4

1 2/L 2/W 2/W

1 2/L 2/W 2/W

1 2/L 2/W 2/W

1 2/L 2/W 2/W

Fh

Tθ

Tφ

Tψ

The masses of the major components from which the vehicle is constructed are listed

in Table 5.1. In order to determine the contribution of each of these masses to the

inertia of the rotational vehicle modes, the distribution of the mass of each component

in a horizontal plane is considered. The heights of the component masses relative to

a horizontal reference plane are considered later.

The centres of gravity of each electromagnet and the induction motor are assumed to

be located at their respective centres of geometry. The electromagnets are located at

the four corners of the rectangular chassis, whilst the motor is located at the geometric

centre of the chassis. The mass of the chassis is concentrated around its periphery, so


it is modelled by four point masses, located at the electromagnet centres, with each

point mass equal to one quarter of the total chassis mass.

The mass of the control system electronics is distributed fairly uniformly over the

rectangular space between the electromagnets. It is therefore modelled by four point

masses, each of one quarter of the total electronics mass, which are located mid-way

between each electromagnet and the geometric centre of the chassis. Equation 5.3

shows the contribution of each component mass to the mass or inertia of each vehicle

mode. The resultant mass and inertias are calculated using the masses and dimensions

given in Table 5.1.

5.3

mh

Iθ

Iφ

Iψ

≈

1 1 1 1

(L/2)2 (L/2)2 (L/4)2 0

(W/2)2 (W/2)2 (W/4)2 0

(W/2)2 (W/2)2 (W/8)2 0

mchassis

mmagnets

mcontrol

mLIM

88 kg

9.7 kgm 2

2.4 kgm 2

2.3 kgm 2

5.4.2 Normalising the vehicle mode motions

For the sake of convenience in considering the design parameters for the vehicle mode

controllers, the rotational vehicle mode motions are normalised so that they are

equivalent to the linear motions which they produce at the electromagnets. The

reformulated transformations are given by Equations 5.4 and 5.5.

In a similar fashion, the inertia of each vehicle mode is normalised so that it is

5.4

zh

zp

zr

zt

≈ 1

4

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

zm1

zm2

zm3

zm4

where zp

Lθ2

, zr

Wφ2

, zt

Wψ2

equivalent to a point mass at the electromagnet centres. Equation 5.6 shows the

normalised mode masses reformulated from Equation 5.3. The angular torsional

stiffness of the vehicle chassis is also normalised using the factors in Equations 5.4

and 5.5 giving an equivalent linear stiffness of 1400 N/mm.


5.5

Fm1

Fm2

Fm3

Fm4

≈ 1

4

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

Fh

Fp

Fr

Ft

where Fp

2Tθ

L, F

r

2Tφ

W, F

t

2Tψ

W

5.6

mh

mθ

mφ

mψ

≈

1 1 1 1

1 1 1/4 0

1 1 1/4 0

1 1 1/16 0

mchassis

mmagnets

mcontrol

mLIM

88 kg

61 kg

61 kg

57 kg

where mθ

4Iθ

L2, mφ

4Iφ

W 2, mψ

4Iψ

W 2

5.4.3 Control of the vehicle mode motions

The centre of mass of the vehicle lies centrally above the horizontal plane linking the

accelerometers which are used to derive the position feedback signals. The pitch and

roll modes are therefore coupled to the heave mode.81 Since the height of the centre

of mass of the vehicle above the accelerometer plane is much smaller than the vehicle

length or width, the degree of mode coupling is low. The vehicle mass distribution

causes negligible cross-coupling between the pitch and roll modes, and from the heave

mode to the pitch and roll modes. Since the amount of vehicle mode cross-coupling

is small, decoupling is considered unnecessary, and the suspension controllers are

applied directly to the vehicle mode motions.

5.4.4 Configuration of the vehicle mode suspension controllers

The functional requirements for the vehicle suspension system are equivalent to those

discussed in Section 4.1. The first requirement is that the suspension should deflect no

more than 30-50% of the maximum allowable air gap deviation, when subjected to a

disturbance or load force equal to 30-50% of the maximum suspended weight. The

second requirement is for a peak acceleration of less than 0.04 g, when negotiating a


1 mm step in track height. In addition, in order to maximise the available air gap

deviation, there must be no steady-state air gap deflection due to disturbance or load

forces for the heave, pitch and roll modes.

The suspension control algorithm developed in Chapter 4 is now configured for each

of the vehicle modes. The electromagnets are rated for a maximum continuous force

of 500 N, so the maximum suspension force for the vehicle is 2000 N. Therefore, in

order to meet the disturbance force rejection requirement at any point on the chassis,

the heave, pitch and roll mode position controllers require a stiffness of 1000 N/mm.

In addition, in order to meet the deflection requirement with the disturbance force

equally divided between two diagonally opposed corners of the vehicle, the torsion

motion also needs a stiffness of 1000 N/mm. Since the chassis has an inherent damped

torsional stiffness of 1400 N/mm, active suspension stiffness for the torsion mode is not

required.

The stiffness of each electromagnet hanger is approximately 1500 N/mm which

translates to a vehicle mode hanger stiffness of 6000 N/mm since the hangers operate

in parallel. This is much larger than the stiffness required for the vehicle mode position

controllers, and it is therefore neglected when designing the suspension controllers.

The design procedure for the position controller described in Section 4.4 is applied for

the vehicle heave, pitch and roll modes. The design procedure uses as its starting point,

the mode stiffnesses determined above, the chassis mode masses calculated in

Equation 5.6, and the maximum passenger load of 110 kg. For the purpose of mode

position controller synthesis, the heave mode is conservatively assumed to have the

same minimum mass as the other modes. The linearity of the suspension control

algorithm and its design procedure, results in vehicle mode feedback gains that are four

times the size of those for the single electromagnet suspension design. Table 5.2 lists

the parameters for the heave, pitch and roll mode position controllers.

The natural frequencies of the various vehicle modes and components are now

considered to check for any potential resonance problems. The predicted closed-loop

suspension poles for the vehicle heave, pitch and roll modes are essentially the same

as those for the single electromagnet suspension, with a resultant position controller

bandwidth of around 10 Hz (see Table 4.4). By assuming a dominantly linear,

second-order behaviour for the vehicle torsion motion, its estimated undamped natural

frequency82 is given by:


Similarly, the estimated undamped natural frequency of each electromagnet and hanger

Table 5.2 Parameters for the heave, pitch and roll mode position controllers.

Parameter Feedback signal Value

Stiffness Position error 1000 N/mm

Damping Velocity 26 N/mm/s

Virtual mass Acceleration 60 kg

Integral time constant Position error integral 1 s

5.7ωtorsion

ktorsion

mtorsion

1400000

57≈ 157 rad/s, f

torsion≈ 25 Hz

is given by:

5.8ωhanger

khanger

mhanger

1500000

7.5≈ 450 rad/s, f

hanger≈ 71 Hz

The predicted natural frequency of the electromagnet hangers is well separated from the

design values for the vehicle heave, pitch, roll and torsion modes, so problems of

resonant coupling are unlikely to occur. However, one of the guideway resonance

modes has a natural frequency of around 20 Hz which is likely to interact with the

torsion mode. In view of this fact, and the open-loop unstable nature of the

electromagnetic force actuators, the use of velocity and acceleration feedback for the

torsion mode is considered prudent.

Therefore, the damping of each electromagnet is made equivalent to that of the

independent electromagnet suspension of Chapter 4 by using the same velocity feedback

gain for the torsion mode as is used for the other mode controllers. The same argument

is applied to the acceleration feedback.

The remaining vehicle suspension system design parameters are the corner frequencies

for the guideway following filters and state integration filters, and the force actuation

time constant. For the vehicle mode suspensions, these are 4 Hz, 0.1 Hz and 3.2 ms


respectively, which are the same as those for the single electromagnet suspension due

to the linearity of the position control algorithm and design procedure. Table 5.3

summarises the parameters for the full vehicle suspension control system.

Table 5.3 Parameters for the vehicle mode suspension controllers

Parameter Name Value

Stiffness* kpos 1000 N/mm

Damping kvel 26 N/(mm/s)

Virtual mass kacc 60 kg

Error integral time constant* Terr 1 s

Force actuation time constant Tforce 3.2 ms

State integration filter frequency ωint 0.6 rad/s

Guideway following frequency ωfollow 25 rad/s

* For the torsion mode suspension controller: kpos = 0 N/mm, Terr = ∞ s.

5.5 Lateral vehicle guidance

The inherent lateral stiffness of the electromagnets provides guidance forces for the

vehicle sway and yaw motions. The undamped natural frequency of the lateral motion

is derived for small lateral perturbations of the electromagnets in Chapter 4, and is

repeated here for convenience. It is approximated by:

where ωn is the undamped natural frequency, mt is the total suspended mass, mc is the

5.9ωn

≈2m

ta

g

mcπp

rigidly coupled suspended mass, ag is gravitational acceleration, and p is the width of

the electromagnet pole pieces. The mass ratio mt /mc is unity for the unloaded vehicle,

and approximately 2 when carrying a passenger. Since the pole width is 9.5 mm, the

undamped natural mode frequencies are around 4 Hz for the unloaded vehicle, and 6 Hz

for the vehicle supporting a passenger. The lateral guidance thus experiences a

second-order guideway following characteristic with a bandwidth very similar to that


of the actively controlled vertical modes, but without the high absolute position

stiffness.

5.6 Performance of the experimental vehicle suspension

In order to verify that the proposed vehicle suspension control system is capable of

meeting the desired suspension performance, an experimental vehicle was developed,

and experimental test results were obtained. The configuration and parameters of the

vehicle control system are described earlier in this chapter, whilst the suspension control

algorithm and electromagnet force control algorithm are described in Chapters 4 and 3

respectively. The implementation of the hardware and software for the experimental

vehicle is described in Chapter 6.

The performance of the experimental vehicle is tested in three phases. The first phase

tests the mode position controllers by examining the reference position step response,

the disturbance force rejection response, and the stability margins. The second phase

tests the degree of cross-coupling between the vehicle mode motions and examines the

load sharing performance. Finally, the third phase tests the suspension ride quality

when subjected to simulated track steps.

Selected tests were performed with the vehicle both stationary and in motion, and no

significant response differences were observed between the two cases.

5.6.1 Performance and stability of the mode position controllers

The performance of the vehicle mode position controllers is tested by analysing the

reference position step response and the force disturbance step response. The stability

margins are then measured by examining the response in the frequency domain using

a Bode plot.83 In addition, a series of position step responses at different air gaps is

presented to gauge the linearity of the suspension control system over the full

operational air gap range.

The position step response of the vehicle heave mode controller for a 1 mm reference

step amplitude is shown in Figure 5.4 for the unladen vehicle, and in Figure 5.5 for the

vehicle with an 80 kg passenger. The transient air gap response should be equivalent

to the simulated and experimental responses for the single electromagnet suspension


presented in Chapter 4 (see Figures 4.6 and 4.7). The 50% rise time for the vehicle is

Figure 5.4 Experimental heave response to a 1 mm position reference step

(no passenger)

Figure 5.5 Experimental heave response to a 1 mm position reference step

(80 kg passenger)

about 10% smaller than that for the single electromagnet suspension, but the responses

are otherwise consistent. The oscillation present on the acceleration signal has a

frequency of just under 70 Hz and is attributed to the electromagnet hangers which have

an estimated undamped natural frequency of 71 Hz (see Equation 5.8). The low

frequency overshoot after the transient part of the response has a peak value of 15%.

This is due mostly to the a.c. coupling of the absolute feedback signals, which causes

reduced position and velocity signal amplitudes at low frequencies (see Section 4.4.6).

Some of the overshoot is also contributed by the position error integral action adjusting


to the new force actuation error that results from the change in electromagnet operating

point.

Since the passenger mass is loosely coupled to the vehicle at the frequencies

encountered in the transient portion of the step response, the loaded and unloaded

responses are very similar. The consistency between the unloaded and loaded responses

also shows that the mode force actuation is dominantly linear over a wide force

actuation range.

A critical performance requirement for the vehicle suspension is disturbance force

rejection. This is tested by measuring the deflection when starting and stopping the

linear induction motor and when adding and removing a 500 N load. The induction

motor produces a lift force of about 200 N at a thrust of 40 N which causes the vehicle

to rise and fall transiently by 0.22 mm. This corresponds well with the theoretical

value of 0.2 mm due to the heave mode stiffness of 1000 N/mm. Operation of the

linear motor has a negligible impact on the heave mode position step response.

Figure 5.6 shows the air gap and acceleration response to a 50 kg vehicle load

reduction. This produces a deflection of 0.61 mm which again corresponds well with

the theoretical value of 0.5 mm. For both test cases, the disturbance response is well

damped with the position error integral feedback restoring the reference air gap in about

2 seconds.

Figure 5.6 Experimental heave response to a 500 N heave disturbance force


The air gap linearity of the mode suspensions is tested by comparing position step

responses at different air gaps. Figure 5.7 shows three position step responses for the

heave mode position controller. The response of each step is essentially the same,

although there is a slight variation in the overshoot recovery characteristic. The latter

effect is most notable at the small and large air gaps because the electromagnet force

controllers are less accurate near the operational air gap limits than they are around the

nominal operational air gap. However, the step responses show that the suspension is

dominantly linear over the full operational air gap range.

For the suspension stability analysis in the frequency domain, the worst case scenario

Figure 5.7 Experimental heave responses to three 1 mm heave position reference steps

is represented by the heave mode controller. The larger mass for the heave mode

relative to the equivalent masses of the pitch and roll modes gives the heave mode a

slightly lower damping ratio. Figure 5.8 and Figure 5.9 show the theoretical and

experimental Bode plots for the position response of the vehicle heave mode controller

for a 1 mm amplitude sinusoidal position reference. The theoretical plots are calculated

using Equation 4.8 which neglects the low frequency effects due to the state integration

filters and the position error integral action. In addition, no allowance is made for the

closed-loop phase delay which is generated by the discrete-time controller. For

example, the average signal processing time delay of 1.6 ms (see Section 6.5.5) causes

an effective feedback loop phase delay of 37° at a frequency 64 Hz.

As with the time domain responses, only a small difference was observed between the

Bode plots for the unloaded and passenger loaded vehicle due to the relatively flexible

coupling of the passenger body to the vehicle. The signal magnification at low


frequencies is due to the low frequency gain roll-off of the state integration filters

which are used to calculate the absolute velocity and position feedback signals.

In practice, the resonance characteristics of the electromagnet hangers and the track are

Figure 5.8 Bode plot (gain) for the position response of the heave mode suspension

Figure 5.9 Bode plot (phase) for the position response of the heave mode suspension

a function of the proximity between the vehicle and the location of the guideway

supports. The worst case vibration conditions were found experimentally and are

marked on the Bode plots by the individual points measured at 42 Hz and 64 Hz.

These are attributed to the track and electromagnet hangers for which the undamped

natural frequencies were measured/estimated to be about 40 Hz and 70 Hz respectively.

The theoretical Bode plots assume that the track is rigidly coupled to the ground.


The gain stability margin84 for the experimental system is 17 dB compared with a

theoretical value of 23 dB. This discrepancy is attributed to the neglection of

discrete-time effects, and also to system model and implementation inaccuracies, in the

theoretical value. The phase stability margin for the experimental system is about 130°

versus a theoretical value which approaches 180°. This difference is mostly due to the

state integration filters and position error integral action which are neglected in the

theoretical calculation. The Bode plots clearly demonstrate a very good correspondence

between the theory and the experimental results from the suspension system, and also

that acceptable stability margins have been achieved.

The time and frequency domain responses for the other position controlled modes,

namely pitch and roll, are dominantly the same as those for the heave mode, and so

they are not presented here. However, the following two test phases present results

which include some time domain responses for all of the vehicle modes.

5.6.2 Decoupling of the vehicle modes and load sharing

Having established a satisfactory performance from the vehicle mode controllers when

tested independently, the cross-coupling between the different modes is now examined.

Figure 5.10, Figure 5.11, Figure 5.12 show the responses of each vehicle mode to a

1 mm reference step input to the heave, pitch and roll position controllers respectively.

The cross-coupling of the heave, pitch and roll modes to the torsion mode is clearly

negligible on all of the experimental test responses. The constant 0.1 mm torsion

position offset reflects the fact that the vehicle chassis and track have a gap

misalignment of +0.1 mm at one diagonal pair of electromagnets, and -0.1 mm at the

other pair.

Low frequency cross-coupling after the transient portion of the step response is apparent

between the heave, pitch and roll modes. This is caused by the electromagnet force

controllers independently adjusting to their new operating points, and it results in a

maximum coupling ratio of 8%. The only significant cross-coupling during the

transient part of the step responses links the pitch to the heave mode, and the roll to the

heave mode. This occurs because the centre of mass of the vehicle is above the

horizontal plane on which the accelerometers are located, and it gives rise to a coupling

ratio of 6%. The low amplitudes of the transient cross-coupling and the low frequency

cross-coupling clearly illustrate the success of the proposed control strategy in terms of


decoupling the vehicle mode motions.

Figure 5.10 Experimental mode responses to a 1 mm heave position reference step

Figure 5.11 Experimental mode responses to a 1 mm pitch position reference step

Figure 5.12 Experimental mode responses to a 1 mm roll position reference step


The lightly damped roll oscillations generated by the roll step are attributed to a small

cross-coupling to the lateral vehicle motion which is undamped. Finally, the very low

amplitude, high frequency oscillation apparent on all of the mode responses is due to

slight vibration of the electromagnet hangers as discussed in the previous test phase.

The load sharing capability of the vehicle suspension is tested by comparing the

operating conditions of the electromagnets at the point on the experimental guideway

where the torsional misalignment between the vehicle and guideway is at its maximum.

Table 5.4 gives a snapshot of the worst case operating conditions at which each

electromagnet deviates from its nominal value by approximately ±0.2 mm. The table

is augmented with another snapshot of the control system, but this time it is

reconfigured with the torsion position error, and error integral feedback gains set equal

to the respective gains for the other vehicle modes. Such a suspension configuration

is equivalent to using four independent electromagnet suspension controllers.

Table 5.4 Effects of worst case experimental vehicle to guideway misalignment

Parameter Electromagnets / Vehicle modes

Electromagnet air gaps (1,2,3,4) /mm 3.21 2.77 3.21 2.81

Vehicle mode gaps (h,p,r,t) /mm 3.00 0.01 -0.01 0.21

Electromagnet force demands /N 279 307 300 275

Electromagnet current demands /A 7.7 7.1 8.0 6.8

Electromagnet power dissipations /W 53 45 58 42

For controller with torsional position error plus integral feedback:

Electromagnet force demands /N 567 20 602 27

Electromagnet current demands /A 10.8 1.8 11.2 2.2

Electromagnet power dissipations /W 105 3 113 4

The air gaps of diagonal pairs of electromagnets (see Figure 5.2) are approximately

2.8 mm and 3.2 mm, thus the larger air gaps are approximately 14% bigger than the

smaller ones. The theoretical current ratio should therefore also be 14%, and the power


dissipation ratio 30%. In fact, due to a slightly uneven load distribution, the highest

electromagnet power dissipation is 38% above the lowest.

The experimental data for the reconfigured suspension with effectively independent

controllers shows that two of the electromagnets carry 96% of the vehicle load. Also,

as the vehicle moves along the guideway, the torsion error changes, and this causes

massive and rapid force fluctuations as the vehicle load is swapped between diagonal

electromagnet pairs. This clearly demonstrates why such a configuration is

unacceptable from both a steady-state and a dynamic viewpoint.

5.6.3 Ride quality of the vehicle suspension

The final test for the vehicle is for suspension ride quality whilst negotiating a step in

track height. The experimental guideway does not have any track steps so they are

simulated by injecting a step into the track position calculation. The experimental

response to a 1 mm step change in track heave position is shown in Figure 5.13 for the

unladen vehicle and in Figure 5.14 for the vehicle loaded with an 80 kg passenger.

Comparison of the experimental responses with the simulated responses for the single

electromagnet suspension (see Figure 4.11) shows that the experimental responses agree

well with the theory. The peak acceleration is 0.025 g which is comfortably below the

ISO target of 0.04 g. The low frequency overshoot (approximately 25%) is again due

to the a.c. coupling of the feedback signals derived from the accelerometers.

Figure 5.15, Figure 5.16, Figure 5.17 show the experimental responses for a 2 mm

simulated track position step for the heave, pitch and roll modes. These show that a

dominantly consistent response is obtained for each of the vehicle modes. The pitch

response overshoot is 16% which is approximately equal to that attributed to the a.c.

coupling of the velocity and position feedback signals (see Section 4.4.6). The heave

response has an additional, slower contribution due to error integral action as the

electromagnet force controllers adjust to the new operating point.

Finally, the roll response overshoot, at 29%, is 13% higher than that of the pitch mode.

Cross-coupling of the roll mode to the undamped sway mode was observed during the

track step response test, and the additional roll mode overshoot is attributed to this

cross-coupling. The cross-coupling occurs because the centre of mass of the vehicle

is located above the plane on which the accelerometers are located. Active control of


the lateral vehicle modes should reduce the roll mode overshoot to a similar level to

that experienced by the pitch mode.

Figure 5.13 Experimental heave response to a 1 mm simulated track step

(no passenger)

Figure 5.14 Experimental heave response to a 1 mm simulated track step

(80 kg passenger)


Figure 5.15 Experimental heave response to a 2 mm simulated track heave step

Figure 5.16 Experimental pitch response to a 2 mm simulated track pitch step

Figure 5.17 Experimental roll response to a 2 mm simulated track roll step


5.7 Conclusions

This chapter describes the development and validation of the multi-electromagnet

vehicle suspension control strategy outlined in Chapter 1. The experimental system

achieved a very good performance in terms of passenger ride quality, disturbance force

rejection, and electromagnet utilisation. The last feature arises from the ability of the

proposed control strategy to support the use of position error integral action which

facilitates accurate control of the nominal air gaps and hence maximises the available

air gap deviations. These benefits accrue from the development of a detailed

electromagnet model for the force control algorithm, and a sophisticated structure for

the vehicle suspension control system.

Finally, it is apparent that existing vehicle control strategies require detailed

consideration of the nonlinear force actuators throughout the design procedure in order

to get predictable results.85 By contrast, the linearity of the proposed electromagnet

force controller permits the assumption of linear force actuation. This fact, coupled

with the modularity of the proposed suspension control strategy, permits a linear vehicle

suspension design procedure, which consists of decoupling the electromagnet motions

and then configuring the mode suspension controllers.

Control system implementation 111

6

Control system implementation

6.1 Introduction

This chapter describes the selection and design of the various components required to

implement the suspension control system for the experimental research vehicle. The

experimental single electromagnet suspension uses a subset of the vehicle suspension

components.

The full set of vehicle control algorithms developed in Chapters 3, 4 and 5 is

computationally complex. Therefore, in order to provide a flexible experimental system

which is capable of being freely modified for current and future research work, a digital

signal processing approach was chosen. Additional benefits of this approach include

the elimination of drift and offsets in the signal processing, and the ability to implement

readily nonlinear functions. The disadvantages of digital processing are primarily those

due to the time and amplitude discretisation of the signals.

The implementation of the experimental control system is described in five sections.

First, the system requirements for the transducers, converters, and signal processors are

identified. Suitable industrial feedback sensors are then selected and an electromagnet

current controller is designed. Next, the signal processing, signal conversion and data

communication subsystems are designed. The design and configuration of the software

which implements the control algorithms is then described, and finally, conclusions are

drawn about the system implementation.


6.2 System requirements

The experimental control system performs three basic tasks. It measures the air gap and

acceleration of each electromagnet, it calculates the electromagnet current demands

according to the control algorithms developed in Chapters 3, 4, and 5, and it adjusts and

maintains the electromagnet currents at the demanded levels.

In order to perform these tasks, the control system is functionally decomposed into five

subsystems, namely feedback sensors, analogue to digital converters, signal processors,

digital to analogue converters, and finally, electromagnet current controllers. The last

subsystem is implemented in the analogue domain to reduce the digital signal

processing load. The critical requirements for each subsystem are specified in terms

of their bandwidth, range, resolution, and accuracy. These specifications are calculated

by first considering the required signal bandwidths and sampling rates, and then the

required signal ranges, resolutions, and accuracies.

6.2.1 Bandwidths and sampling rates

The suspension system has been designed, and the control algorithms developed, using

continuous-time methods. However, since discrete-time, digital signal processing is to

be used, it is necessary to determine an acceptable maximum time interval between the

iterations of the controller. Shannon’s sampling theorem states that a sample rate of

twice the highest frequency component of a signal is theoretically sufficient to describe

that signal completely.86 However, for real-time, closed-loop control applications, the

feedback loop delay introduced by Shannon’s sampling rate would severely disturb the

location of the closed-loop poles, and adversely affect the closed-loop response. In

order to reduce the adverse effects of the feedback loop phase delay and hence obtain

a good response, the sample interval generally needs to be 1/5 to 1/10 of the time

constant of the dominant system pole.87 This results in a sampling rate which is 15

to 30 times Shannon’s theoretical sampling rate.

In addition to determining an acceptable delay for discrete-time sampling, an acceptable

phase delay for the air gap and acceleration sensors must also be determined. The

phase lag due to the electromagnet current controller is already incorporated in the

design of the suspension control algorithms, so it requires no further consideration.


In order to produce a control system design which makes efficient use of the various

subsystem components, a range of acceptable phase delays is calculated, and this is then

apportioned between the feedback sensors and the discrete-time controller.

Table 4.4 lists the location of the dominant poles and zeros of the closed-loop position

control system for the single electromagnet suspension. For the purpose of this

analysis, the poles and zeros of the vehicle mode position controllers can be considered

to be the same as those of the single electromagnet suspension. The suspension

response is dominantly third-order, with a highest pole frequency of around 300 rad/s

for both the unloaded and fully loaded cases, and the calculated minimum phase delay

at this frequency is 3.0 rad (172°). By using the sample rate guideline identified earlier,

an additional phase delay of 10-20% for the position controller processing and sensor

delays is assumed to be acceptable. This permits an additional phase delay

of 0.3-0.6 rad (17-34°) at 300 rad/s, which can also be considered as a time delay

of 1.0-2.0 ms.

Figure 3.4 shows the location of the open-loop poles for the electromagnet at the worst

case operating point. The unstable pole has a worst case value of 167 rad/s, at which

the total open-loop phase delay is 3.4 rad (195°). Using the same assumption as before,

this permits additional force controller and sensor delays of 0.34-0.7 rad (20-39°)

at 167 rad/s, which can be considered as a time delay of 2.0-4.0 ms.

The phase and time delay requirements for the position controller are more exacting

than those for the force controller, so they are used for the baseline requirements. The

approximate equality between the delays occurs because the suspension control

algorithm parameters were designed to make maximum use of the available

electromagnet force actuation bandwidth. It also requires the same overall control

system bandwidth to be used for both the suspension position control algorithm and the

electromagnet force control algorithm.

The phase delays contributed by the air gap and acceleration sensors occur concurrently,

whilst the signal processing time delay follows sequentially. Since attaining high digital

sampling rates is more costly than high sensor bandwidths for this application, the

signal processing should be allocated a larger share of the available time delay than the

feedback sensors. Consequently, the signal processing is allocated roughly two thirds

of the allowable phase delay, with the remaining one third left for the feedback sensors.


The phase delay generated by the first-order phase lag inherent in suitable low-cost

industrial air gap sensors is given by:

where θsensor is the sensor phase delay, ωsensor is the sensor bandwidth, and ω is the

6.1

θsensor

tan 1 ωω

sensor

≈ ωω

sensor

since ω <ωsensor

∴ ωsensor

≈ ωθ

sensor

signal frequency. The air gap sensor bandwidth required to generate a phase delay of

no more than 0.1-0.2 rad at a signal frequency of 300 rad/s is therefore

1500-3000 rad/s. This is equivalent to a sensor pole time constant of 0.3-0.7 ms.

Suitable low-cost industrial accelerometers use a sprung mass as the sensing element

and have a second-order low-pass response. Therefore, in order to generate the same

total phase delay as the air gap sensor, the accelerometer must have a natural resonant

frequency of 3000-6000 rad/s.

The time delay allowed through the signal processors is given by the allowable

processing phase delay (0.2-0.4 rad) divided by the highest closed-loop pole frequency

(300 rad/s). This gives an allowable signal processing time delay of 0.7-1.3 ms.

Assuming that the signal processing takes about 80% of the sampling interval, the

average time delay due to the discrete-time signal conversion and digital processing is

1.3 times the sample interval, and so a sampling interval of 0.5-1.0 ms is required.

The assumptions made about acceptable levels of phase delay were verified using a

detailed simulation of the full closed-loop suspension system (see Appendix C). The

additional phase delay of 10-20% had little impact on the closed-loop response, whilst

phase delays greater than 80% of the original value caused marginal instability

problems at the worst case operating point. Phase delays of less than 5% had a

negligible impact on the suspension response.

Table 6.1 summarises the bandwidth and sampling interval requirements that have been

calculated for the feedback sensors and the digital control algorithms. These figures are

guidelines between which trade-off adjustments can be made as necessary.


6.2.2 Range, resolution and accuracy

Table 6.1 Bandwidth and sampling interval specifications feedback sensors

Component Parameter Value

Accelerometer natural frequency 480-960 Hz

Air gap sensor bandwidth 240-480 Hz

Electromagnet force controller sampling interval 0.5-1.0 ms

Suspension position controller sampling interval 0.5-1.0 ms

The normal operational envelope for the suspension system (see Chapter 3) covers an

electromagnet air gap range of 1-5 mm. However the air gap range required for the air

gap sensors is 0.5-5.5 mm in order to permit operation up to the mechanical limit stops.

Whilst the normal operational acceleration levels are very low, higher levels are

required for testing the suspension position controller. In addition, since the absolute

velocity and position are calculated from the acceleration measurement, it is imperative

that the acceleration signal is not allowed to saturate. Therefore, to accommodate step

response tests, a maximum acceleration measurement range of about ±1 g is required.

Operation of the electromagnets over their full operational envelope requires the current

to be controlled over the range 0-20 A.

The resolution and hysteresis of the transducers and the associated analogue-digital

converters affect the amplitude of the limit cycle oscillations of the control system.88

Since acceleration is the primary quality control factor for the suspension system, an

acceleration limit cycle amplitude of no more than about 20% of the target comfort

threshold of 0.4 m/s2 (see Chapter 4) is considered desirable. This calls for an

acceleration resolution of 80 mm/s2. Assuming an integration interval of 1 ms for the

velocity and position integrators, and sufficient numerical precision, this results in a

velocity resolution of 80 µm/s, and a position resolution of 80 nm.

The required force actuation resolution is calculated next because it is a useful reference

value for calculating the required air gap and current resolutions. The force resolution

is related to the acceleration resolution by:


where ∆force is the force resolution and ∆acceleration is the acceleration resolution.

6.2∆ force mass ∆ acceleration

At the maximum load of 50 kg per electromagnet, an acceleration resolution of

0.08 m/s2 requires a force resolution of 4 N per electromagnet.

The required air gap signal resolution is determined by considering the force resolution

and the air gap feedback gain. The relationship between the air gap resolution and the

force resolution is given by:

The air gap feedback gain for the electromagnet force controller is determined by the

6.3∆airgap∆ force

air_gap_gain

open-loop electromagnet stiffness. This gives rise to a worst case, full load gain of

950 N/mm (see Table 3.2). Since the required force resolution calculated above is 4 N,

this requires an air gap signal resolution of 4 µm. This resolution calls for an

expensive, precision air gap sensor. However, the electromagnet stiffness at the

nominal operating air gap is only one third of the worst case stiffness. The nominal

operating point therefore requires an air gap resolution of 12 µm which is close to that

obtainable by standard industrial sensors which cost significantly less than precision

sensors. Therefore, in order to reduce costs, the lower resolution is preferred and a

slight degradation of the response at small air gaps is anticipated. The use of multiple

air gap sensors to improve the air gap measurement accuracy over the electromagnet’s

full length, would also increase the effective sensor resolution, as well as providing

sensor redundancy. However, in order to reduce the cost and complexity of the

experimental vehicle, only one air gap sensor per electromagnet is employed.

The air gap feedback signal also forms the very low frequency component of the

calculated track position which is filtered and fed into the suspension position

controller. However, since the position feedback gain per electromagnet is 250 N/mm

(see Chapter 4), this presents a less stringent resolution requirement than that identified

above. The position resolution of 80 nm, calculated previously from the acceleration

resolution and integration interval, is clearly satisfactory.

The required current control resolution is determined in a similar manner to the air gap

signal resolution. In this case, the worst case open-loop electromagnet force/current

ratio is 400 N/A (see Table 3.2), which requires a current resolution of 10 mA.


The acceptable accuracy for the air gap and current transducers is determined by

considering a simplified equation for the electromagnet force and the corresponding

error equation. These are given by:

where f is the lift force, i is the coil current, g is the air gap and k is the constant of

6.4f ki 2

g2∴ f

errk

err2 i

err2g

err

proportionality. The error of the nonlinear force controller model is represented by kerr,

which is about ±7% (see Section 2.3.7). By assuming a current controller tolerance of

about ±2% and an air gap sensor tolerance of around ±3%, the resultant force controller

accuracy is expected to be about ±17%.

The accuracy required for the accelerometers is calculated by determining an acceptable

deviation in the characteristic response of the position controller. Under normal

operating conditions, a load change from full load to no load increases the natural

undamped frequency, ω, and damping ratio, ζ, (see Equation 4.7) by a factor of around

50%. Therefore, a further discrepancy of about ±10% to allow for transducer errors is

considered acceptable. Since both the velocity and position signals are calculated from

the acceleration signal, an error in acceleration measurement gain is modelled as an

equivalent error in the position, velocity and acceleration feedback gains. The

suspension parameters and their worst case error equations are approximately given by:

where kpos and kvel are the position and velocity feedback gains, ωerr and ζerr are the

6.5ω ∝ kpos

, ζ ∝k

vel

kpos

∴ ωerr

, ζerr

accerr

2

ferr

2

parameter errors, and accerr and ferr are the acceleration measurement and force actuation

errors respectively. Since the expected force actuation error is ±17% (see above), an

accelerometer accuracy of ±3% is required to achieve a total suspension parameter

deviation of ±10%.

Table 6.2 summarises the range, accuracy and resolution required to measure, convert,

and process the acceleration, air gap, and current signals. The accuracy is an aggregate

figure which should be achieved on an end-to-end basis. In practice, the very high

accuracy of analogue-digital converters and digital signal processing means that the

accuracy figures can be used as transducer requirement specifications.


6.3 Transducers

Table 6.2 Range, accuracy and resolution specifications

Signal Range Accuracy Resolution

Acceleration ±10 m/s2 ± 3% 0.08 m/s2

Air gap 0.5-5.5 mm ± 3% 12 µm

Current 0-20 A ± 2% 10 mA

The suspension position control and the electromagnet force control algorithms require

the measurement of the electromagnet air gaps and accelerations. The control signal

generated by the control algorithms is a current demand, and this is actuated by a

high-gain, closed-loop current controller. In order to complement the benefits of the

electromagnetic suspension, the feedback sensors should not make contact with the

track.

The transducer requirements identified in the preceding section are now used to select

commercial devices for the feedback sensors, and the design of a special purpose

electromagnet current controller is described.

6.3.1 Accelerometer

A large range of industrial accelerometers is available which measure absolute

acceleration. Most accelerometers employ a mass constrained by some stiffness and

damping, and measure deflection of the mass to determine the acceleration. The

sensitivity of such a device is generally a function of the sprung mass divided by the

stiffness. Since the natural frequency is also a function of the sprung mass divided by

the stiffness, the natural frequency and hence the bandwidth of these devices is closely

related to the sensitivity.

The selection of a suitable accelerometer is thus largely determined by the

sensitivity/bandwidth constraint, and this appears to be somewhat more exacting for

electromagnetic suspension control applications than it is for many other industrial

requirements. Other important device parameters include cross-axis sensitivity, thermal


stability, resolution, linearity and hysteresis. The last two factors are particularly

important because the acceleration signal is double integrated within the control system

to calculate the absolute velocity and position.

The basic performance requirements for the accelerometer are the measurement range

and resolution, the natural frequency and the general accuracy, and these are ±10 m/s2,

0.08 m/s2, 480-960 Hz and ±3% respectively (see Table 6.1 and Table 6.2). These

requirements are met most cost effectively by an industrial micro-machined silicon

sensor, which incorporates temperature compensated signal conditioning circuitry on the

silicon substrate. The device has a full scale range of ±20 m/s2, a resolution better than

0.04 m/s2, a natural frequency of 600 Hz, and a maximum error of ±3%. The full

specification is listed in Appendix D.

6.3.2 Air gap sensor

Non-contacting air gap measurement is used extensively in manufacturing processes

where the measurement target is moving. Capacitive, inductive and eddy-current

measurement techniques form the basis of most industrial non-contacting air gap

sensors. Capacitive and inductive devices use the measurement target as part of a

capacitor or inductor and determine the air gap from the measured capacitance or

inductance. Eddy-current devices measure the power loss from a coil due to the power

dissipation associated with eddy-currents circulating in the flux-coupled target.

The maximum performance requirements for the air gap sensor are determined by the

electromagnet force controller which calls for a measurement range of 0.5-5.5 mm, a

resolution of 12 µm, a bandwidth of around 240-480 Hz, and a basic accuracy of ±3%

(see Table 6.1 and Table 6.2). A standard industrial eddy-current loss air gap sensor

has been selected which has a useable air gap measurement range of 3-10 mm, a

resolution of 15-30 µm, a bandwidth of 190 Hz, and a basic accuracy of ±3% when

suitably calibrated. The full specification for the device is listed in Appendix D. The

specification is slightly lower than desired, but precision devices are typically 3-5 times

the price of the standard industrial devices. Since the overall system performance is

affected by both the air gap sensor and the accelerometer, the marginal performance of

the air gap sensor is partly offset by the high resolution and bandwidth of the

accelerometer.


Eddy-current loss devices may not be appropriate for use on a production system

because the output is dependent on the target material resistivity which varies with

target temperature and material. Rapid movement of the sensor along the track may

also cause reduced accuracy due to eddy current losses induced by the motion of the

sensor along the track.

6.3.3 Electromagnet current controller

The electromagnet force control algorithm developed in Chapter 3 incorporates a

closed-loop electromagnet current controller, with a loop gain of 100. The

implementation of the current controller is now described, and it is based on the design

parameters derived in Chapter 3.

The primary dynamic characteristic of the current controller is determined by the

current feedback gain, which has already been calculated. The secondary dynamic

characteristic is the current slew rate, which is determined by the power supply voltage

and the characteristics of the electromagnet. Therefore, a power supply voltage for the

current controller must be calculated which gives an acceptable minimum current slew

rate.

Since the electromagnet is a component of a system designed to provide a comfortable

suspension ride, a force slew rate of 10% of the maximum load force per force

actuation time constant (Tforce ≈ 3.2 ms) is considered to be ample. The worst case

operating point in terms of the force slew rate occurs at the minimum air gap, since it

suffers the highest eddy current lag time constant. At this operating point (1 mm,

500 N), the electromagnet requires an excitation current, Imagnet, of 4 A. Generation of

a ±10% force change requires only a ±5% change in current due to the square law

relationship between the electromagnet current and the force. The required current

change, ∆Imagnet, is therefore ±0.2 A. The required supply voltage, which is the sum of

the voltage needed to slew the current plus the steady-state coil voltage, is given by:

where αphase_lead is time constant ratio of the eddy current phase-lead compensator, and

6.6Vsupply

Lcoils

αphase_lead

∆Imagnet

Tforce

Imagnet

Rcoils

volts

Rcoils is the resistance of the electromagnet coils. The maximum required supply voltage

is therefore:


The supply voltage required at an air gap of 5 mm with a full load is, surprisingly,

6.7Vsupply

0.111 H3×0.2A

0.0032 s4A × 0.8Ω 24V

about the same as that required at 1 mm. At the 5 mm operating point, the reduced

voltage demand due to the smaller eddy current lag time constant, is offset by the

increased voltage demand due to the higher leakage flux and the higher steady-state

current.

The supply voltage required for a given implementation is slightly higher due to the

additional resistance of the power controlling devices, which are connected in series

with the electromagnet. A suitable technique for amplifying the low power control

signal to the high currents required for the electromagnet is now considered.

If a linear mode (class A) power amplifier design were employed, the maximum

theoretical efficiency would be about 33% since the electromagnet would have an

average voltage of 8 V across it, with the remainder dropped across the series current

regulator. This would produce an average amplifier dissipation of 16 V x 10 A =

160 W, and require a power supply rating of 24 V x 20 A = 480 W. Alternatively, the

use of a switch mode (class D) power amplifier typically enables an efficiency of about

85% to be achieved,89 thus reducing the average amplifier dissipation to about 12 W,

and the power supply rating to about 380 W.

A switch mode current controller is therefore desirable and since no suitable

commercial units were available, an electromagnet current controller was designed.90

Figure 6.1 shows the schematic diagram of the controller which incorporates electrical

isolation between the low power signal circuitry, and the high power circuitry.

The current is sensed using an electrically isolated precision current sensor with an

accuracy better than 1%, and a response time of about 1 µs. The measured current is

compared with the reference current demand and amplified to form the current error

signal. This analogue signal is fed into an industry standard pulse width modulator

(PWM) integrated circuit, which drives an electrically isolated H-bridge power

switch.91 Since only a uni-directional current drive is required, two active switches

are used in the bridge configuration,92 with passive devices used in the two remaining

legs.


The time constant of the closed-loop current response is around 1 ms, so a sampling (ie.

Figure 6.1 Schematic design of the electromagnet current controller.

PWM switching) frequency of at least 10 kHz is desirable. In order to introduce a

negligible phase delay due to sampling, and also to make the switching inaudible, a

PWM frequency of 40 kHz is used. This switching frequency favours the use of

MOSFET devices rather than thyristors or bipolar transistors for the bridge switches.93

The selected MOSFET devices have a maximum, temperature derated, drain-source

resistance of 50 mΩ, and are switched in 100-200 ns by a MOSFET driver integrated

circuit. This results in a total power dissipation of about 16 W at the nominal current

rating of 10 A. The sensor accuracy and closed-loop gain combine to give the current

controller an accuracy better than ±2%.

Each current controller module incorporates an on-board PWM oscillator which permits

independent operation of the module. However, for the experimental vehicle which has

four electromagnets, the slight difference in frequency between each of the PWM

oscillators would generate low frequency beat noise. This would be undesirable since

the noise would be within the bandwidth of the feedback control signals. Therefore,

for the vehicle configuration, a single oscillator source is used to ensure synchronous

operation of the four current controller modules, and hence prevent any low beat

frequency noise.

The current controller and electromagnet are also protected by two safety features.

First, the current controller receives an output disable control which is activated if

necessary by a system watchdog timer. In addition, a current limiter set to a level

of 21 A is incorporated in the design.


The high power supply for the current controllers is provided by commercial,

mains-powered switch mode supplies rated at 27 V and 15 A. The circuit diagrams for

the electromagnet current controller are listed in Appendix D.

6.4 Signal processing, conversion and communication

To ensure sufficient flexibility to investigate control algorithms of increased complexity

in the future, a scalable processing resource is desirable. A further requirement for

flexibility and future research is for a distributed processing system. In addition to

these system features, good language support and development system support for

scalable and distributed processor systems are required.

Researchers in the fields of real-time control and simulation94,95,96,97 have found

that parallel processing architectures can offer significant performance and modularity

advantages over single processor architectures.

Three microprocessor development systems were available for use with this work. The

target microprocessors and co-processors were the Intel 80386/80387, the Motorola

68030/68881, and the Inmos T800 transputer. In a study98 of these and other

processors, the transputer came out most favourably in terms of processor performance,

connectivity, parallel language support, and multi-processor development support.

Academic and industrial researchers99,100,101,102 have also found the transputer to

be particularly suitable for implementing parallel processing architectures. Initial work

using one transputer to perform the signal processing for an experimental single

electromagnet suspension103 confirmed that a T800 transputer based system was

capable of meeting the processing requirements at sampling rates around 1 kHz.

6.4.1 Transputers

The Inmos transputer family104 is a range of microprocessors designed for use as

parallel processing components. The family includes 16 bit integer processors (T2xx),

32 bit integer processors (T4xx), and 32 bit processors with a floating point unit (T8xx).

The T800 transputer (Figure 6.2) contains the following system components:


• 32 bit central processing unit (12.5 MIPS at 25 MHz);

• 64 bit floating point processor (1.5 MFLOPS at 25 MHz);

• 4 Kbytes internal memory;

• 2 timers;

• 4 bi-directional 5-20 Mbits/sec inter-processor serial communication links.

The inter-processor communication links facilitate the use of distributed processor

Figure 6.2 T800 transputer architecture.

systems with distributed memory. Unlike conventional bus systems with shared

memory, the inter-process communication bandwidth rises linearly with the number of

processors. Memory and input/output facilities can be readily extended using external

devices.105 Data transfers using the inter-processor links are performed concurrently

with process execution using direct memory access (DMA). This results in a low

performance degradation even when all links are running at full capacity. The

communication links can be connected directly between processors on a single card or

in a chassis unit. For longer distances, twisted pair, coaxial cable or fibre-optic links

can be simply and effectively used. Communication links between transputers and links

to external devices can be configured using a 32-way cross-point switch if necessary.

In order to provide flexibility when configuring software processes onto transputers, a

built-in hardware scheduler is provided which multi-tasks parallel processes on a single

processor.106 This enables algorithms to be coded in parallel without constraining

them to a particular parallel processor configuration. The scheduler performs a task

switch in about 1-4 microseconds which is extremely fast, thus permitting multiple

parallel threads to be efficiently used even with sampling intervals of approximately

1 ms.


Transputer development systems are available for a range of computer platforms. For

this work, an IBM compatible PC/AT is used with a plug-in card which accommodates

the development host processor. The development system software and application

software use the PC as a terminal and file server. Development system support for

many programming languages is available, most of which have parallel programming

extensions. For this work, occam is used because it provides a flexible and robust

environment, with particular regard to the data security issues of parallel processing.

6.4.2 Control system hardware structure

A modular hardware system has been developed to run the vehicle controller and the

single electromagnet rig controller. The system functions are partitioned and

implemented using six cards which can be freely configured. The functional

specification for each card is listed in Table 6.3. The TRAM motherboard takes

standard Inmos TRAM modules which consist of a transputer processor plus random

access memory (RAM).

The cards for the vehicle control system are partitioned between two chassis. One

contains the feedback sensor signal conditioning, the analogue to digital converter and

three TRAM motherboards. The other contains the digital to analogue converter and

the four electromagnet current controller modules. For the single electromagnet control

system, all the cards and modules are located in a single chassis.

In order to prevent earth-loop and other noise problems, the chassis and the

development computer system are all electrically isolated from each other. Digital fibre

optical connections operating at 10 Mbit/s are used for all inter-unit transputer link

communications. Rack input and output interface cards provide the optical input and

output facilities between equipment units. Since the optical link connections are short,

optical signal power loss is not a problem, and so plastic optical fibre cable is used

which is inexpensive.

The issues affecting the design of the analogue signal conditioning and conversion

cards, and the TRAM motherboard and optical interface cards are discussed next. The

chassis configurations and circuit diagrams for all the cards are detailed in

Appendix D.


6.4.3 Analogue signal conditioning and conversion

Table 6.3 Control system hardware function specifications.

CARD FUNCTION

SAP Sensor Analogue Processor:

- Current to voltage conversion for 4 air gap sensors.

- Filtering & amplification for 4 accelerometers.

ADC Analogue to Digital Converter:

- 8 analogue input channels.

- Anti-alias filters.

- Multiplexer selection.

- 12 bit, bipolar ADC, 10 µs conversion time.

- 16 bit, 10 MIPS integer processor (T212).

- Front panel switch input.

TMB Tram MotherBoard:

- Motherboard with 4 size1 TRAM sites.

DAC Digital to Analogue Converter:

- 4 buffered analogue output channels.

- 12 bit unipolar DACs.

- 16 bit, 10 MIPS integer processor (T212).

- Watchdog timer.

- Front panel switch input.

IFIN InterFace INput:

- Optical chassis input interface.

- 1 bidirectional transputer data link.

- Transputer reset & analyse control inputs.

- Transputer error control output.

IFOUT InterFace OUTput:

- Optical chassis output interface.

- 1 bidirectional transputer data link.

- Transputer error control input.

- Transputer reset & analyse control outputs.

The schematic design of the analogue to digital converter card is illustrated in

Figure 6.3. Eight analogue input channels are provided, each of which is buffered and

filtered to remove high frequency noise which would otherwise be aliased107 down to

the frequency bandwidth used by the control system signals. The anti-alias filters have

a critically damped, second-order response, with a corner frequency of 1.6 kHz. The


electromagnet current controllers switch a total of up to 80 A at a frequency of 40 kHz.

They can therefore be expected to be a source of noise which will be inductively and

capacitively coupled to the feedback signals. Such noise (at >40 kHz) is reduced by

56 dB by the anti-alias filters. The anti-aliased signals are selected by an analogue

multiplexer which routes the selected signal to the 12 bit analogue to digital converter,

where it is sampled and converted to the digital domain. A parallel input port is also

provided which is used solely to read the state of a switch mounted on the front panel

of the card.

The sensor analogue processor card applies the appropriate signal conditioning and

Figure 6.3 Schematic diagram of the analogue to digital converter.

amplification to the sensor signals to give a full scale deflection of 0-5 V for the air gap

sensor and ±5 V for the accelerometer. In addition, the accelerometer signals are

filtered by a first-order high pass filter with a corner frequency of 0.1 Hz to remove the

1 g measurement offset due to gravity.

The 12 bit analogue converter produces a channel quantisation amplitude of 2.5 mV

which in turn gives an air gap resolution of 5.0 µm and an acceleration resolution of

5 mm/s2.

The digital to analogue converter card employs four 12 bit converters to drive four

electromagnet current controllers. The schematic design of this card is illustrated

Figure 6.4. In addition, a parallel input port is used to monitor a panel mounted switch.

This card also carries the system watchdog timer which can disable the electromagnet


current controllers. If the watchdog is not triggered at least once every 10 ms, all

connected current controllers will shut down immediately. This feature helps to provide

fail-safe operation and a controlled system startup. The card also includes a 40 kHz

clock generator which is used to synchronise the PWM switching of the electromagnet

current controllers. An optical interface is provided on one of the transputer’s links to

provide electrical isolation for the vehicle chassis containing the DAC card and the

electromagnet current controllers.

The 12 bit digital converters are configured for a unipolar, full scale deflection of 10 V

Figure 6.4 Schematic diagram of the digital to analogue converter.

which produces an output quantisation level of 2.5 mV, and gives a current resolution

of 5 mA. The conversion resolutions for the air gap, acceleration and current signals

are thus at least twice the values specified in Table 6.2 for the end-to-end system signal

resolutions.

6.4.4 Digital signal processing and communication

The TRAM motherboard carries commercial plug-in transputer modules which perform

the signal processing functions. The ADC and DAC cards are provided with 16 bit

integer processors in order to facilitate intelligent analogue interfaces, rather than to

perform signal processing. Figure 6.5 illustrates the schematic design of the TRAM

motherboard. It simply connects four TRAM sites in a pipelined configuration, and


provides buffered interfaces between the processor control signals and the chassis

backplane.

To provide optical fibre connections between the two chassis and the PC, a pair of

Figure 6.5 Schematic diagram of the TRAM motherboard.

optical interface cards are used. One provides a chassis input interface, and the other

is configured as a chassis output interface. Each card provides optical interfaces for one

bi-directional transputer data link, and the processor reset, error and analyse control

signals.

6.5 Software design

The design of the control system software is described in five main parts. The reasons

for using occam are explained first, and some salient features of the language are

outlined. The high-level structure of the control system software is then described.

Next, a method for converting the continuous-time algorithms to the discrete time

domain is selected. The level of numerical precision required for the signal processing

is then calculated. Finally, the configuration of the constituent processes of the control

system software onto the hardware is described.

6.5.1 Occam

The occam programming language108 is a high level language, designed to express the

sequential and concurrent components of algorithms, and their configuration on a

network of processors. Since the transputer has a built-in multi-tasking scheduler,

parallel algorithms can be run on a single processor as well as being distributed over

a network of processors. The strength of this facility is that a parallel algorithm can


be directly coded into a parallel program, with little regard to the processor

configuration. The program can then be configured to execute on one or more

processors, parallelism and communication permitting.

Occam has been used for this work for three main reasons. Firstly, the use of

concurrent processing hardware introduces high level concurrency within the software

whilst control systems typically contain low level concurrency as well.109 Occam is

a rare example of a programming language specifically designed to implement

concurrency at all structural levels in a natural and efficient manner. Secondly, when

using concurrent algorithms, occam can provide a degree of security unknown in

conventional sequential programming languages. Finally, the transputer reflects the

occam structural model and may be considered an occam machine. Programming in

occam is thus almost as efficient as using assembly language on conventional

processors.

The features which differentiate occam from conventional languages are outlined in

Appendix E prior to the listings of the control system software programs.

6.5.2 Control system software structure

The structure of the control system software has been designed using a combination of

functional decomposition as advocated by Wirth,110 and functional partitioning to

minimise the data flow between processes as proposed by DeMarco.111 Figure 6.6

illustrates the structure of the system in terms of tasks which are connected via channels

providing synchronised communication of data signals, control messages and exception

reports. The real-time data flows through the input signal interface, the state

calculators, the control algorithms and the output signal interface. These components

are supported by the sample scheduler, the exception handler, the data monitor and the

user interface.

Table 6.4 gives a functional overview of the software tasks for the vehicle control

system. The specification for the single electromagnet suspension is the same except

for the use of only one electromagnet rather than four.

The control algorithms block depicted in Figure 6.6 is partitioned into two sub-blocks,

which perform the suspension mode force calculation and the electromagnet current

calculation. Figure 6.7 illustrates this arrangement for the single electromagnet


suspension system. The air gap and acceleration signals are supplied by the input signal

Figure 6.6 Control system task structure, and data and control flows.

interface and the current demand is fed to the output signal interface. The parallelism

within the state calculation and the control blocks is fine grained, and is therefore not

well suited to parallel processing using transputers.

The vehicle suspension system illustrated in Figure 6.8 uses four independent

Figure 6.7 Single electromagnet controller data flow

suspension controllers, one for each of the vehicle motions, heave, pitch, roll and

torsion. Each suspension controller, plus its associated transformation and state

calculation, involves a significant amount of computation. This results in a


computational granularity which is sufficiently large to utilise parallel processing

Table 6.4 Vehicle control software functional overview

Input tasks:

• Read in acceleration and air gap sensor signals using ADC.

• Scale signals, check and report errors.

State calculation:

• Calculate velocities and positions from acceleration signals (Chapter 4).

• Calculate track position from magnet positions and air gaps (Chapter 4).

Control tasks:

• Apply the vehicle suspension control algorithm (Chapter 5) and magnet force

control algorithm (Chapter 3) to the input and calculated data sets to produce

a set of current demands for output. Check signals and report errors.

• Send user selected data to PC via asynchronous monitor process.

Output tasks:

• Scale current demand signals and output to the DACs.

• Check signals and report errors.

• Reset watchdog timer if the system is operational.

Miscellaneous:

• Permit user modification of controller parameters (including sample rate).

• Provide user selectable test reference signals (d.c., sine wave, square wave).

• Generate smooth startup and shutdown under normal and error conditions.

effectively using one transputer per suspension controller. After the vehicle motion

force demands have been calculated, they are transformed to electromagnet force

demands which are fed to the electromagnet controllers. The four independent

electromagnet force controllers involve a level of computation similar to that of the

vehicle suspension controllers, and so parallel processing using one transputer per

controller is again efficient.

For the sake of convenience in terms of monitoring and comparing vehicle mode

signals, the input and output decoupling transformations also normalise the signal

amplitudes for the vehicle pitch, roll and torsion modes. This normalisation is

described in Chapter 5, and it converts the angular signals and torques to the equivalent

linear signals and forces at the electromagnet centres. Normalising the signals also

permits a single data monitoring configuration to handle signals for suspension control

algorithms using different decoupling transformations.

The sequencing of the operations of the vehicle control system can be summarised in

terms of the following eight major phases:


1. Delivery of all magnet air gap and acceleration signals to each vehicle

section.

2. Transformation of magnet signals to vehicle mode motions.

3. Calculation of unmeasured vehicle motion states.

4. Computation of each vehicle mode force demand.

5. Distribution of all vehicle mode forces to each magnet control section.

6. Transformation of vehicle mode force demands to magnet force demands.

7. Computation of each magnet current demand.

8. Delivery of each magnet current demand to the current demand signal pool.

These phases occur irrespective of the number and configuration of processors used to

Figure 6.8 Vehicle suspension controller data flow

run the control system. For a balanced processor load distribution, either 1, 2 or 4

processors can be used, each computing either 4, 2 or 1 vehicle motion sections

followed by the same number of electromagnet control sections. The number of

processors required depends on the individual processor power, the algorithm

complexity and the required speed of execution. The multi-processing overheads for


the transputer, including the inter-processor communication overheads are small which

gives a high multi-processor utilisation.

6.5.3 Discrete time domain integration

The control algorithms developed in Chapters 3, 4 and 5 have been designed in the

continuous time domain. Therefore, they must be converted to discrete time domain

representations before they can be implemented in software. The selection of a suitable

technique for implementing discrete-time integrators and filters is considered next.

Filters are used by the state calculation algorithm, the suspension mode control

algorithm and the electromagnet force control algorithm. The first two algorithms use

low pass and high pass filters, with time constants ranging from 1.6 s to 40 ms, whilst

the electromagnet force control algorithm uses a phase lead compensator with a pole

time constant of 1.5 ms. The time constant of the phase lead compensator pole is thus

close to the required control system sampling interval of 0.5-1.0 ms. The discrete-time

representation is therefore a fairly crude approximation to the continuous-time response.

However, the effect of this misrepresentation is to increase the effective phase lead

slightly which is not detrimental to the location of the closed-loop system poles. The

simulation model listed in Appendix C was used to verify this assumption. The high

pass filter and phase lead compensator are formulated in terms of a first-order low pass

filter as described in Table 6.5.

Various integration algorithms are available for approximating the continuous time

Table 6.5 Filter formulations

LowPass(s,T )1

1 sT

HighPass(s,T )sT

1 sT1 LowPass(s,T )

PhaseLead(s,T,N )1 NsT

1 sTN (N 1)LowPass(s,T )

domain in the discrete time domain and each method has its own merits.112 A

primary requirement for this development is that the approximation algorithm must be

cascadable so that each independent functional block within the control system can be

designed and implemented independently. The algorithm must also produce an accurate


d.c. gain, and map stable continuous-time poles to stable discrete-time poles. These

requirements are effectively met by the Tustin algorithm (trapezoidal integration) and

by the first difference algorithm (Euler integration). Since all the filters except for the

phase lead compensator have time constants which are orders of magnitude greater than

the sample interval, the simpler first difference algorithm is sufficiently accurate. The

lower quality approximation that results for the phase lead compensator is acceptable.

Prewarping113 of the filter time constants is unnecessary for the normal control system

parameter settings since the time constants are much larger than the sampling interval,

and so their pole placement accuracy is normally very good. However, frequency

prewarping is required for the phase lead compensator time constant since it is so small.

It is also employed for the other time constants to allow for experimentation with

extreme settings for the filter frequencies and the sampling interval. Table 6.6 lists the

discrete time algorithms which are used for integration and low-pass filtering, and the

prewarping correction factor.

Table 6.6 Discrete-time integrator and filter implementation

Integrator: yk

yk 1

xk

Tsample

Low pass filter: yk

yk 1

1T

sample

Tcutoff

xk

Tsample

Tcutoff

Prewarp correction:

Tsample

Tcutoff corrected

≈ 1 exp

Tsample

Tcutoff

6.5.4 Numerical accuracy

Having determined the discrete time integration algorithm, the level of numerical

accuracy required to implement the control algorithms is now calculated. A

floating-point representation is assumed for all operations in order to have precision

independent of scaling. This eliminates the implementational overhead associated with

the use of fixed-point representations where signals must be re-scaled as necessary to

maintain sufficient precision.


The resolution required for the transducer signals calls for the use of 12 bit

analogue-to-digital and digital-to-analogue converter hardware. The calculation of each

current demand signal involves a total of about 80 mathematical operations, and in

general, a 32 bit floating-point representation (which has a 23 bit mantissa) provides

sufficient numerical accuracy for 12 bit data. However, loss of accuracy can occur

where two values which differ by many orders of magnitude are added or subtracted.

The worst case of this behaviour occurs in the state integration filters for velocity and

position, where the required sample interval of 0.5-1.0 ms can result in very small

values being summed onto the very much larger value of the state integrator.114

Table 6.7 shows the calculation of the relative size of the input and integrator values

for the velocity and position state calculators. The sampling interval, Tsample, is assumed

to be 0.5 ms, and the integration filter corner frequency of 0.1 Hz gives a filter time

constant, Tfilter, of 1.6 s. The decay product term is neglected for this analysis since it

is approximately unity. Accmin is set to half the required quantisation amplitude for the

acceleration measurement and velmin represents the corresponding velocity quantisation.

Velmax and posmax represent the maximum practical values that the velocity and position

signals can have.

The position integration filter provides the most exacting requirement since the

Table 6.7 Signal magnitude calculation for the state integration filter

Euler integration for velocity integration filter (worst case scenario) gives:

vel = velmax + accmin Tsample / Tfilter

= 10-1 m/s + 5x10-2 m/s2 5x10-4 s / 1.6 s

= 10-1 m/s + 1.6x10-5 m/s

Relative size: 10-1/1.6x10-5 = 6.4x103

Euler integration for position integration filter (worst case scenario) gives:

pos = posmax + velmin Tsample / Tfilter

= 5x10-3 m + 1.6x10-5 m/s 5x10-4 s / 1.6 s

= 5x10-3 m + 5x10-9 m

Relative size: 5x10-3/5x10-9 = 106

maximum accumulated position signal can be 106 times larger than the minimum added

value. A 32 bit floating-point representation with a 23 bit mantissa (plus the sign bit)


has a precision of just under 7 decimal digits. Under the worst case scenario of

maximum position signal and minimum velocity, the input value would be summed into

the integrator with a precision of less than one decimal digit. This would cause severe

pole placement inaccuracy, and a significant loss of the summed signal accuracy. The

state integration filters therefore use a 64 bit floating-point representation which has a

52 bit mantissa giving a precision of about 15.5 decimal digits.

A similar argument applies to the guideway following filter which consists of two

Table 6.8 Signal magnitude calculations for the guideway following filter

Euler integration for guideway filter (worst case scenario) gives:

trk = trkmax + gapmin Tsample / Tfilter

= 5x10-3 m + 5x10-6 m 5x10-4 s / 4x10-2 s

= 5x10-3 m + 6.25x10-8 m

Relative size: 5x10-3/6.25x10-8 = 8x104

And similarly for the second (cascaded) guideway filter gives:

Relative size: 5x10-3/7.8x10-10 = 6.4x106

cascaded low-pass filters. Table 6.8 shows the magnitude calculations for the guideway

filter where gapmin is set to half the required air gap signal quantisation and trkmax is the

maximum air gap measurement. Once again, the large relative size between the two

filter terms, almost 7 decimal digits, requires the use of 64 bit floating-point numbers

for the guideway following filters.

The fast time constant used by the phase lead compensator used in the magnet force

control algorithm results in 32 bit floating-point accuracy being sufficient. The

remaining computation for the mode and force transformations and the suspension and

electromagnet control algorithms can all be satisfactorily performed using 32 bit

floating-point arithmetic.

6.5.5 Process configuration

Having determined the requirements for the control system software, the final stage is

to partition the software tasks onto a suitable transputer configuration. For the single


electromagnet suspension controller, this is simple since a single 20 Mhz T800

processor is sufficient to provide a minimum sampling interval of 0.8 ms. Experimental

responses showed this sampling rate to be more than adequate. Responses were

therefore obtained with larger sampling intervals, and they showed that the response

started to become unacceptable for values larger than about 2 ms.

For the vehicle control system software, the execution time for all of the tasks identified

in Figure 6.8 was measured for a single 25 MHz T800 processor. This achieved a

minimum sampling interval of about 2.3 ms, compared with the target range of

0.5-1.0 ms, and experimental responses showed it to be unsatisfactory. It was estimated

that the use of two processors would reduce the sample interval to about 1.2 ms, whilst

employing four processors would reduce it down to about 0.65 ms.

For economic reasons, a two processor configuration was investigated. This

configuration executed with a measured sampling interval of 1.25 ms, and good

experimental responses were obtained. Figure 6.9 shows how the software tasks are

partitioned between the two processors. This configuration and sampling interval is

used for all of the experimental vehicle responses presented in this dissertation. The

lower processing load on the signal processor without the data monitor and its

associated real-time buffer permits the use of a 20 Mhz processor.

A fully controlled vehicle with four suspension electromagnets and four guidance

electromagnets could therefore be controlled by four transputers, since the additional

overheads are minimal. The code listings for all of the software described in this

chapter are listed in Appendix E.

6.6 Conclusions

This chapter has shown that the electromagnetic vehicle suspension strategy developed

in the earlier chapters can be implemented using readily available parallel processing

hardware. The use of transputers for the signal processing and intelligent converter

interfaces enabled a highly modular and inexpensive real-time processing platform to

be constructed. This was complemented by using the occam programming language

which enabled a fairly complex software implementation to be developed very rapidly.


Figure 6.9 Vehicle control system process configuration

Conclusions 140

7

Conclusions

The aim of the research described in this dissertation has been to improve the

performance of electromagnetic secondary suspension for vehicles through the use of

improved control techniques.

The difficulties associated with the control of an electromagnetic vehicle suspension

accrue from the nonlinear and unstable nature of the electromagnetic force

characteristic. In addition, for systems which use electromagnets to provide both the

primary and the secondary suspension functions, the direct coupling of the

electromagnets to the vehicle chassis causes further complications by producing a

multivariable control problem.

The proposed structured design approach requires the electromagnets to be controlled

in such a way that they can be regarded as independent linear force actuators. The

electromagnet force characteristic has therefore been analysed, and a detailed nonlinear

model has been developed. Practical force control schemes employing only linear

feedback techniques have been shown to be unsuitable for providing independent linear

force actuation in an environment where a number of electromagnets are rigidly

coupled. A new force control algorithm has therefore been developed which employs

the detailed electromagnet model, in conjunction with electromagnet air gap and current

feedback, to provide force actuation which is dominantly linear and independent.

Having obtained a suitably linear force actuation, the vehicle suspension is controlled

by using linear transformations to translate between the electromagnet coordinate system

and a vehicle coordinate system based on the heave, pitch, roll and torsion of the

vehicle chassis. Each vehicle mode motion is then controlled by an independent

suspension controller.

Conclusions 141

A sophisticated control algorithm has been developed to control the independent

suspension modes. This consists of an absolute position controller, designed to achieve

the required disturbance force rejection, which receives its position reference from a

filtered version of the track position signal. All feedback signals are derived from

measurements of the absolute acceleration and the air gap of each electromagnet. The

structure of the new suspension control algorithm provides greater design flexibility

than existing algorithms. This is because it permits the algorithm which defines the

guideway following characteristic, to be designed largely independently of other

suspension design considerations.

In order to obtain experimental responses in addition to simulated responses for the

proposed vehicle suspension control scheme, an experimental suspension control system

using digital signal processing has been developed. The signal processing hardware is

based around the transputer family of microprocessors, and the control algorithms have

been implemented using the occam parallel programming language. The simulated and

experimental results presented in this dissertation have indicated the success of the

proposed control method, to an extent where confidence is given in its potential for

development to a full scale system.

The proposed electromagnet force control scheme has been shown to possess significant

advantages compared with existing stabilisation techniques using flux derivative

feedback due to its dominantly linear force actuation. However, it suffers a slight

disadvantage due to an increased reliance on an accurate air gap measurement at small

air gaps (see Chapter 3). If desired, this drawback could be overcome by developing

a hybrid control approach combining flux derivative feedback, for stability, with the

proposed scheme, for force linearity.

In addition, a number of areas which could benefit from further research have been

identified. Firstly, an improvement in the modelling accuracy for the electromagnet

core eddy-current time constant is desirable (see Chapter 2). Secondly, lateral force

control through air gap modulation (see Chapter 4) may permit lateral damping for

research vehicles using only suspension electromagnets. Thirdly, analysis of vehicle -

guideway interaction and the development of suitable damping algorithms may be useful

to permit the use of flexible guideways (see Chapter 1). Finally, and most importantly,

guideway following algorithms need to be researched in order to minimise air gap

deviations at the entrance and exit of gradients (see Chapters 1 and 4).

Conclusions 142

The results of this work suggest that with some additional research and development

in the areas outlined above, electromagnetic suspension may in future provide an

effective method for providing high ride quality, and low cost, for vehicle suspensions

in urban transit applications. For wheel-on-rail transport applications, the wheel may

then become a historical curiosity.

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Documents

Control of an Electromagnetic Vehicle Suspension