nonlinear control of a planar magnetic … · list of figures list of figures 3.12 case 2 response...

NONLINEAR CONTROL OF A PLANAR

MAGNETIC LEVITATION SYSTEM

by

Michel Levis

A thesis submitted in conformity with the requirements

for the degree of Master of Applied Science,

Graduate Department of Electrical and Computer Engineering,

in the University of Toronto.

Copyright c© 2003 by Michel Levis.

All Rights Reserved.

Abstract

This thesis initiates a research aimed at developing tools that may have practical sig-

nificance in contactless position control applications such as, e.g., photolithography.

We describe a simple three-magnet planar positioning device, its mathematical model,

and design a nonlinear controller that stabilizes it about an equilibrium. Specifically,

we derive a feedback transformation mapping the nonlinear system with three posi-

tive inputs into a linear system in Brunovsky normal form with two inputs. Robust,

adaptive and robust adaptive controllers are then designed in the transformed input

domain and their effectiveness in handling uncertainties is compared through simu-

lations. An experimental testbed of the planar magnetic levitation device has been

constructed but due to hardware limitations it cannot yet be used as a benchmark

to test the controllers developed here. However, two simpler experiments give insight

into applied nonlinear control design.

ii

Acknowledgements

The author would first like to thank Manfredi Maggiore for the opportunity to work

on this interesting problem and for all the guidance given over the work period; J.T.

Spooner whom provided inspiration for some of the ideas contained in this document;

Jacob Apkarian for his many helpful remarks on the implementation of the exper-

imental testbed; V.M. Alexander and Al Shabia Engineering, Sharjah, U.A.E., for

supplying and designing the electromagnet cores; Peter Lehn for the help on noise is-

sues and for the practical advice on several implementation aspects; Marcel Levis for

building the platform and the various other objects needed for the implementation.

iii

Contents

Abstract ii

Acknowledgements iii

List of Figures ix

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Contributions of the thesis . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Modelling of the Planar Magnetic Levitation Device 9

2.1 Force Dynamics of Disk from Single Electromagnet . . . . . . . . . . 11

2.2 Vector Analysis and System Dynamics . . . . . . . . . . . . . . . . . 15

2.3 State-Space Representation of Motion Equations . . . . . . . . . . . . 19

2.4 Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Nonlinear Control Design 26

3.1 Ideal Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 37

iv

CONTENTS CONTENTS

3.2 Robust Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . 39


3.3 Adaptive Control Design . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.1 Backstepping Design . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.2 Simulations Results . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4 Robust Adaptive Control Design . . . . . . . . . . . . . . . . . . . . 57

3.4.1 Backstepping Design . . . . . . . . . . . . . . . . . . . . . . . 57


4 Implementation 77

4.1 System Components . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.1.1 Magnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.1.2 Power Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.1.3 Current Controller . . . . . . . . . . . . . . . . . . . . . . . . 83

4.1.4 Disk and Linear Guides . . . . . . . . . . . . . . . . . . . . . 88

4.1.5 Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.1.6 Real-time Controller Software and Interface . . . . . . . . . . 94

4.2 Finding Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . 96

4.2.1 Inductance Models . . . . . . . . . . . . . . . . . . . . . . . . 98

4.2.2 Measuring Inductance . . . . . . . . . . . . . . . . . . . . . . 102

4.2.3 Modelling Procedure . . . . . . . . . . . . . . . . . . . . . . . 104

4.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.3.1 Two-Magnet System . . . . . . . . . . . . . . . . . . . . . . . 110

4.3.2 Three-Magnet System using Equivalent Ideal Controller . . . . 117

4.3.3 Three-Magnet Configuration using 2 DOF Controller . . . . . 124

5 Conclusion 130

v

CONTENTS CONTENTS

A Model Configuration Analysis 132

A.1 Electromagnetics Background . . . . . . . . . . . . . . . . . . . . . . 132

A.2 Superposition Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Appendix 132

Bibliography 139

vi

List of Figures

1.1 Canon high-precision wafer stage. . . . . . . . . . . . . . . . . . . . . 2

1.2 Magnetically levitated stage using linear motors. . . . . . . . . . . . . 3

1.3 Forces acting on disk when at origin. . . . . . . . . . . . . . . . . . . 5

1.4 Roadmap of this thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1 Amperian path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Distances from center of disk to each magnet. . . . . . . . . . . . . . 16

2.3 Flux leakage of an electromagnet. . . . . . . . . . . . . . . . . . . . . 23

3.1 Overview of control system. . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Domain of attraction estimate. . . . . . . . . . . . . . . . . . . . . . . 36

3.3 Position and speed trajectories when using the ideal controller in the nominal system. 37

3.4 Projection of the phase curves on the x1 − x3 plane when using the ideal and linear controllers

3.5 Case 1 response of uncertain system when using ideal and robust nonlinear controllers. 42

3.6 Case 1 response of uncertain system when using a linear controller and the robust nonlinear con

3.7 Currents of robust nonlinear controller in system subject to case 1. . 43

3.8 Case 1 response of uncertain system when using adaptive nonlinear controllers. 53

3.9 Case 2 response of uncertain system when using linear, robust and adaptive nonlinear controllers.

3.10 Control input of adaptive controller in case 2. . . . . . . . . . . . . . 55

3.11 Case 1 response of uncertain system when using the robust adaptive controller. 72

vii

LIST OF FIGURES LIST OF FIGURES

3.12 Case 2 response of uncertain system when using robust, adaptive and robust adaptive nonlinear

3.13 Currents from robust adaptive controller when in case 2. . . . . . . . 74

3.14 Case 3 response of uncertain system when using linear, robust, adaptive and robust adaptive

4.1 Top view of planar magnetic levitation device. . . . . . . . . . . . . . 77

4.2 Side view of planar magnetic levitation device. . . . . . . . . . . . . . 78

4.3 Overview of interfaces in magnetic levitation device. . . . . . . . . . . 79

4.4 Input/Output relationship between PWM and electromagnet. . . . . 82

4.5 Pulse-width modulator wiring. . . . . . . . . . . . . . . . . . . . . . . 83

4.6 Current controller structure. . . . . . . . . . . . . . . . . . . . . . . . 84

4.7 Current controller circuit. . . . . . . . . . . . . . . . . . . . . . . . . 85

4.8 Summer operational amplifier. . . . . . . . . . . . . . . . . . . . . . . 86

4.9 Proportional-integral operational amplifier. . . . . . . . . . . . . . . . 86

4.10 PI control tracking reference current. . . . . . . . . . . . . . . . . . . 87

4.11 Linear guides mounted in an X-Y fashion. . . . . . . . . . . . . . . . 89

4.12 Laser reflected off a curved surface. . . . . . . . . . . . . . . . . . . . 91

4.13 Linear guide and sensor setup. . . . . . . . . . . . . . . . . . . . . . . 93

4.14 Interface between position controller and magnetic levitation device. . 94

4.15 Model B inductance samples using method 1. . . . . . . . . . . . . . 105

4.16 Model B inductance samples using method 2. . . . . . . . . . . . . . 105

4.17 Model A inductance expressions using method 1 and 2. . . . . . . . . 108

4.18 Two-magnet system experimental setup. . . . . . . . . . . . . . . . . 110

4.19 Two-magnet system diagram. . . . . . . . . . . . . . . . . . . . . . . 111

4.20 Run 1: Disk position using the linear controller in the two-magnet system.114

4.21 Run 1: Currents of linear controller in the two-magnet system. . . . . 114

4.22 Run 2: Disk position using the linear controller in the two-magnet system.115

4.23 Run 3: Disk position using the ideal controller in the two-magnet system.116

viii

LIST OF FIGURES LIST OF FIGURES

4.24 Run 3: Currents of ideal controller in the two-magnet system. . . . . 116

4.25 Run 4: Disk position using the ideal controller in the two-magnet system.116

4.26 Run 5: Disk position using the equivalent ideal controller. . . . . . . 122

4.27 Run 5: Currents from the equivalent ideal controller. . . . . . . . . . 122

4.28 Run 6: Disk position using the equivalent ideal controller. . . . . . . 122

4.29 Run 6: Currents from the equivalent ideal controller. . . . . . . . . . 123

4.30 Run 7: Disk position using the 2 DOF ideal controller. . . . . . . . . 126

4.31 Run 7: Currents from the 2 DOF ideal controller. . . . . . . . . . . . 126

4.32 Run 8: Disk position using the 2 DOF ideal controller. . . . . . . . . 127

4.33 Run 8: Currents from the 2 DOF ideal controller. . . . . . . . . . . . 127

4.34 Response of disk from acceleration test. . . . . . . . . . . . . . . . . . 128

4.35 Currents from the acceleration test. . . . . . . . . . . . . . . . . . . . 128

A.1 Fringing between two magnets. . . . . . . . . . . . . . . . . . . . . . 132

A.2 Magnetic flux density plot for narrow magnets. . . . . . . . . . . . . . 134

A.3 Magnetic flux density plot for wide magnets. . . . . . . . . . . . . . . 135

A.4 Magnetic flux density plot for wider sized magnets. . . . . . . . . . . 136

ix

Chapter 1

Introduction

1.1 Motivation

The semiconductor industry is forced to refine the photolithography process to ac-

commodate the increasing need of denser integrated circuits. Photolithography is

a step in semiconductor production where patterns from a mask are drawn on a

photo-sensitive silicon wafer using an optical system. The linewidths of this pattern

are steadily decreasing and are created on silicon wafers of increasing diameter, e.g.

past generation requirements were linewidths of 0.13 µm and 300 mm diameter silicon

wafers [8]. High-precision positioning stages capable of actuating large movements are

used in this process to supply the necessary incremental movement on the wafer such

that the pattern is drawn on correctly. Figure 1.1 is a diagram of a stage developed

by Canon.

Typically, in industry, these positioning systems are comprised of a lower-stage

that actuates large high-speed movements and an upper-stage that delivers high-

precision movements in multiple degrees of freedom [12]. The mechanical contact

between the platen and the stage introduces friction, vibrations, coupling between

1

1.1. MOTIVATION 2

Figure 1.1: Canon high-precision wafer stage.

the axes, and can even introduce impurities in the operating environment. The bot-

tom stage is typically actuated by a linear motor drive such as a ball-screw drive.

The upper-stage in the positioning system described in [12] is a flexure-based design

that is driven by piezoelectric actuators that are capable of fine resolution but possess

severe hysteresis nonlinearity. Mechanical contact problems and the inherent nonlin-

earities of piezoelectric actuators can be avoided by using planar magnetic levitation

technology to move the platen.

Magnetic levitation positioning devices have been investigated in the past. Per-

haps among the most successful research in this direction is the one reported by

Trumper and colleagues in [11], shown in Figure 1.2, where the authors use a linear

controller to actuate a 6 DOF planar magnetic levitation device that achieves planar

motions of up to 50 × 50 mm2 with nm accuracy using linear motors. Linear mo-

tors are indeed particularly suitable for magnetic levitation applications due to their

superior range of operation.

Electromagnets can also be used for magnetic levitation, they are cheaper to

build, easier to control than linear motors, but typically suffer from smaller range

1.1. MOTIVATION 3

Figure 1.2: Magnetically levitated stage using linear motors.

of operation. This drawback becomes particularly evident when controlling them

using linear controllers derived by linearizing the system dynamics about a desired

operating condition, since in this case the range of operation and the robustness

versus uncertainties are affected. Typically when controlling such electromechanical

systems using a linear controller, the focus is directed towards the hardware design of

the system and developing an accurate model of the device. Less emphasis is placed

on the control system design.

Instead nonlinear controllers can be developed that take advantage of the dy-

namics between the magnets and the platen to achieve superior performance and

operating range. In [7], a 6 DOF planar magnetic levitation driven by electromagnets

has planar motions extending to 4 × 4 mm2 when using a nonlinear controller. This

range is limited by the dimensions of the device.

1.2. PROBLEM FORMULATION 4

1.2 Problem Formulation

In this thesis, we focus on a planar magnetic levitation device which employs electro-

magnets to achieve 2 DOF, while keeping a relatively large operating range. To avoid

the limitations mentioned above, we develop a rigorous nonlinear control framework

to solve the stabilization problem over a guaranteed range, and apply various control

techniques to make the closed-loop system robust versus a class of uncertainties. The

electromechanical system considered includes the triangular arrangement shown in

Figure 1.3 which differs from most magnetic levitation positioning devices. Using the

minimum number of electromagnets required to actuate two degrees of freedom in a

triangular arrangement was explicitly chosen to create a challenging and interesting

nonlinear control design problem. Figure 1.3 demonstrates a plan view of the system

and the forces exerted on the disk by each magnet. Each of the rectangles represents

an electromagnet with a ferromagnetic core with N coil windings. The circle in the

middle of the plane is a disk, also of ferromagnetic material, whose position we want

to control.

The full realization of the system would include a fourth magnet suspended above

the three magnets shown. The additional magnet producing a force in the z direction

is independent of the magnets in the xy plane (at least theoretically). That is, the

system is comprised of two decoupled subsystems - the base magnets formed in a

triangle and the suspended magnet above this plane. The focus of this research is on

the xy subsystem.

1.3. CONTRIBUTIONS OF THE THESIS 5

F1

F3

F2

Electromagnet 1

Electromagnet 2

Electromagnet 3

(0,0)

z

y

x

Figure 1.3: Forces acting on disk when at origin.

1.3 Contributions of the thesis

This thesis presents both theoretical and practical aspects of modelling, controlling

and building an electromechanical system. Notably, a complete analysis of a chal-

lenging applied nonlinear control problem is given.

The modelling of the positioning device using standard electromagnetic techniques

is described in detail along with a discussion of the various uncertainties affecting the

model. The nonlinear control design is challenging due to the arrangement of the

magnets and the nonlinear force relationship between each magnet and the disk. By

assuming an uncertainty-free system, a nonlinear control is designed that successfully

stabilizes this nonlinear, coupled system. The control design is then modified by

including robust, adaptive, and robust adaptive techniques to stabilize the system

when uncertainties are present. By comparing the controllers through simulation,

the strengths and hindrances of using the various control techniques in the actual

1.4. ORGANIZATION OF THE THESIS 6

positioning device are outlined.

On the practical side, a prototype of the magnetic levitation device is built and

can potentially be used to test nonlinear controllers and their ability to compensate

for uncertainties. The hardware design of the testbed is described and, although, due

to hardware limitations, the controllers could not be tested, significant modelling and

control results are given and discussed using simpler testbeds.

1.4 Organization of the thesis

The document is divided into three main parts: modelling, nonlinear control design,

and implementation. Chapter 2 develops a mathematical model describing the three-

magnet positioning system. The basic electromagnetic analysis needed to derive the

dynamics of the disk from a single magnet is first derived in Section 2.1. In Section

2.2 this one-dimensional result is expanded in the xy plane to find the equations of the

forces acting on the disk from all three magnets, which we then use to find the state-

space representation of the disk dynamics shown in Section 2.3. The chapter is closed

in Section 2.4 with a discussion of the uncertainties of the model due to some as-

sumptions made. Background information is available in Appendix A which includes

the analysis of the different configurations considered for the triangular arrangement.

The nonlinear control design of the system is presented in the third chapter. In

Section 3.1 we assume the model developed in Chapter 2 is not affected by uncertain-

ties and derive a feedback transformation yielding linear dynamics. In the transformed

domain, an LQR controller is designed to complete the feedback loop and stabilize

the system. In the next three sections, uncertainties in the model are considered and

various control schemes that handle these uncertainties are developed. Specifically,


in Section 3.2 Lyapunov redesign is performed to robustify the ideal controller de-

veloped in Section 3.1. Alternatively, the uncertainties can be compensated using a

classic adaptive controller, developed in Section 3.3. Lastly, Section 3.4 describes a

robust adaptive control that combines the uncertainty handling notions of robust and

adaptive techniques. Simulations comparing the system response of the ideal, robust,

adaptive and robust adaptive controllers are given in each section.

The fourth chapter features the actual hardware implementation of the planar

magnetic device. In Section 4.1, the electromechanical positioning device is described

by explaining the individual components used to build the system and their mu-

tual interactions. Although, due to hardware difficulties, the testbed is not fully

operational, two smaller experiments were designed to reveal the problem in the

three-magnet system and experimentally verify some modelling and control aspects

discussed in the second and third chapters. The results achieved with these testbeds

are fairly encouraging and substantial.

The document is finalized with a summary of the main objectives achieved along

with some recommendations on the hardware design and possible future prospects of

the research.

For readability purposes, a roadmap of this thesis explaining the relationships

between sections in this document is included in Figure 1.4. Sections 2.1 and 2.2

can be considered optional although they give insight into the modelling and some

parts are referenced when analyzing the testbed in Section 4.2. Similarly, Appendix

A gives reasons why this particular magnet configuration is being used and provides

some background on electromagnetic concepts. For the control design, the reader may

skip the uncertainty compensating controllers and focus on the ideal control design

detailed in Section 3.1. In that case, one just needs to read Section 2.3, where the

state-space model is given. If the robust, adaptive, or the robust adaptive controls


Robust Control

Ideal Control

AnalysisAdaptive Control

Robust Adaptive Control

System Components

Model Parameters

1D Dynamics

Model ConfigurationAppendix A

Uncertainties

Model

Vector AnalysisSection 4.2

Section 4.1

Section 3.2

Section 3.1

Section 3.3

Section 3.4

Section 2.2.1

Section 2.2.4

Section 2.2.3

Section 2.2.2

Section 4.3

Figure 1.4: Roadmap of this thesis.

explained in Sections 3.2, 3.3 and 3.4 are of interest then the ideal control design

in Section 3.1 is required, together with the discussion of modelling uncertainties in

Section 2.4. Appendix A is required to understand some concepts in Section 2.4.

As for the last chapter which discusses the implementation of the system, the ideal

control in Section 3.1 is needed to understand the analysis discussion in Section 4.3.

The hardware discussion of the testbed would be necessary for reading the analysis

part along with Section 4.2 which discusses finding model parameters for the testbeds.

The uncertainties in Section 2.4 are referenced by the examinations in Section 4.2.

Chapter 2

Modelling of the Planar Magnetic

Levitation Device

In this chapter, a set of differential equations describing the motion of the disk in our

planar positioning device is developed.

The equations describing the motion of the disk are

x =Fx(x, y, I1, I2, I3)

m

y =Fy(x, y, I1, I2, I3)

m,

(2.0.1)

where m is the mass of the disk. The forces Fx and Fy are the forces generated by

the three electromagnets with currents I1, I2, I3 in the x and y direction, respectively.

In this section we develop a mathematical model of the system depicted in Figure 1.3

using superposition of the forces and neglecting the fringing effect of the magnetic

flux lines. The following assumptions are made throughout the modelling process and

in other parts of the thesis:

Assumption 2.0.1. Fringing of the magnetic flux lines is negligible.

9

CHAPTER 2. MODELLING OF THE PLANAR MAGNETIC LEVITATION DEVICE10

Assumption 2.0.2. Flux leakage through the coils of the electromagnets is negligible.

Assumption 2.0.3. The relationship between the magnetic flux density, ~B, and the

magnetic field intensity, ~H, is linear.

Assumption 2.0.4. Currents vary slowly so a constant-current method can be used.

Assumption 2.0.5. Magnetic flux is contained throughout the Amperian path.

Assumption 2.0.6. The disk remains within the region where superposition holds.

Assumption 2.0.7. The attractive forces acting on the disk point towards the center

of each magnet face.

Assumption 2.0.8. The length of each core is sufficiently larger than the width and

height of the core.

Assumption 2.0.9. The uncertainty term θ2 |x3| in the expression (2.4.2) in Section

2.4 is negligible for the adaptive and robust adaptive control designs.

These assumptions are described later in Section 2.4 in terms of the modelling un-

certainties they may introduce. The dynamics of the disk can be modelled in four

steps.

1. Derive the dynamics of the forces acting on the disk from a single electromagnet.

2. Using the result in step 1, perform vector analysis to construct the force dy-

namics of the disk from all three electromagnets.

3. Use the result from step 2 in motion equations (2.0.1).

4. Find state-space representation.

2.1. FORCE DYNAMICS OF DISK FROM SINGLE ELECTROMAGNET 11

2.1 Force Dynamics of Disk from Single Electro-

magnet

This analysis is standard and the result can be found in, e.g. in [13]. The forces are

calculated by taking the gradient of the system’s magnetic energy. Magnetic energy

can be calculated from the magnetic flux. Thus, the first step is to find the magnetic

flux through the core of the electromagnets. Magnetic flux is found using Ampere’s

Law [1]∮

C

~H · d~l = µoIenc. (2.1.1)

Ampere’s law states that the line integral of H around any closed path C equals

the product of the current Ienc enclosed by the path and the permeability of free space

µo [1]. It can be restated as follows

∮

C

~H · d~l = NI, (2.1.2)

where N is the number of coil windings and I is the current going through the coils.

Magnetic field lines, ~H , and magnetic flux density lines, ~B, have the same direction

in a ferromagnetic material when the core is not saturated [3]. Assuming the core

never enters saturation, the relationship between ~B and ~H is linear,

~H =1

µ~B. (2.1.3)

where µ is the core permeability. The magnetic flux can be calculated in terms of the

magnetic flux density and cross sectional area of the core

Φ =

∫

S

~B · d~s. (2.1.4)


It is common to assume that the magnitude of the magnetic flux is constant, therefore

the expression becomes

Φ = BA, (2.1.5)

where A denotes the area of the cross section for the material. The magnetic flux

density near the electromagnet is ~B = Baz, where az is depicted in Figure 2.1. The

magnetic flux densities in the core, ~B1, in the air gap, ~B2, and in the disk, ~B3 are

given by

~B1 =Φ

A1

az

~B2 =Φ

A1

az

~B3 =Φ

A2

az.

(2.1.6)

C1

C4

C5

C6

W

L2

z

L1

C3

C2

NIA1

A2

az

Figure 2.1: Amperian path.

Taking the closed path shown in Figure 2.1, Ampere’s Law is used with the necessary


substitutions from equations (2.1.3) and (2.1.6) to get magnetic flux. Thus

∮

C

~H · d~l =

∮

C

1

µ~B · d~l

=

∮

C1

1

µ1

~B1 · d~l +∮

C2

1

µo

~B2 · d~l +∮

C3

1

µ2

~B3 · d~l + 0 +

∮

C5

1

µo

~0 · d~l + 0

=L1

µ1

Φ

A1+

z

µo

Φ

A1+L2

µ2

Φ

A2

=

(

L1

µ1A1

+L2

µ2A2

+z

µoA1

)

Φ

= NI.

The line integrals corresponding to C4 and C6 are zero because ~B is perpendicular

to the lines C4 and C6. The line integral corresponding to C5 can be assumed zero by

assuming that the lengths of C4 and C6 stretch to infinity. Thus, the magnetic flux

reads as

Φ =NI

L1

µ1A1+ L2

µ2A2+ z

µoA1

. (2.1.7)

Remark 2.1.1. The last fraction in the denominator of (2.1.7), zµoA1

, is the reluctance

of the air gap between the core and the disk. Since fringing is neglected, it is assumed

that the cross-sectional area of the magnetic flux passing through the air gap is equal

to the cross-sectional area of the core. If fringing were not neglected, the cross-

sectional area of the air gap would be greater than that of the core, A1, and it would

vary in size depending on the length of the air gap, z. Thus the reluctance of the

air gap would instead read as zµoA3(z)

where A3(z) > A1. Fringing is illustrated in

Appendix A.1.

Magnetic energy is defined as

Wm =1

2

∫

V

~B · ~Hdv (2.1.8)


where V is the volume of the object subject to ~B and ~H .

Using the equations (2.1.3) and (2.1.6), the magnetic flux expression (2.1.7) and

(2.1.8), the magnetic energy of the system can be expressed as

Wm =1

2

∫

V

~B · ~Hdv

=1

2

∫

V

1

µB2dv

=1

2

[∫

V1

1

µ1

(

Φ

A1

)2

dv +

∫

V2

1

µo

(

Φ

A1

)2

dv +

∫

V3

1

µ2

(

Φ

A2

)2

dv

]

=1

2

[

L1A1

µ1A21

Φ2 +zA1

µoA21

Φ2 +L2A2

µ2A22

Φ2

]

=Φ2

2

[

L1

µ1A1

+z

µoA1

+L2

µ2A2

]

=(NI)2

2

1L1

µ1A1+ L2

µ2A2+ z

µ0A1

,

where B denotes the magnitude of ~B, h is the height of the disk and A2 = L2h is the

cross-sectional area of the disk.

Using the constant current method (see, e.g. [1]), the force is calculated by taking

the gradient of the magnetic energy

~Fm = 5Wm. (2.1.9)

Relationship (2.1.9) holds true under the assumption that the current is constant.

While the current in the system under consideration is not constant, its time varia-

tion is typically slow and thus (2.1.9) is reasonably accurate (this approximation is

common in the literature (see, e.g. [13])).

2.2. VECTOR ANALYSIS AND SYSTEM DYNAMICS 15

Using equation (2.1.9) the force acting on the disk by one electromagnet is

~Fm = 5Wm

=∂Wm

∂zaz

= − (NI)2

2µ0A1

1(

L1

µ1A1+ L2

µ2A2+ z

µ0A1

)2az.

(2.1.10)

Similarly to the standard result, the one-dimensional force expression is proportional

to the current squared and to the reciprocal of the distance squared.

2.2 Vector Analysis and System Dynamics

Using superposition and the results from the previous section, we next derive the

forces acting on the disk from all three magnets. The force expression for a single

magnet is used in vector analysis to get the force equations of the entire system. The

force model of the system is as follows

~Fx = (F1 cos θ1 + F2 cos θ2 + F3 cos θ3) ax

~Fy = (F1 sin θ1 + F2 sin θ2 + F3 sin θ3) ay.

(2.2.1)

Figure 2.2 shows the angles θ1, θ2, θ3 and the forces acting on the disk when it is at a

location (x, y) inside the triangle 4P1P2P3, which is assumed to be equilateral. The

attractive force from electromagnet i, where i = 1, 2, 3, has the expression

~Fi = −(NIi)2

2µ0A1

1(

L1

µ1A1+ L2

µ2A2+ zi

µ0A1

)2azi. (2.2.2)


1

3

P1

P2

2

P3

w

l

z3

(0,0)

z1

y

x

az3

az2

az1

(x,y)

θ1

z2

θ3

θ2

d

Figure 2.2: Distances from center of disk to each magnet.


The value zi, depicted in Figure 2.2, is the distance between the edge of the disk and

the middle point, denoted Pi, of the face of electromagnet i. The distance between

the disk’s edge when at the origin and the face of each magnet is d. The direction of

the force exerted by each ith electromagnet is the unit vector azi, depicted in Figure

2.2. This unit vector changes with the disk’s position and is defined as

azi=

−~Fi∣

∣

∣

~Fi

∣

∣

∣

.

Next, zi can be expressed as

z1 = ‖(x, y) − P1‖

= ‖(x, y) − (−d, 0)‖

=√

(x+ d)2 + y2

z2 = ‖(x, y) − P2‖

=

∥

∥

∥

∥

∥

(x, y) −(

d

2,−

√3

2d

)

∥

∥

∥

∥

∥

=

√

(

x− d

2

)2

+

(

y +

√3

2d

)2

z3 = ‖(x, y) − P3‖

=

√

(

x− d

2

)2

+

(

y −√

3

2d

)2

.

(2.2.3)

The next step is to express sin θi and cos θi in terms of x and y. From Figure 2.2, it

can be shown that

sin θ1 =−yz1

cos θ1 =x+ d

z1


sin θ2 =y +

√3

2d

z2

cos θ2 =x− d

2

z2

sin θ3 =y −

√3

2d

z3

cos θ3 =x− d

2

z3.

By substituting the variable distances zi and putting the trigonometric functions in

terms of x and y, the Fx equation in (2.2.1) becomes

Fx = F1 cos θ1 + F2 cos θ2 + F3 cos θ3

= − 1

2µoA1

[

(N1I1)2

(

L1

µ1A1+ L2

µ2A2+ z1

µ0A1

)2

x+ d

z1+

(N2I2)2

(

L1

µ1A1+ L2

µ2A2+ z2

µ0A1

)2

x− d2

z2+

(N3I3)2

(

L1

µ1A1+ L2

µ2A2+ z3

µ0A1

)2

x− d2

z3

]

= − 1

2µoA1

[

(N1I1)2

(

L1

µ1A1+ L2

µ2A2+

√(x+d)2+y2

µ0A1

)2

x+ d√

(x+ d)2 + y2+

(N2I2)2

L1

µ1A1+ L2

µ2A2+

s

(

x− d2

)2

+

(

y+√

3

2d

)2

µ0A1

2

x− d2

√

(

x− d2

)2

+(

y +√

32d)2

+

(N3I3)2

L1

µ1A1+ L2

µ2A2+

s

(

x− d2

)2

+

(

y−√

3

2d

)2

µ0A1

2

x− d2

√

(

x− d2

)2

+(

y −√

32d)2

]

,

where N1, N2, N3 are the number of windings of electromagnets 1,2 and 3 shown in

2.3. STATE-SPACE REPRESENTATION OF MOTION EQUATIONS 19

Figure 1.3. Similarly for the force exerted on the disk in the y direction

Fy = − 1

2µoA1

[

(N1I1)2

(

L1

µ1A1+ L2

µ2A2+

√(x+d)2+y2

µ0A1

)2

−y√

(x+ d)2 + y2+

(N2I2)2

L1

µ1A1+ L2

µ2A2+

s

(

x− d2

)2

+

(

y+√

3

2d

)2

µ0A1

2

y +√

32d

√

(

x− d2

)2

+(

y +√

32d)2

+

(N3I3)2

L1

µ1A1+ L2

µ2A2+

s

(

x− d2

)2

+

(

y−√

3

2d

)2

µ0A1

2

y −√

32d

√

(

x− d2

)2

+(

y −√

32d)2

]

.

All that remains is substituting these force expressions into the motion equations

(2.0.1) to find the state space representation.

2.3 State-Space Representation of Motion Equa-

tions

Define the state of the system as

x =

x1

x2

x3

x4

:=

x

x

y

y

. (2.3.1)

Using this definition and substituting the force expressions into motion equations

2.3. STATE-SPACE REPRESENTATION OF MOTION EQUATIONS 20

(2.0.1) gives the dynamics of the entire system

x1 = x2

x2 = − 1

2mµoA1

[

ϕ1(x1, x3)(x1 + d)I21 + ϕ2(x1, x3)

(

x1 −d

2

)

I22

+ ϕ3(x1, x3)

(

x1 −d

2

)

I23

]

x3 = x4

x4 = − 1

2mµoA1

[

ϕ1(x1, x3)(−x3)I21 + ϕ2(x1, x3)

(

x3 +

√3

2d

)

I22

+ ϕ3(x1, x3)

(

x3 −√

3

2d

)

I23

]

(2.3.2)

where

ϕ1(x1, x3) =N2

1(

L1

µ1A1+ L2

µ2A2+

√(x1+d)2+x2

3

µ0A1

)2√

(x1 + d)2 + x23

ϕ2(x1, x3) =N2

2

L1

µ1A1+ L2

µ2A2+

s

(

x1− d2

)2

+

(

x3+√

3

2d

)2

µ0A1

2√

(

x1 − d2

)2

+(

x3 +√

32d)2

ϕ3(x1, x3) =N2

3

L1

µ1A1+ L2

µ2A2+

s

(

x1− d2

)2

+

(

x3−√

3

2d

)2

µ0A1

2√

(

x1 − d2

)2

+(

x3 −√

32d)2

.

The state-space representation of the motion equations for the disk is complete. Table

2.1 lists values of various physical constants in the system used for simulations and

other analysis.

2.4. UNCERTAINTIES 21

Parameter Valueµ0 4π × 10−7

µr 700µ1 2.8π × 10−4

µ2 2.8π × 10−4

L1 0.1000 mL2 0.0167 md 0.0500 mm 0.5000 kgh 0.0083 mN 100A1 0.01 m2

A2 1.39 × 10−4 m2

Table 2.1: Values of physical parameters.

2.4 Uncertainties

The mathematical representation of the planar magnetic levitation device is subject

to uncertainties. Uncertainties are present in this model due to the assumptions made

during the modelling: friction that was not accounted for, force direction estimates

and neglected fringing effects. They are represented in the system as follows

x1 = x2

x2 =Fx(x1, x3, I1, I2, I3)

m+δ2(x)

m

x3 = x4

x4 =Fy(x1, x3, I1, I2, I3)

m+δ4(x)

m,

(2.4.1)

where δ2(x) and δ4(x) represent unknown forces that can be generated from various

electromagnetic modelling assumptions not holding, as well as friction. Such unknown


forces are assumed to have structurally known upper bound as follows

δ2(x) = ∆2(x1, x3) − θ3x2, |∆2(x1, x3)| ≤ θ1 |x1| + θ2 |x3|

δ4(x) = ∆4(x1, x3) − θ6x4, |∆4(x1, x3)| ≤ θ4 |x1| + θ5 |x3|(2.4.2)

where θi ∈ <p, for i = 1, .., 6, are unknown parameters and the terms −θ3x2, −θ6x4

represent viscous friction. Notice that θi above is not to be confused with the force

angles shown in Figure 2.2. The terms ∆2(x1, x3) and ∆4(x1, x3) are upper bounds

to unknown forces that can be generated from one or a combination of Assumptions

2.0.1, 2.0.2, 2.0.3, 2.0.4, 2.0.5, 2.0.6, 2.0.7, and 2.0.8 not holding.

Our model was developed using various assumptions. Assumptions 2.0.1, 2.0.3,

2.0.4, and 2.0.5 were indicated when developing the dynamics of the one-dimensional

result. Although it is necessary to assume that fringing and flux leakage are negligible

to develop this model, assumptions 2.0.1 and 2.0.2, there is definitely fringing and

at least some flux leakage in the actual system. Thus these are regarded as possible

source of uncertainty. Fringing is described in Appendix A.1 and flux leakage is

described later in this section.

The B − H relationship is linear when the material used for the electromagnet

cores is homogeneous, linear and isotropic [1]. Although these properties hold for

soft ferromagnetic materials such as soft iron, Assumption 2.0.3 is still regarded as a

possible source of uncertainty.

Assumption 2.0.4 is realistic as well. The only time the currents vary considerably

is during the transient of the - when the currents begin changing to stabilize the disk

at a position. The current variations during this time could cause problems with the

accuracy of the model.

Magnetic flux is typically assumed constant in literature when, for instance, the


flux lines travel within a high-permeability core through a small air-gap [1]. The cores

used in this case are ferromagnetic and have a large permeability. However, the air

gaps between the cores and the disk are not small and the resulting fringing from this

implies that the magnetic flux at the disk may be significantly less then the Φ going

through the electromagnet. Flux lines follow the path inside the material with least

reluctance and air has a large reluctance. Therefore intuitively speaking, the larger

the air gap the less flux lines will reach the disk. Further, regardless of air gap size

and core permeability, flux leakage between the coils of the magnets decreases the

amount of magnetic flux going through the air gap, as depicted in Figure 2.3. Even

though the coils are wound tightly and wound to the ends of the core, the magnetic

field between the coils may not be entirely cancelled. Fringing and flux leakage make

Assumption 2.0.5 a source of uncertainty.

I

flux leakage

Figure 2.3: Flux leakage of an electromagnet.

The model was built using Assumption 2.0.6, as discussed in Appendix A.2. It is

known that superposition holds when the disk is at the origin. If the disk is moved

at a position where superposition does not hold then the model derived does not

represent the system. Measures are taken in the control design to compensate for the

possibility of superposition not holding. If superposition, or any other uncertainty


for that matter, was not taken into account in the control design then the controller

would output incorrect currents because its results would be based on an incorrect

model. It would therefore be difficult for a controller that assumes the disk never

goes outside the region where superposition holds to stabilize the disk.

The assumption that the forces are directed towards the center of each magnet,

Assumption 2.0.7, is depicted in Figure 1.3. However, as the disk approaches the

face of a magnet, the force exerted on the disk may point towards the part of the

magnet closest to the disk. Realistically, the disk must be confined in an area that is

sufficiently far from all the magnets for this assumption to hold. The mathematical

model we derived does not express the system correctly if the disk is outside of this

region and, thus, the controller must be able to compensate for the presence of this

uncertainty.

Furthermore, the areas where Assumptions 2.0.6 and 2.0.7 hold change in size as

the controller adjusts the currents to stabilize the disk.

Assumption 2.0.8 deals with length of the cores being greater than the both the

height and width of the cores, that is L1 >> A1. The reason is given when discussing

magnets in Section 4.1 of the implementation chapter. Assumption 2.0.9 concerns

the uncertainty term θ2 |x3| in equation (2.4.2). In the adaptive and robust adaptive

control designs it is necessary to impose the requirement that θ2 = 0. This assumption

is discussed further and tested via simulation in Section 3.3, the adaptive control

design, and in Section 3.4, the robust adaptive control design.

The two degrees of freedom of the planar magnetic levitation device are coupled

because of the three-magnet arrangement and hence ∆2 and ∆4 depend on x and y

position of the disk, i.e., ∆2 = ∆2(x1, x3), ∆4 = ∆4(x1, x3). To further illustrate this

point, for a positive small number ε, consider two scenarios: in first case the disk is

initially at (0, 0) and the desired final position is (0, ε); and in the second scenario the


disk starts at (0, ε) and the desired final position is (ε, ε). In the second case, the disk

starts at a position where uncertainties are more significant: the disk may be out of

the region where superposition holds, or perhaps at this position the attractive forces

generated by electromagnet 1 and 3 are not directed towards their centers. In terms

of control, it will be more difficult to move the disk in the x direction from (0, ε) to

(ε, ε) than from (0, 0) to (ε, 0). Uncertainties ∆2 and ∆4 are therefore a function of

(x, y).

In the implementation, the disk is not being levitated by a magnet in the z di-

rection but instead is anchored onto a movable platform that rides on ball bearings.

Although ball bearings enable smooth movement, friction was not included in the

modelling and therefore the last term in the uncertainties δ2 and δ4 includes a viscous

friction expression with unknown coefficients θ3, θ6. However, the friction from the

mechanical contact between the ball bearings and the platform is more accurately

represented by Coulomb friction. This was not considered in the modelling or the

control design and the effect of this is discussed in Section 4.3.3.

Lastly, the platen being moved by the electromagnets was chosen to be a disk

instead of a square so the force acting on the disk from each magnet is equivalent.

Consider if the platen in Figure 1.3 was square shaped. The force exerted on the

platen from electromagnet 1 would differ from those by electromagnet 2 and 3 because

magnet 1 is dealing with a different cross-sectional area than magnets 2 and 3. The

square platform would complicate the model because different force dynamics would

have to be developed for each separate magnet. The disk platen simplifies the model

and avoids possible added uncertainty.

Chapter 3

Nonlinear Control Design

Focussing on the coupled nonlinear xy subsystem at the base, we initiate a research

that aims to develop a rigorous nonlinear control framework to solve the set point

regulation problem for such systems. The controller first converts the magnet-disk

dynamics into a linear system in Brunovsky normal form and then using either robust,

adaptive or robust adaptive techniques to compensate for model uncertainties, the

three-magnet planar magnetic levitation system is stabilized.

Position

Controller

Nonlinear

Transformation

x

xrefPlant

−+

I1, I2, I3u1, u2

Figure 3.1: Overview of control system.

The two major issues dealt with are

1. Constructing a control law that converts the nonlinear model into a simpler

system.

26

3.1. IDEAL CONTROL DESIGN 27

2. Designing a controller that compensates for the uncertainties present in the

system while achieving high performance with minimal control effort.

Figure 3.1 shows the basic structure of the position feedback loop for the planar levi-

ation device. The framework for solving issue 1 is demonstrated in Section 3.1. The

first part develops expressions for three currents needed to transform the nonlinear

system into a linear system. The feedback loop is completed by designing an LQR

controller. The stable response of this ideal control system is shown when used in the

nominal system. In Section 3.2, the ideal controller is robustified to handle uncertain-

ties in the system. Simulations comparing the robust controller with the ideal and

linear controllers are shown when used in the system with uncertainties present. In

Section 3.3, an adaptive controller is designed and simulation results are later shown

comparing it with the linear and robust controllers. In the last part of the chap-

ter, Section 3.4, a robust adaptive controller is developed. Simulations comparing

this controller with the linear, robust and adaptive controller are then shown. The

chapter concludes with a discussion on the most suitable controller to be used in the

actual system.

3.1 Ideal Control Design

In this section, the design of a nonlinear controller that provides asymptotic stabi-

lization to the origin is described. The ideal controller does not take uncertainties

into account, thus we are considering the nominal model when δ2 = 0 and δ4 = 0 in

(2.4.2).


Proposition 3.1.1. There exists a feedback transformation for system (2.3.2),

I21

I22

I23

= T (x,u),

where u = [u1, u2]> and T : <4 × <2 → <3 is well-defined over the set

C =

x ∈ <4,u ∈ <2 : |x1| ≤d

6, |x3| ≤

d

6

, (3.1.1)

such that the dynamics in the transformed input domain read as

x1 = x2

x2 = u1

x3 = x4

x4 = u2,

(3.1.2)

where u1 and u2 are the new control inputs after feedback transformation.

Remark 3.1.1. Note that the feedback transformation in Proposition 3.1.1 is not a

standard feedback linearizing one in that while the original system (2.3.2) has three

positive inputs, the transformed system (3.1.2) is linear with two inputs.

In other words we seek to find a feedback transformation converting (at least on

a suitable compact set) the dynamics (2.3.2) into a linear system where a well-known

linear control technique can be applied to stabilize the entire system. Because the

control enters the system squared, the main difficulty in the design is finding positive

functions for I2i whose combination results in (3.1.2).

Proof. The solution presented here is a generalization of an idea presented in [10],


Section 12.3. Attaining x2 = u1 and x4 = u2 can be more easily achieved by rewriting

x1 = x2

x2 − x4 = u1 − u2 (3.1.3)

x3 = x4

x2 = u1. (3.1.4)

The control design can be broken down into three steps

1. Finding smooth positive functions for I21 , I2

2 and I23 that satisfy equation (3.1.3).

2. Substitute I21 , I2

2 and I23 found in step 1 into (3.1.4) and use the available degrees

of freedom to satisfy equation (3.1.4).

3. Using LQR, design a gain matrix that renders the closed-loop system stable.

In the first part, smooth functions must be found to make x2 − x4 become u1 − u2.

The three currents must cooperate together to supply a function that satisfies (3.1.3).

Similarly to what is done in [10], Section 12.3, we use the following expressions for

the currents

I21 =

−2mµ0A1

ϕ1(x1, x3)(x1 + x3 + d)η1(x1, x3, u1, u2)

I22 =

−2mµ0A1

ϕ2(x1, x3)(

x1 − x3 −√

3+12d)η2(x1, x3, u1, u2)

I23 =

−2mµ0A1

ϕ3(x1, x3)(

x1 − x3 +√

3−12d)η3(x1, x3, u1, u2),

(3.1.5)

where η1, η2 and η3 are degrees of freedom to be defined later. Substituting the


currents (3.1.5) into x2 − x4 gives

x2 − x4 = − 1

2mµ0A1

[

ϕ1(x1, x3)(x1 + x3 + d)I21

ϕ2(x1, x3) +

(

x1 − x3 −√

3 + 1

2d

)

I22

+ϕ3(x1, x3)

(

x1 − x3 +

√3 − 1

2d

)

I23

]

= − 1

2mµ0A1

[

ϕ1(x1, x3)(x1 + x3 + d)−2mµ0A1

ϕ1(x1, x3)(x1 + x3 + d)η1(·)

+ϕ2(x1, x3)

(

x1 − x3 −√

3 + 1

2d

)

−2mµ0A1

ϕ2(x1, x3)(

x1 − x3 −√

3+12d)η2(·)

+ϕ3(x1, x3)

(

x1 − x3 +

√3 − 1

2d

)

−2mµ0A1

ϕ3(x1, x3)(

x1 − x3 +√

3−12d)η3(·)

]

= η1(x1, x3, u1, u2) + η2(x1, x3, u1, u2) + η3(x1, x3, u1, u2), (3.1.6)

The signs of the functions η1, η2 and η3 must be such that I21 , I2

2 and I23 are all

positive. The function ϕi appearing in Ii is positive, for i = 1, .., 3, while the constant

−2mµ0A1 is negative. The denominators x1 + x3 + d and x1 − x3 +√

3−12d in (3.1.5)

are positive over the set C defined in (3.1.1), while the expression x1 − x3 −√

3+12d

in the denominator of I2 is always negative on C. Thus, in order to guarantee that

I21 , I

22 , I

23 are always positive, the signs of η1, η2 and η3 must be as follows

η1 ≤ 0, η2 ≥ 0 and η3 ≤ 0. (3.1.7)

The actual functions are now to be defined. These function must be smooth, obey

the sign constraints given in (3.1.7) and, according to (3.1.6), the sum η1 + η2 + η3


must equal u1 − u2. The following functions satisfy the criteria

η1(x1, x3, u1, u2) =u1 − u2 −

√

(u1 − u2)2 + εb1

4− A(x1, x3, u1, u2)

η2(x1, x3, u1, u2) =u1 − u2 +

√

(u1 − u2)2 + εb1

2+ A(x1, x3, u1, u2) +B(x1, x3, u1, u2)

η3(x1, x3, u1, u2) =u1 − u2 −

√

(u1 − u2)2 + εb1

4− B(x1, x3, u1, u2),

(3.1.8)

where εb1 > 0 and A(x1, x3, u1, u2), B(x1, x3, u1, u2) are positive functions that can be

freely chosen and are used in the next part of the control design. The first step of

the design is now complete, in that equation (3.1.3) has been satisfied. The second

part of the control design involves substituting the expressions I21 , I

22 , I

23 in x2 and

choosing A and B such that the second equality (3.1.4) can be met,

x2 = − 1

2mµ0A1

[

ϕ1(x1, x3)(x1 + d)−2mµ0A1

ϕ1(x1, x3)(x1 + x3 + d)η1(x1, x3, u1, u2)

+ϕ2(x1, x3)

(

x1 −d

2

) −2mµ0A1

ϕ2(x1, x3)(

x1 − x3 −√

3+12d)η2(x1, x3, u1, u2)

+ϕ3(x1, x3)

(

x1 −d

2

) −2mµ0A1

ϕ3(x1, x3)(

x1 − x3 +√

3−12d)η3(x1, x3, u1, u2)

]

=x1 + d

x1 + x3 + d

(

u1 − u2 −√

(u1 − u2)2 + ε

4− A(x1, x3, u1, u2)

)

+x1 − d

2

x1 − x3 −√

3+12d

(

u1 − u2 +√

(u1 − u2)2 + ε

2

+A(x1, x3, u1, u2) +B(x1, x3, u1, u2)

)

+x1 − d

2

x1 − x3 +√

3−12d

(

u1 − u2 −√

(u1 − u2)2 + ε

4−B(x1, x3, u1, u2)

)


=u1 − u2 −

√

(u1 − u2)2 + ε

4

(

x1 + d

x1 + x3 + d

)

+u1 − u2 −

√

(u1 − u2)2 + ε

4

(

x1 − d2

x1 − x3 +√

3−12d

)

+u1 − u2 +

√

(u1 − u2)2 + ε

2

(

x1 − d2

x1 − x3 −√

3+12d

)

+

(

x1 − d2

x1 − x3 −√

3+12d− x1 + d

x1 + x3 + d

)

A(x1, x3, u1, u2)

+

(

x1 − d2

x1 − x3 −√

3+12d− x1 − d

2

x1 − x3 +√

3−12d

)

B(x1, x3, u1, u2).

(3.1.9)

Define

fneg(x1, x3, u1, u2) =u1 − u2 −

√

(u1 − u2)2 + εb1

4

(

x1 + d

x1 + x3 + d

)

fpos(x1, x3, u1, u2) =u1 − u2 −

√

(u1 − u2)2 + εb1

4

(

x1 − d2

x1 − x3 +√

3−12d

)

+u1 − u2 +

√

(u1 − u2)2 + εb1

2

(

x1 − d2

x1 − x3 −√

3+12d

)

fa(x1, x3) =x1 − d

2

x1 − x3 −√

3+12d− x1 + d

x1 + x3 + d

fb(x1, x3) =x1 − d

2

x1 − x3 −√

3+12d− x1 − d

2

x1 − x3 +√

3−12d.

For all (x,u) ∈ C (defined in 3.1.1), fneg, fpos, fa and fb enjoy the properties

fneg(x1, x3, u1, u2) < 0

fpos(x1, x3, u1, u2) > 0

fa(x1, x3) < 0

fb(x1, x3) > 0.


Rewrite (3.1.9) in terms of the defined functions

x2 = fneg + fpos + faA + fbB,

and notice that, since both A(x1, x3, u1, u2) and B(x1, x3, u1, u2) must be positive,

A(x1, x3, u1, u2) can only be used to cancel a positive term while B(x1, x3, u1, u2) can

only be used to cancel a negative term. The identity x2 = u1 is thus satisfied by

choosing A and B as

A(x1, x3, u1, u2) = − 1

fa(x1, x3)

(

fpos(x1, x3, u1, u2) +−u1 +

√

u21 + εb2

2

)

B(x1, x3, u1, u2) = − 1

fb(x1, x3)

(

fneg(x1, x3, u1, u2) +−u1 −

√

u21 + εb2

2

)

,

where εb2 > 0.

Remark 3.1.2. The positive constants εb1 and εb2 in I21 , I

22 , I

23 changes the bias currents

of the controller. When the disk is stabilized the controller still outputs a non-zero

current and this is called the bias current. Currents are reduced by using small values

of εb1, εb2.

The first two parts of the nonlinear control are complete: the original system

(2.3.2) has been transformed into the linear system (3.1.2) by means of the following


feedback transformation

I21

I22

I23

=

−2mµ0A1

ϕ1(x1,x3)(x1+x3+d)

(

u1−u2−

√(u1−u2)2+εb1

4 − A(x1, x3, u1, u2)

)

−2mµ0A1

ϕ2(x1,x3)

(

x1−x3−

√

3+1

2d

)

(

u1−u2+√

(u1−u2)2+εb1

2 + A(x1, x3, u1, u2) + B(x1, x3, u1, u2)

)

−2mµ0A1

ϕ3(x1,x3)

(

x1−x3+√

3−1

2d

)

(

u1−u2−

√(u1−u2)2+εb1

4 − B(x1, x3, u1, u2)

)

= T (x,u).

(3.1.10)

Notice that, while the original system (2.3.2) has three control inputs, (3.1.2) has

two control inputs, u1 and u2. After feedback transformation (3.1.10), system (3.1.2)

reads as

x =

0 1 0 0

0 0 0 0

0 0 0 1

0 0 0 0

x +

0 0

1 0

0 0

0 1

u (3.1.11)

which is in Brunovsky normal form.

To stabilize the origin we can, e.g. employ a LQR controller, u = −Kx, in the

transformed input domain. For our simulations we choose the weighing matrices

Q =

5000 0 0 0

0 100 0 0

0 0 700 0

0 0 0 2000

, R =

5000 1000

1000 5000

. (3.1.12)

Using these, the following Riccati equation solution, P, and gain matrix, K, were


generated

P =

7065.5 4955.6 137.7 340.1

4955.6 7051.7 248.6 847.8

137.7 248.6 2002.6 1866.5

340.1 847.8 1866.5 5349.2

,

K =

1.0183 1.4338 −0.0260 −0.0463

−0.1356 −0.1172 0.3785 1.0791

.

(3.1.13)

The design of a nonlinear stabilizer in the absence of uncertainties is now complete.

Notice that tracking can also be straightforwardly achieved for system (3.1.2). Con-

troller performance for this specific application is measured in two ways, namely the

amplitude of the control input (ie. currents) and the size of the domain of attraction.

The goal is to have small currents and a large domain of attraction. The motivation

of having low currents is to avoid the magnet windings from overheating and, more

importantly, to prevent the actuators from saturating. It is not known exactly at

what current the magnet wire will overheat, however after running some tests on the

actual system, the aim is to have the current peaks below 10 A and the continuous

currents not exceed 6 A. Thus no current should be above 10 A for more then 1

second. These currents are close to the limit of our actuator but if εb1 and εb2 are

made very small, the current can be kept below this limit in most cases.

As for the second performance criterion, an estimate of the domain of attraction

of the origin of the closed-loop system can be obtained by finding the largest level set

of V contained inside the set C defined in (3.1.1), where our controller is guaranteed

to be valid. In other words we want to find the largest value of c > 0 such that

ΩC :=

x ∈ <4∣

∣

∣V (x) = xTPx ≤ c

⊂ C. (3.1.14)


This can be done numerically using a constrained optimization technique. By doing

that, using the plant parameters in Table 2.1, the estimate of the domain of attraction

is

ΩC =

x ∈ <4∣

∣

∣V (x) ≤ 0.0938

. (3.1.15)

To assess whether ΩC is an accurate estimate of the basin of attraction, it would

be valuable to visualize it graphically. Figure 3.2 depicts a slice of the domain of

attraction estimate when the velocities x2 and x4 are set to zero. That is, Figure

3.2 is the level set plot of x ∈ <4|V (x1, 0, x3, 0) ≤ 0.0938. The domain estimate

represents the set of feasible locations where the disk can be initialized at zero velocity

and driven to the origin. This concludes the design of the ideal nonlinear controller.

Simulation results of the system’s response using this controller are shown in the next

section.

−0.01 −0.008 −0.006 −0.004 −0.002 0 0.002 0.004 0.006 0.008 0.01

−0.01

−0.008

−0.006

−0.004

−0.002

0

0.002

0.004

0.006

0.008

0.01

x1

x 3

C

Figure 3.2: Domain of attraction estimate.


3.1.1 Simulation Results

The closed-loop response of the nominal system using the ideal controller is shown in

Figure 3.3. For the given initial condition and εb1 = εb2 = 0.1, the maximum current

attained is 3.5542 A and the steady-state error is 1.2696 × 10−8 m. Small control

inputs are achieved with this controller because of its large settling time and can

be reduced further by making εb1, εb2 smaller. Since the currents between different

controllers will be compared, εb1 = εb2 = 0.1 in all the simulations.

0 5 10 15 20 25−6

−4

−2

0

2

4

6

8x 10

−3

Time (s)

posi

tion

(m),

spe

ed (

m/s

)

x1

x2

x3

x4

Figure 3.3: Position and speed trajectories when using the ideal controller in thenominal system.

To compare our nonlinear designs to the approach, often used in the control of

electromagnetic devices, of linearizing the system about the desired equilibrium and

designing a linear controller, we include in our comparisons a linear controller

[I21 , I

22 , I

23 ]> = KLx,


where the matrix KL is obtained by applying LQR design to the linearization of the

system at the origin, with

Q =

5000 0 0 0

0 100 0 0

0 0 700 0

0 0 0 2000

, R =

1000 0 0

0 100 0

0 0 1000

. (3.1.16)

The projection on the (x1, x3) plane of the phase curves of the uncertainty-free

system using the linear controller and the ideal controller developed in Section 3.1 is

shown in Figure 3.4. It depicts trajectories of the controllers for five initial conditions.

The performance of the ideal and linear controllers in the uncertainty-free system is

comparable.

−0.01 −0.008 −0.006 −0.004 −0.002 0 0.002 0.004 0.006 0.008 0.01−0.01

−0.008

−0.006

−0.004

−0.002

0

0.002

0.004

0.006

0.008

0.01

x1

x 3

linear

ideal

Slice of DOA

Figure 3.4: Projection of the phase curves on the x1 − x3 plane when using the idealand linear controllers in the nominal system.

These are the only simulations on the nominal system. All other simulations

depict various controllers used in the system when uncertainties are present.

3.2. ROBUST CONTROL DESIGN 39

3.2 Robust Control Design

In this section we robustify the controller developed in the previous section to ac-

count for the uncertainties in (2.4.1). To this end, using the feedback transformation

(3.1.10), (2.4.1) is mapped to

x =

0 1 0 0

0 0 0 0

0 0 0 1

0 0 0 0

x +

0 0

1 0

0 0

0 1

(u + δ(x)) (3.2.1)

where δ(x) = [δ2(x1, x2, x3), δ4(x1, x3, x4)]>. Since the uncertainty δ(x) satisfies a

matching condition, Lyapunov redesign is a natural choice for robust stabilization.

Following the standard Lyapunov redesign technique (see, e.g., [2]), we replace the

linear controller (in the transformed input domain) developed in the previous section,

u = −Kx, by u = −Kx+v, where v is an additional control term designed to stabilize

the system subject to uncertainties. In order to do that, we use the inequalities in

(2.4.2) and assume that we know two positive scalars β1 and β2 satisfying

|θi| ≤ β1, i = 1, 2, 4, 5, |θj | ≤ β2, j = 3, 6.

Assume that with u = Ψ(x) + v an upper bound, ρ(x), to the uncertain terms exists

such that ||δ(x)|| ≤ ρ(x),

||δ(x)||2 =

√

|δ2|2 + |δ4|2

= (θ1 |x1| + θ2 |x3| − θ3x2)2 + (θ4 |x1| + θ5 |x3| − θ6x4)

2

= (θ1 |x1| + θ2 |x3|)2 − 2θ3x2 (θ1 |x1| + θ2 |x3|) − 2θ6x4 (θ4 |x1| + θ5 |x3|)

+ (θ4 |x1| + θ5 |x3|)2 + θ23x

22 + θ2

6x24


≤√

2β21(|x1| + |x3|)2 + 2β1β2(|x1| + |x3|)(|x2| + |x4|) + β2

2 (x22 + x2

4)

≤√

2β∗2(|x1| + |x3|)2 + 2β∗2(|x1| + |x3|)(|x2| + |x4|) + β∗2 (x22 + x2

4)

:= ρ(x)

where β∗ = max β1, β2. Next, apply u = Ψ(x) + v to (3.2.1) and perform Lyapunov

analysis

V = xT Px + xT Px

= −xTQx + 2xTPB(v + δ(x))

≤ −λmin(Q)||x||22 + ωT (v + δ(x))

≤ −λmin(Q)||x||22 + ωTv + ||ω||2||δ||2

≤ −λmin(Q)||x||22 + ωTv + ||ω||2ρ(x)

where ωT = 2xTPB, λmin(Q) denotes the minimum eigenvalue of matrix Q and

Q ∈ <2×2 is positive definite and symmetric. Choose v that renders V negative,

v = −η(x) ω||ω||2 , where η(x) ≥ ρ(x), so that

V = −λmin(Q)||x||22 −ωTω

||ω||2η(x) + ||ω||2ρ(x)

= −λmin(Q)||x||22 + ||ω||2(ρ(x) − η(x)) < 0 ∀ x ∈ <4 6= 0.

The redesign of v stabilizes the system with uncertainties. Since v is not smooth at

the origin, we replace v by the following smooth version (see [2]):

v =

−η(x) ω||ω||2 if η(x)||ω||2 ≥ γ

−η(x)2 ωγ

if η(x)||ω||2 < γ(3.2.2)

where γ > 0. The resulting closed-loop trajectories converge to a neighborhood of


order γ about the origin. Since γ can be made arbitrarily small, the asymptotic set-

point regulation error can be made negligible. This completes the robust nonlinear

control design. Simulation results of the robust controller are shown in the next

section.


Introducing the first scenario for simulations:

Case 1:

δ1(x1, x2, x3) ≤ 1.1 |x1| + 0.5 |x3| − 0.01x2

δ2(x1, x3, x4) ≤ 0.5 |x1| + 1.1 |x3| − 0.01x4

x(0) =[

6.67 mm 0 −5 mm 0

]>.

All controllers are tested in this scenario. Figure 3.5 depict the x and y positions

of the ideal nonlinear controller and the robust nonlinear controller when subject to

uncertainties of case 1. The maximum upper bound of the robust controller is set to

β∗ = 1.5 and low steady-state errors are achieved by setting γ = 10−5. In this scenario,

the ideal controller fails to stabilize the system while the robust controller manages to

stabilize the system about the origin. Notice that although the robust redesign does

not guarantee performance improvement, simulations suggest that transient response

is improved with the robust controller.

Figure 3.6 show the x and y positions of the robust nonlinear controller and the LQR

controller designed from the linearized system. Figure 3.6 is the response when the

uncertainties defined in case 1 are present. The linear controller, although robust

about the point, is unable to stabilize the system in this first scenario. As before, the


0 5 10 15 20 25−0.01

−0.005

0

0.005

0.01

0.015

0.02

Time (s)

x po

sitio

n (m

)

0 5 10 15 20 25−0.01

−0.005

0

0.005

0.01

0.015

0.02

Time (s)

y po

sitio

n (m

)

ideal

robust

robust

ideal

Figure 3.5: Case 1 response of uncertain system when using ideal and robust nonlinearcontrollers.

robust nonlinear controller successfully stabilizes the system with reasonable perfor-

mance.

In case 1, the robust controller achieves a steady-state error of 9.4459 × 10−6 m

at a maximum current of 3.7079 A using εb1 = εb2 = 0.1 . The currents of the

robust controller are depicted in Figure 3.7. Although these results indicate that

the robust controller can compensate for large uncertainties while maintaining high

performance, there are some practical issues with using this controller, namely the

chattering effect of the control inputs seen in Figure 3.7 that can excite high-frequency

un-modelled dynamics in the system [9]. Further, these currents are impossible to

obtain in practice because they are too high-frequency.


0 5 10 15 20 25−5

0

5

10

15x 10

−3

Time (s)

x po

sitio

n (m

)

0 5 10 15 20 25−0.01

−0.005

0

0.005

0.01

Time (s)

y po

sitio

n (m

)linear

robust

linear

robust

Figure 3.6: Case 1 response of uncertain system when using a linear controller andthe robust nonlinear controller.

0 2 4 6 8 10 12 14 16 18 202.5

3

3.5

4

Time (s)

Cur

rent

(A

)

I3

I1

I2

Figure 3.7: Currents of robust nonlinear controller in system subject to case 1.

3.3. ADAPTIVE CONTROL DESIGN 44

3.3 Adaptive Control Design

Although the robust controller developed in the previous section guarantees stability

for the system subject to uncertainties δ2 and δ4 in (2.4.1), it does have practical

drawbacks. Specifically, as shown in Section 3.2.1, the currents have a high-frequency

component due to the fact that (3.2.2) is the smooth version of a sliding mode con-

troller. Further, a robust controller may require a large control effort which is not

desirable in the application under consideration because of the saturation limits of

the amplifiers. Both drawbacks above may, in principle, be overcome by designing an

adaptive controller. Thus, assuming that the structure of the uncertainties in (2.4.1)

is known exactly, which is not a realistic assumption, a nonlinear adaptive controller

is designed. The set-point adaptive regulation methodology we employ here is found

in [4] and, depending on the size of the uncertainties, can give an overall better re-

sponse with considerably less control effort. In what follows, we present the control

design procedure and simulation results comparing the performance of the adaptive

control to that of the robust controller.

3.3.1 Backstepping Design

The adaptive regulation presented in [4] globally stabilizes an equilibrium point and

regulates a set-point for a strict-parametric system. A strict-parametric system with


unit control gain reads as

x1 = x2 + ϕ1(x1)>θ

x2 = x3 + ϕ2(x1, x2)>θ

...

xn−1 = xn + ϕn−1(x1, ..., xn−1)>θ

xn = u+ ϕn(x)>θ, (3.3.1)

where θi ∈ <p are unknown parameters and ϕ1ϕ2 . . . ϕn are known smooth nonlinear

functions.

The controller is built according to the recursive design technique of backstepping.

Intermediate control functions, αi, and tuning functions, τi, are built at each step to

stabilize a single subsystem with respect to a Lyapunov function. In the last step, the

update parameter law for the parameter estimate, θ(t), is defined using the tuning

functions designed at each step and the actual control law for u in system (3.3.1) is

constructed using the intermediate control functions. After nonlinear transformation

(3.1.10), the uncertain system reads as

x1 = x2

x2 = u1 + θ1 |x1| + θ2 |x3| − θ3x2

x3 = x4

x4 = u2 + θ4 |x1| + θ5 |x3| − θ6x4,

(3.3.2)

where θ1, . . . , θ6 are unknown scalars. Notice that we are requiring exact knowledge

of the structure of ∆2 and ∆4, thus assuming the uncertainty to be purely parametric.

That is, |∆2(x1, x3)| = θ1 |x1| + θ2 |x3| and |∆4(x1, x3)| = θ4 |x1| + θ5 |x3|. With this


in mind, notice that (3.3.2) matches the strict-parametric system (3.3.1) by letting

u1 := x3

ϕ1 := 0

ϕ2 :=[

|x1| −x2 0 0 0]>

ϕ3 := 0

ϕ4 :=[

0 0 |x1| |x3| −x4

]>

θ :=[

θ1 θ3 θ4 θ5 θ6

]>. (3.3.3)

Remark 3.3.1. The transformed system (3.3.2) is not decoupled because the uncer-

tainty entering u1 and u2 depends on both positions, x1 and x3. Thus the adaptive

scheme cannot be applied to the (x1, x2) subsystem, and to the (x3, x4) subsystem

individually. To avoid this problem we set u1 = x3 and stabilize the entire system

using only u2.

Remark 3.3.2. The adaptive control is based on stabilizing an uncertain system in

strict-parametric form and therefore it cannot compensate for the θ2 |x3| term in

(3.3.2). As shown in the simulations later, the adaptive control is unable to stabilize

the system if this term is too large.

Step 1. In the first step, the error states are

z1 = x1

z2 = x2 − α1,


where α1 is a stabilizing function to be determined later. The z1 error dynamics is

z1 = x2

= z2 + α1.

The origin of the z1-subsystem will be stabilized with respect to Lyapunov function

V1(z1, θ) =1

2z21 +

1

2θ>Γ−1θ, (3.3.4)

where Γ ∈ <5×5 > 0 and θ, the parameter estimation error, is defined as θ := θ − θ.

The time derivative of V1 is

V1 =∂V1

∂z1z1 +

∂V1

∂θ

˙θ

= z1(z2 + α1) − θ>Γ−1 ˙θ.

By setting the stabilizing function, α1, and the tuning function, τ1, to the following

α1 := −k1z1

τ1 := 0,

where k1 > 0, we get

V1 = z1z2 − k1z21 − θ>Γ−1 ˙

θ. (3.3.5)

The term z1z2 in V1 will be eliminated in the next step. Also, the first tuning function

is set to zero because there are no uncertainties present at this point.

Step 2. Uncertainties appear in this step. The error variable is

z3 = x3 − α2,


where α2 is a stabilizing function to be determined later. The (z1, z2)-dynamics are

then given by

z1 = z2 + α1

z2 =∂z2

∂x1

x1 +∂z2

∂x2

x2 +∂z2

∂θ

˙θ

= −∂α1

∂x1x2 + z3 + α2 + ϕ>

2 θ.

Define the Lyapunov function in the second step as,

V2(z1, z2, θ) = V1 +1

2z22 . (3.3.6)

Taking the derivative of V2 along solutions z1, z2 and˙θ gives

V2 = V1 + z2z2

= z1z2 − k1z21 − θ>Γ−1 ˙

θ + z2

(

−∂α1

∂x1x2 + z3 + α2 + ϕ>

2 θ

)

.

The stabilizing function, α2, and the tuning function, τ2, are chosen so that if z3 = 0

and Γ−1 ˆθ = τ2, V2 is negative semi-definite

α2 := −z1 − k2z2 +∂α1

∂x1x2 − ϕ>

2 θ

τ2 := ϕ2z2,

where k2 > 0. The Lyapunov derivative then becomes

V2 = z2z3 − k1z21 − k2z

22 + z2ϕ

>2 (θ − θ) − θ>Γ−1 ˙

θ

= z2z3 − k1z21 − k2z

22 + θ>(ϕ2z2 − Γ−1 ˙

θ)

= z2z3 − k1z21 − k2z

22 + θ>

(

τ2 − Γ−1 ˙θ)

.


Step 3. Introducing the fourth error variable

z4 = x4 − α3,

where α3 will be defined later. The dynamics of (z1, z2, z3)-subsystem are then given

by

z1 = z2 + α1

z2 =∂z2

∂x1

x1 +∂z2

∂x2

x2 +∂z2

∂θ

˙θ

z3 =∂z3

∂x1x1 +

∂z3

∂x2x2 +

∂z3

∂x3x3 +

∂z3

∂θ

˙θ

= −∂α2

∂x1x2 −

∂α2

∂x2(x3 + ϕ>

2 θ) + z4 + α3 −∂α2

∂θ

˙θ

= −∂α2

∂x1

x2 −∂α2

∂x2

x3 + z4 + α3 + w>3 θ −

∂α2

∂θ

˙θ,

where w3 = −∂α2

∂x2ϕ2. Taking the derivative of Lyapunov equation

V3(z1, z2, z3, θ) = V2 +1

2z23 , (3.3.7)

gives

V3 = V2 + z3

(

−∂α2

∂x1x2 −

∂α2

∂x2x3 + z4 + α3 + w>

3 θ −∂α2

∂θ

˙θ

)

.

For any k3 > 0, using the stabilizing and tuning functions

α3 := −z2 − k3z3 +∂α2

∂x1x2 +

∂α2

∂x2x3 − w>

3 θ +∂α2

∂θΓτ3

τ3 := τ2 + w3z3


gives the Lyapunov function derivative

V3 = z3z4 − k1z21 − k2z

22 − k3z

23 + θ>

(

τ2 − Γ−1 ˙θ)

+ z3w>3 θ + z3

∂α2

∂θΓ(

τ3 − Γ−1 ˙θ)

= z3z4 − k1z21 − k2z

22 − k3z

23 + θ>

(

τ2 + z3w3 − Γ−1 ˙θ)

+ z3∂α2

∂θΓ(

τ3 − Γ−1 ˙θ)

= z3z4 − k1z21 − k2z

22 − k3z

23 +

(

θ> + z3∂α2

∂θΓ

)

(

τ3 − Γ−1 ˙θ)

.

Step 4. In the final step, the parameter estimate update law is finalized and the

control law entering u2 in (3.3.2) is built using the previous stabilizing functions.

The final Lyapunov function is

V4 = V3 +1

2z24 ,

where z4 is the error variable defined in the step 3. The complete error dynamics are

z1 = z2 + α1

z2 =∂z2

∂x1

x1 +∂z2

∂x2

x2 +∂z2

∂θ

˙θ

z3 =∂z3

∂x1x1 +

∂z3

∂x2x2 +

∂z3

∂x3x3 +

∂z3

∂θ

˙θ

z4 =∂z4

∂x1x1 +

∂z4

∂x2x2 +

∂z4

∂x3x3 +

∂z4

∂x4x4 +

∂z4

∂θ

˙θ

= −∂α3

∂x1x2 −

∂α3

∂x2(x3 + ϕ>

2 θ) −∂α3

∂x3x4 + u2 + ϕ>

4 θ −∂α3

∂θ

˙θ

= −∂α3

∂x1

x2 −∂α3

∂x2

x3 −∂α3

∂x3

x4 + u2 +

[

ϕ>4 − ∂α3

∂x2

ϕ>2

]

θ − ∂α3

∂θ

˙θ

= −∂α3

∂x1x2 −

∂α3

∂x2x3 −

∂α3

∂x3x4 + u2 + w>

4 θ −∂α3

∂θ

˙θ,

where w4 = ϕ4 − ∂α3

∂x2ϕ2. Taking the derivative of V4 along solutions gives

V4 = V3 + z4

(

−∂α3

∂x1x2 −

∂α3

∂x2x3 −

∂α3

∂x3x4 + u2 + w>

4 θ −∂α3

∂θ

˙θ

)

.


The final control law, u2, and the parameter update law,˙θ, can now be defined to

stabilize (z, θ), where z :=[

z1 z2 z3 z4

]>, with respect to V4

u2 := −z3 − k4z4 +∂α3

∂x1x2 +

∂α3

∂x2x3 +

∂α3

∂x3x4 +

∂α3

∂θ

˙θ − w>

4 θ

+z3∂α2

∂θΓw4

τ4 := τ3 + w4z4

˙θ := Γτ4 = Γ

[

ϕ2z2 −∂α2

∂x2z3 +

(

ϕ4 −∂α3

∂x2ϕ2

)

z4

]

.

Applying the control law, update law, and tuning function just defined to V4 gives

V4 = −k1z21 − k2z

22 − k2

3z23 − k4z

24 +

(

θ> + z3∂α2

∂θΓ

)

(

τ3 − Γ−1 ˙θ)

+z3∂α2

∂θΓw4z4 + z4w

>4 θ

= −k1z21 − k2z

22 − k2

3z23 − k4z

24 +

(

θ> + z3∂α2

∂θΓ

)

(

τ3 − Γ−1 ˙θ)

+

(

θ> + z3∂α2

∂θΓ

)

w4z4

= −k1z21 − k2z

22 − k2

3z23 − k4z

24 +

(

θ> + z3∂α2

∂θΓ

)

(

τ4 − Γ−1 ˙θ)

= −k1z21 − k2z

22 − k2

3z23 − k4z

24 . (3.3.8)

Thus V4 is negative semidefinite and, by LaSalle’s theorem, trajectories (z(t), θ(t))

converge to the largest invariant set, M, inside

E =

(z, θ) ∈ <9

∣

∣

∣

∣

z = 0

. (3.3.9)

This means that z(t) → 0 as t → ∞ and from this we can find the set M. By

substituting the stabilizing functions α1, α2, α3, and α4 into z1, z2, z3 and z4, the


closed-loop error dynamics can be represented in matrix form

z =

−k1 1 0 0

−1 −k2 1 0

0 −1 −k3 1 + ϕ>2 Γw4

0 0 −1 −k4

z +

0 0 0 0

0 1 0 0

0 −∂α2

∂x20 0

0 ∂α3

∂x20 −1

ϕ>1

ϕ>2

ϕ>3

ϕ>4

θ.

Since z = 0 and therefore z = 0, the following system of equations arise

0 0 0 0

0 1 0 0

0 −∂α2

∂x20 0

0 ∂α3

∂x20 −1

ϕ>1

ϕ>2

ϕ>3

ϕ>4

θ = 0,

where it can be concluded that ϕ>2 θ = 0 and ϕ>

4 θ = 0. Using this result, the error

states become

x1 = 0

x2 = z2 + α1(z1, θ) = 0 + α1(0, θ) = 0

x3 = α2(0, 0, θ) = −ϕ>2 θ = 0

x4 = α3(0, 0, 0, θ) = −∂α2

∂x2ϕ2θ = 0.

The largest invariant set contained in (3.3.9) is

E =

(x, θ) ∈ <9

∣

∣

∣

∣

x = 0, ϕ>2 θ = ϕ>

2 θ, ϕ>4 θ = ϕ>

4 θ

. (3.3.10)

In conclusion, the closed-loop adaptive system is globally stable at equilibrium point

(x, θ) = (0, θ) and is set-point regulated, x(t) → 0 as t → ∞. Notice however that


the equilibrium point is not globally asymptotically stable because the parameter

estimates do not converge to the actual parameters. The reader is referred to [4] for

more information on stability results.

3.3.2 Simulations Results

Figure 3.8 shows the closed-loop system response when using the adaptive controller

in the first scenario. The adaptive control design parameters are: k1 = 2, k2 = 3.5,

k3 = 2, k4 = 3.5, θ(0) =[

0 0 0 0 0

]>and Γ = diag

[

3, 3, 3, 3, 3

]

.

As shown in Figure 3.8, the adaptive control fails to handle the uncertainties in case

1. The adaptive controller is unable to stabilize the system under conditions of case

1 because the term 0.5 |x3| in δ2 is not handled by the adaptive scheme.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.005

0.01

0.015

Time (s)

x po

sitio

n (m

)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.01

−0.008

−0.006

−0.004

−0.002

0

Time (s)

y po

sitio

n (m

)

Figure 3.8: Case 1 response of uncertain system when using adaptive nonlinear con-trollers.

The adaptive will now be tested with uncertainties that it can fully compensate


for. Introducing the second scenario:

Case 2:

δ2(x1, x2, x3) ≤ 1.1 |x1| − 0.01x2

δ4(x1, x3, x4) ≤ 0.5 |x1| + 1.1 |x3| − 0.01x4

x(0) =[

5 mm 0 8.33 mm 0

]>

Notice that 0.5 |x3| in δ2 is not present in this case and the initial conditions have

changed. Figure 3.9 depicts the responses of the linear, robust and adaptive controller

in the second scenario.

0 2 4 6 8 10 12 14 16 18 20−5

0

5

10x 10

−3

Time (s)

x po

sitio

n (m

)

0 2 4 6 8 10 12 14 16 18 20−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

Time (s)

y po

sitio

n (m

)

linear

adaptive robust

linear robust

adaptive

Figure 3.9: Case 2 response of uncertain system when using linear, robust and adap-tive nonlinear controllers.

The responses under the linear, robust and adaptive controllers is shown in Figure

3.9. The robust controller achieves a steady-state error of 1.7005 × 10−5 m at a

maximum current of 4.0696 A. The steady-state error and maximum current for the


adaptive controller are 2.5163×10−4 m and 6.3447 A. Control inputs of the adaptive

controller are shown in Figure 3.10. The adaptive controller does not out-perform the

robust controller but, unlike the linear controller, manages to stabilize the system.

The adaptive controller does, however, calculate currents that do not chatter like

those from the robust control system. From a practical viewpoint, the hardware can

provide smooth currents from the adaptive controller but the high-frequency currents

from the robust controller cannot be realized.

0 2 4 6 8 10 12 14 16 18 201

2

3

4

5

6

7

Time (s)

Cur

rent

(A

)

I3

I2

I1

Figure 3.10: Control input of adaptive controller in case 2.

Remark 3.3.3. Note that the x3 trajectory using the adaptive controller in Figure 3.9

goes outside the domain C defined in (3.1.1) where our controller is valid. However, in

this particular response the signs of the denominators x1 +x3 +d and x1−x3 +√

3−12d

in (3.1.5) are still positive and the expression x1 − x3 −√

3+12d in the denominator

of I2 is still negative. In simulation it occurs that the trajectories go outside C but

rarely does the control become undefined. This shows that domain C is somewhat

conservative.


It was found that the adaptive control system performs very well with low currents

when uncertainties ∆2 and ∆4 are in the lower range. It was also observed that the

adaptive controller performs even better when the frictions applied to the system

are high. Thus, the hypothesis of a better performance and control effort tradeoff is

confirmed through simulation for lighter uncertainties.

The adaptive control cannot be used in the actual system because in practice

the functional forms of the uncertainties are not known. Thus both the robust (be-

cause of the chattering) and the adaptive controllers cannot be used if the simulated

uncertainties in case 1 or 2 are large (and therefore realistic).

3.4. ROBUST ADAPTIVE CONTROL DESIGN 57

3.4 Robust Adaptive Control Design

The tradeoff between control effort and steady-state error is only improved with the

adaptive controller if the uncertainties present in the system are small. Further,

the adaptive control requires the uncertainties in (2.4.1) to be structured, i.e., to

exactly match the structure the adaptive control design calls for, and to be linear with

respect to unknown parameters. Uncertainties δ2 and δ4 in (2.4.2) cannot be assumed

to be structured and to be linear functions of unknown parameters (although, this

assumption is reasonable for the viscous friction terms). In the light of the above,

we choose to develop a robust adaptive control that handles friction in the classic

adaptive manner but compensates uncertainties ∆2 and ∆4 in (2.4.2) using adaptive

upper bounds. Thus, even if the uncertainty of the model assumptions outlined in

Chapter 2 do not match the strict-parametric structure in the adaptive controller,

the system can still be stabilized. To this end, we employ the methodology developed

in [6]. The robust adaptive controller enjoys better performance than the robust and

adaptive controllers at the expense of high peak currents. Simulations that compare

these controllers will be shown after the control design is illustrated.

3.4.1 Backstepping Design

We employ the technique developed by Polycarpou and Iannou in [6], which en-

sures that the trajectories of the closed-loop system are globally uniformly ultimately

bounded (GUUB) with a small ultimate bound. The methodology in [6] applies to

the class of nonlinear systems that satisfy the triangular bounds condition. Consider

the following n-dimensional single-input nonlinear system

xi = xi+1 + θ>ϕi(x1, ..., xi) + ∆i(x, t), 1 ≤ i ≤ n− 1

xn = u+ θ>ϕn(x) + ∆n(x, t),(3.4.1)


where θ1, . . . , θn are unknown scalar parameters, ϕ1, . . . , ϕn are known smooth func-

tions and ∆1, . . . ,∆n are unknown functions that satisfy Assumption 3.4.1.

Assumption 3.4.1. There exists, possibly unknown, parameters ψi ∈ < ≥ 0 and

known smooth functions pi : <i → <+\0 such that for all x ∈ <n and t ∈ <+

|∆i(x, t)| ≤ ψipi(x1, ..., xi), 1 ≤ i ≤ n. (3.4.2)

Recall that, after applying the nonlinear input transformation (3.1.10) to the uncer-

tain system (2.4.1), we get

x1 = x2

x2 = u1 +δ2(x1, x2, x3)

m

x3 = x4

x4 = u2 +δ4(x1, x3, x4)

m.

(3.4.3)

Recalling, from (2.4.2), that

δ2(x) = ∆2(x1, x3) − θ3x2,

δ4(x) = ∆4(x1, x3) − θ6x4,

where ∆2(x1, x3) ≤ θ1|x1| + θ2|x3| and ∆4(x1, x3) ≤ θ4|x1| + θ5|x3|, and letting

u1 := x3

ϕ1 :=[

0 0 0 0

]>

ϕ2 :=[

0 −x2 0 0]>


ϕ3 :=[

0 0 0 0

]>

ϕ4 :=[

0 0 0 −x4

]>,

we have that (3.4.3) fits the structure in (3.4.1). Next, in order for condition (3.4.1)

to be satisfied we need to impose the requirement that θ2 = 0, and we need to find

a smooth upper bound to |∆2(x1, x3)| (the function θ2|x3| is not smooth). Noticing

that for any c1, c2 > 0, there exist scalars ψ2, ψ4 > 0 such that

θ1|x1| ≤ ψ2(x21 + c1),

θ4|x1| + θ5|x3| ≤ ψ4(|x1| + |x3| + c2),

and setting

p2(x1) = x21 + c1 (3.4.4)

p4(x1, x3) = |x1| + |x3| + c2, (3.4.5)

we have that (3.4.1) is satisfied. The imposed requirement that θ2 = 0 (i.e. that ∆2

be independent of x3) poses a limitation to the generality of the solution presented

here. The impact of this requirement on the performance of the experimental testbed

under construction will be the subject of future investigation in other research.

Similarly to the adaptive control design, the robust adaptive controller is con-

structed using the backstepping technique. At each step we define an error variable

zi, an intermediate feedback control, αi, a tuning function, τi, and a update law,

˙Ψ, made to stabilize the (z1, . . . , zi) subsystem, for i = 1, . . . , n, with respect to a

Lyapunov function Vi. The intermediate feedback controls are used to define the final


control u2 in system (3.4.3), while the tuning functions are used to construct the pa-

rameter update law˙θ that estimates the coefficients of friction. The final control law,

u2, employs these parameter estimates to compensate for the effect of friction. The

robust adaptive controller includes an additional update law,˙Ψi, that compensates

the effects of ∆i. This law updates the parameter estimates, Ψi, that multiply the

upper bounds pi defined in (3.4.1). In effect, the control law uses the upper bound

estimates generated by this update law to handle the effects of uncertainties caused

by various electromagnetic modelling assumptions not holding.

Step 1. The error variables are

z1 = x1

z2 = x2 − α1,

and the z1-dynamics are

z1 = x2 = z2 + α1,

where α1 is a stabilizing function that is yet to be defined. The z1 system will be

stabilized with respect to V1, defined as

V1(z1, θ, ψ1) =1

2z21 +

1

2θ>Γ−1θ +

1

2γ1

ψ21 (3.4.6)

where θ = θ(t) − θ, ψ1 = ψ1(t) − ψ1, Γ ∈ <4×4 > 0 and γ1 > 0. The time derivative

of V1 is

V1 =∂V1

∂z1z1 +

∂V1

∂θ

˙θ +

∂V1

∂ψ

˙ψ

= z1(z2 + α1) + θ>Γ−1(˙θ − τ1) + θ>Γ−1τ1 +

1

γ1ψ1

˙ψ1(t).


For z1 and z2 to be stabilized with respect to V1, set α1, τ1 and˙ψ1 to

α1 := −k1z1

τ1 := −σθΓ(θ − θ0)

˙ψ1(t) := −σ1γ1(ψ1 − ψ0

1)

where k1, σθ, σ1, > 0 and θ0, ψ01 ≥ 0. Thus

V1 = z1z2 − k1z21 + θ>Γ−1(

˙θ − τ1) − σθθ

>(θ − θ0) − σ1ψ1(ψ1 − ψ01).

By completing the squares, the two last terms in the previous expression are written

as

−σθ θ>(θ − θ0) = −1

2σθθ

>θ − 1

2σθ

∣

∣

∣θ − θ0

∣

∣

∣

2

+1

2σθ

∣

∣θ − θ0∣

∣

2

−σ1ψ1(ψ1 − ψ01) = −1

2σ1ψ

21 −

1

2σ1(ψ1 − ψ0

1)2 +

1

2σ1(ψ

M1 − ψ0

1)2

where ψM1 = maxψ1, ψ

01. Using these equalities we obtain

V1 = z1z2 − k1z21 + θ>Γ−1(

˙θ − τ1) −

1

2σθ θ

>θ − 1

2σθ

∣

∣

∣θ − θ0

∣

∣

∣

2

+1

2σθ

∣

∣θ − θ0∣

∣

2

−1

2σ1ψ

21 −

1

2σ1(ψ1 − ψ0

1)2 +

1

2σ1(ψ

M1 − ψ0

1)2

≤ z1z2 − k1z21 −

1

2σθθ

>θ − 1

2σ1ψ

21 + θ>Γ−1(

˙θ − τ1) +

1

2σθ

∣

∣θ − θ0∣

∣

2

+1

2σ1(ψ

M1 − ψ0

1)2

≤ z1z2 − c1V1 + λ1 + θ>Γ−1(˙θ − τ1)

where

c1 := min

2k1, σ1γ1,σθ

λmin(Γ−1)

> 0


λ1 :=1

2σθ

∣

∣θ − θ0∣

∣

2+

1

2σ1(ψ

M1 − ψ0

1)2.

Step 2. The new error variable is

z3 = x3 − α2,

where α2 is a stabilizing function to be chosen later. The error dynamics of the

(z1, z2)-subsystem are

z1 = z2 + α1

z2 =∂z2

∂x1

x1 +∂z2

∂x2

x2 +∂z2

∂θ

˙θ +

∂z2

∂ψ2

˙ψ2

= −∂α1

∂x1x2 + x3 + δ2

= −∂α1

∂x1x2 + z3 + α2 + θ>ϕ2 + ∆2.

Using the function

V2(z1, z2, θ, ψ1, ψ2) = V1 +1

2z22 +

1

2γ2ψ2

2,

where γ2 > 0 and for ψM2 = max

ψ2, ψo2

, ψ2 := ψ2 − ψM2 , we can find an α2, τ2 and

˙ψ2 such that the (z1, z2) subsystem is stabilized. The derivative of V2 gives

V2 = V1 + z2z2 +1

γ2ψ2

˙ψ2

= V1 + z2

[

−∂α1

∂x1x2 + z3 + α2 + θ>ϕ2 + ∆2

]

+1

γ2ψ2

˙ψ2

≤ z1z2 − c1V1 + λ1 + θ>Γ−1(˙θ − τ1)

+z2

[

−∂α1

∂x1x2 + z3 + α2 + θ>ϕ2

]

+ z2ψM2 p2 +

1

γ2ψ2

˙ψ2.


For ε2, σ2 > 0 and ψ02 ≥ 0, set

α2 := −z1 − k2z2 +∂α1

∂x1

x2 − θ>ϕ2 − ψ2ω2

τ2 := τ1 + Γz2ϕ2

˙ψ2(t) := γ2

(

z2ω2 − σ2(ψ2 − ψ02))

ω2 := p2 tanh

[

z2p2

ε2

]

.

Applying intermediate control law α2, tuning function τ2 and parameter update law

˙ψ2(t) to V2 gives

V2 ≤ z2z3 − c1V1 − k2z22 + λ1 + θ>Γ−1(

˙θ − τ1) − z2θ

>ϕ2 + z2ψM2 p2 − z2ψ2ω2

+z2ψ2ω2 − σ2ψ2(ψ2 − ψ02)

= z2z3 − c1V1 − k2z22 + λ1 + θ>Γ−1(

˙θ − τ1) − z2θ

>ϕ2 + z2ψM2 p2 − z2ψ2ω2

+z2(ψ2 − ψM2 )ω2 − σ2ψ2(ψ2 − ψ0

2)

= z2z3 − c1V1 − k2z22 + λ1 + θ>Γ−1(

˙θ − τ1) − z2θ

>ϕ2 + z2ψM2 p2 − z2ψ

M2 ω2 −

σ2ψ2(ψ2 − ψ02)

≤ z2z3 − c1V1 − k2z22 + λ1 + θ>Γ−1(

˙θ − τ1) − z2θ

>ϕ2 + ψM2 (|z2|p2 − z2ω2) −

σ2ψ2(ψ2 − ψ02)

The Lyapunov analysis continues with some simplifications and various inequalities.

In this case term, −z2θ>φ2 is combined with θ>Γ−1(˙θ − τ1) to give θ>Γ−1(

˙θ − τ2).

Completing the squares, the last term becomes

−σ2ψ2(ψ2 − ψ02) = −1

2σ2ψ2

2 − 1

2σ2(ψ2 − ψ0

2)2 +

1

2σ2(ψ

M2 − ψ0

2)2,


yielding

V2 ≤ z2z3 − c1V1 − k2z22 −

1

2σ2ψ2

2+ λ1 + θ>Γ−1(

˙θ − τ2) + ψM

2 (z2p2 − z2ω2)

−1

2σ2(ψ2 − ψ0

2)2 +

1

2σ2(ψ

M2 − ψ0

2)2

≤ z2z3 − c1V1 − k2z22 −

1

2σ2ψ2

2+ λ1 + θ>Γ−1(

˙θ − τ2) + ψM

2 (z2p2 − z2ω2)

+1

2σ2(ψ

M2 − ψ0

2)2.

Using the following inequality introduced in [6]

0 ≤ |u| − u tanh(u

ε

)

≤ δε ≤ 1

2ε,

where δ = e−(δ+1) = 0.2785, ε > 0 and u ∈ <, we obtain

ψM2

(

|z2|p2 − z2p2 tanh

[

z2p2

ε2

])

≤ δψM2 ε2 ≤

1

2ψM

2 ε2.

The Lyapunov derivative then becomes

V2 ≤ z2z3 − c1V1 − k2z22 −

1

2σ2ψ2

2+ λ1 +

1

2ψM

2 ε2 +1

2σ2(ψ

M2 − ψ0

2)2 +

θ>Γ−1(˙θ − τ2)

≤ z2z3 + −c2V2 + λ2 + θ>Γ−1(˙θ − τ2)

where

c2 := min

2k1, 2k2, σ1γ1, σ2γ2,σθ

λmin(Γ−1)

> 0,

λ2 :=1

2ψM

2 ε2 +1

2σθ

∣

∣θ − θ0∣

∣

2+

1

2σ1(ψ

M1 − ψ0

1)2 +

1

2σ2(ψ

M2 − ψ0

2)2.


Step 3. Consider the error variable

z4 = x4 − α3,

and the error dynamics of the (z1, z2, z3)-subsystem given by

z1 = z2 + α1

z2 = −∂α1

∂x1x2 + z3 + α2 + θ>ϕ2 + ∆2

z3 =∂z3

∂x1x1 +

∂z3

∂x2x2 +

∂z3

∂x3x3 +

∂z3

∂θ

˙θ +

∂z3

∂ψ1

˙ψ1 +

∂z3

∂ψ2

˙ψ2

= −∂α2

∂x1x2 −

∂α2

∂x2(x3 + θ>ϕ2 + ∆2) + x4 −

∂α2

∂θ

˙θ − ∂α2

∂ψ1

˙ψ1 −

∂α2

∂ψ2

˙ψ2

= −∂α2

∂x1

x2 −∂α2

∂x2

x3 + z4 + α3 −∂α2

∂x2

θ>ϕ2 −∂α2

∂x2

∆2 −∂α2

∂θ

˙θ − ∂α2

∂ψ1

˙ψ1 −

∂α2

∂ψ2

˙ψ2

= −∂α2

∂x1x2 −

∂α2

∂x2x3 + z4 + α3 −

∂α2

∂x2θ>ϕ2 + Λ3 −

∂α2

∂θ

˙θ − ∂α2

∂ψ1

˙ψ1 −

∂α2

∂ψ2

˙ψ2,

where α3 is a stabilization function to be chosen later and Λ3 = −∂α2

∂x2∆2. A triangular

upper bound must be defined for Λ3 such that

|Λ3| ≤ ψ3p3 ≤ ψM3 p3 (3.4.7)

where ψM3 = max

ψ3, ψo3

. Besides satisfying the above inequality, the function p3

must also be strictly positive and smooth. Choose

p3 :=

[

∂α2

∂x2tanh

(

1

ε

∂α2

∂x2

)

+ ε

]

ψ2p2,


where ε > 0 and ψ2 is the smallest upper bound satisfying (3.4.7). For γ3 > 0 and

ψ3 = ψ3 − ψM3 , the (z1, z2, z3)-subsystem will be stabilized with respect to

V3(z1, z2, z3, θ, ψ1, ψ2, ψ3) = V2 +1

2z23 +

1

2γ3ψ2

3 .

Take the derivative of V3

V3 = V2 + z3z3 +1

γ3ψ3

˙ψ3

= V2 + z3

[

− ∂α2

∂x1

x2 −∂α2

∂x2

x3 + z4 + α3 −∂α2

∂x2

θ>ϕ2 + Λ3 −∂α2

∂θ

˙θ −

∂α2

∂ψ1

˙ψ1 −

∂α2

∂ψ2

˙ψ2

]

+1

γ3

ψ3˙ψ3

≤ z2z3 − c2V2 + λ2 + θ>Γ−1(˙θ − τ2) +

1

2ψM

2 ε2

+z3

[

−∂α2

∂x1x2 −

∂α2

∂x2x3 + z4 + α3 −

∂α2

∂x2θ>ϕ2 −

∂α2

∂θ

˙θ − ∂α2

∂ψ1

˙ψ1 −

∂α2

∂ψ2

˙ψ2

]

+|z3|ψ3p3 +1

γ3

ψ3˙ψ3.

Set the intermediate control law α3 and tuning functions to

α3 = −z2 − k3z3 +∂α2

∂x1

x2 +∂α2

∂x2

x3 +∂α2

∂x2

θ>ϕ2 − ψ3ω3 +∂α2

∂θτ3 +

∂α2

∂ψ1

˙ψ1

+∂α2

∂ψ2

˙ψ2

τ3 = τ2 − Γz3∂α2

∂x2ϕ2

˙ψ3(t) = γ3

(

z3ω3 − σ3(ψ3 − ψ03))

ω3 = p3 tanh

[

z3p3

ε3

]

,


where γ3, ε3 > 0, to obtain

V3 ≤ z3z4 − c2V2 − k3z3 + λ2 + θ>Γ−1(˙θ − τ2) + z3

∂α2

∂x2

θ>ϕ2 + z3∂α2

∂θ(τ3 − ˙

θ)

+z3(ψM3 p3 − ψ3ω3) + ψ3z3ω3 − σ3ψ3(ψ3 − ψ0

3)

= z3z4 − c2V2 − k3z3 + λ2 + θ>Γ−1

(

˙θ − τ2 + Γz3

∂α2

∂x2

ϕ2

)

+ z3∂α2

∂θ(τ3 − ˙

θ)

+z3

(

ψM3 p3 − ψ3ω3 + ψ3ω3

)

− σ3ψ3(ψ3 − ψ03)

= z3z4 − c2V2 − k3z3 + λ2 +

(

θ> − z3∂α2

∂θΓ

)

Γ−1(

˙θ − τ3

)

+z3(

ψM3 p3 + ψM

3 ω3

)

− σ3ψ3(ψ3 − ψ03)

≤ z3z4 − c2V2 − k3z3 + λ2 +

(

θ> − z3∂α2

∂θΓ

)

Γ−1(

˙θ − τ3

)

+ψM3 (|z3|p3 − z3ω3) − σ3ψ3(ψ3 − ψ0

3).

By completing the square for −σ3ψ3(ψ3 − ψ03) we get

−σ3ψ3(ψ3 − ψ03) = −1

2σ3ψ

23 −

1

2σ3(ψ3 − ψ0

3)2 +

1

2σ3(ψ

M3 − ψ0

3)2.

Since |Λ3| ≤ ψM3 p3, for all ε3 > 0 the term ψM

3 (|z3|p3 − z3ω3) is upper bounded as

follows

ψM3 (|z3|p3 − z3ω3) ≤ ψM

3

(

|z3| p3 − z3p3 tanh

[

z3p3

ε3

])

≤ 1

2ψM

3 ε3.

Thus

V3 ≤ z3z4 − c2V2 − k3z3 −1

2σ3ψ

23 + λ2 +

1

2ψM

3 ε3 +1

2σ3(ψ

M3 − ψ0

3)2

+

(

θ> − z3∂α2

∂θΓ

)

Γ−1(

˙θ − τ3

)

≤ z3z4 − c3V3 + λ3 +

(

θ> − z3∂α2

∂θΓ

)

Γ−1(

˙θ − τ3

)


where

c3 := min

2k1, 2k2, 2k3, σ1γ1, σ2γ2, σ3γ3,σθ

λmin(Γ−1)

> 0,

λ3 :=1

2ψM

2 ε2 +1

2ψM

3 ε3 +1

2σθ

∣

∣θ − θ0∣

∣

2+

1

2σ1(ψ

M1 − ψ0

1)2 +

1

2σ2(ψ

M2 − ψ0

2)2

+1

2σ3(ψ

M3 − ψ0

3)2.

Step 4. In the final step of the backstepping process, the parameter update laws

for θ and ψ4, and the control input, u2, are found. The complete error dynamics are

given by

z1 = z2 + α1

z2 = −∂α1

∂x1x2 + z3 + α2 + θ>ϕ2 + ∆2

z3 = −∂α2

∂x1x2 −

∂α2

∂x2x3 + z4 + α3 −

∂α2

∂x2θ>ϕ2 + Λ3 −

∂α2

∂θ

˙θ − ∂α2

∂ψ1

˙ψ1

−∂α2

∂ψ2

˙ψ2

z4 =∂z4

∂x1x1 +

∂z4

∂x2x2 +

∂z4

∂x3x3 +

∂z4

∂x4x4 +

∂z3

∂θ

˙θ +

∂z4

∂ψ1

˙ψ1 +

∂z4

∂ψ2

˙ψ2

+∂z4

∂ψ3

˙ψ3

= −∂α3

∂x1

x2 −∂α3

∂x2

(x3 + θ>ϕ2 + ∆2) −∂α3

∂x3

x4 + u2 + θ>ϕ4 + ∆4

−∂α3

∂θ

˙θ − ∂α3

∂ψ1

˙ψ1 −

∂α3

∂ψ2

˙ψ2 −

∂α3

∂ψ3

˙ψ3

= −∂α3

∂x1

x2 −∂α3

∂x2

x3 −∂α3

∂x3

x4 + u2 + θ>ϕ4 −∂α3

∂x2

θ>ϕ2 + ∆4 −∂α3

∂x2

∆2

−∂α3

∂θ

˙θ − ∂α3

∂ψ1

˙ψ1 −

∂α3

∂ψ2

˙ψ2 −

∂α3

∂ψ3

˙ψ3

= −∂α3

∂x1x2 −

∂α3

∂x2x3 −

∂α3

∂x3x4 + u2 + θ>

(

ϕ4 −∂α3

∂x2ϕ2

)

+ Λ4

−∂α3

∂θ

˙θ − ∂α3

∂ψ1

˙ψ1 −

∂α3

∂ψ2

˙ψ2 −

∂α3

∂ψ3

˙ψ3


where Λ4 := ∆4 − ∂α3

∂x2∆2. The triangular bound for Λ4 is

|Λ4| ≤ ψ4p4 ≤ ψM4 p4,

where ψM4 = max

ψ4, ψo4

and

p4 := |∆4| +∣

∣

∣

∣

∂α3

∂x2

∣

∣

∣

∣

|∆2| .

The z dynamics are stabilized with respect to the following Lyapunov equation

V4(z, θ, ψ1, ψ2, ψ3, ψ4) = V3 +1

2z24 +

1

2γ4

ψ4,

where γ4 > 0. The time derivative of V4 is bounded as follows

V4 = V3 + z4z4 +1

γ4

ψ4ψ4

= V3 + z4

[

− ∂α3

∂x1x2 −

∂α3

∂x2x3 −

∂α3

∂x3x4 + u2 + θ>

(

ϕ4 −∂α3

∂x2ϕ2

)

+ Λ4

−∂α3

∂θ

˙θ − ∂α3

∂ψ1

˙ψ1 −

∂α3

∂ψ2

˙ψ2 −

∂α3

∂ψ3

˙ψ3

]

+1

γ4ψ4

˙ψ4

≤ z3z4 − c3V3 + λ3 +

(

θ> − z3∂α2

∂θΓ

)

Γ−1(

˙θ − τ3

)

+ z4

[

− ∂α3

∂x1x2 −

∂α3

∂x2x3

−∂α3

∂x3x4 + u2 + θ>

(

ϕ4 −∂α3

∂x2ϕ2

)

− ∂α3

∂θ

˙θ − ∂α3

∂ψ1

˙ψ1 −

∂α3

∂ψ2

˙ψ2 −

∂α3

∂ψ3

˙ψ3

]

+z4ψM4 p4 +

1

γ4ψ4

˙ψ4.

Choosing k4, σ4 > 0 and setting

u2 := −z3 − k4z4 +∂α3

∂x1x2 +

∂α3

∂x2x3 +

∂α3

∂x3x4 − θ>

(

ϕ4 −∂α3

∂x2ϕ2

)

− ψ4ω4

+∂α3

∂θ

˙θ +

∂α3

∂ψ1

˙ψ1 +

∂α3

∂ψ2

˙ψ2 +

∂α3

∂ψ3

˙ψ3 + z3

∂α2

∂θΓ

(

ϕ4 −∂α3

∂x2ϕ2

)


˙θ := τ3 + Γz4

(

ϕ4 −∂α3

∂x2ϕ2

)

˙ψ4(t) = γ4

(

z4ω4 − σ4(ψ4 − ψ04))

ω4 := p4 tanh

[

z4p4

ε4

]

,

we obtain

V4 ≤ −c3V3 − k4z24 + λ3 +

(

θ> − z3∂α2

∂θΓ

)

Γ−1(

˙θ − τ3

)

+z4

[

θ>(

ϕ4 −∂α3

∂x2

ϕ2

)

− θ>(

ϕ4 −∂α3

∂x2

ϕ2

)

+ z3∂α2

∂θΓ

(

ϕ4 −∂α3

∂x2

ϕ2

)]

+z4

(

ψM4 p4 − ψ4ω4

)

+ ψ4ω4z4 − σ4ψ4(ψ4 − ψ04)

= −c3V3 − k4z24 + λ3 +

(

θ> − z3∂α2

∂θΓ

)

Γ−1(

˙θ − τ3

)

−z4(

θ> − z3∂α2

∂θΓ

)(

ϕ4 −∂α3

∂x2ϕ2

)

+ z4

(

ψM4 p4 − ψ4ω4 + ψ4ω4

)

−σ4ψ4(ψ4 − ψ04)

= −c3V3 − k4z24 + λ3 +

(

θ> − z3∂α2

∂θΓ

)

Γ−1

[

˙θ − τ3 − Γz4

(

ϕ4 −∂α3

∂x2ϕ2

)]

+ψM4 (z4p4 − z4ω4) − σ4ψ4(ψ4 − ψ0

4)

≤ −c3V3 − k4z24 + λ3 + ψM

4 (|z4|p4 − z4ω4) − σ4ψ4(ψ4 − ψ04).

Similarly to steps 2 and 3, completing the squares of −σ4ψ4(ψ4 − ψ01) gives

−σ4ψ4(ψ4 − ψ01) = −1

2σ4ψ

24 −

1

2σ4(ψ4 − ψ0

4)2 +

1

2σ4(ψ

M4 − ψ0

4)2

ψM4 (|z4|p4 − z4ω4) ≤ 1

2ψM

4 ε4

where ε4 > 0. Using these results, the Lyapunov function becomes

V4 ≤ −c3V3 − k4z24 −

1

2σ4ψ4

2+ λ3 +

1

2ψM

4 ε4 +1

2σ4(ψ

M4 − ψ0

4)2

≤ −c4V4 + λ4 (3.4.8)


where

c4 := min

2k1, 2k2, 2k3, 2k4, σ1γ1, σ2γ2, σ3γ3, σ4γ4,σθ

λmin(Γ−1)

> 0,

λ4 :=1

2ψM

2 ε2 +1

2ψM

3 ε3 +1

2ψM

4 ε4 +1

2σθ

∣

∣θ − θ0∣

∣

2+

1

2σ1(ψ

M1 − ψ0

1)2

+1

2σ2(ψ

M2 − ψ0

2)2 +

1

2σ3(ψ

M3 − ψ0

3)2 +

1

2σ4(ψ

M4 − ψ0

4)2.

Inequality (3.4.8) implies that trajectories z(t), θ(t), Ψ(t) and x(t) are globally uni-

formly ultimately bounded to a small region around the origin [6]. Specifically, there

exists a finite T > 0 such that for all t ≥ T , trajectory z(t) is contained within the

compact set

F :=

z ∈ <4 : |z(t)| ≤ µ

where µ >

√

2λ4

c4,

which can be made arbitrarily small by adjusting the design parameters found in the

definition of c4 and λ4.


Figure 3.11 and 3.12 depict the position trajectories of the robust controller, the adap-

tive controller and the robust adaptive controller. The following design parameters

were used when simulating the robust adaptive control for case 1 and case 2:

θ(0) =[

0 0 0 0]>, ψ(0) =

[

0 0 0 0]>,

ε1 = 0.01, ε2 = 10−9, ε3 = 0.01, ε4 = 10−9,

k1 = 1.5, k2 = 1.75, k3 = 1.5, k4 = 1.75,

γ1 = γ2 = γ3 = γ4 = 3.5,

σ1 = σ2 = σ3 = σ4 = σθ = 1,


θo =[

0 0 0 0

]>, ψo

1 = ψo2 = ψo

3 = ψo4 = 0

p1(x1) = p1(x1) = 0, p2(x1) = p2(x1) = x21 + 1.65,

p3(x1, x2, x3) =

[

∂α2

∂x2tanh

(

10∂α2

∂x2

)

+ 0.1

]

(

x21 + x2

3 + 0.6)

,

p4(x) = (|x1| + |x3|)(

1 +

∣

∣

∣

∣

∂α3

∂x2

∣

∣

∣

∣

)

.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.005

0.01

0.015

Time (s)

x po

sitio

n (m

)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.01

−0.008

−0.006

−0.004

−0.002

0

Time (s)

y po

sitio

n (m

)

Figure 3.11: Case 1 response of uncertain system when using the robust adaptivecontroller.

In Figure 3.11, the robust adaptive closed-loop system cannot handle the uncer-

tainties of case 1 and becomes unstable. The robust adaptive controller was designed

under the assumption that δ2 depends only on states x1 and x2. It cannot stabilize the

system under case 1 conditions because its simulated uncertainties depend on state

x3. On the other hand, since the robust controller is not based on the triangularity

condition it does compensate for this term and as a result is the only controller that

stabilizes the system under the conditions of case 1.


0 2 4 6 8 10 12 14 16 18 20−0.01

−0.005

0

0.005

0.01

Time (s)

x po

sitio

n (m

)

0 2 4 6 8 10 12 14 16 18 20−0.04

−0.03

−0.02

−0.01

0

0.01

Time (s)

y po

sitio

n (m

)

robust adaptive

robust adaptive

robust adaptive

robust

adaptive

Figure 3.12: Case 2 response of uncertain system when using robust, adaptive androbust adaptive nonlinear controllers.

The robust adaptive controller stabilizes the system when case 2 is considered, as

shown in Figure 3.12, and achieves a steady-state error of 1.7184 × 10−6 m with a

peak current of 6.2250 A. The current plot is show in Figure 3.13. Comparatively

speaking, this controller has the best steady-state error but with the drawback that its

peak current is 2 A higher than the robust controller. The idea that adaptive upper

bounds would lower peaking currents is not confirmed when the system is subject to

case 2 uncertainties.

Another case is introduced for the controllers to be tested when the uncertainties

have smaller magnitude. Figure 3.14 is the closed-loop system response of the linear,

robust, adaptive, and robust adaptive controllers subject to uncertainties described

in the case defined below.


0 2 4 6 8 10 12 14 16 18 201

2

3

4

5

6

7

Time (s)

Cur

rent

(A

)I3

I1

I2

Figure 3.13: Currents from robust adaptive controller when in case 2.

Case 3:

δ2(x1, x2, x3) ≤ 0.4 |x1| + 0.2 |x3| − 0.01x2

δ4(x1, x3, x4) ≤ 0.2 |x1| + 0.4 |x3| − 0.01x4

x(0) =[

8.33 mm 0 8.33 mm 0

]>

In this case, the uncertainties caused by assumptions (2.0.3), (2.0.4), (2.0.5),

(2.0.6) and (2.0.7) not holding are made smaller. Performance of each controller

will now be analyzed.

Using design parameters β∗ = 0.5 and γ = 10−4 for the robust controller; k1 =

k2 = k3 = k4 = 2, Γ = 3 and θ(0) =[

0 0 0 0 0]>

for the adaptive controller

and the following changes to the design parameters used earlier in the robust adaptive


controller

ε2 = 10−4, ε4 = 10−4,

k1 = 1.5, k2 = 1.5, k3 = 1.5, k4 = 1.5,

γ1 = γ2 = γ3 = γ4 = 3,

ψo1 = 0, ψo

2 = 0.5.ψo3 = 0.5, ψo

4 = 0,

p2(x1) = p2(x1) = x21 + 0.02,

p3(x1, x2, x3) =

[

∂α2

∂x2tanh

(

10∂α2

∂x2

)

+ 0.1

]

(

x21 + x2

3 + 0.02)

,

we get the responses shown in Figure 3.14. The performance of each control system

is as follows

0 2 4 6 8 10 12 14 16 18 20−10

−5

0

5x 10

−3

Time (s)

x po

sitio

n (m

)

0 2 4 6 8 10 12 14 16 18 20−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

Time (s)

y po

sitio

n (m

)

linear robust adaptive

adaptive robust

robust adaptive

linear

adaptive

robust

Figure 3.14: Case 3 response of uncertain system when using linear, robust, adaptiveand robust adaptive nonlinear controllers.


Controller Steady-State Error (m) Max. Current (A)

Linear 2.4963 × 10−4 1.4933

Robust 6.9753 × 10−5 5.7069

Adaptive 3.1737 × 10−4 6.5066

Robust Adaptive 2.6172 × 10−5 6.1068

The disk’s final position should be at least within 0.1 mm. The linear and adaptive

controllers do not meet this performance requirement while both the robust and

robust adaptive controllers surpass it. The robust adaptive control system has a

better error but its maximum current is higher then that of the robust controller.

However, as already mentioned before, chattering of the robust controller currents

may cause practical difficulties. From a practical viewpoint, the robust adaptive

controller is the most desirable of the feedback systems as long as the actuators can

support the higher currents it tends to use.

Chapter 4

Implementation

The planar magnetic levitation device is built and shown in Figure 4.1 and Figure

4.2. Due to hardware limitations of the device, which are described in the following,

the nonlinear controllers developed in the previous chapter cannot be tested on the

system. However, some modelling and control principles of this system are experi-

mentally confirmed with two smaller testbeds.

Figure 4.1: Top view of planar magnetic levitation device.

77

CHAPTER 4. IMPLEMENTATION 78

Figure 4.1 shows the three electromagnets surrounding the disk in a triangular

configuration. Each electromagnet is then attached to an analogue current controller

that feeds a voltage to an amplifier powering the magnet. Reference currents are sent

from the computer through the data acquisition card shown in the bottom left-hand

corner of Figure 4.1. Currents in the coils of the magnets and the x and y positions of

the disk are read through the data acquisition card from current and position sensors.

Figure 4.2: Side view of planar magnetic levitation device.

The disk is not being levitated by a magnet. Including an additional magnet in the

z direction for vertical levitation would, from an implementation viewpoint, increase

expenses while, from a control design viewpoint, would not make the problem more

interesting because the force in the z direction is decoupled from the forces acting

in the xy plane. Instead, the disk is anchored by a wooden dowel mounted on a

small platform of a linear guide, as shown in Figure 4.2. Combining two such guides

together enables the disk to move in both the x and y directions.

Figure 4.3 depicts the individual components of the system and the input/output

relation between each of them. The position controller is run real-time on a computer

CHAPTER 4. IMPLEMENTATION 79

ControllerPosition

ControllerCurrent

Amplififer Electromagnet Disk

CurrentSensor

SensorPosition

IrefVref Vm

(x, y)

Icoil

Figure 4.3: Overview of interfaces in magnetic levitation device.

using WinCon designed by Quanser Consulting. The controller calculates a current

based on the position of the disk measured by the position sensors, and the currents

running through the coils measured by the current sensor. The reference current is

an input to a current controller that computes a voltage based on the present current

in the coils of the magnet. The power amplifier outputs a high-voltage signal to the

magnet depending on the reference voltage from the current control system. Next are

the nonlinear dynamics between the magnet and the disk. The force acting on the

disk depends on the current in the magnet coils, the air gap, and various constants

such as the dimensions and permeabilities of the disk and the cores. As the disk

moves in the x and y directions along the two linear guides, the position sensors

continuously measure the position of disk. The position of the disk and the currents

in the magnets measured by the sensors are inputs to the controller in the next loop.

The individual components of the system are described in detail in Section 4.1.

Descriptions of how the components operate, reasons why a certain technology was

chosen, consideration taken and problems encountered are explained in each section.

The discussion begins with the electromagnets and steps back to the current con-

troller. The guides and position sensors are then explained, followed by the software

and interface used to implement the controller. This completes the description of the

4.1. SYSTEM COMPONENTS 80

system. In Section 4.2, different procedures considered for finding the model param-

eters are examined. The section concludes with a definitive modelling methodology

that is used later to achieve some fairly impressive control results on actual systems.

Section 4.3 describes two simpler testbeds that are primarily designed to verify ex-

perimentally the principles of the three-magnet system and isolate the problem that

is causing it to operate only partially. Individually, the setups also offer some sub-

stantial modelling and applied nonlinear motion control results as well. This section

closes with a discussion on the issues encountered in the full three-magnet system.

4.1 System Components

4.1.1 Magnet

The electromagnet cores, shown in Figure 4.1, have a cross sectional area of 5×5 cm2

and are 40 cm long. These dimensions were chosen to obtain substantial strength

from inefficient rectangular magnets (see e.g., [1]). In particular, the cross sectional

area of the cores should be considerably smaller than the length of the core. The cores

were originally 10 × 10 × 10 cm3 and the attractive magnetic force was only exerted

along the outer edge of the core’s face. In [1], the magnetic flux density inside an ideal

solenoid is compared with the magnetic flux density of two solenoids of finite length.

It is shown that the solenoid with a length to diameter ratio of 1, Ld

= 1, has half the

magnetic flux density of an ideal solenoid. On the other hand, a solenoid with Ld

= 10

is about the same strength as the ideal solenoid. This observation is consistent with

the numerical experiments done in FEMLAB for bar magnets and is briefly discussed

in Appendix A.2. Narrow magnets are needed to attain sufficient force to move the

disk but this increases fringing. Thus, to avoid the problems discussed above and

increase the strength of each electromagnet, the original magnets were each cut into


Magnet No. of Windings Resistance (Ω) Inductance (mH)1 490 5.3 40.52 482 5.0 42.43 503 5.8 50.4

Table 4.1: Measured resistance and inductance of electromagnets.

four 10 × 5 × 5 cm3 rectangles and stacked lengthwise to make a long bar magnet

with a length to diameter ratio of about

L

d=

40 cm

5 cm= 8. (4.1.1)

Magnet strength is further increased by using laminated cores made of low-copper

soft steel. Pure iron and permalloy 80 were considered first but no supplier was found

that could furnish pure iron of the dimensions needed and permalloy is too expensive.

The cores are wound with 22-Gauge magnet wire which can withstand current bursts

of 10 A and continuous currents of 6 A without the insulator coating melting. This

limit becomes a controller constraint - current must now be both positive and less

then 10 A.

The resistance and inductances measured for each magnet is shown in Table 4.1.

They are described by the following transfer functions

Icoil1 =1

0.041s+ 5.3Vm1

Icoil2 =1

0.042s+ 5.0Vm2

Icoil3 =1

0.050s+ 5.8Vm3

(4.1.2)

where for i = 1, 2, 3, Vmi is the voltage delivered by the amplifier connected to magnet

i.


4.1.2 Power Amplifier

The power amplifier is the component driving current in the electromagnet coils. The

12A8 Advance Motion Controllers Brush Type PWM Servo Amplifier is a compact

pulse-width modulator capable of outputting continuous currents of ±6 A and peak

currents of ±12 A. This output range is suitable for the currents required by the

position controller. The servo amplifier operates in open-loop mode, thus it amplifies

its input voltage by a fixed gain. Figure 4.4 shows the signal transition between the

PWM input and the magnet output. The power amplifier also features a current

monitor that measures the current in the coils of the electromagnet. This sensor is

used by the current controller and the position controller.

Vm 1Ls+R

IcoilKm

Vref

PWM Electromagnet

Figure 4.4: Input/Output relationship between PWM and electromagnet.

Grounding and sensor noise issues are avoided using the PWM wiring setup shown

in Figure 4.5. The internal signal grounds of the PWM are isolated from the power

ground terminals. The - ref terminal is connected to the shield of the signal wire

that is connected to the ground of the data acquisition card. Noise from an unused

input is prevented by attaching the - tach to the signal gnd connector. The power

wires and motor leads are twisted so the inductance formed between the wires is

cancelled in successive twists. Thus, the power ground and high-voltage terminals of

each PWM are twisted together and attached to the 0 V and 55 V terminal of the

power supply and the motor leads carrying the voltage supplied from the PWM to

the electromagnet are also twisted (not shown in diagram). Finally, the 0 V terminal

of the power supply is connected to the ground of the data acquisition card.


curr monitor out

- ref in

+ ref in

power gnd

high voltage

signal gnd

+ tach (gnd)

Data AcquisitionCard

PowerSupply

PWM- tach in

signal

return

shield

DAC gnd

0 V

55 V

Figure 4.5: Pulse-width modulator wiring.

According to the installation notes, the majority of the noise comes from the

high-voltage switching of the motor leads. As different voltages are sent though the

leads, a capacitive coupling is formed between the leads and the signal wires. This is

significantly attenuated by appropriately shielding the signal wires.

4.1.3 Current Controller

The current controller is a feedback loop that supplies a reference voltage to the am-

plifier to achieve a desired current. In this case, the desired current is given by the

position controller (see Figure 4.3) and the current control output is the PWM refer-

ence signal, Vref. The output voltage of the PI controller is amplified by the PWM to

achieve the desired current in the coils of the electromagnet. The PWM unit actually

features its own current control system. However, the amplifier current control gives

an unsuitable steady-state error because the inductances of the electromagnets are

too large and the capacitor in the current loop cannot be varied (the capacitor de-

termines the steady-state error). To overcome this problem, the current control was


designed separately and is a simple proportion-integral controller with the structure

D(s) = Kp +KI

s, (4.1.3)

where Kp, KI > 0. Figure 4.6 shows the PI compensator with the power amplifier,

the electromagnet and a sensor gain Ks.

KI

Kp

+++

−Iref 1

Ls+R

IcoilKm

Vref Vm

Ks

1s

PI Controller

PWM Electromagnet

Figure 4.6: Current controller structure.

The closed-loop system that successfully tracks a reference current is thus

Icoil =KmKs (Kps+KI)

Ls2 + (R +KmKsKp) s+KmKsKI

Iref. (4.1.4)

The proportional gain, Kp, and integral gain, KI , were tuned using the magnets

transfer functions (4.1.2) such that the best tracking could be achieved while keeping

overshoot low and ensuring the output voltage is confined within the ±15 V limit of

the + ref connector of the PWM. The current going through the coils is twice the

current monitor reading. Thus the sensor gain, Ks, from the PWM’s current monitor

is Ks = 2 A/V.

The PI current controller was implemented in both an analogue circuit and digi-

tally in WinCon. The original reason to design an analogue circuit, shown in Figure


Re

Re

R1

Re

Re

Ci

V1

Rf

Ks

Sensor Gain

Icoil

−

+

+

−

Electromagnet and Amplififier

Km

Ls+R

Vref

Vin

Figure 4.7: Current controller circuit.

4.7, was to reduce the computational complexity of the software component of the

controller, thus saving bandwidth in WinCon. Since the analogue circuit is also much

faster than the software-implemented current controller, it can potentially track the

currents much better. The first operational amplifier acts as a summer, see Figure

4.8. Thus

V1 = K1Vin +K2Vcoil

=−Re

Re

Vin +Re +Re

Re

Re

Re +Re

Vcoil

= −Vin + Vcoil,

where Vcoil is the voltage measured from the current monitor pin (not multiplied by

2) and Vin = Iref2

. The output signal from the position controller, although called

the reference current, is actually a voltage with respect to the ground of the data

acquisition card. This signal is divided by half instead of adding circuitry to double


the current monitor output to get Icoil.

Re

Re

Re

Re

+

−

V1+

+V1

K1

K2

Vin

VcoilVcoil

Vin

Figure 4.8: Summer operational amplifier.

Vref

R1

CiRf

−

+

V1

V1 VrefK3(s)

Figure 4.9: Proportional-integral operational amplifier.

Figure 4.9 depicts the second operational amplifier that implements the gains of

the PI controller. The circuit analysis from the input, the reference signal Vin, to the


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1.5

−1

−0.5

0

0.5

1

1.5

Time (s)

Cur

rent

(A

)

I refI coil

Figure 4.10: PI control tracking reference current.

PI output, Vref, is

Vref = K3(s)V1

=

(

−(

Rf + 1Cis

)

R1

)

(−Vin + Vcoil)

=

(

Rf

R1

+1

CiR1s

)

(Vin − Vcoil)

=

(

Kp +KI

s

)

(Vin − Vcoil) ,

where Kp =Rf

R1and KI = 1

CiR1. Table 4.2 lists typical resistance and capacitance

values used. Figure 4.10 depicts the PI analogue control tracking a square reference

current using the parameter values listed in Table 4.2.

It was noted that the operational amplifiers saturate at approximately 4.7 V when

using a 30 V power supply, therefore preventing large reference currents from being

tracked. Further, gains used in simulation that yielded good tracking would occa-

sionally output a high-frequency chattering signal in the circuit. Thus, it was finally


Parameter ValueRe 10kΩRf 20kΩR1 5kΩCi 50nF

Table 4.2: Typical resistance and capacitance values used for PI control.

decided to implement digitally in WinCon. This makes it convenient to test differ-

ent Kp and KI gains. In some cases, it was desired to minimize the PI voltage at

the expense of steady-state error. For instance, this is done when using controllers

with large current transients or high current biases and when testing with large step

inputs. The digital PI controller is used in all the results shown later in this chapter.

4.1.4 Disk and Linear Guides

Due to budget constraints and because of a sensing issue, discussed in the next section,

the disk is not being levitated by a fourth magnet or by an air bearing system. Instead,

linear guides are used to provide the disk with movements in the x and y directions.

A linear guide consists of a small platform that rides on one or more rails over ball

bearings for low-friction movement. Linear guides are often used in motion control

but usually in these applications the platform is actuated by a motor. In this case,

we require a guide that is unactuated since the magnets will be moving the disk. The

disk is constrained to move about a small region in the xy plane. Thus, linear guides

that are compact and that can be placed one on top of the other perpendicularly are

needed. Further, the guide must be as light as possible to minimize the current needed

to move the disk and the guide it is attached to. The Thomson Microstage linear

guide was chosen because it is constructed of a lightweight alloy, provides low-friction

precise motions and, as shown in Figure 4.11, can be mounted in an X-Y fashion.


Figure 4.11: Linear guides mounted in an X-Y fashion.

Using epoxy glue, the disk is fastened onto a wooden dowel that is anchored onto

a wood piece acting as a platform extension. The platform extension is then screwed

tightly onto the top platform, see Figure 4.2. The dowel is needed to move the disk

through the hole of the platform. Although there are not many parts of the guide

that are magnetizable, precaution was taken to ensure that the magnets would be a

fair distance away from the guides.

The viscous friction of the guides is low. According to the manufacturer, the

coefficient of viscous friction is 0.002. However, the Coulomb friction is quite large

for a motion control application. Friction, in the general sense, between the platform

and the rails of these guides is minimized by having a significantly heavy load on the

platform. Heavy loads increases the contact between the platform, the ball bearings,

and the rails. The load on the top platform includes the disk, dowel, carriage extension

and screws and weighs 1.56 kg while the load on the bottom guide is 2.30 kg, since

it also includes the weight of the top linear guide, 0.74 kg. Increasing the weight of

the load would minimize at least the viscous friction but perhaps not the Coulomb

friction and further heavier loads require higher currents. The dimensions of the cores


Parameter Valuel1 40 cml2 10.15 cmm 2.30 kgh 2.4 cmA1 0.0025 m2

Ar 2.4 × 10−3 m2

Table 4.3: Values of physical parameters in actual planar magnetic levitation device.

and the disk, as well as the total movable mass are listed in Table 4.3.

As briefly mentioned, besides solving the levitation issue, the guides are part of a

sensing solution to measure the x and y position of the disk.

4.1.5 Sensor

Sensors are needed to measure the x and y positions of a metal disk that is subject

to electrical and magnetic fields. The sensor must be non-contact, have a resolution

of ≤ 0.1 mm, response time ≤ 0.002 s (sampling frequency of ≤ 500 Hz) and should

be reliable.

Optical solutions such as laser interferometers or infrared may give false readings

as the disk moves away from the origin because the laser reflects off the tangent of the

disk’s curved surface and therefore the beam sent back to the sensor is not parallel

to the incident beam. This phenomenon is depicted in Figure 4.12. Optical sensors

are usually used on a target with a flat surface. However, there are reflectors where

the beam reflected is parallel to the incident beam. Perhaps one of these reflectors is

available in an adhesive tape form that can be fastened around the disk. The beam

reflected off the disk is parallel to the beam sent towards the disk from the sensor

and a correct position can be measured. However, no such commercial product was

found and, further, high-resolution optical solutions such as laser interferometers are


too expensive for our budget.

Incident Beam

Reflected Beam

Tangent

Disk

Figure 4.12: Laser reflected off a curved surface.

Sensing technologies such as ultrasonic sensors function by emitting a sound wave

and recording the time it takes for the wave to return after it bounces from a surface.

However, the time to receive the sound wave back to the receiver would be extended

when the disk moves off the origin. Consider the following example for explanation,

the x position sonic sensor measures a distance ε when the disk is at the origin, x = 0

and y = 0. If the disk is moved 1 cm up along the y axis but remains along the x

axis, that is x = 0 and and y = 1 cm, the sensor would read a value larger than ε

because the sound wave hits an edge of the disk that is further away than when the

disk was at the origin. That is, the sensor would read x > 0 and y = 1 cm, when the

actual location of the disk is x = 0 and y = 1 cm. The return time of the wave is

extended and therefore the position read is greater. For this reason ultrasonic sensors

are not used.

Inductive and capacitive sensors were considered because they are used in high-

precision applications. Inductive sensors supply a magnetic field and measure the

inductance of a metal object which is proportional to the distance of the object.

Capacitive sensors function in a similar manner, they supply an electric field and

measure the capacitance of an object with a significant dielectric constant, such as

metal, which is proportional to the distance. However, because these sensors rely


on magnetic or electric fields to get their measurements, the electromagnetic field

generated by the magnets moving the disk may interfere with those of the sensors

and cause false readings. As a result, they were discarded as a reliable source of

reading position.

Another option was tracking the position of the disk by mounting a CCD camera

looking down on the xy plane of the experiment. This sensing technology was used

in a magnetic suspension system in [5] with a resolution of 0.43 mm. The camera

alternative requires substantial effort in designing the signal processing to get the

positions. Further, its response time is too slow and the resolution is inadequate,

> 0.1 mm.

Fast high-resolution measurements of the disk’s (x, y) position can be obtained by

mounting optical encoders on the platform of each linear guide and attaching their

shafts with a fishing line that is fastened at each end of the guides. Figure 4.13 shows

how the encoder is anchored on the linear guide. As the disk moves about the xy

plane, the platform on each guide moves and the shafts on the encoders rotate. The

fishing line is fastened tightly to prevent slippage of the line. As the shaft rotates,

the encoder outputs a count that is translated in a linear distance using the sector

formula

x = r∆θn

=0.125 in

2× 1 m

39.37 m× 2π rad

4 · 1024 counts× n

= 4.8704 × 10−6n m (4.1.5)

where r is the radius of the optical encoder’s shaft, ∆θ is the change in angle of

the shaft and n is the number of counts the encoder is reading. The resolution is

four times the counts per revolution of the sensors because the data acquisition card


decodes in quadrature mode. Thus assuming ideal operation, the sensor achieves

about 5µm resolution instead of 20µm.

Figure 4.13: Linear guide and sensor setup.

Tests were made on each encoder by moving the linear guide platform at different

points and comparing the output of the encoder with a manual reading off a ruler

fastened on the side of the guide. It was found that there was a consistent error

of 0.5 mm for every 1 cm of distance. For instance, when the platform was moved

to 4 cm the encoder measured 3.80 cm and when moved to 1 cm it read 0.95 cm.

However, when the platform was moved at a certain distance and moved back to its

original position the encoder would correctly measure a distance of 0 at the platform’s

final position. For example, if the platform was moved 5 cm and moved back the same

distance, the encoder would measure 0 cm. Thus it appears from the second tests that

there is no slippage of the fishing line on the shaft. However, the 5% error in the first

test could not be rooted to any other problem. Since the error is consistent and the

resolution is still at least 0.1 mm, a 1.05 adjustment was made in the transformation

that calculates linear position from the optical encoder’s count.

Transducers in place of the optical encoders could also be used but concerns were

made on the amount of friction they would introduce.


4.1.6 Real-time Controller Software and Interface

All the nonlinear controllers designed are implemented in the Simulink environment

and are run using the Wincon software platform by Quanser Consulting. WinCon

is a realtime software that extends Simulink and offers a variety of Simulink blocks

to interact with the Quanser MultiQ PCI data acquisition card. Such blocks that

communicate externally are A/D input, encoder input and D/A output. WinCon

generates the code from a Simulink model and the code is run on the computer. The

controller can be started locally on the machine the code was built from or, since this

is a client-server program, can be run from a remote machine via the Internet. The

controller can be run at a sampling rate of at least 1 kHz and control parameters

can be adjusted in real-time, that is, the changes are applied immediately while the

controller is running.

Nss+N

x

x

y

y

Input

Encoderto Pos.Count

Input

Analogue2 A/V

Output

Analogue

Icoil 1

Icoil 2

Icoil 3

Iref 1

Iref 2

Iref 3

Nss+N

Position

Controller

Figure 4.14: Interface between position controller and magnetic levitation device.

Figure 4.14 shows the various interfaces between the position controller and sen-

sors. The three reference currents computed by the position controller are sent to the

analogue output box. These signals are outputted from the the D/A outputs on the

DAC’s terminal board, shown in Figure 4.1. The actual currents going through the

electromagnets coils are taken from the current monitor pins on the power amplifiers.

In the software, current measured in the coils of the electromagnets is received from


the analogue input block. The x and y positions of the disk are measured from the

optical encoders using the encoder inputs. The encoder input gives the count and

then the sector formula described in (4.1.5) calculates the positions from the count.

The disk velocity is attained using a high-gain observer on the x and y displacements,

as shown in Figure 4.14.

4.2. FINDING MODEL PARAMETERS 96

4.2 Finding Model Parameters

Consider again the system model (2.3.2), included here for convenience

x1 = x2

x2 = − 1

2mµoA1

[

ϕ1(x1, x3)(x1 + d)I21 + ϕ2(x1, x3)

(

x1 −d

2

)

I22

+ ϕ3(x1, x3)

(

x1 −d

2

)

I23

]

x3 = x4

x4 = − 1

2mµoA1

[

ϕ1(x1, x3)(−x3)I21 + ϕ2(x1, x3)

(

x3 +

√3

2d

)

I22

+ ϕ3(x1, x3)

(

x3 −√

3

2d

)

I23

]

(4.2.1)

where

ϕ1(x1, x3) =N2

1(

L1

µ1A1+ L2

µ2A2+

√(x1+d)2+x2

3

µ0A1

)2√

(x1 + d)2 + x23

ϕ2(x1, x3) =N2

2

L1

µ1A1+ L2

µ2A2+

s

(

x1− d2

)2

+

(

x3+√

3

2d

)2

µ0A1

2√

(

x1 − d2

)2

+(

x3 +√

32d)2

ϕ3(x1, x3) =N2

3

L1

µ1A1+ L2

µ2A2+

s

(

x1− d2

)2

+

(

x3−√

3

2d

)2

µ0A1

2√

(

x1 − d2

)2

+(

x3 −√

32d)2

,

and notice that the only unknown parameters in (4.2.1) is the permeability of the

cores, µ1, and of the disk, µ2. These permeabilities are found by measuring the

inductance of the electromagnet for a given position of the disk. Then, using an


inductance expression describing the inductance of the electromagnet with respect

to the air gap between the core and the disk, the permeabilities can be found. The

inductance expression is developed by using the flux linkage formula

λ = NΦ, (4.2.2)

where N is the number of windings and Φ is the magnetic flux, and the relationship

between inductance and flux linkage

λ = L(z)I. (4.2.3)

Combining equations (4.2.2) and (4.2.3) with the one-dimensional flux relationship

described in (2.1.7),

Φ =NI

l1µ1A1

+ l2µ2A1

+ zµoA1

, (4.2.4)

where l1 is the core’s length, l2 is the disk diameter, µ1 = µr1µ0, µ2 = µr2µ0 and

µ0 = 4π× 10−7 H/m is the permeability of free space, gives the relationship between

inductance, L, and the air gap z,

L(z) =1

l1µ1A1N2 + l2

µ2A1N2 + zµoA1N2

. (4.2.5)

Assuming the permeability of the core, µ1, and of the disk, µ2, are equal and measuring

the zero air gap inductance, L(0), the relative permeability µr1 = µr2 can be found.

Note that assumption µ1 = µ2 is realistic because the disk is constructed of the same

material as the cores. Using the resistance and inductance measurements in Table 4.1

and the various properties in Table 4.3, the relative permeability of electromagnet 1


is

µ1 = µ2 =L10

N21

(

l1

A1

+l2

A2

)

µr1 = µr2 =L10

µ0N21

(

l1

A1+

l2

A2

)

=40.2 × 10−3 H

(4π × 10−7 H/m)(490)2

(

0.4 m

(0.05m)2+

0.10 m

(0.10 m)(2.4 × 10−2 m)

)

= 26.9.

Similarly, the relative permeabilities of magnets 2 and 3 are

µr2 = 29.4, µr3 = 31.9.

The model parameters of the three-magnet dynamics in (4.2.1) are completely known.

However, instead of measuring the inductance and finding the relative permeabilities,

the dynamics in (4.2.1) can be expressed directly in terms of the measured inductance.

This is discussed in the next section.

4.2.1 Inductance Models

Expressing the model in terms of inductance yields a model that is more intuitive

in a practical sense and results in dynamics that are presented in a more compact

form by grouping various parameters together. In the first part of this section, the

force between an electromagnet and a disk in the one-dimensional case is developed

using the new presentation of the inductance relationship in (4.2.5). The resulting

dynamics are expressed in terms of inductance and are entirely equivalent to the one-

dimensional result obtained earlier in Section 2.1. Due to the modelling uncertainty

concerns outlined in Chapter 2, an alternative expression of inductance is discussed

in the second part of this section that is developed experimentally. This, on the


other hand, is not equivalent to force expression in Section 2.1. The reasons why

another expression for inductance is considered along with the problems associated

with deriving it are discussed.

Model A

The magnetic force between a single electromagnet and a disk in the one-dimensional

case is now found using inductance as opposed to the analysis done in Section 2.1.

Change inductance relationship (4.2.5) into

L(z) =1

l1µ1A1N2 + l2

µ2A1N2 + zµoA1N2

=N2

(

l1µ1A1

+ l2µ2A1

)

(

1 + z

µoA1

“

l1µ1A1

+l2

µ2A1

”

)

=L0

1 + za

(4.2.6)

where

L0 =N2

l1µ1A1

+ l2µ2A1

a = µ0A1

(

l1

µ1A1+

l2

µ2A1

)

=µ0A1N

2

L0.

(4.2.7)

The inductance of the electromagnet when the disk is attached to the face of the core

is L0 = L(0), and the decay parameter, a, is based on the reluctance principle in [1].

Inductance expressed in this manner is called model A.

Alternatively to the relationship in (2.1.8), magnetic energy can be expressed in

terms of inductance as

E(z, I) =1

2L(z)I2, (4.2.8)

where z is the air gap distance between the magnet and the disk, L(z) is the mutual

inductance of the electromagnet and the disk as a function of the air gap, and I is


the current in the coils of the electromagnet. By taking the derivative with respect

to the air gap we obtain the force exerted by the electromagnet on the disk

F (z, I) =∂E(z, I)

∂z. (4.2.9)

Using relationships (4.2.8) and (4.2.9), the magnetic force between an electromagnet

and a disk is

F (z, I) =∂

∂z

(

1

2L(z)I2

)

=1

2I2∂L(z)

∂z

= −L0

2a

1(

1 + za

)2 I2

=−b

(

1 + za

)2 I2,

(4.2.10)

where b = L0

2a. We stress that the force expression in (4.2.10) is equivalent to that

in (2.1.10). The three-magnet dynamics (4.2.1) can be developed following the same

procedure illustrated in Section 2.2 and Section 2.3.

Model B

An alternative way of expressing inductance in terms of air gap, herein called model

B, is described in [13] for a 1 DOF magnetic levitation apparatus. Model B has a

similar hyperbolic expression as in (4.2.6) but one difference is that parameter a is

found experimentally rather than being calculated using the relationship in (4.2.7).

This model can therefore be entirely derived using experimental results.

The motivation for a purely experimental model is due to some concern on the

accuracy of the model developed in Chapter 2 (see Section 2.4). Namely the main

assumptions when modelling such an electromechanical system are that the air gaps

be small and the permeability of the cores be much larger then the permeability of air.


Under these conditions it is fair to assume that fringing can be neglected. Fringing

is illustrated in Appendix A.1. These assumptions are used in [1] and even in many

of the problems illustrated in [13], except when an actual magnetic levitation device

is being considered and the dynamics are instead found experimentally to take into

account fringing and other uncertainties. Various examples in [13] assume that the

air gap, z, is much smaller then the width of the core, w, and the height of the core,

h, thus z w and z h, and further consider the permeability of the cores to be

infinite, µ = ∞.

In our case, the core used has a width and height of 5 cm and the disk is usually

at least z > 1 cm, so at best the air gap is only smaller then the magnet width or

height, z < w or z < h, and not much smaller. The cores have a relative permeability

between 27 and 32, thus µ is far from being infinite. The material of our cores have a

low relative permeability compared to other materials often used in electromagnets.

Pure iron has a relative permeability of at least 700 and iron-steel has at least about

200. In conclusion, both major modelling assumptions do not hold. Physically, low

permeability and large air gaps promote excess fringing.

Another concern is the effect of mutual inductance. Mutual inductance between

electromagnets is present when the changing flux of one magnet induces a voltage

in the coils of another magnet. This cross-coupling behaviour could potentially be a

source of major uncertainty in the dynamics.

The inductance of the disk as it moves from the face of the electromagnet, z = 0,

to z = ∞, can be described by the following hyperbolic function

L(z) = L1 +L0

1 + za

, (4.2.11)

where L0 + L1 is the inductance of the magnet when the disk is at z = 0, L1 is the


electromagnet inductance when the disk is at z = ∞, or in practical terms is distant,

and a > 0 is the factor that affects the rate at which the inductance decreases as

the disk moves away from the magnet. By determining the parameters L0, L1, and

a experimentally, one obtains a model of the system that can be compared to model

A which is based on theoretical principles. To experimentally find L0, L1, and a,

we measure the inductance for different positions of the disk and fit the function in

(4.2.11) to the resulting data points.

In model A the inductance of the core decreases to 0 as the disk moves away,

that is L(∞) = 0, while in model B (and in practice) this is not the case. This is

the first difference between the models. The second difference is that in model A

the inductance of the magnet is measured once when the air gap is zero, L(0) = L0

and the decay function a is calculated using the expression in (4.2.7) rather than,

as in model B, being found on the basis of a best-fit curve mapped to a series of

experimental data points. So while model B is entirely experimental, model A is not.

4.2.2 Measuring Inductance

In model A, the inductance at z = 0 must be measured and in model B the inductance

must be measured for different disk locations. Inductance of the electromagnet for a

given air gap can be measured using two techniques: using reference step signals and

doing analysis in the time-domain, method 1, or using a sinusoidal reference and

doing analysis in the frequency-domain, method 2. Both measurements methods

are described followed by a discussion of the preferred method in Section 4.2.3.

Method 1

In method 1, a step input is given to the electromagnet and the response is saved on

a computer using WinCon and the DAC that reads the currents in the coils. Thus,


for a given disk location, a step input is given and the response is observed. The

response of the magnet to a step of amplitude A in the time-domain is

y(t) =KmA

R

(

1 − e

(

−RL

(t−to))

)

= c1(

1 − ec2(t−to))

, t ≥ to,

(4.2.12)

where Km is PWM’s amplification, R is the coil resistance, to is the time the reference

is given, L is the inductance, c1 = KmAR

and c2 = −RL. Using a curve fitting optimiza-

tion algorithm we find c1 = KmAR

and c2 = −RL

that best fit the experimental data.

Since Km, A, and R are known, the inductance L can be found. The inductance is

measured four times with step inputs V = 1 . . . 6 V. The mean of the four samples

is the inductance of the electromagnet for a certain current step amplitude when the

disk is at a certain position. Prior to taking the average, the variance of the four

samples is analyzed to omit any inaccurate results from being taken into account.

Method 2

In method 2, the voltage and current in the electromagnet are measured using a

multimeter when using a 60 Hz sinusoidal signal as the input. In the frequency-

domain, the electromagnet frequency response is

I(jω) =1

jωL+RV (jω), (4.2.13)

where I is the current response, V in the input voltage, ω is the frequency of the

input signal, L is the inductance, and R is the coil resistance. The current, voltage,

and resistance are measured and the angular frequency is ω = (2π rad/Hz)(60Hz) =

120π rad. For a certain disk location, the inductance of the electromagnet is then


found by taking the gain of (4.2.13) and solving for L

L =1

ω

√

( |V (ω)||I(ω)|

)2

− R2. (4.2.14)

4.2.3 Modelling Procedure

Four possible procedures are available to find the model parameters - measuring the

inductance using method 1 or 2 to build model A or model B. In this Section, it is

shown that model B cannot be practically developed because of the low permeability

of the cores and the disk. The model parameters can however be described accu-

rately by model A using either inductance measuring method. Even though both

methods yield accurate model A parameters, method 2 is chosen over method 1 and

the reasons are discussed. The results shown in this section were performed on the

magnets when they were positioned in the two-magnet setup shown in Figure 4.18. In

this configuration, the electromagnets are placed on opposite sides of the disk along

a straight line. This two-magnet testbed is a control experiment designed to test

modelling accuracy, discussed later in Section 4.3.1.

Model B Issues

Finding the parameters of model B, L0, L1, a, using method 1 and 2 will now be

discussed. For model B, the inductance of the electromagnets is measured at different

disk positions with respect to each core - beginning at zi = 0 and going towards zi = 2d

for magnet i = 1, 2. This model is entirely derived from a series of experimental runs.

For magnet 1, Figure 4.15 depicts the results using method 1 and Figure 4.16 shows

the results using method 2. Similar results were found using other magnets in the

two-magnet setup and when the magnets were used in the three-magnet configuration.


0 1 2 3 4 5 6 7 889.4

89.6

89.8

90

90.2

90.4

90.6

90.8

91

91.2

91.4Magnet 1

z (cm)

indu

ctan

ce (

mH

)

Figure 4.15: Model B inductance samples using method 1.

0 1 2 3 4 5 6 7 835.2

35.3

35.4

35.5

35.6

35.7

35.8

35.9

36Magnet 1

z (cm)

indu

ctan

ce (

mH

)

Figure 4.16: Model B inductance samples using method 2.


Magnet Method 1 Inductance (mH) Method 2 Inductance (mH)1 91.2 36.32 85.2 48.5

Table 4.4: Inductance measurements of magnets 1 and 2 using the two differentmethods.

Thus, using either method, the hyperbolically decaying function cannot be fitted

to the set of points, or more specifically, the decay parameter a cannot be found. The

changes in inductance with respect to the disk are very small because the magnetic

permeability of the cores and the disk are quite low. The instruments used in method

1, the current sensor on the PWM, and in method 2, the multimeters, do not have

the necessary resolution for measuring the changes in inductance as the disk is moved

from z = 0 to z = 2d. In conclusion, model B using either method 1 or 2 is a discarded

modelling procedure.

Model A using Method 1 and 2

In model A, the inductance when the disk is at z = 0, L0, is measured using either

method 1 or 2 and the hyperbolic decay constant, a, is calculated using a = µ0A1N2

L0.

Table 4.4 lists the inductances of magnets 1 and 2 measured with method 1 and

method 2, again in the two-magnet configuration. Method 2 yields significantly lower

inductances because the measurements are taken at a higher frequency then that of

method 1.

According to method 1, the inductance relationship of magnet 1 and magnet 2

are

L1(z1) =L10

1 + z1

a1

=91.2

1 + z1

a1

mH

L2(z2) =L20

1 + z2

a2

=85.2

1 + z2

a2

mH


where z1 is the air gap with respect to magnet 1, z2 is the air gap with respect to

magnet 2, L10 = L1(0) is the zero air gap inductance of magnet 1, L20 = L2(0) is the

zero air gap inductance of magnet 2, and the decay factors a1 and a2 are

a1 =µ0A1N

21

L10=

(4π × 10−7)(0.05 m)2(419)2

91.2 × 10−3= 6.05 × 10−3 m−1

a2 =µ0A1N

22

L20=

(4π × 10−7)(0.05 m)2(482)2

85.2 × 10−3= 8.57 × 10−3 m−1.

Alternatively, using method 2 the expressions become

L1 =36.3

1 + z1

a2

mH

L2 =48.5

1 + z2

a2

mH,

where

a1 =µ0A1N

21

L10=

(4π × 10−7)(0.05 m)2(419)2

36.3 × 10−3= 1.52 × 10−2 m−1

a2 =µ0A1N

22

L20=

(4π × 10−7)(0.05 m)2(482)2

48.5 × 10−3= 1.50 × 10−2 m−1.

Figure 4.17 compares the inductance expressions from method 1 and 2 for each mag-

net.

The model parameters developed using model A with either method 1 or 2 were

used in a controller designed for the two-magnet system and the disk was successfully

stabilized to the origin. This seems to indicate that model A accurately represents the

magnetic force of each electromagnet, at least when the disk travels within a certain

range. More specifically, using the hyperbolic decay based on the air gap reluctance

principle, a = µ0A1N2

L0, offers satisfactory control performance in two-magnet configu-

ration. Even though each method measures different values of inductance, the reason

why both methods yields successful control results is perhaps that the inductances of


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

20

40

60

80

100Method 1

z (m)

indu

ctan

ce (

mH

)

magnet 1magnet 2

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

10

20

30

40

50Method 2

z (m)

indu

ctan

ce (

mH

)

magnet 1magnet 2

Figure 4.17: Model A inductance expressions using method 1 and 2.

the magnets converge as the disk moves away from the magnets, as shown in Figure

4.17.

Method 2 is chosen over method 1 because the measuring procedure is much

simpler than the lengthy analysis of method 1, and using the parameters from method

2 seems to yield better control results. This procedure is used to find the model

parameters in the various testbeds discussed in Section 4.3. In particular, Section

4.3.1 shows the actual control results when using the model parameters developed

with this procedure on the two-magnet system. The electromagnets used to achieve

the results in the two-magnet system of Section 4.3.1, however, are not the same

electromagnets used for the results just shown in this section. As a result, the model

parameters found in Section 4.3.1 are different than those in Table 4.4.

4.3. ANALYSIS 109

4.3 Analysis

The controllers designed for the planar magnetic levitation system could not be tested

due to hardware limitations that are discussed later. The exact problem in the 2 DOF

case was isolated using two smaller experiments. While these experiments were used

to troubleshoot the difficulties present in the planar magnetic levitation configuration,

the results obtained are significant and deserve close attention.

In Section 4.3.1, a two-magnet system that moves the disk along the x axis is

considered. The dynamics of this one-dimensional device are derived and the design

of a nonlinear control is summarized. Using the model parameters developed with

the modelling procedure discussed in Section 4.2, it is shown that the disk can be

stabilized with good performance. Thus, the nonlinear control result validates the

modelling procedure for this particular setup. Further, the control results using the

nonlinear controller are compared to that of a linear controller when using identical

model parameters.

The control results in Section 4.3.1 show that accurate model parameters can be

found for a one-dimensional system when the disk is aligned with the axes of the

magnets. In Section 4.3.2, a three-magnet testbed is used to show that the disk can

also be stabilized using a controller when the disk goes off the axes of the magnets.

The experiment is designed to test the two-dimensional model and, in particular, the

notion of air gaps when the disk is not aligned with a magnet. The disk however,

is still constrained to move only in the x direction so this is simpler than the full

three-magnet setup. The dynamics of this three-magnet system are given followed by

a nonlinear control design that enables the controller developed for the two-magnet

system to be used in the three-magnet system. The section ends with some results.

Lastly, Section 4.3.3 deals with the full three-magnet system using the controller

4.3. ANALYSIS 110

developed in Section 3.1. The dynamics of this system using the modelling procedure

in Section 4.2 and the 2 DOF ideal nonlinear controller are summarized. Then, the

problem encountered in the planar magnetic levitation device is discussed along with

results illustrating its behaviour.

4.3.1 Two-Magnet System

Figure 4.18: Two-magnet system experimental setup.

The experimental setup of the two-magnet system is shown in Figure 4.18. This

experimental testbed eliminates the asymmetry of the x and y guides and allows the

focus to be on the modelling of the magnetic forces. Thus the main motivation of

the two magnet system is to eliminate the y guide that had noticeably more friction

and focus on the one-dimensional problem of validating that the procedure discussed

in Section 4.2.3 obtains accurate model parameters. The dynamics of the system as

well as a nonlinear controller are given. Results comparing a linear controller and the

nonlinear controller designed using the model parameters conclude this section.

4.3. ANALYSIS 111

Dynamics

I2I1

O

MAGNET 1 MAGNET 2

Disk

d d

x

Figure 4.19: Two-magnet system diagram.

Figure 4.19 depicts the setup of the two-magnet 1 DOF system. The dynamics of

the system can be represented as

x1 = x2

x2 =−1

m

b1I21

(

1 + x1+da1

)2 − b2I22

(

1 + d−x1

a2

)2

,

(4.3.1)

where m is the mass of the system and d = 2.48 cm is the air gap between the magnets

and the disk when the disk is at the origin. Magnets 3 and 2 in Table 4.1 are magnets

1 and 2, respectively, in Figure 4.19 and the inductances of electromagnets 3 and 2

when the disk is at z1 = 0 and z2 = 0, respectively, are

L10 = 50.4 mH, L20 = 42.4 mH,

4.3. ANALYSIS 112

giving the model parameters

a1 =µ0A1N

23

L10=

(4π × 10−7)(0.05 m)2(503)2

50.4 × 10−3= 1.58 × 10−2 m−1

a2 =µ0A1N

22

L20=

(4π × 10−7)(0.05 m)2(482)2

42.4 × 10−3= 1.72 × 10−2 m−1

b1 =L10

2a1

= 1.59 H/m

b2 =L20

2a2

= 1.23 H/m.

(4.3.2)

This completes the model of the two-magnet system.

Controller

Recall that the three-magnet ideal nonlinear controller constructed in Section 3.1 is

based on the nonlinear controller developed in [10], Section 12.3. In fact, [10] considers

precisely the setup in Figure 4.19. This nonlinear controller is summarized as

I1 =

√

√

√

√

m(

1 + x1+da1

)2

b1ηa

I2 =

√

√

√

√

m(

1 + d−x1

a2

)2

b2ηb

(4.3.3)

where

ηa =−u+

√u2 + ε

2+ b

ηb =u+

√u2 + ε

2+ b,

b ≥ 0 is the bias current and u is a control variable to be defined later. Using the

control I1, I2 in the two-magnet system (4.3.1) yields the linear dynamics

x1 = x2

x2 = u.

4.3. ANALYSIS 113

The origin of the uncertainty-free system is stabilized by closing the feedback loop

using LQR with the weighing matrices

Q =

100 0

0 100

, R = 0.35,

that generates the gain matrix

K =[

16.9031 17.8751

]

. (4.3.4)

Results

Similarly to the simulations in the three-magnet system, the ideal nonlinear stabi-

lizer for the two-magnet system is compared with a linear controller. The system is

linearized about the origin and we obtain a gain

[I21 , I

22 ]> = KLx =

−336.5913 −165.8485

322.4925 158.9016

,

using LQR with the weighing matrices

Q =

500 0

0 500

, R =

1 0

0 1

. (4.3.5)

Figure 4.20 depicts the closed-loop response of the linear controller when the disk

is initialized at −0.51 cm. The steady-state position error in this particular case is

−0.12 cm. The currents from the linear controller in this response are shown in Figure

4.21 and the maximum current is 6.46 A. Another result when the linear controller

is used in the two-magnet system is shown in Figure 4.22. In this case, the disk

starts at 1.05 cm and ends up at 0.52 cm with a maximum peak current of 9.17 A

4.3. ANALYSIS 114

0 2 4 6 8 10 12 14 16 18

−0.5

−0.4

−0.3

−0.2

−0.1

0

Time (s)

x po

sitio

n (c

m)

Figure 4.20: Run 1: Disk position using the linear controller in the two-magnetsystem.

0 2 4 6 8 10 12 14 16 18

0

0.5

1

1.5

2

2.5

3

3.5

Time (s)

Cur

rent

(A

)

I ref 1I coil 1

0 2 4 6 8 10 12 14 16 180

1

2

3

4

5

6

7

Time (s)

Cur

rent

(A

)

I ref 2I coil 2

Figure 4.21: Run 1: Currents of linear controller in the two-magnet system.

4.3. ANALYSIS 115

(the currents are not shown). The steady-state error of the linear controller may be

improved using an integrator anti-windup. This shall be subject to future research.

0 2 4 6 8 10 12 14 16 18

0

0.2

0.4

0.6

0.8

1

Time (s)

x po

sitio

n (c

m)

Figure 4.22: Run 2: Disk position using the linear controller in the two-magnetsystem.

The ideal controller is now tested with the model parameters given in (4.3.2) and

the controller design parameters b = 0, ε = 1× 10−5 and the LQR gain stated earlier

in (4.3.4). Figure 4.23 is the position trajectory under the ideal controller and Figure

4.24 are the currents. The nonlinear control manages to stabilize the disk to −0.02 cm

when the disk begins at −0.81 cm with a maximum current of 8.70 A. In the another

run when the disk’s initial position is 1.99 cm, the ideal controller moves to disk at

0.06 cm from the origin with a peak current of 9.89 A. The currents are appropriately

tracked by the current controller ever though there is considerable chattering.

The ideal controller outperforms the linear controller in terms of steady-state error

even when the initial state is further away from the origin. Further, this experiment

shows that the procedure used to develop the model parameters is sufficient to sta-

bilize the disk with at least 1 mm precision. Many modelling assumptions discussed

earlier and in Chapter 2 (see Section 2.4) are now verified to be negligible with respect

to stabilizing the disk using the nonlinear controller. For instance, it has now been

shown that it is reasonable to ignore fringing and mutual inductance.

4.3. ANALYSIS 116

0 5 10 15

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

Time (s)

x po

sitio

n (c

m)

Figure 4.23: Run 3: Disk position using the ideal controller in the two-magnet system.

0 5 10 150

1

2

3

4

5

Time (s)

Cur

rent

(A

)

I ref 1I coil 1

0 5 10 150

2

4

6

8

10

12

Time (s)

Cur

rent

(A

)

I ref 2I coil 2

Figure 4.24: Run 3: Currents of ideal controller in the two-magnet system.

0 5 10 15 20 25

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time (s)

x po

sitio

n (c

m)

Figure 4.25: Run 4: Disk position using the ideal controller in the two-magnet system.

4.3. ANALYSIS 117

4.3.2 Three-Magnet System using Equivalent Ideal Controller

The previous experiment confirmed that the modelling procedure finalized in Section

4.2.3 is sufficiently accurate, at least when the disk travels along the magnets axes.

However, the two-magnet system is a one-dimensional setup and therefore we cannot

test the effects on the model when the disk does not travel along the axes of the

magnets. That is, the notion of air gaps in the 2 DOF case, described in the equations

(2.2.3), may not hold in the actual system. There is some concern that fringing

between magnets 2 and 3 (see Figure 1.3) would create some difficulties and also that

the disk may have been attracted to the corners of the magnets where the flux is

stronger, rather then being attracted towards their centers. Some of these concerns

were expressed in the modelling uncertainties outlined in Chapter 2 (see Section 2.4).

In this section we develop a control that enables the 1 DOF control system in

Section 4.3.1 to be used in the three-magnet system. Using the vector air gap ex-

pressions (2.2.3) in the two-dimensional case, we find a current that is used in both

magnets 2 and 3 so that they act as a single magnet. That is, by supplying this

current to magnets 2 and 3 they together supply a force that is equivalent to having

a single magnet aligned with magnet 1, as in the two-magnet system. If the control

is successful, then the force from an electromagnet on the disk when it is off the axes

of the magnet is represented correctly by our model.

The first part describes the dynamics of the three-magnet system along with some

simplifications since the disk is constrained in the x direction. In the second part, the

equivalent current control is developed, and the last part shows the results obtained

using this controller on the actual testbed.

4.3. ANALYSIS 118

Dynamics

The dynamics of the three-magnet system in terms of inductance is

x1 = x2

x2 = − 1

m

[

ϕ1(x1, x3)(x1 + d)I21 + ϕ2(x1, x3)

(

x1 −d

2

)

I22 + ϕ3(x1, x3)

(

x1 −d

2

)

I23

]

x3 = x4

x4 = − 1

m

[

ϕ1(x1, x3)(−x3)I21 + ϕ2(x1, x3)

(

x3 +

√3

2d

)

I22 + ϕ3(x1, x3)

(

x3 −√

3

2d

)

I23

]

(4.3.6)

where

ϕ1(x1, x3) =b1

(

1 +

√(x1+d)2+x2

3

a1

)2

1√

(x1 + d)2 + x23

ϕ2(x1, x3) =b2

1 +

s

(

x1− d2

)2

+

(

x3+√

3

2d

)2

a2

2

1√

(

x1 − d2

)2

+(

x3 +√

32d)2

ϕ3(x1, x3) =b3

1 +

s

(

x1− d2

)2

+

(

x3−√

3

2d

)2

a3

2

1√

(

x1 − d2

)2

+(

x3 −√

32d)2.

Note that under the modelling procedure of model B using method 2 detailed in

Section 4.2, the only difference between the dynamics of the three-magnet system

(4.2.1) developed in Chapter 2 are the expressions ϕ1, ϕ2 and ϕ3.

In this configuration, we assume magnets 2 and 3 to be identical, which is realistic

given that the force exerted from either electromagnet when the disk is at a certain

distance is quite close, as shown in Figure 4.17. Also since the disk is restricted to

4.3. ANALYSIS 119

move along the x axis, we consider only the x-subsystem dynamics of (4.3.6)

x1 = x2

x2 =−1

m

b1I21

(

1 + z1

a1

)2

x1 + d

z1+

b2I22

(

1 + z2

a2

)2

x1 − d2

z2+

b3I23

(

1 + z3

a3

)2

x1 − d2

z3

(4.3.7)

where the air gap expressions are

z1 =√

(x1 + d)2 + x23

z2 =

√

(

x1 −d

2

)2

+

(

x3 +

√3

2d

)2

z3 =

√

(

x1 −d

2

)2

+

(

x3 −√

3

2d

)2

,

and using the inductances

L10 = 50 mH, L20 = 40 mH, L30 = L20

we obtain the model parameters

a1 =µ0A1N

23

L10=

(4π × 10−7)(0.05 m)2(503)2

50 × 10−3= 1.59 × 10−2 m−1

a2 =µ0A1N

22

L20=

(4π × 10−7)(0.05 m)2(482)2

40 × 10−3= 1.82 × 10−2 m−1

a3 = a2

b1 =L10

2a1

= 1.57 H/m

b2 =L20

2a2

= 1.10 H/m

b3 = b2.

(4.3.8)

4.3. ANALYSIS 120

Controller

The equivalent current that is supplied to magnets 2 and 3 so they perform as a single

magnet is now derived. Since the disk is only allowed to move along the x axis, there

is no movement in the y direction and therefore x3 := 0. The air gap calculations

become

z1 = x1 + d

z2 = z3 =√

x21 − dx1 + d2,

and the x-subsystem simplifies to

x1 = x2

x2 =−1

m

b1I21

(

1 + z1

a1

)2 +

(

b2I22

(

1 + z2

a2

)2 +b3I

23

(

1 + z3

a3

)2

)

x1 − d2

z2

.

Under the assumption that the properties of magnets 2 and 3 are the same, we can

state that I2 = I3. The x-subsystem becomes

x1 = x2

x2 =−1

m

b1I21

(

1 + z1

a1

)2 +2b2I

22

(

1 + z2

a2

)2

x1 − d2

z2

,

(4.3.9)

and now an expression for I2 can be defined such that the x-subsystem is mapped to

the two-magnet system in (4.3.1), re-written as

x1 = x2

x2 =−1

m

b1I2v1

(

1 + x1+da1

)2 − b2I2v2

(

1 + d−x1

a2

)2

,

(4.3.10)

4.3. ANALYSIS 121

where Iv1, Iv2 are the currents supplied from the ideal nonlinear controller in (4.3.3).

Using the following currents

I1 = Iv1

I2 =1 + z

a2

1 + d−x1

a2

√

z2

2(

d2− x1

)Iv2

I3 = I2,

(4.3.11)

the three-magnet system when constrained in the x axis, described by (4.3.9), and

assuming magnets 2 and 3 are identical, we get the dynamics of the two-magnet

system (4.3.10) in terms of the currents being supplied. Thus, by using (4.3.11),

the same ideal nonlinear controller used in the two-magnet system can be utilized in

the three-magnet configuration. This controller is referred to as the equivalent ideal

nonlinear controller .

Results

The equivalent ideal controller uses the model parameters defined earlier in (4.3.8)

and the same controller parameters as in the 1 DOF case in the three-magnet system

when d = 2 cm. The disk is positioned at a point inside the range where the 2 DOF

controller is valid, defined by the set (3.1.1), in this case being |x| ≤ d6

= 0.33 cm.

Figure 4.26 depicts the response of the 1 DOF controller with the equivalent

current transformation when the disk begins at 0.25 cm off the origin in the positive

x direction. The steady-state error is 0.02 cm. The three currents from the equivalent

ideal controller reach a maximum peak of 8.27 A and are shown in Figure 4.27.

In another run, the disk is initially positioned at −0.26 cm. The controller man-

ages move the disk at a final position of −0.03 cm, shown in Figure 4.28, with a

maximum current of 8.04 A, depicted in Figure 4.29.

4.3. ANALYSIS 122

0 2 4 6 8 10 12 14 16 18 20

0

0.05

0.1

0.15

0.2

0.25

Time (s)

x po

sitio

n (c

m)

Figure 4.26: Run 5: Disk position using the equivalent ideal controller.

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

Time (s)

Cur

rent

(A

)

I ref 1I coil 1

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

Time (s)

Cur

rent

(A

)

I ref 2I coil 2I coil 3

Figure 4.27: Run 5: Currents from the equivalent ideal controller.

0 5 10 15 20 25

−0.25

−0.2

−0.15

−0.1

−0.05

0

Time (s)

x po

sitio

n (c

m)

Figure 4.28: Run 6: Disk position using the equivalent ideal controller.

4.3. ANALYSIS 123

These results verify that the disk can be stabilized with three-magnets even when

the disk goes off the axes of the magnets. The air gap expressions (2.2.3) hold in the

actual system and thus the concerns expressed earlier can be discarded, at least when

the disk remains inside set C defined in (3.1.1).

0 5 10 15 20 25

0

0.5

1

1.5

Time (s)

Cur

rent

(A

)

I ref 1I coil 1

0 5 10 15 20 250

2

4

6

8

Time (s)

Cur

rent

(A

)

I ref 2I coil 2I coil 3

Figure 4.29: Run 6: Currents from the equivalent ideal controller.

Although the controller performs well, it was found that the LQR gain used had

to be quite large. In earlier tests, a smaller gain K than (4.3.4) was used and the

disk remained motionless. This indicates that the Coulomb friction from the linear

guides becomes a significant factor when the magnets are on angles pulling the disk.

Further, the equivalent ideal controller performed much better when the magnets are

closer to the disk, as in the present case when d = 2 cm, rather than when the setup

was using d = 3 cm. These observations indicate the actuators are pushed to their

limit even when the disk is being moved in 1 DOF using two inputs. Regardless of

these problems however, the controller is robust enough to stabilize the disk.

4.3. ANALYSIS 124

4.3.3 Three-Magnet Configuration using 2 DOF Controller

It has been shown that model (4.3.6) represents the actual three-magnet testbed

dynamics with sufficient accuracy in terms of the control results achieved. This

has been shown when the disk is in the C range where the 2 DOF ideal nonlinear

controller is valid, defined in (3.1.1), and when the disk is constrained to move along

the x axis. In this section, we test the ideal nonlinear control in (3.1.10). Due to

hardware difficulties, the robust, adaptive, and robust adaptive controllers developed

in Section 3.2, 3.3 and 3.4 cannot be tested.

The model parameters derived and the control parameters used are first listed.

This section ends with a discussion of the results obtained when using the nonlinear

control in the three-magnet testbed along with the hardware issues encountered that

prevented full testing.

Dynamics

The dynamics for the three-magnet device is already shown in (4.3.6), except the

model parameters are derived from the inductance measurements in Table 4.1

L10 = 50.4 mH, L20 = 42.4 mH, L30 = 40.5 mH

giving

a1 =µ0A1N

23

L10

=(4π × 10−7)(0.05 m)2(503)2

50.4 × 10−3= 1.58 × 10−2 m−1

a2 =µ0A1N

22

L20=

(4π × 10−7)(0.05 m)2(482)2

42.4 × 10−3= 1.72 × 10−2 m−1

a3 =µ0A1N

21

L30=

(4π × 10−7)(0.05 m)2(490)2

40.5 × 10−3= 1.86 × 10−2 m−1

4.3. ANALYSIS 125

b1 =L10

2a1= 1.59 H/m

b2 =L20

2a2= 1.23 H/m

b3 =L30

2a3= 1.09 H/m.

Control

Recall the ideal nonlinear control in (3.1.10),

I21

I22

I23

=

−2mµ0A1

ϕ1(x1,x3)(x1+x3+d)

(

u1−u2−

√(u1−u2)2+εb1

4 − A(x1, x3, u1, u2)

)

−2mµ0A1

ϕ2(x1,x3)

(

x1−x3−

√

3+1

2d

)

(

u1−u2+√

(u1−u2)2+εb1

2 + A(x1, x3, u1, u2) + B(x1, x3, u1, u2)

)

−2mµ0A1

ϕ3(x1,x3)

(

x1−x3+√

3−1

2d

)

(

u1−u2−

√(u1−u2)2+εb1

4 − B(x1, x3, u1, u2)

)

(4.3.12)

where

A(x1, x3, u1, u2) = − 1

fa(x1, x3)

(

fpos(x1, x3, u1, u2) +−u1 +

√

u21 + εb2

2

)

B(x1, x3, u1, u2) = − 1

fb(x1, x3)

(

fneg(x1, x3, u1, u2) +−u1 −

√

u21 + εb2

2

)

,

εb1 = 1 × 10−9, εb2 = 1 × 10−9 and using LQR with the weighing matrices

Q =

100 0

0 100

, R =

0.35 0

0 0.35

, (4.3.13)

we obtain the gain

K =

16.90 17.88 0 0

0 0 16.90 17.88

. (4.3.14)

This finalizes the model parameters and control parameters used in the nonlinear

4.3. ANALYSIS 126

controller on the testbed.

Results

When the disk is initialized in the negative region of the x axis, the controller has no

difficulty stabilizing the disk to the origin. Figure 4.30 shows the evolution of the disk

position when the disk is initialized at −0.28 cm. The steady-state error is −0.03 cm.

The currents for this response are depicted in Figure 4.31 and the maximum current

is 7.76 A.

0 5 10 15 20 25

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

Time (s)

x po

sitio

n (c

m)

Figure 4.30: Run 7: Disk position using the 2 DOF ideal controller.

0 5 10 15 20 250

2

4

6

Time (s)

Cur

rent

(A

)

I ref 1I coil 1

0 5 10 15 20 250

2

4

6

8

Time (s)

Cur

rent

(A

)

I ref 2I coil 2

0 5 10 15 20 250

2

4

6

8

Time (s)

Cur

rent

(A

)

I ref 3I coil 3

Figure 4.31: Run 7: Currents from the 2 DOF ideal controller.

4.3. ANALYSIS 127

When the disk is initialized anywhere in the positive x axis however, the controller

is unable to stabilize the disk. Figure 4.32 shows the response when the disk is

initialized at 0.05 cm and Figure 4.33 depicts the resulting currents.

0 5 10 15 20 25−0.01

0

0.01

0.02

0.03

0.04

0.05

Time (s)

x po

sitio

n (c

m)

Figure 4.32: Run 8: Disk position using the 2 DOF ideal controller.

0 5 10 15 20 250

1

2

3

4

Time (s)

Cur

rent

(A

)

I ref 1I coil 1

0 5 10 15 20 250

1

2

3

Time (s)

Cur

rent

(A

)

I ref 2I coil 2

0 5 10 15 20 250

1

2

3

Time (s)

Cur

rent

(A

)

I ref 3I coil 3

Figure 4.33: Run 8: Currents from the 2 DOF ideal controller.

The problem is the 2 DOF ideal nonlinear controller is unable to compensate

for the Coulomb friction of the guides without saturating the actuators. In more

practical terms, consider the 1 DOF controller. It could supply a relatively large

difference between the currents I1 and I2 to move the disk. However, this kind of

discrepancy is only attainable in the 2 DOF ideal nonlinear control by raising all the

4.3. ANALYSIS 128

three currents. That is, because the 2 DOF ideal controller is coupled, it cannot raise

a current without raising the overall bias of all the currents. For example, I1 cannot

be increased by 2 A without increasing currents I2 and I3 as well. As a result, the

control system cannot establish the necessary current discrepancy without increasing

the overall currents above the capability of the power amplifiers.

0 5 10 15 20 25

−2

0

2

4

6

8

10x 10

−3

Time (s)

x po

sitio

n (c

m)

Figure 4.34: Response of disk from acceleration test.

0 5 10 15 20 250

2

4

6

Time (s)

Cur

rent

(A

)

I ref 1I coil 1

0 5 10 15 20 250

2

4

Time (s)

Cur

rent

(A

)

I ref 2I coil 2

0 5 10 15 20 250

2

4

Time (s)

Cur

rent

(A

)

I ref 3I coil 3

Figure 4.35: Currents from the acceleration test.

For further clarification consider the following test. Instead of trying to stabilize

the disk to the origin, another method of testing whether the ideal nonlinear controller

4.3. ANALYSIS 129

operates is to manually assign an acceleration to the degrees of freedom left available

from the transformation (4.3.12), as shown in the linear dynamics yielded in (3.1.2)

x1 = x2

x2 = u1

x3 = x4

x4 = u2.

Thus the test is to observe the behaviour of the disk when assigning an acceleration

manually to u1, u2 in (4.3.12). In this case, an acceleration in the negative x direction

is set, u1 = −1.2 cm s−2 and u2 = 0, and the disk should at least move in that

direction. However, as shown in Figure 4.34 the disk remains motionless even when

this is almost the largest acceleration that can be assigned without exceeding the

limits of the power amplifiers. The pulse-width modulators can supply a maximum

continuous current of +/- 6 A and the currents supplied to attain this acceleration

in an uncertainty-free system are depicted in Figure 4.35. This type of behaviour is

not seen in either of the testbeds described earlier for an acceleration this large.

Chapter 5

Conclusion

We developed nonlinear controllers stabilizing the model of a planar magnetic levita-

tion device and discussed their implementation. The motivation for the construction

of an experimental testbed was to investigate the limitations of the controllers pre-

sented here. However, due to saturation in the actuators, the testing of the various

nonlinear controllers could not be realized. Thus future research will focus on improv-

ing the hardware of the device. Specifically, larger power amplifiers with individual

power supplies and better electromagnets are required. It is suggested that com-

mercial high-permeability bar magnets be used with a relatively large cross-sectional

area. Lower currents can then be used to actuate movements in the disk. These two

recommendations are probably sufficient for testing the nonlinear control system.

For greater performance, however, linear guides should not be used to suspend the

disk. The disk could be decoupled mechanically by using a magnet mounted vertically

that levitates the disk, effectively eliminating the Coulomb friction from the guides.

After all, one of the main motivations for these magnetic levitation systems is to

eliminate friction. In this setup however, another position sensing solution will have

to be developed. Once the device is rebuilt the nonlinear controllers designed can be

130

CHAPTER 5. CONCLUSION 131

tested.

Future directions in the control design would be enlarging the domain where the

control is well-defined, investigating constrained control techniques, and tracking. In

the simulations it is shown that the trajectories go outside set C, defined in (3.1.1),

and the control remains defined, implying that the domain where the control holds

is quite conservative. Research on ways to increase the size of this domain would be

an interesting direction. For instance, the region where the control holds could be

made dynamic and dependent on the current position of the disk. Once the range of

guaranteed operation is enlarged, constrained control should be adopted to restrict

the trajectories inside that set to prevent the possibility of the control becoming un-

defined. Tracking while still compensating for uncertainties would be another future

direction in the control design. Further, with a fully-operational testbed, all these

control notions can be used on the actual system.

Appendix A

Model Configuration Analysis

A.1 Electromagnetics Background

The forces in the x and y direction acting on the disk, Fx and Fy respectively, from

the electromagnets are difficult to model once fringing becomes a factor. As depicted

in Figure A.1, fringing happens when the magnetic flux density streamlines, ~B, bend

in an air gap between two relatively high permeability cores.

µ µ

µo

Figure A.1: Fringing between two magnets.

In order to derive a mathematical model, we need to verify whether superposition

holds, i.e., whether the sum of the forces produced by each individual electromagnet

equals the force generated when all three electromagnets are activated at the same

time. If superposition holds, the force equations of the disk for all three magnets

132

A.2. SUPERPOSITION ANALYSIS 133

can be constructed from the force model of the disk and one electromagnet, and thus

modelling is greatly simplified.

A.2 Superposition Analysis

The geometry of the model and the current used in the arrangement are factors

that affect how the magnetic field lines interact with the disk. The goal is to find

the simplest implementable electromagnet configuration that satisfies the superposition

condition without saturating the actuator or the coils overheating.

Simulations using FEMLAB were performed to test whether superposition holds

for different geometries and currents. FEMLAB is a finite element analysis tool that

enables magnetic fields to be calculated numerically. This tool is used to verify the

principle of superposition in the three-magnet configuration. Let the force created by

each magnet 1, 2 and 3 corresponding to Figure 1.3 be F1, F2 and F3. Superposition

exists if the following holds

Fall = F1 + F2 + F3. (A.2.1)

Some geometrical features that were considered included

• Varying the magnets’ width

• Changing between paramagnetic and ferromagnetic materials for the magnets

and the disk

• Adding a core that connects all the electromagnets together

Fringing is minimized by using high-permeability cores and a small air gap. More

specifically, the air gap between the disk and the core, z, should be much less the


width of the core, w, and the height of the core, h [13]. Narrow magnets result in

a small width to air gap ratio, wz, and therefore allows for much fringing. Figure

A.2 demonstrates this model. The arrows on the plane represents the magnetic flux

density. Because of excess fringing, it was found through simulation that the condition

for superposition is not satisfied for narrow magnets.

Figure A.2: Magnetic flux density plot for narrow magnets.

Making the magnets wider to increase the wz

relationship minimizes fringing and

results in condition (A.2.1) being satisfied. See Figure A.3 for the magnetic flux

density line plot of the model.

The potential problem of the previous case is the area where superposition holds

may not be large. This would minimize the distance the disk can travel and would

therefore restrict testing of the controller. Although no simulations were ran to verify

this, increasing the width of the magnets should expand the region where super-

positon holds. Figure A.4 shows this arrangement. The results from each of these


Figure A.3: Magnetic flux density plot for wide magnets.

configurations is shown later.

The material of the magnets and disk was chosen to be ferromagnetic because it

is the most widely used material for electromagnets. In particular, iron was chosen in

the simulations. Using a ferromagnetic material for the core of a magnet wrapped in

coils generates a magnetic field whose intensity is unmatched by other conventional

paramagnetic materials. Using a permanent magnet as the core strengthens the field

generated but complicates the model without improving superposition.

The final geometrical feature investigated entails adding a ferromagnetic core that

connected all the electromagnets together. This strengthens the magnetic fields pro-

duced by each magnet but complicates the model. Superposition with a reasonable

current can be achieved without the core.

Current, although not a geometrical feature, is another element of the model that

was considered. It is desired to keep current at a minimum to prevent the magnet’s


Figure A.4: Magnetic flux density plot for wider sized magnets.

windings from overheating. Magnet wire is able to sustain a certain amount of current

and exceeding the limit causes the wire’s insulation coating to melt. Contacting wires

or wires touching the core may cause short circuits that would have adversely affect

the strength of the electromagnet (ie. the magnet could lose strength). Small currents

are also desired to avoid saturating the actuators. The actuators have a limit on the

amount of current they can supply. Thus it is required that the electromagnet is able

to exert a substantial force on the disk with a current that does not go beyond the

actuator limit.

Figure A.4 illustrates the most suitable model. The current passing through each

of the coils is I0 = 100N

Amps, where N is the number of windings. The current density


is given by

Js =Io

Area

=100 A

(

0.01 × 0.1m0.6

)

·(

0.5 × 0.1m0.6

)

0.01 × 0.01m

0.6

= 720000 A/m2.

The model scale in FEMLAB does not have any units. Therefore, it is necessary

to convert the distances from the model into a unit scale, such as SI, to calculate

the current density. The conversion factor is 0.1m0.6

and is based on the length of the

magnets being 10 cm.

The force values and how close they are to superposition was analyzed for the

three arrangement variations described earlier - narrow magnets, wide magnets and

wider sized magnets. The vector sum of the forces from each magnet F1, F2 and F3

is compared to the force acting on the plate when all magnets are on, Fall. In each

case, both the magnitude and the direction difference between these two vectors is

analyzed. Superposition holds when the amplitude and direction discrepancies are

small. The results are as follows

Case 1) Using Narrow Magnets

F1 + F2 + F3 (N) Fall (N) Magnitude Diff. (%) Direction Diff. (o)

0.4678 × 10−4 −0.2368 × 10−5 2.0750 × 103 163.5669

0.3908 × 10−4 −0.3553 × 10−5 1200

Case 2) Using Wide Magnets



−0.9355 × 10−4 0 0.0000 0.0000

−0.7283 × 10−4 0 0.0000

Case 3) Using Wider Magnets


0.4619 × 10−4 0 0.0000 0.0000

−0.5684 × 10−4 0 0.0000

In conclusion, superposition holds in case 2 and case 3. Although not tested, the

wider magnet model probably results in a larger region where superposition holds.

Therefore, case 3 is chosen as the best configuration for the implementation. However,

the force exerted on the disk by the magnets becomes smaller as the width of magnets

is increased. Thus, the size of the superposition region may have to be sacrificed so

that low currents can provide a sufficient force to move the disk without saturating

any of the cores or going over the magnet wire’s current threshold. The dimensions

of the cores chosen for implementation will be explained in Chapter 4.

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nonlinear control of a planar magnetic … · list of figures list of figures 3.12 case 2 response...

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