DEPARTMENT OF MECHANICAL ENGINEERING & MECHANICS

COLLEGE OF ENGINEERING & TECHNOLOGY

OLD DOMINION UNIVERSITY

NORFOLK, VIRGINIA 23529

DYNAMICS AND CONTROL OF A FIVE DEGREE-OF-FREEDOM

MAGNETIC SUSPENSION SYSTEM

By

Anwar Mohammed Haj, Graduate Research Assistant

Principal Investigator: Colin P. Britcher

Progress Report

For the period November I, 1991 to April 30, 1992

Prepared for

National Aeronautics and Space Administration

Langley Research Center

Hampton, VA 23665

Under

Research Grant NAG-I-1056

Nelson J. Groom, Technical Monitor

GCD-Spacecraft Controls Branch

Submitted by the

Old Dominion University Research Foundation

P.O. Box 6369

Norfolk, Virginia 23508-0369

June 1992
DYNAMICS AND CONTROL OF A FIVE DEGREE-OF-FREEDOM

MAGNETIC SUSPENSION SYSTEM

ABSTRACT

By

Anwar Mohammed Haj 1

Principal Investigator:. Colin P. Britcher 2

A large-gap magnetic suspension system with five degrees-of-freedom is presented. The

system is multi-input multi-output with coupling between degrees-of-freedom. Simulation was

performed on this multi degree-of-freedom system in order to control each degree-of-freedom

separately. Two types of controllers are considered by adding white noise to a single degree-offreedom system in order to test their behavior and determine which is the best choice for the

system. The responses of the system are produced in continuous and discrete time where a

sample interval and delay time was introduced. Using these responses, a comparison between

each degree-of-freedom was made and the maximum value of the delay time was determined.

1Graduate Research Assistant, Department of Mechanical Eng/neering and Mechanics,

Old Domin/on Un/versity, Norfolk, VA 23529.

2Assistant Professor, Department of Mechan/cal Engineering and Mechan/cs, Old

Dominion University, NorfoLk, Va 23529.
ACKNOWLEDGEMENTS

This is a thesis being submitted in lieu of a progress report for the research

project entitled "Large Angle Magnetic Suspension Test Fixture" for the period

November 1, 1991 through April 30, 1992. This work was partially supported by the

NASA Langley Research Center through research grant NAG-l-1056 and monitored by

Nelson J. Groom, of the GCD, Spacecraft Controls Branch, NASA Langley Research

Center, Mail Stop 161.

ii
TABLE OF CONTENTS

Page

ABSTRACT .................................................. i

ACKNOWLEDGEMENTS ........................................ ii

TABLE OF CONTENTS ......................................... iii

LIST OF SYMBOLS ............................................ v

LIST OF FIGURES ........................................... viii

LIST OF TABLES ............................................. x

NOMENCLATURE ............................................ x

Chapter:

1. INTRODUCTION .......................................... 1

2. BACKGROUND ........................................... 6

2. I Magnetic Suspension and Balance Systems ................. 6

2.2 The Large Gap Magnetic Suspension System ............... 7

3. GOVERNING EQUATIONS ................................... 9

3.1 Single Degree of Freedom ........................... 9

3.2 Five Degrees of Freedom .......................... 11

3.2. I Equations of Motion ....................... 11

3.2.2 Magnetic Forces and Torques ................. 14

3.2.3 System Equations ........................ 16

4. SINGLE DEGREE OF FREEDOM SIMULATION .................... 21

4.1 System Selection and Transfer Functions ................. 21

4.2 Step Responses ................................. 28

iii
5. COMPARISON OF CONTROLLERS ............................ 31

5.1 White Noise .................................. 31

5.2 Dual Phase Advance Controller ....................... 33

5.3 PD Controller .................................. 35

5.4 Comparison of Results ............................ 37

6. FIVE DEGREE OF FREEDOM SIMULATION ...................... 38

6.1 The System ................................... 38

6.2 Power Supplies ................................. 39

6.3 The Plant ..................................... 44

6.4 Controllers .................................... 44

6.5 Continuous-Time Step Respomes ...................... 45

6.6 Discrete-Time Step Responses ........................ 48

7. DISCUSSION ........................................... 52

8. CONCLUSIONS .......................................... 54

REFERENCES .............................................. 56

APPENDICES ............................................... 58

A. Matrices of the System .................................. 58

B. M Files ........................................... 66

C. Program ........................................... 74

iv
LIST OF SYMBOLS

a,b

B

C

d

dt

f

F

F^

F,

Fv

g

fI

i

I

K

K8

Ko

constants

fluxdensityff_la)

damping coefficient

derivative with respect to time

external input force

force (N)

attraction force

gravitational force

damping force

force on core (model) in core's coordinates

electromagnet forces

external disturbance forces

acceleration due to gravity (m/s 2)

momentum of the model

current (Amps)

equilibrium current

S-domain small change in current

product of inertia

gain

matrix representing the values of B

coil constant

linearization constant
IL

L

m

rn_

M

R

S

t

Tm

L

U

V

Vo

V D

AV

V,V'

X

AX

Y

8

linearization constant

inductance(H)

mass(kg)

the core (model) mass

core (model) magnetization (A/m)

resistance (fl)

laplace variable

time (seconds)

total torque about core (model) center of mass

electromagnets torque

external disturbance torques

input

voltage

core (model)velocity

equilibriumvoltage

power supply input

S-domain smallchange involtage

outputfrom controller

position,separationdistance(m)

firstderivativeofposition,velocity

second derivativeof position,acceleration

S-domain smallchange inposition

state space output

small variation in parameter

vi
P

I'

a

Ox

a

#i

variance

angular velocity (rad/sec)

permeability of material in free space, equal 4_" x 10-7 (Hertrys/m)

standard deviation

partial derivative with respect to x

partial derivative with respect to i

vii
LIST OF FIGURES

Figure

I

2

3.2.1

4.1.1

4.1.2

4.1.3

4.1.4

4.3.1

4.3.2

5.1.1

5.1.2

5.2.1

5.2.2

5.3.1

5.3.2

6.1.1

6.2.1

6.2.2

6.2.3

6.2.4

6.2.5

Page

Magnetic Suspension controller arrangement .......................... 8

Schematic of five coil Large-Gap Magnetic Suspension System .............. 8

Schematic of single degree-of-freedom Magnetic Suspension System ......... 10

Bode plot of the state-space matrices with the yaw mode ................. 23

Bode plot of the transfer function of the yaw mode .................... 23

Root-locus with the Dual Phase Advance controller .................... 27

Root-locus with the Dual Phase Advance controller ................... 28

Step response curve with Dual Phase Advance controller ................ 30

Step response curve with PD controller ........................... 30

Magnetic suspension controller arrangement with noise ................. 31

White noise power spectral density plot ........................... 32

Power spectral density with Dual Phase Advance controller ............... 34

Power spectral density with Dual Phase Advance controller (shorter freq.) ..... 35

Power spectral density plot with PD controller ....................... 36

Power spectral density plot with PD controller (shorter freq.) ............. 36

The five degree-of-freedom system block diagram ..................... 38

Power supply block diagram .................................. 40

Power supply step response, "Pitch" degree-of-freedom ................. 41

Power supply step response of yaw as the degree-of-freedom .............. 42

Power supply step response of X degree-of-freedom ................... 42

Power supply step response of Y degree-of-freedom ................... 43

.°°

VIII
Figure Page

6.2.6

6.4.1

6.5.1

6.5.2

6.5.3

6.5.4

6.5.5

6.6.1

6.6.2

6.6.3

6.6.4

6.6.5

Power supply step response of Z degree-of-freedom ................... 43

Dual Phase Advance controller step response ........................ 45

Continuous-time step response, "Pitch" degree-of-freedom ............... 46

Continuous-time step response of Yaw degree-of-freedom ................ 46

Continuous-time step response of the axial degree-of-freedom ............. 47

Continuous-time step response of the lateral degree-of-freedom ............ 47

Continuous-time step response of the vertical degree-of-freedom ........... 48

Discrete-time step response, "Pitch" degree-of-freedom ................. 49

Discrete-time step response of Yaw degree-of-freedom .................. 50

Discrete-time step response of the axial degree-of-freedom ............... 50

Discrete-time step response of the lateral degree-of-freedom .............. 51

Discrete-time step response of the vertical degree-of-freedom .............. 51

ix
LIST OF TABLES

Table

1.I

4.1

6.2

A.1

B.1

B.2

Page

Wind tunnel Magnetic Suspension and Balance Systems ................... 5

Eigen-values and Eigen-vectors of the overall system .................... 22

Current values of the five degree-of-freedom system .................... 41

Field gradients contributions values ............................... 51

List of variables of the single degree-of-freedom M files ................. 59

List of variables of the five degree-of-freedom M files ................... 61

DPA

M1T

MSBS

LGMSS

NAL

NASA

ONERA

TsAGI

PD

PID

SISO

UVa

NOMENCLATURE

Dual Phase Advance

Massachusetts Institute of Technology

Magnetic Suspension and Balance System

Large-Gap Magnetic Suspension System

National Aerospace Laboratory, Japan

National Aeronautics and Space Administration, United States

Office National d'Etudes et de Recherches A&ospatiales, France

The Central Aero-Hydrodynamics Institute, Soviet Union

Proportional Derivative

Proportional Integral Derivative

Single Input Single Output

University of Virginia

X
CHAPTER 1

INTRODUCTION

An actively stabilized magnetic suspension system was first used at the University of

Virginia, USA, in 1937 (Ref. 1). Such systems are now being used in a variety of applications

including world-wide investigation for advanced ground transportation schemes and also for

application in contactless bearings for high and very low speeds. Additional applications include

rotors of high speed centrifuges required in the fields of biology and medicine; testing bursting

speeds of spheres such as ball bearings; testing adhesion of metal films; turbomolecular pumps

for high vacuums free of bearings requiring lubricants. The same principle has been used to

suspend aircraft models in wind tunnels which helps the investigation of more subtle aerodynamic

details and improves techniques for studying aero-vehicle stability.

The researchers at the Office National d'Etudes et de Recherches A6rospatiales

(ONERA), France were first to achieve magnetic suspension of models in a wind tunnel in 1957

CRef. 2). The ONERA system controlled models in five degrees-of-freedom in test sections up

to 30 cm in diameter. So far it is believed that about 17 wind tunnel magnetic suspension

systems have been built since then, with six currently in operation (Ref. 3,4). Of these six wind

tunnels, two are at NASA Langley Research Center in the USA. The others are at Oxford

University and the University of Southampton in England, the National Aerospace Laboratory

(NAL) in Japan, and the Central Aero-Hydrodynamics Institute (TsAGI) in the Soviet Union.

Table 1.1 gives the detailed listings of those wind tunnel systems.

Most magnetic suspension and balance systems require the use of controlled dc

electromagnets acting on suspended body composed of a ferromagnetic material. Using this

approach, a feedback controller is required for the stabilization of the position and attitude of the

I
suspended body. Other approaches generally cannot generate high damping, especially with large

air-gaps. The following is a list of these systems (Ref. 14)

1) Levitation using forces of repulsion between permanent magnets.

2)

3)

Stable suspension or

levitation is impossible with a system of permanent magnets (or fixed current

electromagnets) unless part of the system contains either diamagnetic material or a

superconductor. Developments in fabricating permanent magnet material have raised interest

in the idea of using such magnets. The most common application of such magnets is in the

suspension of shafts or spindles.

Levitation using forces of repulsion between diamagnetic materials. Levitation can be

achieved in static magnetic fields by employing diamagnetic material. However, even the

two materials which exhibit the most pronounced diamagnetic properties, i.e. bismuth and

graphite, are so weakly diamagnetic that only small pieces can be levitated.

Levitation using superconducting magnets. The superconducting state is indicated by the

complete absence of electrical resistance, and once initiated a current will continue to flow

without the presence of a voltage source in the circuit. One of the primary applications is

an electrodynamically levitated vehicle lifted and guided by repulsion forces between

superconducting magnets on the vehicle and secondary circuits on the track (eddy currents

if the track circuits are passive). The stiffness and damping of the suspension are low, and

the vehicle must be in motion in order to generate lift. Therefore, there is a minimum

velocity which must be exceeded before the vehicle becomes levitated. Some of the

problems and drawbacks that remain unsolved are that in addition to the aerodynamic drag

on such vehicles there is an eddy current drag which is large at low speeds. More recently,

however, due to developments in linear induction motors, particularly of the transverse flux

type, it has been claimed that such machines might be used for combined levitation and
4)

propulsion of high speed vehicles.

Suspension using a tuned LCR circuit and the electromagnetic force of attraction between

two plates. The variation of the inductance of an electromagnet in the proximity of a

ferromagnetic object, depending on the separation between the two, is utilized in this method

to regulate the current and hence the attraction force. This is achieved by incorporating the

electromagnet within an LCR circuit tuned in such a way that when the object to be

suspended moves away from the electromagnet the circuit tends to become resonant, thus

increasing the current and hence the force acting on it. The main disadvantages stem from

the fact that at the equilibrium point the circuit is predominantly inductive and hence the

reactive power input is rather large. Also the iron structure including the object to be

suspended must be laminated to reduce eddy current losses.

There is a wide range of applications associated with large-gap magnetic suspension

technology, including microgravity and vibration isolation systems, magnetically suspended

pointing mounts, large-angle magnetic suspension systems for advanced actuators, wind tunnel

magnetic suspension systems, and remote manipulation/control/positioning of objects in space.

The Large-Gap Magnetic Suspension System (LGMSS) is a hypothetical design for a groundbased experiment that could be used to investigate the technology issues associated with magnetic

suspension systems at large gaps, accurate suspended element control at large gaps, and accurate

position sensing at large gaps.

The uncontrolled and unknown aerodynamic loads on model in wind tunnel have to be

magnetically opposed by the MSBS while the absence of these loads in the LGMSS causes a

reduction in the magnetic field intensity and the dynamic force and moment requirements. A

model in a MSBS is never stationary, since the systems are open-loop unstable and rely on
positionand attitude error feedback for stabilization.

The two controllers that are most commonly used with the magnetic suspension systems are

the phase-advance and proportional-integral-derivative (PID) controllers.

This report will concentrate on the development of a simulation of the five degree-offreedom Large-Gap Magnetic Suspension System. The behavior of the system is observed when

each degree-of-freedom is activated and the coupling of certain degrees-of-freedom is observed.

For this simulation, continuous-time step responses and discrete-time step responses including the

effect of time delay were used to monitor the behavior of the system.

This report will also examine the effect of random noise when added to the single-degree-offreedom system with either the Phase Advance controller or the PID controller to indicate which

controller is best suited for this system.
Organization

NAL

NASA Langley

NASA Langley

Oxford University

TsAGI

University of Southampton

Degrees of

Freedom

5

5

5/6

3

5

516

Size, cm

10 x 10

26.7 x 31.8

15 oct.

12 x 12

40x60

18 oct.

Controller

digital

digital

analog

analog

analog

digital

Table 1.1 Wind Tunnel Magnetic Suspension and Balance Systems.
CHAPTER 2

BACKGROUND

2.1 Magnetic Suspension and Balance System (MSBS)

Until recently, the development of multi-degree-of-freedom magnetic suspension systems has

been hindered by the limited flexibility of control systems. Magnetic suspension devices of the

type under consideration can be made stable with a feedback control system (Fig. 2.1). In order

to achieve stable suspension it is necessary to devise means of regulating the current in one or

more electromagnets using position feedback of the object to be suspended. Further, it can be

shown that feedback of some form of rate information is necessary. In magnetic suspension

applications this cannot usually be achieved by direct sensing. Instead, rate information is

synthesized in the controller by differentiation of the position data (PID controller) or by a DPA

controller (usually referred to as "lead"). Assuming that the suspended object position is the

controlled variable, the "controller" can be placed ahead of a position demand input (in the

feedback path, H(s) in or after this input (feedforward path, G(s)) (Fig. 1). Closed loop stability

is the same in both cases, whereas the response to demanded changes in object position is quite

different. Position error integrators are often added and located in the forward path.

Two important variables of relevance to the digital controller are the sample interval and the

time delay. These quantities are not necessarily equal but will be regarded as such in analysis.

Complex multi-degree-of-freedom systems tend to force these two quantities to be roughly equal

since all the degrees-of-freedom are sampled together to produce input to the controller. Once

all the degrees-of-freedom are processed and command output, the sampling procedure starts

again without allowing any further delay time.

6
2.2 Large-Gap Magnetic Suspension System 0LGMSS)

The Large-Gap Magnetic Suspension System is composed of five electromagnets and a

suspended ferromagnetic object. The suspended ferromagnetic object contains a core which is

a cylinder composed of permanent magnet material, and the magnetic actuators are air core

electromagnets mounted in a planar array. The core is suspended or levitated by repulsive forces

produced by the five electromagnets which are mounted horizontally. The permanent magnet

material provides the needed magnetic field which interacts with the fields of the suspension

electromagnets to produce suspension and positioning forces and torques.

The LGMSS, which has an air-gap of about 1 meter, provides five degree-of-freedom control

where the suspended element is a cylinder composed of permanent magnet material. In the

LGMSS (Fig. 2), the model's own weight is the principal force acting on the levitated model, in

addition to some other loads that may be anticipated. The model's spatial position and orientation

are to be maintained to high accuracy, with very small allowable undemanded motion around any

particular location. However, the position and attitude must be variable under operator control,

preferably over a wide range. The rate of movement from one position or attitude to another

may be slow.

The uncontrolled degree of freedom is rotation about the long axis of the suspended cylinder.

The analytical model consists of an open-loop representation of the suspension system with

electromagnet currents as inputs and displacements and rates in inertial coordinates as outputs.

The proper description of the power supply is actually a power amplifier, since controlled

currents must be supplied to the load electromagnet by varying the amplifier output voltage.
kF

O

mg

Figure 1

\

L,R v,i ___ <__

0 Fx- H(s)

Magnetic suspension controller arrangement.

X o

>-

X

Figure 2

Z

!

O

Schematic of five coil Large-Gap Magnetic Suspension System.

Y

8
CHAPTER 3

GOVERNING EQUATIONS

3.1 Single Degree-of-Freedom System (MSBS)

Equations for a one degree-of-freedom magnetic suspension system (Ref. 6) are derived

starting with Newton's second law of motion. The resulting equations are converted to the

S-domain using Laplace transforms and then the state-space form of these equations is obtained.

The equation of motion for the body is

mg=F s- F_(x,i) - Fo+ f (3.1.1)

where F s is the weight of the body, F^ is the magnetic force exerted on the body by the

electromagnet, FD is the damping force acting on the body, and f is an external force disturbance

see (Fig. 3.2.1). The linearized equation of motion for the suspended body is

m 8J_(t) =K, dix(t) -K_Si(t) -C6$(t) +f (3.1.2)

where

and

K ,, -_x(FA) (3.1.3)

K_ ,, _/(FA) (3.1.4)

9
i

©

Power

©

A

X

FA

Fn

()

f

Figure 3.1.1 Schematic of Single Degree-of-Freedom

Magnetic Suspension and Balance System.

The linearized governing equation of the electromagnetic coil is the sum of the voltage

drop across the coil resistance and the voltage across the inductance (Ref. 6).

$V(t) =Si(t)R +L d [5i(t)] +K_X(t) (3.1.5)

where

i d

K ,, o_(L) (3.1.6)

Transforming equations 3.1.2 and 3.1.5 to the Laplacian S-domain, they become

ms_ax = g ax - K,at - CSaX + f

(3.1.7)

10
AV = RAI + LSAI + KSAX (3.1.8)

Combining equation 3.1.7 and 3.1.8, the Plant Transfer Function (Ref. 6) becomes

AX=

_ KtAv + R I+L ]

mL m"-L _ S

I

S 3 + S 2

f

•4" _ + S _ - m - -

m mL m mL Lm

(3.1.9)

Due to the distance of the model from the electromagnets, I_ is very small, therefore it is equal

to zero, and since the aerodynamic forces are negligible, C is very small and it is equal to zero,

therefore, equation 3.1.9 simplifies to

AX- _-_ I+_S f

[ s. T

(3.1.10)

3.2 Five Degree-of-Freedom System

3.2.1 Equations of Motion

Five electromagnets is the minimum number of actuators required since five degrees-offreedom are being controlled. A representation of this system is shown in figure 3.2.1. The

motion of the core (cylinder) is defined by the body fixed axes x, y, z that define the motion of

the core with respect to fixed inertial axes x, y, z. The electromagnet array is also defined by

fixed inertial axes xb, Yb, zb. The x, y axes are parallel to the xb, Ybaxes; z and zb are coincident.

11
The origins of the two axes are separated by the distance h (Fig. 3.2.1). Deriving the following

equations (Ref. 8), the following assumptions where used: (1) The core is a rigid body, (2) the

core has negligible products of inertia, and (3) there is no motion about the _ axis. _ is an axis

of symmetry so that:

/; = /_ = / (3.2.1.1)

In core coordinates, the momentum of the core is

[h]. (t) [01 (3.2.1.2)

where

and

(I) =

0 0

o /_ o

o o

[fl] = [0 t2_ fl_]

(3.2.1.3)

(3.2.1.4)

The total torque, [7], about the core center of mass is

[TI " dfHl = (I) _dIfi] + (fi] × { (I) [_1 } (3.2.i.5)

dt dt

the total torque can be written as

= +[LJ (3.2.1.6)

where [T.] are the control torques produced by the electromagnets, [T'd are the external

disturbance torques. From equation 3.2.1.2 and with the assumption that fl_ = 0, the cross

12
product term becomes zero and

(3.2.1.7)

The core angular rates are obtained by integrating equation 3.2.1.7.

The forces on the core, in core coordinates are

[F] =m_{ d[_']-dt+ [_] [_']}

where me is the core mass, [lq is the core velocity. [_ can be written as

(3.2.1.8)

[F] ,, [._,1 + [Fd] (3.2.1.9)

where [F.] are the control forces produced by the electromagnets, [F,t] are the disturbance forces.

The core translational rates become

[V] = IT] -1 IV] (3.2.1.10)

[T,,]4 gives a transformation matrix from core coordinates to inertial space. Integrating equation

3.2.1.8, the displacement of core center of mass is obtained. The assumption is made that the

rates will be small and their products can be neglected since the core is actively controlled,

therefore, the result is

13
X,X

Y,Y

Figure 3.2.1 Initial coordinate system alignment for Large-Gap Magnetic Suspension

System.

3.2.2 Magnetic Forom and Torques

In x, y, z coordinate system, the torque on a magnetic core in a nonuniform magnetic

field (see Ref. 8) can be written as

[T] = L {[M]x[BI}d(Vol) (3.2.2.1)

where the integral is taken over the core volume, M is the core magnetization in Amp/meter, B

is the flux density in Tesla, Vol is the volume in cubic meters and the torque T is in Newton14
meters. The force on the core is written as

[FI -- L {[MI .V) [B] d(Vol)

where F is in Newtons and V is the gradient operator that is defined as

(3.2.2.2)

ai+a/+ak (3.2.2.3)

The size of the core is small relative to the electromagnets and the air-gaps, therefore,

assuming that the field gradient components are uniform over the volume of the core, an

approximation of the forces and torques can be obtained, so the torque becomes

[T] - Vol [M] x [B] (3.2.2.4)

where [M] is the magnetization. After some simplification and notation change, the force becomes

where

[F] ,, Vol [aB] (3.2.2.5)

B, z B B, 1

[#B] = [By, B B[ (3.2.2.6)

LB,, B,,.I

Maxwell equations apply in the core region, therefore V x B = 0 which results in: B_y = Br, ,

B_,=B_,,By. =B_, AlsoV.B =0 results in: B.,+Brj + Bn=0

In equations 3.2.2.5 and 3.2.2.6, [if-/] is defined in core coordinates while [B] is in

inertial coordinates. In order to calculate the torque in core coordinates, [B] has to be

15
transformed into the same coordinate system, this results in

it] = rot[,_l [rl x[B]

and the force transformed back into core coordinates

(3.2.2.7)

[F] = Vol [T] [OB] [T] -t [_r] (3.2.2.8)

3.2.3 System Equations

The magnetic forces and torques are combined with the permanent magnet core equations

of motion to produce a model of the magnetic suspension system. The following are the resulting

system equations

d--r- "

Equation 3.2.3.1 and 3.2.3.2 are the equations of motion in core coordinates in terms of [B] and

[aB]. In inertial coordinates [B] can be written as

(3.2.3.3)

16
where [Ks] is a 3 x 5 matrix whose dements represent the values of [B] produced by a

corresponding coil driven by the maximum current, I,_. By arranging the elements of laB] as

a column vector, the gradients can be put in the same form which results in

where [c3B] is a nine element column vector containing the gradients of [B], [Kan] is 9x5 matrix

whose elements represent the values of [aB] produced by a corresponding coil driven by the

maximum current, each element can be written as

where [Kffi] is a 1 x 5 matrix containing values of B= produced by a corresponding coil.

This model is nonlinear because of the combination of states resulting from the coordinate

transformations and has the following form

X = f(x,u) (3.2.3.6)

where x is given by

X r _[O O 0, 0_ V V r V x y z] (3.2.3.7)

and the inputu isgiven by

u r ffi [11 12 13 I, Is] (3.2.3.8)

The equations of motion can be linearized around the nominal operating point Xo, Io by

performing a Taylor Series Expansion. Ignoring second order terms and subtracting out Xo

17
results in

where

(3.2.3.9)

and

(3.2.3.10)

where X has the following form

(3.2.3.11)

The A matrix of Eq. 3.2.3.10 has the following form

(3.2.3.12)

gS

T_k

o,k

m

F,k

_k

v k

v_,

v k

D

r k T;

_k

k,,

(3.2.3.13)

which reduces to the following

18
oo T,. r,.O00 r r T

00 _,. _o.000 _. _, _.

I0 0 0 000 0 0 0

01 0 0 000 0 0 0

o o _,, _o, o o o F _, _,

o o _,, _,ooo F_._, _,

O0 0 0 l O0 0 0 0

00 0 0 010 0 0 0

O0 0 0 001 0 0 0

(3.2.3.14)

where the following notation has been used for simplification

(3.2.3.15)

At initial alignment and equilibrium, F, = Fy = Tr = L = O.

Equation 3.2.3.11 takes the following form

L L,

Oyl, (3.2.3.16)

Terms involving 9 and V are neglected.

19
Talcing the first term of the B matrix

O_ _ (vol)u_ [ -o OB, r_ _ B•CgOy I-7- OB }

(3.2.3.17)

002

The term -- = 0 also, when evaluating Eq. 3.2.3.11 at Xo, the terms with 0y and O, drop 0tl

Out.

The coefficients of the above A and B matrices are given in more detail in Appendix A along

with their numerical values.

20
CHAFFER 4

SINGLE DEGREE OF FREEDOM SEVIULATION

4.1 System Selection and Transfer functions

To assist in determination of the gains of the five degree-of-freedom system an

approximate value needed to be found prior to full simulation. Therefore, a transfer function was

found for a single-input single-output (SISO) system using values from the yaw component of the

complete system. The eigen-values and eigen-vectors that represent the mode shapes and

frequencies are presented in Table 4.1.1. A DPA controller was added to compensate this

unstable system. An approximate gain that can be used to estimate the gains of the remaining

degrees-of-freedom can then be determined. This uncontrolled SISO system has eight poles and

eight zeros that cancel each other out due to the other uncoupled degrees-of-freedom, leaving two

poles that compose the system's transfer function denominator. Using MATLAB, a transfer

function of the yaw degree-of-freedom was obtained from the overall system, setting the

derivatives equal to zero, a constant value equal to .3019 was obtained which is equal to k/c

where k is the gain, therefore .3019 was multiplied by c to obtain k equal to 49.9453. With

these results, the transfer function is second order and takes the following form

o(s) - 49.9453 (4.1.1)

aS 2 + bS + c

where a = 1, b = 0, and c = -165.4233. These values of the yaw degree-of-freedom were

obtained from the system matrix of the overall system shown in Appendix A. The following

bode plots were achieved from the overall state-space matrices for the yaw degree-of-freedom and

the above transfer function which prove that they are identical.

21
Mode 1 2 3

Eigenvalue 13.6918 -13.6918 1.7843i -1.7843i 12.86177 -12.8617

Oz

0y

O_

Vx

Vy

V_

X

Y

Z

44.9957

0

44.0727

0

0.058

0

0

0.0042

0

0

0.9957

0

0.0727

0

0.058

0

0

43.0042

0

0

44.7968

0

0.4466i

0

44.3551

0

0

0.1990i

0

0

44.7968

0

44.4466i

0

44.3551

0

0

44.1990i

0

0

0

0.9970

0

0.0775

0

0

0

0

0

0

0

0.9970

0

-0.0775

0

0

0

0

0

0

Mode 4 5

Eigenvalue 5.0313i -5.0313i 2.5058 -2.5058

o,

o,

V.

Vy

V_

X

Y

Z

0

0

0

0

0

0

0.9808

0

0

44.1949i

0

0

0

0

0

0

0.9808

0

0

44.1949i

0

0

0

0

0

0.9288

0

0

0.3707

0

0

0

0

0

0

0.9288

0

0

0.3707

0

Table 4.1 Eigen-values and Eigen-vectors of the overall Magnetic Suspension System.

Observing table 4.1, the modes are identified as follows: The translational divergence

in the y direction is Mode 5. The undamped vertical motion (mass + spring stiffness type) is

Mode 4. Mode 3 is the "compass needle" term, in this case relating to divergent rotation about

the z axis. Mode 1 arises due to the compass needle term about the y axis, but couples into

translation in the x-direction. Mode 2 is due to the translational divergence in the x-direction,

but couples into pitching moment strong enough to produce an oscillatory mode.

22
-20

-30

-40

-60

-60

-70

-80

-gO

-100

-110

Figure

BOO[ PLOT

'_ _'o _'o , , , ;o , , 1 40 50 80 8o 90 10(3

Ir ellq. "HZ"

4. I. I Bode plot of the yaw degree-of-freedom from the

overall state-space matrices.

-2O

-30

-4.0

-_(

-e(

-70

-80

-gO

-100

-110

Door PLOT

1o' ;o ;o 60' 50' 80' ;o 80' 9

'_

_'ellq. "MI"

Figure 4.1.2 Bode plot of equation 4.1.1.

100

23
The two types of controllers that were used in order to determine which one is better

suited for this system are the DPA and the PID controllers. The DPA transfer function has the

following form

G(S) - (1 + n TS): (4.1.2)

(1 ÷ IS):

Equations 4.1.1 and 4.1.2 are combined to form the following SISO open-loop compensated

transfer function without the power supply.

G(S) - (I + nTS) 2 (4.1.3)

(1 + TS) 2 (aS 2 + bS + c)

The transferfunctionofthe PID takesthe followingform

G(S) = aS2 + bS + c (4.1.4)

S

to obtain PD control, c is set equal to zero. The value of a and b were taken to equal 1 and 18

respectively. If the value of b is increased, the system becomes stiffer. Multiplying (Eq. 4.1.1)

with (Eq. 4.1.4) also produces a compensated open-loop system but with the PD controller

instead.

The values from the coupling matrix (Ref. 4) of the field gradients for the five-degree-offreedom LGMSS were used to calculate the current distribution. It is found that the required B,=

to suspend equal 0.0962 T/m, therefore solving the following equation (Eq. 3.2.3.5)

24
r .. r,

_zl ....

(xx),

K(xy)L ....

f(xz), •

r"

il l

15

,-i

By

B,[

B_x[

By I

.B**J

(4. I. 5)

substituting for By; B,; B=; B_ = 0 and B= = 0.0962, gives the current distribution required as

-0.7753"

-0.2417

0.6293

0.6293

-0.2417

(4.1.6)

Using the equations of the forces and torques (Appendix A), the values of By and B=

were found with the following procedure. In equilibrium, the only force on the core is along the

z axis and is equal to the core weight

F_ ffi mc g

where g is the acceleration of gravity, and from the equations in Appendix A

(4.1.7)

B = mc g (4.1.8)

*: ( Vol) M_

The power supply transfer funaion is

G(S) - 1 (4.1.9)

S + 130

where the power supply pole located at -130.

25
In the Single Degree-of-Freedom transfer function Eq. 4.1.1, the large negative number

in the denominator indicates considerable spring stiffness and shows the presence of instability.

A B_ value of 1.0001 is multiplied by Eq. 4.1.1 and Eq. 4.1.9 to produce the following

uncompensated system transfer function

G(S) = .6293 (4.1.10)

.0077S 3 + S 2 - 1.2738S- 165.42

The value of Byd was obtained by multiplying each value of the second column of the decoupling

matrix by Eq.4.1.9, the five resulting values were multiplied by the values of By for each coil

(Table A. 1), and the sum of the results produced the value of Ra.

The DPA transfer function is

G(S) = "0014S2 + .0741S + 1 (4.1.11)

.0002S 2 + .0074S + 1

Eq. 4.1.10 and Eq. 4.1.11 are multiplied to produce the following open-loop compensated

transfer function with DPA controller

G(S) = 0.637 S 2 +3.5675 S +49.9453 (4.1.12)

9.81 x 10-sS _ +6.77 x 10-5S4 + .015S 3 +.989S 2 -2.4541S-165.42

The following is the PD controller equation

G(S) = S + 18 (4.1.13)

multiplying equation 4.1.10 and 4.1.13, the open-loop compensated transfer function with the PD

controller is

26
G(S) " 12.4863S + 224.754 (4.1.14)

.0077S _ + S_ - 1.2725S - 165.42

Equations 4. i. 12 and 4.1.14 are the sources for calculating the root-locus of the closedloop system in order to determine the approximate gain that produces stability.

The determination of the stability of the system was accomplished by the use of the rootlocus method. The compensated open-loop transfer function had to be derived by Eq. 4.1.1 and

Eq. 4.1.2. In Eq. 4.1.2, the value of n was taken to be equal 10, and the optimal value of T that

produced a root-locus which gives a stable system with sufficient amount of gain is equal to

0.0036. This value of T helped producing a root-locus of a system that becomes stable with

sufficient amount of gain. The location of the single power supply pole was determined by

multiplying the maximum open-loop plant eigenvalue of 13 by 10. After the location of the poles

and zeros of the controller and the power supply were established, the appropriate root-locus was

plotted in (Fig. 4.1.3 and Fig. 4.1.4). To achieve the stability of the system with the desired

damping value, a

50

3(=

20

10

--10

gain of about 7 was chosen.

SlnQle--Oegrllll _ _'_)e(:loer_ ROO_--IocuII

/

-20

-30

-50 i

-250

Figure 4.1.3

l I

' - 1'50 ' ' -200 - 1O0 -50 0

Real Axle

Root-Locus plot of the overall single degree of freedom

system. The value of the gain K = 7.

27
J

!

iS

Slngle-Oegroe ¢_ lreoedom Root-loau|

Real _fo

Figure 4.1.4 Root-locus plot of the overall system with zeros and poles

closest to the origin. The value of the gain K = 7.

4.2 The Step Response

To obtain the step response of the yaw mode, the closed-loop transfer function has to be

evaluated. This closed loop transfer function has the following form

G(S) = KH(S) (4.2.I)

1 + KH(S)

where K is the gain.

Using Eq. 4.2.1, the closed-loop transfer functions of the DPA and PD controller respectively

were obtained and took the following form

G(S) = 6.37 x 10"4S 2 + .3568S + 49.9453 (4.2.2)

9.8 x 10-aS 5 +.0001S' +.0148S _ + 1.435S 2 + 22.529S + 184.194

28
G(S)

49.9453

.00769S 3 + S 2 + 36.187S + 508.838

(4.2.3)

where the denominator of the above two transfer functions has the gain already included.

The step response curve corresponding to Eq. 4.2.2 using the DPA Controller is Fig.

4.3.1. Restricting the overshoot to five percent, the amount of gain (K) required to reach this

objective was 7, and the settling time was obtained to equal .24 seconds which is quite rapid.

This value of gain is the target quantity that satisfies the damping criteria of .6963 obtained from

the root-locus plot (Fig. 4.1.1), which is close to the optimum value of .707. Increasing the

above value of gain, the overshoot becomes smaller than the five percent criteria above but the

damping value acquires very small change, eventually, the system becomes overdamped.

Looking at Fig. 4.2.2 which corresponds to the system with PD control, It is observed

that the overshoot falls within the five percent criteria and has a rapid settling time also. This

PD controller appear to be a good reliable alternative for this system since it exhibits a

satisfactory behavior.

29
0 0.5 1 1.5 2 2.5 3 3.5 4

Time '*llOOfldl"

Figure 4.2.1 Step response curve of single degree-of-freedom

system with Dual Phase Advance controller.

Slnglo-D4gr_-of-lrr_edorn S_p Ruponee

I °'°'rl °.°,I/

_ I i i I i f

I

0 0.S 1 1.5 2 /,S 3 3.5

Time "romantic"

Figure 4.2.2 Step response curve of single degree-of-freedom

system with PD controller.

3O
CHAPTER 5

COMPARISON OF CONTROLLERS

5.1 White Noise

The single degree-of-freedom Magnetic Suspension and Balance System simulation was

used to examine the effect of random noise on the Phase Advanced and PID controllers. The

existing simulation program that was used to compare different kinds of controllers (Ref. 5) was

modified to accommodate the noise equations and extra graphic capability. Random numbers

between zero and one were inserted in the normal or Gaussian function (Ref. I 1) to generate

uniformly distributed white noise, which was taken and added to each of the controller's inputs

where the sensor is located, Figure 5.1.1 shows where the noise was introduced. The output data

generated above for each of the controllers was used to calculate their root mean square (RMS)

value.

L,R _"

kF

mg Noise

Figure 5.1.1

H(s)

Magnetic suspension controller arrangement with noise.

Xo

31
TheGaussianfunctionis

1 e_-----yjr-o/ (5.1.1)

where _ is the variance and _ is the median.

To demonstrate the noise distribution graphically, a Fourier Transform algorithm to

estimate the power spectra of signals was used with the assistance of Matlab (Ref. 13). Eq.

5.1.1 was incorporated in the single degree of freedom program (Appendix C) where the white

noise data was produced. This data was then used in the Fourier transform algorithm to plot the

power spectral density (Fig 5.1.2) which shows reasonably good white noise, even though only

one thousand points were used.

10 _

Power Spectrol DBneTty

103

102

E

2

10'

10o

10-1

10-2'

0 50 1O0 150 200 250 300

Frequency "Hz"

350 AO0 4.50 500

Figure 5.1.2 White noise power spectral density plot.

32
Therootmeansquarewascalculated using the following equation (Ref. 12)

T

(5.1.2)

where T is the time.

deviation equation

Equation 5.1.2 produces the same results as the following standard

(5.1.3)

where I' is the standard deviation, _ is the mean and n is the number of points.

To evaluate the In of the DPA and the PID controllers, the program of the SingleDegree-of-Freedom system (Appendix C) was used. The standard deviation equation (Eq. 5.1.3)

was incorporated in this program to perform this operation.

5.2 Dual Phase Advance (DPA) ControllerTo test the effect of the random white noise on this controller, random numbers were

generated in the program (Appendix C), processed by equation (Eq. 5.1.1) and were incorporated

in the DPA subroutine, the resulting output was used to produce the power spectral density plot

(Fig. 5.2.1) which shows the presence of noise. The single-degree-of-freedom magnetic

suspension system simulation program (Ref. 5) tested the behavior of various types of controllers

on the suspension of spherical metal object. In this simulation, the aim was to obtain the value

of gain that produced five percent position over-shoot. For the DPA controller, the value was

33
found to equal 2784. This value of gain is then used in the modified program (Appendix C) to

evaluate the RMS of this particular controller, and the resulting value is. 145. In the simulation

program (Appendix C), the output data including the noise had a large numbers, therefore it was

multiplied by .0001 for scaling. Looking at Fig. 5.2.2, it is observed that the resonant frequency

of the order of 10 Hz.

Power Speotral Denel_

105 , ,

10 =

I I0_I

I0-4

I0-7

J

I i i i i I I I i

50 100 150 200 250 300 350 400 450 500

Figure 5.2.1

F'_lq uency "Hz"

Power spectral density of the single degree-of-freedom system with

Dual Phase Advance controller.

34
I0-I

_ 10__

ek

10-1

10-_o

0

Figure 5.2.2

Power Speotrel Donol_

I I i i I

0 1O0 150 200 250 _00

Frequency "Hz"

Power spectral density of single degree-of-freedom system with Dual

Phase Advance controller, and smaller frequency range.

5.3 PD Controller

Applying the same principles as the DPA controller, the gain value used for the PD

controller to accomplish the five percent overshoot is 396. Using this value, the resulting RMS

is .6136 for this controller where the data that included the noise was also multiplied by .0001

for scaling. This RM$ value is about four times bigger than the DPA controller RMS value.

(Fig. 5.3.1) shows the power spectral density plot with noise present. This plot shows that the

resonant frequency is also of the order of 10 Hz.

35
Powor SDeCcroi DenO_'y

103

101 _,

10°

i '°-'r '°-'F

10 -7 ' ' ' _ ' ' ' _ '

0 _0 100 1._0 200 2'_0 300 .3_0 400 450 SO0

Fr@Quency "Hz"

Figure 5.3.1 Power spectral density plot of single degree-of-freedom system

with PD controller.

101

leower Spe_ml DoneK'y

I 0 o

E

10-1

10" z

_ 10 "-a

10--=

10-4

I0-7

@

Figure 5.3.2

SI0 I I I I O0 _SO 200 2_0

F_ w_cy "Hz"

Power spectral density of single degree-of-freedom system

with PD controller and smaller frequency range.

36
5.4 Comparison:

Executing the single-degree-of-freedom simulation program (Appendix C) with the noise

generation, the output of the PD and the DPA controllers was Fourier Transformed to generate

power spectral density plots. Examining the plots, it is observed from Fig. 5.2.1 and Fig. 5.3.1

that the resonant frequency of both the DPA and the PD controllers are of the same order of

about 10 Hz. Within the simulation program, the RMS value both controllers was evaluated

where the PD controller produced a value about four times larger than the DPA controllers.

Considering the over-shoot and settling-time behavior of both controllers in chapter 4, the DPA

controller produced a quick settling-time while maintaining the five percent over-shoot criteria.

Mean while, the PD controller also had very quick settling-time while maintaining the five

percent over-shoot criteria.

Therefore due to the results mentioned above, the two controllers have proved to be

prime candidates for the magnetic suspension system. But the DPA controller has the edge with

its noise performance (RMS value) and permits much higher gain before it becomes unstable.

Based on this criteria, the DPA controller was the choice for this system.

37
CHAPTER 6

FIVE DEGREE-OF-FREEDOM SYSTEM

6.1 The System

The systems is composed of the following blocks: A decoupling matrix, the power

supplies, the plant, and the five controllers with the gain blocks.

_:(

I1

h

- Ill v v

[_ --m, Power Is _

Decoupling _ _ Plant --_

___ Supply 16 w --_

Y

Figure 6.1.1 The five-degree-of-freedom system block diagram.

f],

×

Z

The doc.oupling matrix was created by computing the inverse of the magnetic field

gradients matrix given in (Appendix A).

38
6.2 PowerSupply:

Thepowersupplygoverningdifferentialequationtakesthe following form

V. = I_R, + L..--dlj + .... where i=1...5 • j=l...5 (6.2.1)

u dt '

where V is the voltage, I is current, R is the resistance, and L is the inductance. If i = j, the

result is self-inductance, but ifi ;_ j, the outcome is mutual inductance L.,. Figure 6.2.1 shows

the power supply diagram where the following equations apply

_=RI+LI+L (6.2.2)

vo . K(V_ - kt) (6.2.3)

= Voj - k/: (6.2.4)

Combining equations 6.2.2-4, the result is the following

[L] [11 - [K Vo] - [Kk] It] - [R] It] 6.2.5

39
K

vo

//////

Figure 6.2.1 Power supply block diagram.

Following equation 6.2.5, the power supply matrices, for which values are given in

Appendix A, have the following form

a-- [(Kk+R)[,] Z

(6.2.6)

where L is 5 x 5 matrix, (Kk + R) are constants and [_I,] is an identity matrix.

The B matrix takes the following form

a -x[%] (6.2.7)

The resulting matrices have the following dimensions

A is a 5 x 5 system dynamics matrix

B is a 5 x 5 input matrix

C is a 5 x 5 output matrix

D is a 5 x 5 matrix of zeros

4O
Thecurrentdistributionproduced by the five electromagnets varies for each activated

degree-of-freedom. The following graphs show which currents are produced for each degree-offreedom. The table below represents the value of each of these figures.

I Pitch Yaw X Y Z

-0.1611

-0.0098

-0.1038

-0.1038

-0.0098

-0.0000

0.1352

0.0835

-0.0835

-0.1352

0.1537

-0.1249

0.0480

0.0480

43.1249

-0.0000

0.0899

-0.1458

0.1458

-0.0899

-0.0621

-0.0192

0.0503

0.0503

-0.0192

Table 6.2 Current values of the five degree-of-freedom system.

Figure 6.2.2

Powor SupDly St4D Rooponoe

50 I O0 150 200 250

Time m ecloon¢lo x 10A3

Power supply step response, "Pitch" degree-of-freedom.

41
Power Supply Stud RellDOnee

0.15

]

0.1

Q.OS

-0.05

-0.1

12

t

5O

_3

I&

15

-0.15 _ i

0 100 150 200 230

"nine m e4©onde x 10-3

Figure 6.2.3 Power supply step response of yaw as the degree-of-freedom.

0.2

Power SuDply StM_ RetDoneQ

0.15

0.1

"_ 0.05

-0.05

-0,1

13,14

,I

O0

12.,15

-0,15 i i i

0 100 150 200 250

TTm4 a, ee¢onde x 10-3

Figure 6.2.4 Power supply step response of X as the degree-of-freedom.

42
Powor SuDply $t:0D ReODonoe

0.15 t , ' 1¢

0,1

-0.05

-0. I

#2

I

_0

15

13

-0.15 _ ' '

0 100 ISO 200 250

Time m Noond$ x I0..,3

Figure 6.2.5 Power supply step response of Y as the degree-of-freedom.

Power SuDp_ StJlp Reeponoe

0.08

0.0_,

0.02

-Q.02

-0.04.

-0,08

1.1,14

12,15

-0.08 ' ' ' '

0 _,0 I O0 150 200 250

Time m ee¢onde x 10,,3

Figure 6.2.6 Power supply step response of Z as the degree-of-freedom.

43
6.3 The Plant:

The governing equations which lead to the development of the plant's matrices axe given

in Chapter 3. Those plant matrices were incorporated with the power supply matrices, and the

five DPA matrices in order to produce the simulation of the entire system.

The derivation and numerical values of the plant's A, B, C, and D matrices axe given in

Appendix A. The following axe the plant dimensions

A is a 10 x 10 system dynamics matrix

B isal0x5 input matrix

C is a 10 x 10 output matrix

D is a 10 x 5 matrix of zeros

6.4 The Controllers:

A DPA controller transfer function was converted to a state-space form using MATLAB.

Appending the state-space matrices of the DPA controller five times, state-space system matrices

of the five controllers were created. The dimensions of these matrices are

A is5X5

B is5X5

C is5X 10

D is5 X 10

As stated in Chapter 5, the numerical values of the DPA controller transfer function

numerator and denominator were determined when the stability of the single-degree-of-freedom

was accomplished.

44
Figure 6.4.1 shows a step response, which is identical for each of the five controllers.

I OC

80

60

C

°_

o 4.0

20

-20

0 0.02

Figure 6.4.1

I:'ive P._. Controilere Sf.ep ReIponle

All flvllCOnLroIlerl have

I/re|lot Irtllp eliponsi.

i I I I I I i I

0.04. 0.06 O.OB 0.1 0.12 0.14 0.16 0.1S

Time "Secon till"

Step response of Dual Phase Advance controller.

0.2

6.5 Continuous-time Step Responses

The following step response graphs correspond to the above state-space system in a closed

loop form. Each graph represents the behavior of the system when a certain degree-of-freedom

is activated. To non dimensionalize the X degree-of-freedom, its values were divided by the

length of the model which is equal to .3048 m.

45
0.3

0.25

0.2

0.15

0.1

0.0S

0

-0.0_

0

Cor_IMuouw--'rlme S'_p RNponme

"P I t o h"

Set'kli'ng Llme --, .4 nec.

- "... Noe--O|_lel|OeOI X

I I I I i

0.5 I 1.5 2 2.5 ,1

"l_mg "ig_:O I'1 ¢:1l"

Figure 6.5.1 Continuous-time step response, "Pitch" degree-of-freedom.

0.25

Contlnuoul-'rlme Step RNponme

0.2

i 0.I_

_ 0.1

B

O.OS

0 0.5

Settling time ,, .28 X¢.

I I I I

1 t .5 2 2.5

"i=r m,ll "1,1100 rl(::ll"

Figure 6.5.2 Continuous-time step response, Yaw degree-of-freedom.

46
x10-a Contlnuoue-TIme StoD Reeponee

8

¢l I

2

0

I

$

(

I

t

l

Non-drrnQnllionol X

Se((Irnq t;me - ,2 mec.

x

x -2

|

0.5 1 1.5 2 2.5

'rlme "llecond|"

Figure 6.5.3 Continuous-time step response, Axial degree-of-freedom.

xlO -'I Contfnuoue--Tlme 5ted ReeponlNi

=/ I

,.e_ ;

l

I Settling time = .18 |e¢,

'[i

°"V!

o.e_:

0"4I!

0 0.5 I 1,5 2 2.5 3

1"free "lleCOn@ll"

Figure 6.5.4 Continuous-time step response, Lateral degree-of-freedom.

47
×10 -,1 CCreklnuoull--T1me 5tip RIIDonII

jl _ _ _

14._ I --

• / _ Settling tl_e ,, .18 nc.

2 o:f/

°"C!

l_.l.p

o.'II

0_

0 0.5 1 1.5 2 2.5

Time "locc3ncll"

Figure 6.5.5 Continuous-time step response, Vertical degree-of-fi'eedom.

6.6 Discrete-Time Step Responses:

To examine the behavior of the system, its state-space equations were converted to the

discrete-time equations where a sample interval and a delay time was added. The sample interval

and the delay time supposed to have an equal magnitude since all the degrees-of-freedom are

sampled together to produce input. Once those degrees-of-freedom are processed and command

output, the sampling procedure starts again without allowing any further delay time. The

maximum value of the plant's eigenvalues (13.6986) was chosen since it represents the maximum

frequency. The reciprocal of twice of this value (.0365) was taken to be the sample interval.

Testing the behavior of the system with different values of delay time, the discrete-time

step responses remained virtually unchanged for low values, but for each degree-of-freedom, the

48
maximum value of delay time allowed before the system becomes unstable are given in the

figures. Those values are at the critical point before instability occurs.

Comparing the step response curves Figures 6.6.1-5 before reaching the critical point of

instability and Figures 6.5.1-5 for each degree of freedom, their behavior is virtually identical

using the same gain value of each degree-of-freedom. This result shows that the system is well

behaved and has a fast settling time of approximately .25 seconds which may increase or decrease

by .01 seconds for each mode.

0.4.

0..._

0.3

i 0.2_

0.2

O._

.t=

_. 0.1

0.0_ 1

0

-0.05

0

DIo,or_e-l"l'me St;eo RooD©nee

|

"p I t. c h"

h40xlmum det0y Umo ,, .0307 eoo.

NO I_ --¢1] I_ lrl li'O l_¢11 X ..... *"

I I I I I

20 40 80 OO '1O0

Numblr of I_O;niW

120

Figure 6.6.1 Discrete-time step response, "Pitch" degree-of-freedom.

49
im

0.0_

0

0 20 4.0 80 80 10Q

Number of pointe

120

Figure 6.6.2 Discrete-time step response of Yaw degree-of-freedom.

|

|

8

8

4.

2

x10-_ OIl¢l-ltl-'rrme S_IID J_OlaOnlll

'_ Non-drmenelonol X

J

s Moxlmum dllOy tlmo ,m .0243 |oc.

;

.!

x

:1(

"Plto h"

i

20 40 80 80 100

Number of Dolnbl

120

Figure 6.6.3 Discrete-time step response, Axialdegree-of-freedom.

5O
x10 -3 D1mcr_tm-'rlmo $_el_ _mmDonmo

2

I .B t I_" " ................................

| I M@xfmum delay _lm0 n .020.300C.

o.:ti

°"Ii

°"r:

0

0 20 40 O0 80 I O0 120

Numl_Ir o4 Dolntl

Figure 6.6.4 Discrete-time step response of the Lateral degree-of-freedom.

x10-; OlllCrl_l--l"_ml $1_ID RlllD@.lle

1.8

l"e t Ill,, .................................

1,41- ,

| i Moxfmum deloy tlrn. = ,0'le3 sec.

/

1.21-,

!',,°"rl

o.e[,,

0 20 4,0 BO BO 1 O0 _ 20

Numb4r o¢ I=ol,t_

Figure 6.6.5 Discrete-time step response, Vertical degree-of-freedom.

51
CHAPTER 7

DISCUSSION

The focus of this thesis was the analysis of the five degree-of-freedom system simulation

and testing the effect of white noise on the DPA and the PD controllers.

The single degree-of-freedom simulation program was executed separately for each of the

two controllers. In the controllers subroutines, the output values were multiplied by .0001 for

scaling purposes in order to be able to see its behavior in the graphics window created by the

program. The previously determined gain values of both controllers were used since the five

percent over-shoot criteria was adopted. The output data that included the white noise generated

by the Gaussian function evaluated the noise performance (RMS) of both controllers. The values

•145 and .6136 for the DPA an PD controllers respectively indicate that the DPA controller noise

tolerance is about four times better. This data was also incorporated in a Fourier Transformation

M-file to plot the power spectral density of both controllers where the resonant frequency of the

controllers was approximated to about 10 Hz. Those spectral density plots do not present a

smooth behavior of the noise due to insufficient number of points obtained from the simulation

program. Considering the noise performance of both controllers, the DPA controller was chosen

for the system while the PD controller is another very good alternative.

The resulting Dual Phase Advance controller was used in the single degree-of-freedom

simulation in order to have an approximate value of the gain required to tune the overall five

degree-of-freedom system. The gain value of this simulation (7) produced a good damping ratio

of about .7 which is close to the target value of .707. Also, this gain value satisfied the criteria

of achieving the five percent overshoot with the step response graphs.

Using this approximate gain value, the simulation of the five degree-of-freedom system

52
was executed in order to determine the gain matrix which provides a stable system that satisfies

the step response criteria of five percent overshoot for each degree-of-freedom. The resulting

system produced a fast settling time of about .24 seconds in addition to maintaining a damping

ratio for each of the degrees-of-freedom of about .7 and above.

In order to determine the maximum value of the delay time required for each degree-offreedom before they become unstable, a second order Pade Approximation was used with the

closed-loop discrete system. A sampling time of .0365 was used which is the reciprocal of twice

of the maximum frequency of the system. The resulting values of the delay time were between

50 and 75 percent of the sampling time which are a very good results.

Approximations and their effects

Approximations were made deriving the MSBS system governing equation. These

equations are considered to be reasonable approximations of the non-linear system equations of

the real MSBS where literature has also shown the linear approximation is a good representation

of the system's dynamics (Ref. 4). For the bearing and the wind tunnel systems, these

approximations apply well while they are operating on their equilibrium points. But during large

position changes, these equations do not represent satisfactorily the dynamics away from the

equilibrium point.

The availability of power supply is the most important factor for any MSBS system. It

is possible to operate systems with their power supply capability is low, but when certain

commands or loads are applied, great attention is required. This power supply limitation is not

usually a problem for magnetic bearings since they only requires low amount of current, but for

large gap MSBS, the power supply availability is a serious concern.

53
CHAPTER 8

CONCLUSIONS

The overall system is a feedback loop that contains a decoupling matrix, power supply,

the plant, and five controllers. The values in each of the matrice_ in the above system were

calculated using the system's parameters given in Appendix A.

Two controllers were tested to determine which is better suited for the system by

generating white noise using the Gaussian equation with the single degree-of-freedom system.

The output produced was used to plot the power spectral density of both controllers where a

comparison was made between them. The outcome was that the two controllers have

approximately similar resonant frequency but their gains were different.

The distinguishing result of both controllers was the noise performance (RMS), where

the DPA controller has better tolerance to noise than the PD controller and this was the dominant

condition of choosing the controller of the system. In addition to this result, the single degree-offreedom system step responses were examined with both controllers where they both showed a

rapid settling time and an over- shoot which met the criteria value of five percent error and

lower. Due to these results, the DPA controller was chosen for this system.

With the determination of the controller type, a single degree-of-freedom transfer function

of the yaw mode was acquired using the overall system's matrix (Appendix A). The new

resulting system's transfer function was used to determine the location of the poles and zeros of

the DPA controller by using root-locus plot. Different locations of these poles and zeros were

examined until the appropriate root-locus plot that shows stability with acceptable amount of gain

was achieved. This resulting gain was used to obtain the step response of the system.

After determining the desired location of the poles and zeros of the DPA controller, this

54
controller was appended four times in order for the five resulting controllers be incorporated in

the overall five degree-of-freedom system.

The basic system provides five degree-of-freedom control of a suspended cylinder that

contains a core composed of permanent magnet material. The magnetization vector is along the

horizontal axis of the cylinder. This system uses five electromagnets which is the minimum

configuration for a five degree-of-freedom control. The highest frequency open-loop mode is the

compass-needle mode that result from the magnetization vector trying to align itself with the

applied field and it is caused by the presence of B_. Since those high frequencies are caused by

Bx and since B_ is uncontrolled with five.coil, increasing the number of coils might make it

possible to independently control I_ and, therefore, control the highest open-loop frequencies.

This system is a multi-input multi-output system, some of the modes in this system are

coupled, therefore, simulation of the system was performed where the strategy to control each

degree of freedom was adopted. Using MATLAB, the gain matrix (Appendix A) produced a

stable system by using each degree-of-freedom separately starting with the vertical followed by

the lateral, yaw, the axial, and pitch. The order of difficulty to stabilize each of those degrees-offreedom is different, therefore the order of simplicity to achieve stability was followed. The

continuous step response plots of the five modes show that the system is well behaved since the

overshoot of all of the degrees-of-freedom fell bellow the criteria of five percent error value and

has a quick settling time. Discrete-time step responses of all the degrees-of-freedom were also

produced and showed promising results as the above continuous step responses.

At_er examining the results produced by the simulations and observing the behavior of

the system, it is concluded that the overall system is well behaved and exhibits a very good

stability.

55
REFERENCES

,

.

°

4o

.

o

,

°

°

10.

11.

12.

Holmes, F. T.: Axial Magnetic Suspension. Review of Scientific Instruments, Vol

8, November 1937.

Tournier, Marcel; and Laurenceau, P.: Suspension Magnetique d'une Maquette en

Soufflerie. (Magnetic Suspension of a Model in a Wind Tunnel.) La Recherche

Aeronautique, No. 59, July - August 1957.

Boyden, R. P.: A Review of Magnetic Suspension and Balance Systems. AIAA

15th Aerodynamic Testing Conference, San Diego, California, May 18-20, 1988.

Tuttle, Marie H.; kilgore, Robert A.; and Boyden, Richmond P.: Magnetic

Suspension and Balance Systems. A selected, Annotated Bibliography. NASA

Technical Memorandum 84661, February 1988.

Kilgore, William, A.: Comparison of Digital Controllers Used in Magnetic

Suspension and Balance Systems. NASA Contractor Report 182087, December

1989.

Britcher, Colin, P.: Technical Background for a Demonstration Magnetic

Levitation System. Department of Mechanical Engineering and Mechanics, Old

Dominion University, Norfolk, VA. Grant NAG1-716, May 1987.

Groom, Nelson, J.: Analytical Model of Five Degree of Freedom Magnetic

Suspension and Positioning System. NASA Technical memorandum 100671,

March 1989.

Britcher, Colin, P.; Groom, Nelson, J.: Stability Considerations for Magnetic

Suspension Systems Using Electromagnets Mounted in a Planar Array. NASA

Conference Publication 10066, Part 1.

Groom, Nelson, I4 Britcher, Colin, P.: Open-Loop Characteristics of Magnetic

Suspension Systems With Electromagnets Mounted in a Planar Array. NASA TP-3229,

May 1992.

Takahashi, Y.; Rabins, M. J.; Auslander, D. M.: Control and Dynamic Systems.

Addison-Wesley Publishing, Co., 1970.

Maciejowski, J. M.: Multivariable Feedback Design. Addison-Wesley Publishing,

Ltd., 1989.

Buckingham, M. J.: Noise in Electronic Devices and Systems. John WHey &

Sons, Inc., 1983.

56
13.

16.

17.

18.

19.

20.

23.

24.

Lewis, T. G.: Distribution Sampling for Computer Simulation. Heath and

Company, 1975.

MATLAB. The MathWorks, Inc.

layawant, B. V.: Electromagnetic Levitation and Suspension Techniques. Edward

Arnold ('Publishers) Ltd., 1981.

Parker, D. and Britcher, C. P.: Progress Toward Extreme Attitude Testing with

Magnetic Suspension and Balance Systems. AIAA 15th Aerodynamic Testing

Conference, San Diego, California, May 18-20, 1988.

Beaussler, J. nd Zakneim, J.: Telemetric Multivoies pour Maquettes en Suspension

Magnetique. (Multichannel Telemetering Device for Magnetically Suspended Models)

Presented at the Colloquium on the Properties and behavior of Electronic Components

and Assemblies Submitted to Strong Acceleration. Saint-Louis, France. October 1968.

Franklin, G. F.; Powel, J. D.; and Emami-Naeini, Abbas: Feedback Control of

Dynamic Systems. Addison-Wesley Publishing Company Inc., Massachusetts, 1986.

Matsuda, R.; Nakagawa, M.; and Yamada, I.: Multi Input-Output Control of a

Magnetically Suspended Linear Guide. IEEE Transactions on Magnetics, MAG-20, No.

5, September 1984.

Hornbeck, Robert W.: Numerical Methods. Prentice-Hall, Inc., New Jersey, USA,

1975.

Microsoft QuickBASIC Version 4.5. Microsoft Corporation 1988.

Britcher, C. P.: User Guide for the Digital Control System of the NASA/Langley

Research Center's 13-inch Magnetic Suspension and Balance System. Old Dominion

University Research Foundation, NASA CR 178210, March 1987.

Sawada, Hideo; Kanda, Hiroshi; and Suenaga,Hisashi: Apositionand Attitude Sensing

Camera in NAL's MSBS. MSBS Newsletter published by the Experimental Techniques

Branch, NASA Langley Research Center, No. 4, January 1988.

Bradley, Fennimore N.: Materials for Magnetic Functions. Hayden Book Company,

Inc., 1971.

57
APPENDIX A

The development of the plant state-space matrices has the following procedure:

Using the small angle assumptions and assuming that the magnetization is along the long

(x) axis of the core, that is

[fir]. [M_ 0 O]

Assuming O, is zero, the torque and force equations reduce to

T = (vol) Mi (-O, B, - B)

Tz ,,(vol) Mi (- 0 B,,+ B )

_ . (,,oz) :% (B + o, B,, - o B + o, _ - o B.)

F' ,, (vol)Mi(-O B= +B + 8_B. + _B )

_'z " (vol ) Mi (Oy B,_ + B + 0z Br/ - OyB,.)

Using the electromagnet specifications (Ref. 9)

Inner radius

Outer radius

Depth

Location radius

Maximum current density

Datum orientation

= 0.173 m

= 0.386 m

= 0.493 m

= 0.7 m

= 1535.87 A/cm 2

= Positive _ axis vertically over coil 1
-thefield andfield gradientcontributionsforthe full design current

_

By

B_

B_

B_

B_,

B_,

B_

COIL

1 2 3 4 5

0.0216

0

-0.0198

0.0092

0

-0.0497

-0.0306

0

0.0067

0.0206

-0.0198

-0.0269

0.0118

-0.0152

0.O054

-0.0472

-0.0175

0.0127

-0.0198

-0.0046

-0.0191

0.04

-0.017

-0.029

-0.0175

-0.0127

-0.0198

-0.0046

0.0191

0.04

-0.017

0.029

0.0067

-0.0206

-0.0198

-0.0269

-0.0118

-0.0152

0.0054

0.0472

Table A. 1 Field gradients contributions values.

The model specifications are (Re(. 9)

Magnet length = 0.3048 m

Magnet diameter = 0.1016 m

Magnetization = 954930 A/ra/m (1.2 Tesla)

Total mass = 23.11 kg

Moment of inertia = 0.6 kg m:

Using the expression for torque and force developed earlier, this means that

B_,= Br_= Bg =B, = By=0 and F_=F r =T r=T, =0. This results in

_L,- B _L o

_" - -(vol) _ B _ - -(vot) M_s

_ - -(vot)M, s _,; - o
?r', ,. (vet) M_ a. 7"_,= (vol) M_ a,,

_,,, . -(vol) M_ B_,

Yzo = 0

IZ y = (vol) M i B<_)y

_y, = - (vet) Mr By,

_,,. . (vol) Mr B,,

_: = (rot)Mr B,._.

_: = (vol)Mr B<.._y

fzy, = (vol) Mr B_y,>,

_" , ,, - (vol) MT,B,,

Byz

Due to symmetry, for a five coil configuration

= B. = 0 B. = -B. = 0 B_,_ = B_.). ffi 0

B(_y = 0

B_._ = Bo.)7 ffi 0 B¢_o_ ffi 0 B(.,_, = B{_._. = 0 B_, = Bc_). = 0

To obtain some of the above Br. and B0,_ relationships, Maxwell's equations were used. The

non-zero terms are B=, Bo_,, Bv,_, B,, and B_,. The related non-zero elements of the A matrix

are
_';= -(vot)M_ Bx _" = -(rot)M_ axz

T n = -(vol) M_ B, Fxo, = -(vol) M_ Bx,

F,,,, = (vol)M_ B_,,,,>,, _'yy= (vol) M r B_,oy

Fe, = (vol) M r Bc,,,)_

Using the values of the above coefficients, the resulting A matrix is

a z

0 0 165.424

O0 0

1 0 0

0 1 0

0 0 -9.8225

O0 0

O0 0

O0 0

O0 0

O0 0

0 0 0 0 -378.359 0 0

165.424 0 0 0 0 0 0

0 000 0 0 0

0 000 0 0 0

0 0 0 0 18.858 0 0

0 0 0 0 0 6.279 0

0 0 0 0 0 0 -25.314

0 100 0 0 0

0 010 0 0 0

0 O0 1 0 0 0

The resulting B matrix is
B m

77. 8679 77. 8679 77. 8679 77. 8679 77. 8679

0 81.014 49.9455 -49.9455 -81.014

0 0 0 0 0

0 0 0 0 0

•9394 -2.7466 -.4697 -.4697 -2.7466

0 1.2048 -1.9502 1.9502 -1.2048

-5.0746 -1.552 4.0842 4.0842 -1.552

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

C x

1 0 0 0 0 0 0 0 0 O"

0 1 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 1

O m

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0
The power supply

Factoring terms in equation 6.2.5, the result is

I(R + LD + kK) - KV v

in a matrix form, the above equation becomes

[L,j][I] _, -[(R ÷ kK)][/,] ÷ [KlVo

To produce the A and B matrix, [(R + kK)] and [K] were multiplied by the inverse of

the inductance matrix respectively.

The inductance matrix is

L

2.9668 0.1082 0.0247

0.1082 2.9668 0.1082

0.0247 0.1082 2.9668

0.0247 0.0247 0.1082

0.1082 0.0247 0.0247 0.1082

0.0247 O. 1082"

0.0247 0.0247

O.1082 0.0247

2.9668 O. 1082

2.9668

a I

-43.9375 1.5759 0.2849 0.2849 1.5759

1.5759 -43.9375 1.5759 0.2849 0.2849

0.2849 1.5759 -43.9375 1.5759 .2849

0.2849 .2849 1.5759 -43.9375 1.5759

1.5759 0.2849 0.2849 1.5759 -43.9375

B I

•3380 -0.0121 -0.0022 -0.0022 -0.0121 1

]

-.0121 .3380 -0.0121 -0.0022 -0.0022

-.0022 -.0121 .3380 -0.0121 -0.0022

-0.0022 -.0022 -.0121 .3380 -0.0121

-0.0121 -0.0022 -0.0022 -0.0121 0.3380

The decoupling matrix was evaluated by taking the inverse of the field gradients matrix

and is equal to
-20.95

-1.27

-13.5

-13.5

-1.27

0.0 19.985 0.0 -8.074

17.58 -16.24 11.69 -2.498

10.86 6.245 -18.96 6.535

-10.86 6.245 18.96 6.535

-17.58 -16.238 -II.69 -2.498

The B matrix is multiplied by the decoupling matrix to produce the following

BCI

-6.9918 0.00 7.1208 -0.00 -2.6968

0.0194 5.8714 -5.7841 4.1640 -0.8345

-4.3354 3.6273 2.2237 -6.7531 2.1829

-4.3354 -3.6273 2.2237 6.7531 2A829

0.0194 -5.8714 -5.7841 -4.1640 -0.8345

The above BC matrix is used as the B matrix since it includes dec_upling.

C I

10000

01000

00100

00010

00001

D

00000 °

00000

00000

00000

00000

The five Dual Phase Advance controllers matrices

A = I0_

-.044 -4.84 0 0 0 0

.0001 0 0 0 0 0

0 0 -.044 -4.84 0 0

0 0 .0001 0 0 0

0 0 0 0 -.044 -4.84

0 0 0 0 .0001 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

-.O44 -4.84 0 0

.0001 0 0 0

0 0 -.044 -4.84

0 0 .0001 0
B

O01000000 O"

0000000000

0001000000

0000000000

00000001 O0

0000000000

0000000010

0000000000

000000000 l

0000000000

C = 106

-.04 -4.79

0 0

0 0

0 0

0 0

0 0 0 0

-.04 -4.79 0 0

0 0 -.04 -4.79

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

-.04 -4.79 0 0

0 0 -.04 -4.79

D I

001_ 0 000 0 0 0

00 0 1_000 0 0 0

00 0 0 0001_ 0 0

0 0 0 0 0 0 0 0 1_ 0

O0 0 0 000 0 0 I_

The five degree-of-freedom gain matrix

The following gain matrix was formed by restricting the step response of each of the five

degrees-of-freedom to five percent overshoot.

Gain =

9 0 0 0 0

0 II 0 0 0

0 0 572 0 0

0 0 0 586 0

0 0 0 0 618
APPENDIX B

SINGLE DEGREE-OF-FREEDOM M FILES

The "MATLAB" M files below evaluate the numerator and denominator of single degree

of freedom system combined with the Dual Phase Advance controller to produce the compensated

system transfer function. The files may be named as (filename.m).

To achieve a stable system, the poles and zeros of the controller were examined in

different locations. Once those value were produced, a root-locus graph was plotted and the

value of gain that produces stability was determined. For this purpose, the root-locus M file was

used.

The step responses of the single-degree-of-freedom system with either the Dual Phase

Advance or the PD controllers are plotted aRer evaluating and producing the closed loop transfer

function.

The remaining M files evaluate the five-degree-of-freedom step responses for the

continuous and discrete-time, they also evaluate the five percent over-shoot and under-shoot, and

the settling-time.
nump =

denp =

nauru --

nden =

numpa =

denpa =

num2pa =

den2pa =

snum -

sden --

chum -

eden =

¢inum --

olden -

t

dt

time --

zfreq =

setime =

cnt

errup --

errdn --

DEFINITIONS OF VARIABLES

i

Single-Degree-of-Freedom power supply numerator.

...... denominator.

....... plant numerator.

.... plant denominator.

Phase Advanced Controller numerator.

.... denominator.

Dual Phase Advanced Controller numerator.

..... denominator.

Uncompensated system numerator.

" " denominator.

Compensated system numerator.

" " denominator.

Closed loop numerator.

Closed loop denominator.

Time.

Time increment.

Time vector.

Zero frequency.

Settling time.

Counter.

The value of the error above zero frequency.

..... below " "

Table B. 1 List of variables of the single degree-of-freedom M files.
Root Locus:

% Evaluating the system's and the

% controller's transfer functions.

snum = conv(nump,nnum)*.O126; % Combining the plant and the power

sden = conv(denp,nden) % supply transfer functions.

num2pa = conv(numpa, numpa); % Creating Dual Phase Advance

den2pa = conv(denpa, denpa); % controller transfer function.

chum = conv(snum, num2pa); % Creating the system's compensated

cden= conv(sden, den2pa);

vv= [xl,x2,yl,y2];

axis(w)

rlocus(cnum,cden)

rlocfind(cnum,cden)

% transfer function.

% Scaling the minimum and maximum

% values of the axis.

Dual Phase Advanced Step Responses:

snum = conv(nump,nnum)*.0126;

sden = conv(denp,nden);

num2pa = conv(numpa, numpa);

den2pa = conv(denpa, denpa);

gnum = conv(num2pa, snum);

gden= conv (den2pa, sden);

clnum=conv(snum,den2pa);

n=t/dt;

errup = 1;

ymax = 1e-15;

while ymax < errup,

gain = gain+ g 1;

if ymax < errup,

ymax = 0;

¢Iden = gden + gnum*gain;

zfreq = (clnum(1, 6) / olden(l, 6))

errdn = zfreq - .05 * zfreq;

errup = zfreq + .05 * zfreq;

gain

y = step(clnum,¢lden,time);

forcnt= l:n,

ifcnt > 1,

if y(cnO > y(cnt-1), %

tmp=y(cnt);

else

end

if map > ymax,

ymax=tmp;

else

% Closed-loop transfer function numerator.

% Calculating the zero Frequency value.

% Error of five percent below zero frquency.

% .... above " "

Finding the maximum over-shoot.
end

else

end

if y(cnt) < errdn, % Evaluating the settling time.

setimel -- cnt*dt;

elseif y(cnt) > errup,

setime2 = cnt*dt;

else

end

end

if setime2 > setimel,

setimel

errdn

else

setime2

errup

end

ymax

pause

plot(y);title('Single-Degree-of-Freedom Step Response')

xlabel('Time = seconds x 100')

else

end

end

PD Step Response

snum = eonv(nump,nnum)*.0126;

glen = conv(denp,nden);

gnum = conv(numpd,snum);

clnum-- snum;

n = t/dt;

ermp= 1;

flag -0;

while flag - = 1,

gain = gain+gl;

if ymax < errup,

if ymax > (errup - errup * .01),

flag = 1;

else

end

else

end

ymax = 0;

olden = glen + gnum*gain;

zfreq = (clnum(1,3)/clden(1,4))

% With these statements, the over-shoot that falls

% between four and five percent is getting evaluated.

% The rest, the same explanation as above.
errdnffi zfreq - .05 * zfreq;

errup = zfreq + .05 * zfreq;

gain

y = step(clnum,clden,time);

forcnt- 1 :n,

if cnt > I,

if y(cnt) > y(cnt-1),

tmp = y(cnt),

else

end

if trap > ),max,

ymax-tmp,

else

end

else

end

if y(cnt) < errdn,

setimel - cnt*dt;

elseif y(cnt) > errup,

setime2 = cnt*dt;

else

end

end

if setime2 > setimel,

setimel

errdn

else

setime2

errup

end

pause

plot(y);title('Single-Degree-of-Freedom Step Response')

xlabel('Time = seconds x 100')

end
FIVE DEGREE OF FREEDOM SYSTEM

DEFINITIONS OF VARIABLES

a,b,c,d

sa,sb,sc,sd

pa,pb,pc,pd

da,db,dc,dd

fa,fb,fc,fd

invl

pbc

= Plant's state space matrices.

= System's state space matrices.

= Power supply state space matrices.

= Discrete time state space matrices.

= The feedback state space matrices of the entire system.

= Inverse of the inductance matrix.

= The power supply B matrix multiplied by the

decoupling matrix.

apa,apb,apc,apd = The appended five dual phase advanced controllers

state space matrices.

gain = 5 x 5 gain matrix.

fivprovr = Five percent over-shoot.

fivprund = Five percent under-shoot.

tt = Time vector.

Table B.2 List of variables of the five degree-of-freedom M files.

Continuas time step response:

pa = -eye(5) * fq * invl;

[sa,sb,sc,sd] = series(pa,pbc,pc,pd,a,b,c,d);

ap3 = ape;

ap4=apd;

nn=t/dtt;

ymax= le7;

fivprovr=O;

while fivprovr < ymax,

gl =gl +g2;

gain(ui,ui)=gl;

ape=gain*ape;

apd =gain*apd;

[fa,to,fc,fd] =feedback(sa,sb,sc,sd,apa, apb,apc,apd);

y = step(fa,fb,fc, fd,ui,tt);

ymax-O;

for 11 ffi l:nn+ 1,

ifuc =ffi 8,

y(U,uc)ffiy(ll,uc)/.3048; % .3048m is the model's length

elseif uc = = 3, % 3 & 8 are the degree-of-freedom numbers

y(U,8)=y(ll,8)l.3048;

end

if 11 > 1,

if y(ll,uc) > y(U-l,uc),
sir--y(U,uc);

else

end

if str> ymax,

ymax = str;

else

end

ysOl) = y(ll,uc);

if y(ll,u¢) = = ys(ll),

tmp=y(ll,uc);

else

end

else

end

end

err = .05*trap;

fivprund = map-eft;

fivprovr = trap + err;

for Ik= l:nn+ I,

if ys(Ik) < fivprund,

stimet = lk*dtt;

elseif ys(Ik) < fivprovr,

stime2 = lk*dtt;

else

end

end

if stimel < stime2,

setltime = stimel

else

setltime = stime2

end

gl

),max

fivprovr

apc=ap3;

apd=ap4;

end

plot(y);title('Continuous-Time Step Response')

end
Discrete with time delay M file:

This M file converts the continuas closed loop system matrices to discrete with time delay

system matrices, then, the step responses of those matrices is plotted. Pade command was used

since it returns an nth order pade approximation to a time delay.

for lam=sl :s2:s3,

ap3- gain*apc;

ap4 = gain*apd;

nn- time/samp;

[lma,lmb,lmc,lmd] =pade(lam,ord); %

[la,lb,lc,ld] = append(lma,lmb,lmc,lmd,lma,lmb,lmc,lmd);

for i= 1:3,

[la,lb,lc,ld] = append(la,lb,lc,ld,lma,lmb,lmc,lmd);

end

[dsa,dsb,dsc,dsd] =series(apa,apb,ap3,ap4,1a,lb,lc,ld);

Ira,fb,fc, fd] = feedback(sa, sb,sc,sd,dsa,dsb,dsc,dsd);

[da,db,dc,dd] = c2dt(fa, fb, fc,samp,lam2);

sy = dstep(da,db,dc,dd,ui,n_n);

ord is the order system.

% Samp is the sampling time.

% lain is the delay time.

for 11- l:nn+l,

if uc ---- 8,

sy(ll,uc)=sy(ll,uc)/.3048;

elseif uc = - 3,

sy(ll, 8) ffisy(ll, 8)/. 3048;

else

end

end

plot(sy),title('Discrete-Time Step Response')

xlabel('Number of points')

ap3=O;

ap4ffiO;

end

% .3048m is the model's length.

% 3 & 8 are the mode numbers.

% nn is the number of points.

% t is the time vector with

% interval = sampling time.
APPENDIX C

This program generates random numbers to be used as noise in order to test the behavior

of the Dual Phase Advanced and PID controllers. This program which was written originally by

W, A, Kilgore (Ref. 5) compares different controllers with Single-Degree-of-Freedom Feedback

magnetic suspension system.

CLS

CLEAR

DIM M(2000), GS(2000), XA(2000)

COLOR 12

PI = 3.14159

N = I000

LOCATE 12, 22: PRINT "Generating random numbers (noise)"

FOR I = I TO N

M(1) = RND

SUM = SUM + RND

NEXT I

MN = SUM / N

FOR L = 1 TO N

SG = M(L) ^2-MN'2

GS(L) -- (1 / (2 * PI * SG)) * EXP(-(M(L) - MN) " 2 / 2 * SG)

NEXT L

"OPEN "a:pid,mat" FOR OUTPUT AS #1

'Sampling Time

T = .01

'The MSBS plant variables

got = -1

Kc = -.1

Ki= .1

M=I

R-1

L= .1

C=0

'The MSBS plant coefficients

aO -- -Ki l M I L

al = R/MIL + 2/M/T

a2= 3*RIM/L + 2/M/T

a3 = 3 *RIMIL-21M/T

a4-R/M/L-2/M/T
+Kx/

L+Kx

+Kx

Kx/

'Screen layout

XMAX +

'Borders

b0 = (2/T)"3 + (2/T)"2*(R/L + C/M) + 2/T*(C*R/M/L

M-Ki*Kc/L/M) + R*Kx/L/M

bl =-3*(2/T)"3-(2/T)"2*(R/L+ C/M) + 2/T*(C*R/M/

/M-Ki*Kc/L/M) + 3*R*Kx/L/M

b2 = 3*(2/T)"3-(2/T)A2*(R/L + C/M)-2/T*(C*R/M/L

/M-Ki*Kc/L/M) + 3*R*Kx/L/M

b3 =-(2/T)'3 + (2/T)"2*(R/L + C/M)-2/T*(C*R/M/L +

M-Ki*Kc/L / M) + R*Kx/L/M

tmax ---10

tmin - 0

XMAX - 2

xmin= 0

SCREEN 9

COLOR 14, 3

'VIEW (30, 125)-(610,320),8 'Original

VIEW (30,105)-(610,300),8

WINDOW (train- .01 * tmax, xmin - .02 * XMAX)-(tmax + .01 * tmax,

.02* XMAX)

LINE (train,

LINE (train,

LINE (train,

LINE (tmax,

'Horizontal lines

LINE (train,

LINE (train,

LINE (train,

x n)-(tmin,XMAX), 13

xmin)-(tmax, xmin),13

XM/OC)-(tmax,XMAX), 13

xmix)-(tmax, XMAX), 13

.25 * XMAX')-(tmax, .25 * XMAX), I0,, &HFF00

.5 * XMAX)-(tmax, .5* XMAX), I0,, &HFF00

.75 * XMAX)-(tmax, .'/5* XMAX), 10, ,&HFF00

'Verticallines

'Label

LINE (tmax * . 1,

LINE (tmax * .2,

LINE (tmax * .3,

LINE (tmax * .4,

LINE (tmax * .5,

LINE (tmax * .6,

LINE (tmax * .7,

LINE (tmax * .8,

LINE (tmax * .9,

xmin)-(tmax*. I,XMAX), 10,, &HFF00

xmin)-(tmax* .2,XMAX), 10,, &HFF00

xmin)-(tmax* .3,XMAX), I0,, &HFF00

xmin)-(tmax* .4,XMAX), I0,, &HFF00

xmin)-(tmax* .5,XMAX), I0,, &HFF00

xmin)-(tmax * .6, XMAX), 10,, &HFF00

xmin)-(tmax * .7, XMAX), 10,, &HFF00

xmin)-(tmax * .8, XMAX), 10,, &HFF00

xmin)-(tmax* .9,XMAX), I0,, &HFF00

LOCATE 12, 2: PRINT

LOCATE 13, 2: PRINT

LOCATE 14, 2: PRINT

LOCATE 15, 2: PRINT

LOCATE 16, 2: PRINT

LOCATE 17, 2: PRINT

LOCATE 18, 2: PRINT

LOCATE 19, 2: PRINT

LOCATE 2, 30: PRINT

-p.

NOM

WiM

NtN

him

mOB

mnN

"MSBS Simulation"

75
"Input step of position

'ref= 1

LOCATE 3, 2: PRINT TIMES

52 Total = Total + T

"GOSUB 100 "DPA, Tustin's Method (Appendix B)

GOSUB 1100 'PID, Tustin's Method (Appendix B)

'Total Error Sum

' SUError = ABS(Xpl - X) / ref + SUError

'Max. Overshoot or undershoot and Peak Time for Position Input

IF Total > 5 THEN 72

IF X > MAXX THEN 61

GOTO 63

61 IF MX1 = 0 THEN 62

IF X < MX1 THEN 73

62 MX1 = X

IF MX11 > MX1 THEN 73

MXll = MXI

MAXX - X

PCNT = X - 1

PTIME = Total

GOTO 73

63 IF MX2 = 0 THEN 64

IF X > MX2 THEN 65

64 MX2 =X

MXDI = ABS0- MX2)

IF MXD2 > MXD1 THEN 65

MXD2 --- MXD1

X1-X

TOT = Total

GOTO 73

65 MAXX = X1

PCNT - Xl - I

PTIME = TOT

72 IF Total < 5 THEN 73

AXX - PCNT

73

Feedback (Done)

Feed forward (Done)

"MX1 AND MXll ARE STORAGE POINTS'

'MX2 IS A STORAGE POINT

'MXD1 AND MXD2 ARE DISTANCES

'BETWEEN THE TWO PEAKS

'Max. Overshoot and Peak Time for Force Input

IF Total > 5ANDX > MAX2 THEN MAX2 = X

IF X - MAX2 THEN PTIME2 = Total

'Rise Time

RISE

IF X < - (.1 * ref) THEN RT1 -- Total

IF jj = 1 THEN GOTO 98

IF X > = (.9 * ref) THEN jj = 1

IF X > - (.9 * ref) AND jj = 1 THEN RT2 = Total

RISE = RT2- RT1

LOCATE 6, 15: PRINT "Rise Time = ": LOCATE 6, 27: PRINT USING "##.###";

76
98

MAXX

' LOCATE

PTIME

' LOCATE

PTIME2

• LOCATE

' LOCATE

• LOCATE

MAX2

'Position Input

-x) <

• LOCATE 9, 13: PRINT "Overshoot 1 ="' LOCATE 9, 27: PRINT USING "##.###';

7, 13: PRINT "Peak Time 1 -': LOCATE 7, 27: PRINT USING "##.###";

7, 43: PRINT "Peak Time 2 = ": LOCATE 7, 57: PRINT USING "##.###";

5, 20:

5, 46:

9, 43:

PRINT "Time =': LOCATE 5, 26: PRINT USING " ##.###'; Total

PRINT "Position =": LOCATE 5, 57: PRINT USING "##.###"; X

PRINT "Overshoot 2 =": LOCATE 9, 57: PRINT USING "##.###";

Settling Time

p = .001

IF iii = 1 THEN GOTO 59

IF (ABS(Xpl - X) < p * X) AND (ABS(Xp2 - X) < p * X) AND (ABS(Xp3

p * X) AND (ABS(Xp4 - X) < p * X) AND (ABS(Xp5 - X) < p * X) AND

(ABS(XP6 - X) < p * X) AND (ABS(Xp7 - X) < p * X) AND (ABS(Xp8 - X) < p *

X) THEN SETYIME = Total

IF SETI'IME = Total THEN jjj = 1

' LOCATE 8, 9: PRINT "Settling Time 1 = ": LOCATE 8, 27: PRINT USING "##.###

"' SETHME

59 •Force Input Settling Time

pp = .00O5

IF jjjj = 1 THEN GOTO 70

IF fd > 1 THEN GOTO 60 ELSE GOTO 70

60 IF ABS(X - ref) / ref < pp AND ABS(Xpl - ref) / ref < pp AND ABS(Xp2 - re0 / ref

< pp AND ABS(Xp3 - ret) / ref < pp AND ABS(Xp4 - ref) / ref < pp AND

ABS(Xp5 - ref) / ref < pp AND ABS(XP6 - ref) / ref < pp AND ABSCXp7 - ref) / ref

< pp AND

ABS(Xp8 - re0 / ref < pp THEN SETHME2 = Total

IF SETHME2 = Total AND Total > 6 THEN jjjj = 1

' LOCATE 8, 39: PRINT "Settling Time 2 = ": LOCATE 8, 57: PRINT USING "##.###

"" SETHME2 9

'Shift the variables back in time

7O fdp3 = fdp2

fdp2 = fdpl

fdpl -- fd

Xp8 = Xp7

Xp7 = XP6

XP6 - Xp5

Xp5 = xp4

Xp4 = xp3

Xp3 = Xp2

Xp2 = Xpl

XplfX

Ep3 = Ep2

Ep2 = Epl

EplfE

Vp3 = Vp2

77
Vp2= Vpl

Vpl = V

PSET (Total, X + 1), 12:

COUNT - COUNT + I

LOCATE 23, 40: PRINT COUNT

PRINT #1, USING "###.#######"; X

'Input step of force

' IF Total > 5 THEN fd = 10

8O

SUMX = SUMX + X

XA(COUm3 = X

IF Total > tmax AND Total < tmax + T THEN GOTO 80 ELSE GOTO 52

LOCATE 24, 37: PRINT "Time"

AVG = SUNIX I COUNT

FOR LK = 1 TO COUNT

XTX = (XA(LK) - AVG) " 2

SUNIXTX = SUMXTX + XTX

NEXT LK

RMS = SQR(SUMXTX / COUNT) ' < ----- The standard deviation

88

LOCATE 6, 25: PRINT "The RMS value = ";USING "##.####";RMS

CLOSE #I

END

"Subrountines

100 "DualPA, Tustin'sMethod (Appendix B)

'Thiscontrollerislocatedinthe feedbackpath

IF first= I GOTO II0

K = -2784

LOCATE 3, 21: PRINT "Tustin'sMethod, Dual Phase Advance"

LOCATE 4, 35: PRINT "K= ";K

gain- 1

Kin - .5

A = .01

N= 10

c0-- (T'T+ 4*N*A*T+4*N*N*A*A)/(T*T + 4*A*T +

4*A *A)

cl = (2*T*T-8*N*N*A*A)/(T*T + 4*A*T + 4*A'A)

c2 = (T*T-4*N*A*T + 4*N*N*A*A)/(T*T + 4*A*T +

4*A *A)

c3 = (2 *T'T- 8 *A * A) / (T* T + 4 *A *T + 4 *A *A)

c4- (T*T-4*A*T + 4*A*A)/(T*T + 4*A*T + 4*A'A)

dl = Kin * T

78
110

a4 * fdp3

CNT = CNT + 1

E = ref*gain-G

Etotal = E + Etotal

Z =E +dl*Etotal

V=K*Z

X = (a0* (3/+ Vpl + Vp2 + Vp3) + al *fd + a2* fdpl + a3* fdp2 +

-bl*Xpl-b2*Xp2-b3*XpB)/bO

X = X + GS(CNT) * .0001

G = c0*X + cl*Xpl + c2*Xp2-c3*Gpl-c4*Gp2

Gp2 = Gpl

Gpl = G

first = 1

RETURN

1100 'PID Tustin's Method, (Appendix B)

"This controller is located in the feed forward path.

IF first - 1 GOTO 1110

K = -396

LOCATE 3, 28: PRINT "PID Tustin's Method"

LOCATE 4, 35: PRINT "K--"; K

gain = 1

Kp=l

Kd = .4

Kin = .5

cl = Kp + 2*Kd/T+ T'Kin/2

c2 = T* Kin-4* Kd/T

c3 = T'Kin/2 + 2*Kd/T-Kp

1110 CNT = CNT + 1

E = ref * gain - G

Z = c l*E + c2*Epl + c3*Ep2 + Zp2

VffiK*Z

a4 * fdp3

G=X

X= (aO*(V +Vpl + Vp2 + Vp3) + al*fd + a2*fdpl + a3*fdp2 +

-bl*Xpl-b2*Xp2-b3*Xp3)/bO

X = X + GS(CNT) * .0001

Zp2 = Zpl

Zpl = Z

first= 1

RETURN

79