Intelligent Control Using Confidence Interval Networks: Applications to Robust Control of Active Magnetic Bearings

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Abstract

CHOI, HEEJU. Intelligent Control Using Confidence Interval Networks: Appli-cations to Robust Control of Active Magnetic Bearings. (Under the direction of Dr. Gregory D. Buckner.)

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by

Heeju Choi

A dissertation submitted to the Graduate Faculty of North Carolina State University

in partial satisfaction of the requirements for the Degree of

Doctor of Philosophy

Department of Mechanical and Aerospace Engineering

Raleigh March 2005

Approved By:

Dr. Richard F. Keltie Dr. Paul I. Ro

Dr. Stephen L. Campbell Dr. Fen Wu

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ii

To my wife Jinkyung Heo and my son Aaron Jichang Choi

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Biography

Heeju Choi was born in Gwangju, South Korea on November 23, 1973. He earned

a B.S. degree in Mechanical Engineering in 1998 from Chonnam National University

and a M.S. degree in Mechatronics in 2000 from Gwang-Ju Institute of Science and

Technology, South Korea. He began his doctoral work at North Carolina State

Uni-versity in 2001, and earned the Ph.D. degree in 2005. He enjoys spending most time

in sports, such as swimming, jogging, golf, tennis, racquetball, Taekwondo, archery,

rock climbing, etc....

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Acknowledgements

I respectfully and gratefully acknowledge some of those who made this work possible:

To Dr. Gregory Buckner, my advisor, I would like to extend my sincere thanks. You

are the man who guided me through the dark tunnel and helped me discover the light at its

end. I especially appreciate your help and encouragement during my difficulties of August

2003. I will always remember you not only as a great advisor, but a wonderful friend.

To my committee members Dr. Richard Keltie, Dr. Paul Ro, Dr. Stephen Campbell,

and Dr. Fen Wu, who graciously afforded me their time and wisdom. I would like to thank

each of them for their guidance and service. I would also like to thank Dr. Helen Zhang

and Dr. Bei Lu for their contributions to this research.

To the National Science Foundation and Dr. Paul Werbos, who generously sponsored

this research.

To Nathan Gibson, my best friend and respected colleague, who helped my every need

and answered my questions. Without your friendship and support, life in the lab would

have been unbearable. I will invite you to South Korea sometime.

To my longtime friends Aaron Kiefer (triathlete), Michael Craft (car racer), Donald

Caulfield (guitarist and bug’s life), Jason Stevens (bluegrass musician), Soheil Saadat

(in-tellectual), the Carmon family (international friendship), Jeongbeom Ma (God Father), and

Junemo Koo (comedian) my thanks and best wishes.

And most of all, to my wife Jinkyung Heo, my son Aaron Choi, and my parents who

provided love and support during this demanding journey. It’s over. I’m done. Be happy!

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7 Conclusions 80 List of References 82 A MATLAB codes: Modal Analysis 88 B MATLAB codes: System Identification 91 C MATLAB codes: Robust Control Synthesis 94 D Proof of Network Convergence Using Bayes Estimation Theory 97 E Proof of Confidence Interval Networks 99 E.1 Training Convergence Using The Quadratic Error Cost Function . . . 100

E.2 Training Convergence Using The Bilinear Error Cost Function . . . . 102

E.2.1 Markov Chain Framework . . . 103

E.2.2 Stationary Distribution . . . 104

F MATLAB codes: Intelligent Uncertainty Identification 109 G MATLAB codes: LTI Intelligent Robust Control Synthesis 114 H MATLAB codes: LPV Intelligent Robust Control Synthesis 117

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List of Tables

2.1 Natural frequency comparison according to the number of FEM nodes 13

2.2 Comparison of natural frequency versus model types . . . 20

3.1 Tracking performance (SSE) vs. rotational speed: Nominal robust

control . . . 27

4.1 Information of intelligent uncertainty identification . . . 39

5.1 Tracking performance (SSE) vs. rotational speed: LTI intelligent

ro-bust control . . . 43

5.2 Tracking performance (SSE) vs. rotational speed: Robustness tests

using nominal robust control . . . 50

using LTI intelligent robust control . . . 50

6.1 Tracking performance (SSE) vs. rotational speed: LPV intelligent

robust control . . . 67

using LPV intelligent robust control . . . 73

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List of Figures

2.1 The Combat Hybrid Power System flywheel alternator, with “inside

out” AMB topology [17,18]. . . 6

2.2 AMB operating principle . . . 7

2.3 Radial AMB . . . 8

2.4 Revolve MBRotor AMB system . . . 9

2.5 Generalized rigid rotor supported by two radial bearings . . . 10

2.6 Finite Element Method model . . . 12

2.7 System identification diagram . . . 17

3.1 k-step ahead prediction for k = 10 . . . 22

3.2 k-step ahead prediction for k = 20 . . . 23

3.3 H∞ system interconnection . . . 24

3.4 Nominal weighting functions . . . 25

3.5 Tracking performance (SSE) vs. rotational speed: Nominal robust control . . . 27

3.6 Nominal robust control based on analytical model: non-driven end AMB orbit versus rotational speed . . . 28

3.7 Nominal robust control based on analytical model: driven end AMB orbit versus rotational speed . . . 28

3.8 Nominal robust control based on FEA model: non-driven end AMB orbit versus rotational speed . . . 29

3.9 Nominal robust control based on FEA model: driven end AMB orbit versus rotational speed . . . 29

3.10 Nominal robust control based on SysId model: non-driven end AMB orbit versus rotational speed . . . 30

3.11 Nominal robust control based on SysId model: driven end AMB orbit versus rotational speed . . . 30

4.1 Control and model error modeling diagram . . . 32

4.2 Architecture of a radial basis function network . . . 35

4.3 Multiple-time sampled data with quantiles . . . 36

4.4 Intelligent model error identification using a non-recurrent RBFN . . 38

4.5 CIN estimate of 99.9% uncertainty bound at 6.0 krpm . . . 40

5.1 Comparison of nominal W∆ and CIN W∆ . . . 42

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5.2 Tracking performance (SSE) vs. rotational speed: LTI intelligent

ro-bust control . . . 44

5.3 LTI intelligent robust control based on analytical model: non-driven

end AMB orbit versus rotational speed . . . 45

5.4 LTI intelligent robust control based on analytical model: driven end

AMB orbit versus rotational speed . . . 45

5.5 LTI intelligent robust control based on FEA model: non-driven end

5.6 LTI intelligent robust control based on FEA model: driven end AMB

orbit versus rotational speed . . . 46

5.7 LTI intelligent robust control based on SysId model: non-driven end

AMB versus rotational speed . . . 47

5.8 LTI intelligent robust control based on SysId model: driven end AMB

5.9 Balancing disk with 1 g additional mass for robustness test . . . 48

5.10 Tracking performance (SSE) vs. rotational speed: Robustness tests . 49

5.11 Robustness test using nominal robust controller based on analytical

model: non-driven end AMB orbit versus rotational speed . . . 51

5.12 Robustness test using nominal robust controller based on analytical

model: driven end AMB orbit versus rotational speed . . . 51

5.13 Robustness test using nominal robust controller based on FEA model:

non-driven end AMB orbit versus rotational speed . . . 52

5.14 Robustness test using nominal robust controller based on FEA model:

driven end AMB orbit versus rotational speed . . . 52

5.15 Robustness test using nominal robust controller based on SysId model:

non-driven end AMB versus rotational speed . . . 53

5.16 Robustness test using nominal robust controller based on SysId model:

driven end AMB orbit versus rotational speed . . . 53

5.17 Robustness test using LTI intelligent robust controller based on

ana-lytical model: non-driven end AMB orbit versus rotational speed . . . 54

5.18 Robustness test using LTI intelligent robust controller based on

ana-lytical model: driven end AMB orbit versus rotational speed . . . 54

5.19 Robustness test using LTI intelligent robust controller based on FEA

5.20 Robustness test using LTI intelligent robust controller based on FEA

5.21 Robustness test using LTI intelligent robust controller based on SysId

model: non-driven end AMB versus rotational speed . . . 56

5.22 Robustness test using LTI intelligent robust controller based on SysId

5.23 Updated CIN estimate of uncertainty bound at 6.0 krpm . . . 57

5.24 Comparison of uncertainty weighting functions at 6.0 krpm . . . 58

5.25 Tracking performance (SSE) vs. rotational speed: FEA model . . . . 59

5.26 Robustness test using updated LTI intelligent robust controller based

on FEA model: non-driven end AMB orbit versus rotational speed . . 59

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5.27 Robustness test using updated LTI intelligent robust controller based

on FEA model: driven end AMB orbit versus rotational speed . . . . 60

6.1 XlrotorT M_{: Varying eigenvalues versus rotor speed . . . .} ₆₂

6.2 CINs uncertainty bounds . . . 63

6.3 LPV gain-scheduled H∞ problem . . . 65

6.4 Tracking performance (SSE) vs. rotational speed: LPV intelligent

robust control . . . 68

6.5 LPV intelligent robust control based on Analytical model: non-driven

end AMB orbit versus rotational speed . . . 68

6.6 LPV intelligent robust control based on Analytical model: driven end

6.7 LPV intelligent robust control based on FEA model: non-driven end

6.8 LPV intelligent robust control based on FEA model: driven end AMB

6.9 LPV intelligent robust control based on SysId model: non-driven end

6.10 LPV intelligent robust control based on SysId model: driven end AMB

6.11 Tracking performance (SSE) vs. rotational speed: Robustness tests . 72

6.12 Robustness test using LPV intelligent robust controller based on

ana-lytical model: non-driven end AMB orbit versus rotational speed . . . 73

6.13 Robustness test using LPV intelligent robust controller based on

ana-lytical model: driven end AMB orbit versus rotational speed . . . 74

6.14 Robustness test using LPV intelligent robust controller based on FEA

6.15 Robustness test using LPV intelligent robust controller based on FEA

6.16 Robustness test using LPV intelligent robust controller based on SysId

model: non-driven end AMB versus rotational speed . . . 75

6.17 Robustness test using LPV intelligent robust controller based on SysId

6.18 Comparison of updated CIN uncertainty weighting functions . . . 77

6.19 Tracking performance (SSE) vs. rotational speed: Analytical model . 78

6.20 Robustness test using updated LPV intelligent robust controller based on analytical model: non-driven end AMB orbit versus rotational speed 78 6.21 Robustness test using updated LPV intelligent robust controller based

on analytical model: driven end AMB orbit versus rotational speed . 79

E.1 Quadratic error cost function. . . 100

E.2 Symmetric bilinear error cost function (c1 =c2 = 1). . . 103

E.3 Asymmetric bilinear error cost function (c1 = 10, c2 = 1). . . 104

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List of Symbols

Nomenclature for Chapters 2,3,5, and 6

Scalars

x, y longitudinal displacement of radial active magnetic bearings

z axial displacement of thrust active magnetic bearing

i coil current

m rotor weight

Ix, Iy polar mass inertia of rotor

Iz axial mass inertia of rotor

ks displacement stiffness

ki current stiffness

k_c coupling stiffness

Ω nominal rotor speed

ω natural frequency

λ eigenvalues

n number of nodes

K_P proportional control gain

KI integral control gain

K_D derivative control gain

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q number of input in system identification

p number of output in system identification

us input of system identification

y_s output of system identification

e prediction error in system identification

η neural network learning rate, 0< η <1

N data points during one cycle of rotor orbit

T discrete sampling time

p(t) rotor speed according to time

γ finite positive number between 0 and 1

Vectors and matrices

x state vector of state space representation

y output vector of state space representation

z_B rotor displacements

A system matrix

B input matrix

C output matrix

Ks displacement stiffness matrix

K_i current stiffness matrix

G gyroscopic matrix

M global mass matrix

K global stiffness matrix

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r displacement vector in physical coordinates

F force vector in physical coordinates

P modal matrix

q displacement vector in generalized coordinates

Q generalized forces

Am system matrix obtained from modal analysis

Bm input matrix obtained from modal analysis

Cm output matrix obtained from modal analysis

θ parameter vector in system identification

Ao nominal system matrix

A_p₍_t₎ speed-dependent system matrix

z controlled output vector

w external disturbances or noise

A(p(t)) speed-dependent system matrix

G(p(t)) speed-dependent gyroscopic matrix

K(p(t)) speed-dependent controller matrix

Functions

Go nominal transfer function

G∆a additive uncertainty function

G_∆_m multiplicative uncertainty function

W∆ uncertainty weighting function

W_in control input weighting function

Wp performance weighting function

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Nomenclature for Chapter 4 and proofs in Appendix only

Variables and Constants

c1, c2 penalties of positive and negative errors

e prediction error of a neural network, also: exponential constant

np number of basis functions (hidden layer neurons)

q order of a quantile, 0< q <1

cp activation center of thepth hidden layer neuron

sp activation width of thepth hidden layer neuron

U random number

w neural network weight vector with components wp

c RBFN normalized activation center

x input of a neural network

y(x) observation ofy corresponding tox

ˆ

y(x) output (estimate) of a neural network corresponding to x

η neural network learning rate, 0< η <1

µ(x) mean function ofy(x)

ν(x) quantile function ofy(x)

σ(x) square root of variance function

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Functions

B Bayes cost

Bin Binomial distribution

E mathematical expectation

f probability density function

F distribution function

J error cost function

N Normal distribution

P transition function

V variance

φ characteristic function

π stationary distribution

Ψ vector of radial basis functions with componentsψp

Other Notation

R set of real numbers

Z set of integers

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Chapter 1 Introduction

1.1 Background and Motivation

The development of robust control systems for Active Magnetic Bearings (AMBs) is an

attempt to provide guaranteed stability despite uncertainties and disturbances associated

with the plant. Robust control synthesis requires an explicit description of the plant

dy-namics (a model) and uncertainty bounds associated with that model. Model uncertainty

may result from under-modeled dynamics, parameter variations, and disturbances. The

requisite uncertainty bounds, however, are usually chosen arbitrarily and conservatively to

guarantee stability at the expense of performance.

There are two common robust control approaches for AMB systems: fixed-gain

con-trol and gain-scheduling. Fixed-gain µ-control of a flexible rotor system was investigated

by Nonami, et al. [1], who demonstrated stability robustness to variations in rotor mass.

However, fixed-gain controllers for gyroscopic rotors can experience significant performance

degradations when the rotor speed differs from the fixed design condition. Gain-scheduled

H∞control of a rigid rotor system was investigated by Matsumura and Mohamed [2] [3], but

switching and interpolation schemes eliminate the robust stability and performance

guar-antees. Additionally, gain-scheduling approaches suffer from a “slow varying parameter”

requirement [4][5]. Gain-scheduled controllers can render the closed-loop system unstable

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Chapter 1. Introduction 2

when the rotor speed changes rapidly [3].

Recently, a robust stability analysis and systematic gain-scheduling design technique for

linear systems affected by time-varying parametric uncertainties has been developed in the

form of Linear Parameter Varying (LPV) control theory [6][7][8][9][10]. LPV models can

be characterized as convex sets specifying both structured uncertainty and linearly varying

parameters (usually associated with rotor rotational speed). This LPV approach provides

guaranteed stability and performance and simplifies the interpolation and realization

prob-lems associated with conventional gain-scheduling. LPV control has been applied to AMB

systems using rigid rotor models [11][12] and flexible rotor models [13][14]. However,

gyro-scopic effects are frequently ignored and uncertainty weighting functions are usually specified

in an arbitrary and overly conservative manner.

The uncertainty associated with linear models (especially LPV models) can be difficult

to quantify, and few formal techniques exist for estimating the uncertainty bounds so critical

for robust control synthesis. Recently, however, research has focused on improving the

ac-curacy of uncertainty bounds associated with dynamic models. Specific approaches include

stochastic embedding, which relies on the statistical properties of an identified model [15];

set membership, an identification procedure that estimates a set of feasible models which

in turn provides the uncertainty quantification related to the nominal model [16][17][18];

and model error modeling, which identifies a linear mapping from inputs to modeling

er-ror [17][19][20].

The goal of this research is to demonstrate an alternative approach to estimating

uncer-tainty bounds, an “intelligent” approach, that can be used with LPV models to synthesize

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1.2 Scope of Work

This research demonstrates the Confidence Interval Network (CIN), a unique artificial

neu-ral network (ANN) that utilizes asymmetric bilinear error cost functions, to estimate

un-certainty bounds for the synthesis of robust AMB controllers. A high-speed flexible rotor

supported by AMBs is modeled using analytical approaches, finite element analysis (FEA),

and system identification. CINs “learn” the statistical bounds of model uncertainty

result-ing from unmodeled dynamics and parameter variations. These bounds are incorporated

into the robust control synthesis process. Experimental results on a multivariable AMB

test rig reveal the benefits of this combination of intelligent system identification and

ro-bust control: significant performance improvements vs. conventional roro-bust control with

and without mass imbalance (process disturbances).

This dissertation is structured as follows:

Chapter 2 - Modeling of Active Magnetic Bearing Systems. Three dynamic mod-els are investigated in this chapter. The first model is a low-order, lumped-parameter

model which is obtained analytically. The second is a flexible rotor model obtained

using finite element method and modal analysis. The last is a flexible rotor model

obtained from experimental system identification.

Chapter 3 - Robust Control of Active Magnetic Bearings. A detailed description of the “nominal” robust control synthesis process is presented. Experimental results

using all three models are presented. Nominal performance results motivate the need

for the quantification of model error.

Chapter 4 - Intelligent Uncertainty Identification Using Confidence Interval Networks.

The statistical characteristics and convergence properties of CINs are presented.

Un-like conventional ANNs, networks trained using the bilinear error cost function

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conditional mean. The use of CINs to estimate uncertainty bounds for robust control

synthesis is presented.

Chapter 5 - Intelligent Control: Linear Time Invariant Approach. A conceptually new approach for bounding model uncertainty for robust control is presented.

Exper-imental results using this “intelligent robust” control approach are compared to the

“nominal” robust control case.

Chapter 6 - Intelligent Control: Linear Parameter Varying Approach. Systematic tools to design gain-scheduled controllers for speed-dependent AMB systems is

pre-sented. Experimental results are compared to the “nominal” robust control case.

Chapter 7 - Conclusions. Summarizes the contributions of this dissertation and com-ments on future research directions extending from it.

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Chapter 2 Modeling of Active Magnetic Bearing

Systems

2.1 Overview of Active Magnetic Bearings

AMBs provide non-contacting support of rotors, eliminating concerns with lubrication and

wear and making them attractive for a variety of high-speed applications [21]. One

poten-tial application is the Flywheel Energy Storage Systems (FESS), which may soon replace

batteries on Earth-orbit satellites. Based on the current payload premiums ($15,000 perlb

[22]), FESS could ultimately save millions of dollars in satellite launch costs. Commercial

satellites require 2,400lbsof chemical batteries, which could be replaced with an equivalent

FESS weighing only 720 lbs [22]. FESS applications for terrestrial vehicles include trains,

buses, and combat vehicles. The Combat Hybrid Power Systems (CHPS) flywheel

alterna-tor (Figure 2.1) developed at the University of Texas Center for Electromechanics, salterna-tores

25.0M J of energy at its peak operating speed of 20,000rpmand delivers up to 5.0M W of

electrical power (350kW continuous) to meet the needs of future combat systems [23][24].

Its 318 kg (700 lb) rotor is supported using an innovative “inside out” AMB topology in

which the flywheel rotor is located outside the stationary bearing components.

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Chapter 2. Modeling of Active Magnetic Bearing Systems 6

Figure 2.1: The Combat Hybrid Power System flywheel alternator, with “inside out” AMB topology [17,18].

Despite their advantages, AMBs are inherently unstable devices that require

sophisti-cated real-time controllers. The modal characteristics and gyroscopic nature of high-speed

rotors further complicate the control issues and have hindered commercial adoption of this

technology. For this reason, the development of robust control algorithms for AMBs has

been an active field of research in recent years.

A representation of a single-input, single-output (SISO) magnetic bearing is shown in

Figure 2.2. The control objective is to manipulate the coil currenti(t) so that the vertical

position of the rotor x(t) tracks the desired trajectory(or regulates to center the bearing

at x = 0). When multiple actuators of this type are arranged circumferentially, as shown

in Figure 2.3, the assembly becomes a radial magnetic bearing, which controls the rotor motion in the radial direction. This bearing does the majority of the stabilization and

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N-turn coil

Control Current i(t)

Rotor

Displacement x(t)

Displacement Sensor

High-Permeability Core

High-Permeability

Lamination

Magnetic Flux

Figure 2.2: AMB operating principle

centered so that the radial bearings can transmit forces appropriately.

The experimental research described in this dissertation required several critical AMB

system capabilities, most importantly the ability to implement and test customized

con-trollers (requiring access to all input and output signals). Figure 2.4 shows the MBRotor

Research Test Stand from Revolve Magnetic Bearings, Inc. which was the only commercial

AMB system to satisfy these requirements. The steel rotor has a mass of 1.549 kg and

is 0.457 m in length, and has two steel disks that can be positioned to modify the modal

characteristics at high speeds. A thrust AMB regulates the axial position (z direction) of

the rotor using PID control. Two radial AMBs are located at the ends of the rotor,

orthog-onally aligned in the x and y directions (Figure 2.5), and two orthogonal pairs of sensors

measure rotor displacements from the bearing centerline. These radial AMBs comprise a

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ROTOR

x

y

Figure 2.3: Radial AMB

application focus of this research.

The following sections introduce three dynamic models of the multi-input, multi-output

(MIMO) radial rotordynamic system derived using analytical approaches, FEA, and system

identification.

2.2 Analytical Modeling

This modeling approach used to derive rigid-body state equations is presented in detail in

Gibson, et al. [25][26]. The linearized system dynamics for the AMB system, obtained using

a Lagrangian analysis of Figure 2.5, can be represented using a state vectorxcomposed of the rotor displacementszB and their time derivatives

˙

x=Ax+Bu y=Cx

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Figure 2.4: Revolve MBRotor AMB system

where

x=

 

 zB

˙

zB

 

, A=

 

 0 I

M−_B1Ks −M−_B1GB

 

, B=

 

 0

M−_B1Ki

 

, C=

·

I 0

¸ ,

z_B =

           xa x_b ya yb           

, K_s=

          

ks 0 0 0

0 k_s−k_c 0 0

0 0 ks 0

0 0 0 ks−kc

          

, K_i=

          

ki 0 0 0

0 k_i 0 0

0 0 ki 0

0 0 0 ki

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v=TZzB, MB=T_F−1MTZ, GB =T−_F1GTZ,

v=            x β y α           

, TZ= _l 1

a+lb

          

l_b la 0 0

1 −1 0 0

0 0 l_b l_a

0 0 −1 1

          

, TF =

          

1 1 0 0

la −lb 0 0

0 0 1 1

0 0 −la lb

           , M=           

m 0 0 0

0 Iy 0 0

0 0 m 0

0 0 0 Ix

          

, G= Ω

          

0 0 0 0

0 0 0 −Iz

0 0 0 0

0 Iz 0 0

           (2.2)

with system parametersm= 1.54kg,Ix =Iy = 2.39×10−2kg·m2,Iz = 4.12×10−4 kg·m2,

la = 0.153 m, lb = 0.170m, ks = −96.5×103 N/m, ki = 29.9 N/A, kc = 2.6×103N/m,

and Ω = 627.0rad/sec(nominal speed: 6.0 krpm).

l_a l_b

g

F_g z

F_ax

F_ay Fby

F_bx k cy a b x y k_cx

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For a nominal rotational speed of 6.0 krpm, the resulting model is:

˙ x=              

04×4 I4×4

1.7×105 ₋₅_.₀_×₁₀4 ₀ ₀ ₀ ₀ ₋₅_.₂ ₅_.₂

−5.0×104 ₁_.₈_×₁₀5 ₀ ₀ ₀ ₀ ₅_.₆ ₋₅_.₆

0 0 1.7×105 ₋₅_.₀_×₁₀4 ₅_.₂ ₋₅_.₂ ₀ ₀

0 0 −5.0×104 1.8×105 −5.6 5.6 0 0

              x +              

04×4

4.9×101 ₋₁_.₅_×₁₀1 ₀ ₀

−1.5×101 ₅_.₅_×₁₀1 ₀ ₀

0 0 4.9×101 −1.5×101

0 0 −1.5×101 ₅_.₅_×₁₀1

              u y= ·

I4×4 04×4

¸

x+

·

04×4

¸

u

(2.3)

This eighth-order continuous-time model is unstable, with eigenvalues (at 6.0 krpm)

λ=−471±3.6i,−351±0.006i,351±0.006i,471±3.6i (2.4)

Note that these eigenvalues are dependent on rotor speed due to the gyroscopic terms in

(2.1).

2.3 Finite Element Analysis Modeling

A flexible rotor model was constructed using ANSYS version 7.1. The rotor geometry and

material properties of Figure 2.4 are modeled using three-dimensional Solid45 elements,

as shown in Figure 2.6. Each element was constructed of steel with an elastic modulus

of 2.06E11[Pa], a density of 7783 kg/m3_{, and a Poisson’s ratio of 0.32. Each node has 3}

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rotor was constrained to zero displacement in all directions to prevent rigid body motion.

A static analysis was performed to create mass and stiffness matrices for the modal analysis

to follow.

Figure 2.6: Finite Element Method model

The number of FEA nodes was determined by trial and error to optimize the

accu-racy/computational tractability tradeoff. Cases of 187, 411, 761, and 952 nodes were solved

and compared to examine the effect of mesh density on model accuracy. As expected,

increasing the number of nodes improved the accuracy of the model’s eigenvalues

(deter-mined through experimental impact hammer testing), as shown in Table 2.1. However, as

the number of nodes increase, the computational burden increases significantly (particularly

in subsequent modal analysis calculations, which require mass and stiffness matrices). The

model with 761 nodes was chosen to optimize this trade-off, as its first bending mode was

close enough to the experimental measurement. As an aside, the 11th _{and 16}th _{modes not}

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Table 2.1: Natural frequency comparison according to the number of FEM nodes

Modes Impact test 187 nodes 411 nodes 761 nodes 952 nodes 2496 nodes

– (Hz) (Hz) (Hz) (Hz) (Hz) (Hz)

1∼6 0 0 0 0 0 0

7∼8 107.5 439.2 210.1 121.5 118.5 113.7

9∼10 293.0 516.4 544.6 339.2 320.7 293.1

12∼13 527.5 1021.5 749.1 422.4 435.4 354.5

14∼15 946.5 1191.3 1201.7 803.7 797.9 699.0

17∼18 1182.0 1884.6 1766.8 932.4 1011.1 1161.4

Modal Reduction

From the ANSYS data, the global mass [M] and stiffness [K] matrices were initially very large, 2283×2283 elements because each of the 761 nodes have 3 associated DOF. For

com-putational tractability, it was necessary to reduce the size of these matrices using modal

analysis techniques. Modal analysis is well-known technique used to characterize elastic

structures according to their modal parameters, i.e., their natural frequencies and

vibra-tion modes. In this method, the expansion theorem is used, and the displacements of the

masses are expressed as linear combinations of the normal modes of the system. The linear

transformation uncouples the equations of motion so that a set of n uncoupled differential

equations of second order is obtained. Using the FEA data and modal analysis, a reduced

order model containing three rigid body modes and one (the lowest natural) flexible mode

can be obtained by truncating the negligible higher order modes. The modal analysis

tech-nique is briefly summarized below, details can be found in [27][28].

A mathematical model of a multi DOF flexible structure subject to external forces can

be represented:

[M]¨r+ [K]r=F (2.5)

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·,0,0]T

3n×1, andn= 761. By solving the eigenvalue problem of (2.5), a 3n×1 modal matrix

Pcan be obtained and the modal transformation matrix becomes:

r(t) =Pq(t)

q(t) = [q1(t), q2(t),· · ·, q3n(t)]T

(2.6)

whereq1(t), q2(t),· · ·, q3n(t) are time-dependent generalized coordinates, also known as the

principal coordinates or modal participation coefficients. And PT ₌ _P−1 _since _P _{is an}

orthogonal matrix. Using (2.6), (2.5) becomes:

PT[M]Pq¨+PT[K]Pq=PTF (2.7)

If the normal modes are mass normalized:

PT[M]P= [I] (2.8)

Ω=PT[K]P=

       -w2 &        (2.9)

(2.7) can be expressed, using (2.8) and (2.9), as

¨

q+Ωq=Q (2.10)

where Q=PT_F _{is the vector of generalized forces associated with the generalized} coordi-nates q. (2.10) denotes a set of 3nuncoupled differential equations of second order:

¨

qi(t) +w2iqi(t) =Qi(t), i= 1,2, ...,3n (2.11)

Note thatwi is usually sorted in ascending order.

Finally, the state space representation based on 4-input, 4-output, 761 node model is

A_m=

 

 02283×2283 I2283×2283

−Ω2283×2283 02283×2283

  

4566×4566

B_m=

 

 02283×4

Q2283×4

  

4566×8

(2.12)

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of an undamped single DOF system. The 3nresponses in the principal coordinate system

can be transformed back to the physical coordinate system using:

q(t) =PTr(t) (2.13)

wherePT =P−1 sinceP is an orthogonal matrix. Thus, the output matrix becomes:

Cm =

·

P₂₂₈₃_×₂₂₈₃ 0₂₂₈₃_×₂₂₈₃

¸

2283×4566

Dm=02283×8 (2.14)

Note that (2.12) does not contain acceleration terms, thus it is not accurate for rapid

changes in rotational speed. Also note that, because the reduced model is backtransformed

(represented in the physical coordinate system), the same gyroscopic dynamics appearing

in the analytical model (2.1) can be included in (2.12).

Because the MIMO AMB test rig exhibits only one flexible mode in its speed range

(0-10,000 rpm, 0-166.67Hz), it is only necessary that the reduced-order model exhibit modal

characteristics up to 166.67 Hz. By truncating the higher frequency modes in (2.12) and

(2.14), an eighth-order model with 3 rigid body modes and 1 flexible mode can be obtained.

This reduced-order FEA model (2.16) is unstable, as expected, with eigenvaluesλi at:

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Chapter 2. Modeling of Active Magnetic Bearing Systems 16 ˙ x=              

04×4 I4×4

−4.8×105 ₋₈_.₇_×₁₀−4 ₋₈_.₄_×₁₀−4 ₃_.₀_×₁₀−4 ₀ ₀ ₋₅_.₂ ₅_.₂

−9.2×10−4 ₉_.₈_×₁₀4 ₃_.₇_×₁₀−3 ₋₆_.₈_×₁₀−4 ₀ ₀ ₅_.₆ ₋₅_.₆

−8.1×10−4 ₃_.₈_×₁₀−3 ₁_.₀_×₁₀5 ₁_.₂_×₁₀−3 ₅_.₂ ₋₅_.₂ ₀ ₀

3.0×10−4 −6.7×10−4 1.2×10−3 1.0×105 −5.6 5.6 0 0

              x +              

04×4

3.1×101 ₋₆_.₈_×₁₀−1 ₋₈_.₉_×₁₀−1 ₅_.₃_×₁₀−1

−1.5×10−1 ₃_.₁_×₁₀1 ₋₅_.₄_×₁₀−1 ₋₈_.₈_×₁₀−1

−6.1×10−1 ₂_.₇_×₁₀−1 ₃_.₁_×₁₀1 ₄_.₄_×₁₀−1

−5.2×10−1 ₂_.₈_×₁₀−1 ₃_.₅_×₁₀−1 ₂_.₈_×₁₀1

              u y=           

1.2×100 ₆_.₅_×₁₀−1 ₋₆_.₁_×₁₀−1 ₋₅_.₂_×₁₀−1

5.0×10−2 ₁_.₂_×₁₀0 ₂_.₇_×₁₀−1 ₂_.₈_×₁₀−1

9.2×10−2 ₋₅_.₄_×₁₀−1 ₁_.₂_×₁₀0 ₃_.₅_×₁₀−1

1.4×10−2 ₋₈_.₈_×₁₀−1 ₄_.₄_×₁₀−1 ₁_.₅_×₁₀0

04×4

           x+ ·

04×4

¸

u

(2.16)

2.4 System Identification

System identification is useful for building accurate, simplified models of complex systems

from noisy time-series data. System identification techniques are useful for applications

ranging from control system design and signal processing to time-series analysis and

vi-bration analysis. The basic procedure involves specifying a model structure, estimating its

parameters from observed input-output data, and evaluating this model’s properties to see

if they are satisfactory [29][30]. The cycle can be itemized as follows:

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The AMB plant is open loop unstable; thus to perform system identification it must first

be stabilized. PID control was implemented (Figure 2.7) using MATLAB’s xPC Target,

two PC-type desktop computers in a host-target configuration, and a NI PCI6024E data

acquisition card. Four single-input, single-output controllers were manually tuned to

sta-bilize the radial bearings (albeit with marginal performance over limited speed ragnes) for

purposes of system identification. Using gains ofKP=-145, KI=-120, KD=-0.3, and total

gain 0.00003, and a fixed timestep of 0.1 msec, this controller stabilized the AMB test rig at a rotor speed of 6.0krpm. Uniform random noise was injected into the control signal to

ensure persistent excitation, and input-output data was collected.

PID Controller

AMB System

t

y

ˆ

_s

t ys

t

u

_s

System Identification

)

(

t

ref

-+

Figure 2.7: System identification diagram

Step 2: Define a model structure (a set of candidate system descriptions).

Based on knowledge gained from hammer impact testing, lumped-parameter modeling, and

FEA modeling, an 8th-order model was selected.

Step 3: Compute the best model in the model structure.

System identification is the process of estimating a linear mapping from system input(s)

to system output(s). The resulting model is then a description of the system dynamics,

useful for purposes such as control design or determination of system parameters. Classical

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system model using a finite data history of system inputs and outputs [30][31]. A typical

“black box” approach is shown in Figure 2.7.

To develop the parametric system model, the system inputs and outputs can be related

using a linear ARX (autoregressive with exongenous input) model

ys(t) +a1ys(t−1) +a2ys(t−2) +. . .+apys(t−p) =b1us(t−1) +b2us(t−2) +. . .+bqus(t−q)

(2.17)

wheresoand siare the number of output and input, respectively. There exists an optimal

parameter vector

θ= [a1a2 . . . ap b1 b2 . . . bq]T (2.18)

that when used in (2.17),will best predict the current system output y_s(t) given a history

of inputs and outputs, orregressors, expressed as

ψ(t) = [−ys(t−1) −ys(t−2) . . . −ys(t−p)us(t−1)us(t−2) . . . us(t−q)]T (2.19)

Then (2.17) can be written

ˆ

y_s(t) =ψT(t)ˆθ(t) (2.20)

The recursive least squares (RLS) estimate of the optimal system parameter vector ˆθ(t) can

be determined using the algorithm [31]

ˆ

θ(t) = ˆθ(t−1) +ηR_t−1ψ ³

t,θˆ(t−1)

´ e

³

t,θˆ(t−1)

´

e(t, θ) =ys(t)−yˆs(t|θ)

Rt=Rt−1+η

h ψ

³

t,θˆ(t−1)

´

ψT

³

t,θˆ(t−1)

´

−Rt−1

i

(2.21)

whereη is the learning rate.

Step 4: Validate the estimated model.

If the model is good enough, then stop; otherwise go back to Step 2 to try another model

structure.

MATLAB’s System Identification Toolbox was used to obtain an eighth-order model

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resulting “SysId model” automatically includes the gyroscopic effects at its nominal speed,

6.0krpm. As expected, the SysId model is unstable with eigenvaluesλi at:

λ=−30.8±670i, −315, −317, −325, 323, 324, 332 (2.22)

˙ x=                        

3.1×101 ₁_.₂_×₁₀−1 ₋₅_.₅_×₁₀−4 ₅_.₄_×₁₀−4 ₉_.₉_×₁₀−1 ₋₃_.₉_×₁₀−4 ₁_.₂_×₁₀−4 ₋₁_.₂_×₁₀−4

1.3×10−1 ₃_.₉_×₁₀0 ₋₃_.₈_×₁₀−4 ₃_.₇_×₁₀−4 ₃_.₉_×₁₀−4 ₉_.₉_×₁₀−1 ₅_.₅_×₁₀−5 ₋₅_.₂_×₁₀−5

5.7×10−2 ₁_.₅_×₁₀−2 ₃_.₁_×₁₀0 ₁_.₂_×₁₀−1 ₂_.₆_×₁₀−3 ₋₂_.₉_×₁₀−3 ₉_.₉_×₁₀−1 ₋₇_.₁_×₁₀−4

5.3×10−2 ₁_.₄_×₁₀−2 ₁_.₂_×₁₀−1 ₃_.₇_×₁₀0 ₂_.₄_×₁₀−3 ₋₂_.₇_×₁₀−3 ₆_.₂_×₁₀−4 ₉_.₉_×₁₀−1

−4.5×105 ₋₂_.₄_×₁₀3 ₋₉_.₆_×₁₀−1 ₁_.₁_×₁₀0 ₃_.₁_×₁₀1 ₋₉_.₀_×₁₀−3 ₋₁_.₇_×₁₀−1 ₁_.₇_×₁₀−1

−2.5×103 ₁_.₀_×₁₀5 ₁_.₀_×₁₀0 ₋₁_.₁_×₁₀0 ₋₂_.₁_×₁₀−3 ₃_.₉_×₁₀0 ₁_.₉_×₁₀−1 ₋₁_.₉_×₁₀−1

5.4×100 ₋₁_.₂_×₁₀0 ₁_.₁_×₁₀5 ₋₂_.₄_×₁₀3 ₁_.₆_×₁₀−1 ₋₁_.₆_×₁₀−1 ₃_.₁_×₁₀0 ₃_.₈_×₁₀−3

−6.6×100 ₁_.₁_×₁₀0 ₋₂_.₅_×₁₀3 ₁_.₁_×₁₀5 ₋₁_.₉_×₁₀−1 ₂_.₀_×₁₀−1 ₋₃_.₂_×₁₀−3 ₃_.₇_×₁₀0

                        x +                        

−2.3×10−3 ₇_.₄_×₁₀−4 ₋₃_.₈_×₁₀−7 ₄_.₁_×₁₀−7

7.4×10−4 ₋₂_.₇_×₁₀−3 ₋₁_.₇_×₁₀−7 ₁_.₇_×₁₀−7

−8.4×10−6 ₉_.₇_×₁₀−6 ₋₂_.₅_×₁₀−3 ₇_.₄_×₁₀−4

−7.9×10−6 ₉_.₀_×₁₀−6 ₇_.₄_×₁₀−4 ₋₂_.₇_×₁₀−3

4.9×101 ₋₁_.₅_×₁₀1 ₅_.₅_×₁₀−4 ₋₅_.₉_×₁₀−4

−1.5×101 ₅_.₅_×₁₀1 ₋₆_.₀_×₁₀−4 ₆_.₅_×₁₀−4

−4.9×10−4 ₅_.₄_×₁₀−4 ₄_.₉_×₁₀1 ₋₁_.₅_×₁₀1

6.6×10−4 ₋₇_.₁_×₁₀−4 ₋₁_.₅_×₁₀1 ₅_.₅_×₁₀1

                        u y=          

1.19×100 ₆_.₄₅_×₁₀−1 ₋₆_.₀₅_×₁₀−1 ₋₅_.₁₉_×₁₀−1

5.02×10−2 ₁_.₁₆_×₁₀0 ₂_.₇₁_×₁₀−1 ₂_.₇₈_×₁₀−1

9.18×10−1 ₋₅_.₄₃_×₁₀−1 ₁_.₂₀_×₁₀0 ₃_.₄₆_×₁₀−1

1.36×10−2 ₋₈_.₇₇_×₁₀−1 ₄_.₄₄_×₁₀−1 ₁_.₄₉_×₁₀0

04×4

          x+ ·

04×4

¸

u

(2.23)

2.5 Model Comparison

Table 2.2 compares the natural frequencies of all three models (analytical, FEA, SysId) with

experimental hammer impact test data. All models include the actuator natural frequency

at approximately 53 Hz('3.0krpm). Obviously, because the analytical model is a

rigid-body model, it doesn’t include any flexible bending modes. The FEA and SysId models

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the experimental impact test data. These models will be used for the multivariable robust

control synthesis presented in subsequent chapters.

Table 2.2: Comparison of natural frequency versus model types

Modes (Hz) Impact test Analytical model FEA model SysId model

rigid body 0 0 0 0

actuator frequency – 55.9 50.6 52.8

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Chapter 3 Robust Control of Active Magnetic

Bearings

Minimizing the adverse effects of modeling uncertainty, steady-state error, noise, and

dis-turbances is a fundamental goal of control synthesis. H∞ control is a multivariable robust

control technique that seeks to calculate a controller K such that these signals are

mini-mized according to performance specifications. H_∞ control allows for frequency-dependent

bounds to be placed on each of these signals during controller synthesis to ensure admissible

levels of these undesirable effects [32]. For stability considerations, the synthesis objective

is to specify a nominal linear model and a bound (weighting function) on the uncertainty

between that model and the actual system. TheH∞ controller will then be guaranteed to

stabilize the set of all plants bounded by the nominal model plus the uncertainty.

3.1 Representation of Modeling Uncertainties

The analytical model (2.3) is based upon a linearization of electromagnetic nonlinearities

about a nominal setpoint (the bearing centerline). As such, this model can never fully

capture the system dynamics. Frequently, modeling error can be revealed in the time

domain by comparing the outputs of the actual system with those of its model, given

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Chapter 3. Robust Control of Active Magnetic Bearings 22

the same inputs. Because the model (2.3) is unstable, model validation involves comparing

modeled and measured outputsktime steps ahead using measured control inputs, a process

known ask-step ahead prediction [31]. Figures 3.1 and 3.2 compare the measured outputs

r(θ) at 6.0 krpm to the 10-step ahead and 20-step ahead modeled predictions, r10(θ) and

r₂₀(θ), respectively.

2e−005 4e−005

6e−005 8e−005

0.0001

30

210

60

240

90

270 120

300 150

330

180 0

r(θ) r₁₀ (θ)

Angle [deg] Position [m]

Figure 3.1: k-step ahead prediction fork = 10

The prediction performance is noticeably reduced with the larger horizon, indicating

the presence of modeling error. This uncertainty must be accounted for in the design of

robust controllers.

For H∞ control synthesis, a representation of model uncertainty is required to bound

the differences between the actual plant (which has an infinite number of states) and the

linear model used for control design (which has a small, finite number of states). These

unmodeled dynamics typically result in high-frequency uncertainties that are best

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2e−005 4e−005

6e−005 8e−005

0.0001

30

210

60

240

90

270 120

300 150

330

180 0

r(θ) r

20 (θ) Angle [deg] Position [m]

Figure 3.2: k-step ahead prediction fork = 20

parameterized forms [32].

Two common unstructured representations of modeling uncertainty are additive

uncer-tainty

G∆a(s) =G0(s) +W∆a(s)∆a(s)

∆a(s) = G∆a_W(s)−G0(s)

∆a(s)

(3.1)

and multiplicative uncertainty

G_∆_m(s) =G₀(s)(1 +W_∆_m(s)∆_m(s))

∆_m(s) = G∆m(s)−G0(s)

G0(s)W∆m(s)

(3.2)

HereW∆aandW∆m(generalized byW∆) are weighting functions that describe the

frequency-dependent characteristics of the uncertainty and define a neighborhood about the nominal

model G0 inside which the actual infinite-order plant resides [32]. In this research, the

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3.2 Nominal Robust Control Synthesis

The AMB system (Figure 2.4) was used to experimentally validate the robust stability

and performance of H_∞ control. Anominal H_∞ controller was designed using MATLAB’s LMI Control Toolbox (msfsyn) and the system interconnection in Figure 3.3. Herenominal

controller means the robust controller designed withnominaluncertainty weighting function

W∆.

o

G

K

'

W

in

W

y

r

p

W

-+ +

+

e

u

Figure 3.3: H∞ system interconnection

Typically, uncertainty weighting functions are chosen arbitrarily or through lengthy

trial and error procedures that can result in overly conservative error bounds to

guaran-tee stability. Textbooks on robust control provide general guidelines for specifying bounds

on modeling uncertainty W∆, performance Wp, and control input Win as functions of

fre-quency [32].

For this application, all three weighting functions were constructed based on a priori

knowledge of the AMB system dynamics and trial and error adjustments. The shape of

W∆ (Figure 3.4), a high-pass filter, was selected based on the assumption that unmodeled

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that |W∆(jω)| ≥ |G∆(jω)−G0(jω)| for all frequencies. Wp penalizes steady-state error,

which generally has low-frequency properties. By making its shape a low-pass filter, the

controller will focus on minimizing low-frequency errors. Similarly, it is desirable to reduce

the high-frequency content of the control signal, so Win has the shape of a high-pass filter

(Figure 3.4).

Three “nominal” robust controllers were designed using the dynamic models described

in Chapter 2 (analytical, FEA, and SysId), all based on a nominal rotation speed of 6.0

krpm. Each controller incorporated thenominal weighting functions of Figure 3.4.

10−2 100 102 104 106 108

10−5 10−4 10−3 10−2 10−1 100

Frequency (Hz)

Magnitude

W_∆ Wp Win

Figure 3.4: Nominal weighting functions

3.3 Experimental Results

All three nominal controllers were implemented using MATLAB’s xPC Target at a fixed

timestep 0.1 msec. To investigate the robustness of these controllers, the rotor speed was varied from 0.0 to 10.0krpm, away from the nominal speed (6.0krpm) at which gyroscopic

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quantified by the sum of squared errors (SSE) for both bearings:

SSE=

N

X

k=1

[rd(kT)−rm(kT)]2 (3.3)

where rd is the desired orbital position (rd=0) and rm is the measured distance from the

center, since the control objective is to regulate the rotors to the centers of the bearings.

HereN and T are the number of data points in one cycle and the discrete sample time, 0.1

msec, respectively.

Figure 3.5 reveals a clear trend; as the rotational speed of the system moves away from

the nominal rotation speed (6.0krpm), stability was maintained but performance

deterio-rated. This deterioration in performance (Table 3.1) is especially evident near 2.5krpm(due

to the actuator natural frequency) and 8.0 krpm (due to the rotor’s first bending mode).

This value doesn’t correspond exactly to the experimental impact test data (Table 2.2) due

to the coupling which connects the rotor and drive motor.

Figures 3.6 - 3.11 present 3-dimensional orbit plots for each bearing as a function of rotor

speed for the analytical, FEA, and SysId nominal controllers. These figures indicate that the

FEA and SysID models both result in superior tracking performance, despite the relative

convenience and simplicity of system identification compared to FEA modeling and modal

reduction. From these plots, the rotor’s first bending mode is evident at approximately

8.0 krpm. Note that the improved performance near 6.0 krpm illustrates the benefits of

operating near the nominal speed of the linear model; it also illustrates the need for model

Intelligent Control Using Confidence Interval Networks: Applications to Robust Control of Active Magnetic Bearings

Abstract

Biography

Acknowledgements

Table of Contents

List of Tables

List of Figures

List of Symbols

Chapter 1

Introduction

1.1

Background and Motivation

1.2

Scope of Work

Chapter 2

Modeling of Active Magnetic Bearing

Systems

2.1

Overview of Active Magnetic Bearings

x

y

2.2

Analytical Modeling

2.3

Finite Element Analysis Modeling

2.4

System Identification

t

y

ˆ

t

u

)

(

t

ref

2.5

Model Comparison

Chapter 3

Robust Control of Active Magnetic

Bearings

3.1

Representation of Modeling Uncertainties

3.2

Nominal Robust Control Synthesis

o

G

K

'

W

W

y

r

W

e

u

3.3

Experimental Results