modeling and feedback control of a mems electrostatic actuator

200 Pages·2009·2.16 MB·English

by  Jason Edwards

MODELING AND FEEDBACK CONTROL OF A MEMS
ELECTROSTATIC ACTUATOR

JASON M. EDWARDS

Bachelor of Electrical Engineering

Cleveland State University

May, 2004

submitted in partial fulfillment of requirements for the degree

MASTER OF SCIENCE IN ELECTRICAL ENGINEERING

at the

CLEVELAND STATE UNIVERSITY

December, 2008
This thesis has been approved
for the Department of Electrical and Computer Engineering
and the College of Graduate Studies by

________________________________________________
Thesis Committee Chairperson, Lili Dong

________________________________
Department/Date

________________________________________________
Dr. Zhiqiang Gao

________________________________
Department/Date

________________________________________________
Dr. Dan Simon

________________________________
Department/Date
ACKNOWLEDGEMENTS

First of all, I would like to thank my employer for being patient and supporting
the furthering of my education. Without their support (temporal and financial) this thesis
wouldn’t be possible. Secondly, I would like to thank my advisor, Dr. Dong, for not only
helping me choose my thesis topic, but also for the tedious hours proof reading it. I
would also like to thank my thesis committee, Dr. Dong, Dr. Gao and Dr. Simon for
helping instill my knowledge of controls, and in the end, letting me know that I have only
begun my journey to understanding. Finally, I would like to thank my family for putting
up with me while I finish this stage of my academic career.
MODELING AND FEEDBACK CONTROL OF A MEMS
ELECTROSTATIC ACTUATOR

JASON M. EDWARDS

ABSTRACT
This thesis describes the mathematical modeling and closed-loop voltage control
of a MEMS electrostatic actuator. The control goal is to extend the travel range of the
actuator beyond the open-loop pull-in limit of one third of the initial gap. Three
controller designs are presented to reach the control goal. The first controller design
utilizes a regular fourth order Active Disturbance Rejection Controller (ADRC) and is
able to achieve 97% of the maximum travel range. The second design also uses a fourth
order ADRC, while additional modeling information is included in an Extended State
Observer (ESO), which is part of the ADRC, to improve control performance. This
controller achieved 99% of the travel range. The third design is a multi-loop controller
with a second order ADRC in an inner loop and a Proportional-Integral (PI) controller in
an outer loop. This design achieved 100% of the travel range. Transfer function
representations of the three controller designs are developed. The controllers are
successfully applied and simulated in a parallel-plate electrostatic actuator model. The
simulation results and frequency domain analyses verified the effectiveness of the
controllers in extending the travel range of the actuator and in noise attenuation.
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TABLE OF CONTENTS
Page
ABSTRACT ...................................................................................................................... iv
LIST OF TABLES ........................................................................................................... ix
LIST OF FIGURES ......................................................................................................... xi
NOMENCLATURE ...................................................................................................... xvii
I. INTRODUCTION................................................................................................. 1
1.1 Electrostatic Actuators ................................................................................ 2
1.2 Applications of MEMS Electrostatic Actuators ......................................... 4
1.3 Control of Electrostatic Actuators .............................................................. 6
1.3.1 Open-loop Control of Electrostatic Actuator .................................. 7
1.3.2 Linear Feedback Control ................................................................. 9
1.3.3 Nonlinear Feedback Control ......................................................... 12
1.4 Proposed Control Methods ....................................................................... 14
1.5 Overview ................................................................................................... 15
II. MATHEMATICAL MODELING .................................................................... 17
2.1 Mathematical Modeling ............................................................................ 18
2.1.1 First Principles Modeling ................................................................. 24
2.1.2 Lagrange’s Equations .................................................................... 26
v
2.2 Equation Normalization ............................................................................ 29
2.2.1 Normalization Results of the Nonlinear Model of the Electrostatic
Actuator..................................................................................................... 29
2.3 Model Linearization .................................................................................. 30
2.3.1 Standard Model ............................................................................. 30
2.4 Transfer Function Derivation .................................................................... 36
2.4.1 Linearized Actuator Model ........................................................... 36
2.5 Electrostatic Actuator Model Behavior .................................................... 40
2.5.1 Pull-in Phenomenon and Noise Amplification of the Actuator .... 40
2.5.2 Linear Models ............................................................................... 44
2.6 Closed Loop Transfer Functions ............................................................... 47
III. CONTROL TECHNIQUES ............................................................................... 50
3.1 Feedback Control for MEMS.................................................................... 51
3.2 State Space Representation of ADRC....................................................... 52
3.3 Frequency Domain Observers................................................................... 56
3.3.1 Frequency Domain Observer Principles ....................................... 56
3.3.2 Classic Luenberger Observer ........................................................ 63
3.3.3 Transfer Function Representation of a Fourth-order ESO ............ 67
3.4 Transfer Function Description of ADRC .................................................. 71
IV. CONTROLLER DESIGNS AND PERFORMANCE ANALYSES ............... 81
vi
4.1 Classic LADRC Design ............................................................................ 82
4.1.1 Controller Framework ................................................................... 82
4.1.2 Classic ADRC Design ................................................................... 83
4.1.2.1 Design 1...................................................................................... 86
4.1.2.2 Design 2...................................................................................... 89
4.1.2.3 Design 3...................................................................................... 92
4.1.2.4 Design Comparison .................................................................... 94
4.2 LADRC 4th Order Alternative Design ...................................................... 97
4.2.1 Controller Framework ................................................................... 98
4.2.2 Comparison Study of Classic and Alternative ADRC Designs .. 103
4.2.2.1 Classic ADRC Design .............................................................. 104
4.2.2.2 Alternative ADRC Design ....................................................... 104
4.2.2.3 Classic and Alternative Controller Comparison....................... 105
4.3 Controller Noise Analyses ...................................................................... 111
4.3.1 Electrostatic Actuator Noise Analysis......................................... 111
4.3.1.1 Classic ADRC Comparison ...................................................... 112
4.3.1.2 Comparison between Alternative and Classic ADRCs ............ 123
4.4 Multi-loop Control .................................................................................. 125
4.4.1 Transfer Function Derivation ...................................................... 125
vii
4.4.2 Linear Controller Design ............................................................. 129
4.4.2.1 Secondary Loop Design ........................................................... 129
4.4.2.2 Primary Loop............................................................................ 131
4.4.3 Controller Analysis ..................................................................... 131
V. SIMULATION RESULTS ............................................................................... 139
5.1 Classic ADRC Simulation Results ......................................................... 140
5.2 Alternative ADRC Simulation Results ................................................... 148
5.3 Multi-loop Controller Simulation Results .............................................. 154
5.4 Controller Comparison............................................................................ 158
VI. CONCLUSIONS ............................................................................................... 164
6.1 Future Research ...................................................................................... 166
REFERENCES .............................................................................................................. 168
APPENDICES ............................................................................................................... 171
Appendix A ......................................................................................................... 172
Equation Normalization .......................................................................... 172
Appendix B ......................................................................................................... 178
Frequency Domain Representation of a 2nd Order ESO ......................... 178
viii
LIST OF TABLES

Table Page
TABLE I: Partial Derivatives ..................................................................................... 35
TABLE II: Unstable Pole Phase Lag ........................................................................... 43
TABLE III: Equilibrium Points Part I .......................................................................... 45
TABLE IV: Equilibrium Points Part II ......................................................................... 45
TABLE V: Plant Variation .......................................................................................... 46
TABLE VI: Three sets of tuning parameters ................................................................ 84
( )
TABLE VII: Stability Margins for Design 1 w =50 ............................................ 87
o
( )
TABLE VIII: Stability Margins for Design 2 w =30 ............................................ 90
o
( )
TABLE IX: Stability Margins for Design 3 w = 20 ................................................ 93
o
TABLE X: Integral of the Squared Errors for the Three Designs ............................... 95
TABLE XI: Overshoot Percentages of the Three Designs ........................................... 96
TABLE XII: Alternate ADRC Comparison ............................................................. 107
TABLE XIII: Controller Noise Comparison ............................................................. 112
TABLE XIV: ISE for three LADRC designs at 10% of full gap .............................. 143
TABLE XV: ISE for three LADRC designs at 90% full gap ................................... 144
TABLE XVI: ISE of classic ADRC and alternate ADRC ......................................... 151
ix
TABLE XVII: ISE Error ............................................................................................. 158
TABLE XVIII: Controller Comparison ISE............................................................. 159

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