Quasilinear Control Theory for Systems with Asymmetric Actuators and Sensors

222 Pages·2013·2.8 MB·English

Quasilinear Control Theory for Systems
with Asymmetric Actuators and Sensors
by
Hamid-Reza Ossareh
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
(Electrical Engineering: Systems)
in the University of Michigan
2013
Doctoral Committee:
Professor Pierre Kabamba, Co-Chair,
Professor Semyon Meerkov, Co-Chair,
Professor Jessy Grizzle,
Professor Ilya Kolmanovsky,
Professor Demosthenis Teneketzis.
c Hamid-Reza Ossareh 2013
(cid:13)
All Rights Reserved
To my family.
ii
ACKNOWLEDGEMENTS
First and foremost, I would like to thank my advisors, Professors Pierre Kabamba
and Semyon Meerkov, for all their efforts and guidance. They have been there with
me throughout the course of my graduate studies and, for that, I am very grateful.
Next, I wish to thank Professors Jessy Grizzle, Ilya Kolmanovsky, and Demos-
thenis Teneketzis for their interest in my work and for serving as members of my
dissertation committee.
I also want to thank Professor Domitilla Del Vecchio for her support and mentor-
ship.
During my doctoral studies I collaborated with the following scholars, to whom I
am grateful: Dr. Mads Almassalkhi, Professor Shinung Ching, Professor Yongsoon
Eun, Dr. Eric Gross, Dr. Yi Guo, and Professor Choon Yik Tang.
Moreover, I want to thank my family and friends whose love and support made
this achievement possible.
Finally, I acknowledge financial support from the National Science Foundation
that made this work possible.
iii
TABLE OF CONTENTS
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . iii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
LIST OF APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
CHAPTER
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation and Approach . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Technical approach . . . . . . . . . . . . . . . . . . 3
1.2 Definition of S- and A-LPNI Systems . . . . . . . . . . . . . . 4
1.3 Problems Considered . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Problem 1: Formalism of stochastic linearization for
A-LPNI systems . . . . . . . . . . . . . . . . . . . . 7
1.3.2 Problem 2: Performance analysis of A-LPNI systems 7
1.3.3 Problem 3: Time-domain design of A-LPNI systems 8
1.3.4 Problem 4: Design of step-tracking controllers for
LPNI and A-LPNI systems . . . . . . . . . . . . . . 9
1.3.5 Problem 5: Performance recovery in A-LPNI systems 10
1.3.6 Problem 6: LQR approach for A-LPNI systems . . . 10
1.4 Original Contributions . . . . . . . . . . . . . . . . . . . . . . 11
1.4.1 Contributionstoformalismofstochasticlinearization
in A-LPNI systems . . . . . . . . . . . . . . . . . . 12
1.4.2 ContributionstoperformanceanalysisinA-LPNIsys-
tems . . . . . . . . . . . . . . . . . . . . . . . . . . 12
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1.4.3 Contributions to time-domain controller design in A-
LPNI systems . . . . . . . . . . . . . . . . . . . . . 13
1.4.4 Contributionstostep-trackingcontrollerdesigninA-
LPNI systems . . . . . . . . . . . . . . . . . . . . . 13
1.4.5 Contributions to performance recovery . . . . . . . 14
1.4.6 Contributions to the A-SLQR technique . . . . . . . 14
1.4.7 Application: Wind farms controller design . . . . . 14
1.4.8 QLC toolbox . . . . . . . . . . . . . . . . . . . . . . 15
1.5 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5.2 Performance analysis and design in A-LPNI systems 16
1.5.3 Stochastic linearization . . . . . . . . . . . . . . . . 17
1.6 Statement of Impact . . . . . . . . . . . . . . . . . . . . . . . 18
1.7 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . 19
II. Stochastic Linearization for A-LPNI Systems . . . . . . . . . . 20
2.1 Open Loop Environment . . . . . . . . . . . . . . . . . . . . 20
2.1.1 General equations . . . . . . . . . . . . . . . . . . . 20
2.1.2 Stochastic linearization of common nonlinearities . . 23
2.2 Closed Loop Environment . . . . . . . . . . . . . . . . . . . . 26
2.2.1 Reference tracking with nonlinear actuator . . . . . 28
2.2.2 Disturbance rejection with nonlinear actuator . . . . 30
2.2.3 Reference tracking with nonlinear sensor . . . . . . 31
2.2.4 Disturbance rejection with nonlinear sensor . . . . . 32
2.2.5 Reference tracking with nonlinear actuator and non-
linear sensor . . . . . . . . . . . . . . . . . . . . . . 32
2.2.6 Disturbance rejection with nonlinear actuator and
nonlinear sensor . . . . . . . . . . . . . . . . . . . . 33
2.2.7 Simultaneous reference tracking and disturbance re-
jection with nonlinear actuator and nonlinear sensor 33
2.3 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.1 Statistical experiment . . . . . . . . . . . . . . . . . 35
2.3.2 Filtering hypothesis and accuracy of stochastic lin-
earization for filtering plants . . . . . . . . . . . . . 38
2.4 Measure of Asymmetry . . . . . . . . . . . . . . . . . . . . . 38
III. Performance Analysis in A-LPNI Systems . . . . . . . . . . . . 44
3.1 Analysis of Tracking Performance . . . . . . . . . . . . . . . . 45
3.1.1 Motivating example . . . . . . . . . . . . . . . . . . 45
3.1.2 Trackable domains for A-LPNI systems . . . . . . . 48
3.1.3 The quality indicators and the diagnostic flowchart . 52
3.2 Analysis of Disturbance Rejection Performance . . . . . . . . 63
v
3.3 Analysis of Noise-Induced Loss of Tracking in Systems with PI
Control and Anti-Windup . . . . . . . . . . . . . . . . . . . . 65
IV. Time Domain Design of Tracking Controllers in A-LPNI Sys-
tems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.1 Performance Loci . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . 71
4.1.2 The AS-root locus . . . . . . . . . . . . . . . . . . . 73
4.1.3 TE locus . . . . . . . . . . . . . . . . . . . . . . . . 79
4.1.4 Effect of asymmetry on the performance loci . . . . 82
4.2 Design using the performance loci . . . . . . . . . . . . . . . 85
4.2.1 Design for required dynamic performance . . . . . . 85
4.2.2 Design for required steady state performance . . . . 88
V. Design of Step-Tracking Controllers in LPNI Systems . . . . 91
5.1 Design of Step-Tracking Controllers . . . . . . . . . . . . . . 91
5.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . 91
5.1.2 Necessary and Sufficient Condition for Existence of
StepTrackingControllersSatisfyingSteadyStateSpec-
ifications . . . . . . . . . . . . . . . . . . . . . . . . 95
5.1.3 Calculating the Adjoint Bandwidth . . . . . . . . . 96
5.1.4 Examples of QLC-based controller design . . . . . . 97
5.1.5 ComparisonofQLC-basedandanti-windup-basedde-
sign methodologies . . . . . . . . . . . . . . . . . . 107
5.1.6 Step-tracking design for the asymmetric case . . . . 112
5.2 Analysis and Design of Systems With Integrator Anti-windup
Using Stochastic Linearization . . . . . . . . . . . . . . . . . 114
5.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . 114
5.2.2 Design strategy . . . . . . . . . . . . . . . . . . . . 117
VI. Linear Performance Recovery in A-LPNI Systems . . . . . . . 122
6.1 Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.2 Computing the Boosting Gains . . . . . . . . . . . . . . . . . 125
6.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
VII. LQR-based Design of A-LPNI Systems . . . . . . . . . . . . . . 129
7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.2 The A-SLQR Problem . . . . . . . . . . . . . . . . . . . . . . 131
7.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
VIII. Application: QLC-based Design of a Wind Farm Controller 137
vi
8.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
8.3 Problem formulation and controller design . . . . . . . . . . . 141
8.4 Performance evaluation . . . . . . . . . . . . . . . . . . . . . 145
IX. Conclusions and Future Work . . . . . . . . . . . . . . . . . . . 150
9.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
9.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
A.1 Proofs for Chapter II . . . . . . . . . . . . . . . . . . . . . . 157
A.2 Proofs for Chapter III . . . . . . . . . . . . . . . . . . . . . . 162
A.3 Proofs for Chapter IV . . . . . . . . . . . . . . . . . . . . . . 169
A.4 Proofs for Chapter V . . . . . . . . . . . . . . . . . . . . . . 174
A.5 Proofs for Chapter VII . . . . . . . . . . . . . . . . . . . . . 175
B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
B.2 QLC Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 177
B.2.1 stochlinearize . . . . . . . . . . . . . . . . . . . . . 177
B.2.2 stochlinearizeMIMO . . . . . . . . . . . . . . . . . . 179
B.2.3 SRS . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
B.2.4 trackingind . . . . . . . . . . . . . . . . . . . . . . . 182
B.2.5 admissibledomain . . . . . . . . . . . . . . . . . . . 183
B.2.6 srlocus . . . . . . . . . . . . . . . . . . . . . . . . . 184
B.2.7 boosting . . . . . . . . . . . . . . . . . . . . . . . . 185
B.2.8 slqr . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
B.2.9 slqg . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
B.2.10 ilqr . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
B.2.11 ilqg . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
B.2.12 stepTracker . . . . . . . . . . . . . . . . . . . . . . 192
B.2.13 graphicalStochLinearize . . . . . . . . . . . . . . . . 194
B.2.14 stochlinearizeAsym . . . . . . . . . . . . . . . . . . 195
B.2.15 srlocusAsym . . . . . . . . . . . . . . . . . . . . . . 197
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
vii
LIST OF FIGURES
Figure
1.1 SISO linear system and LPNI system. . . . . . . . . . . . . . . . . . 2
1.2 LPNI system with nonlinear actuator and sensor with the original
and translated operating points. . . . . . . . . . . . . . . . . . . . . 4
1.3 Saturation function satβ(u). . . . . . . . . . . . . . . . . . . . . . . 6
α
1.4 A-LPNI system and its stochastic linearization. . . . . . . . . . . . 8
1.5 System considered for time domain design. . . . . . . . . . . . . . . 9
1.6 System considered for performance recovery. . . . . . . . . . . . . . 11
2.1 Stochastic linearization of an isolated nonlinearity. . . . . . . . . . . 21
2.2 Alternative representation of Figure 2.1. . . . . . . . . . . . . . . . 22
2.3 Common piece-wise differentiable functions. . . . . . . . . . . . . . 24
2.4 Closed loop LPNI system. . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 LPNI system and its stochastic linearization. . . . . . . . . . . . . . 27
2.6 Histograms of e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1
2.7 Histograms of e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2
2.8 Histograms of v and y. . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.9 Accuracy as quantified by e as a function of the midpoint of satura-
2
tion, i.e., (α+β)/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
viii
2.10 Degree of asymmetry A as a function of µ and σ . . . . . . . . . . 41
u u
2.11 Quasilinear gain N and quasilinear bias m as a function of degree of
asymmetry A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1 LPNI and quasilinear systems for the tracking problem of Section 3.1.1. 45
3.2 Traces of r(t), y(t), and yˆ(t) for the example of Subsection 3.1.1. . . 47
3.3 The standard deviations σ and σ , average values µ and µ , and
e eˆ e eˆ
the square root of the second moments E[e2] and E[eˆ2] as a
function of the midpoint of nonlinearity, for the tracking problem of
(cid:112) (cid:112)
Subsection 3.1.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4 System for studying the trackable domain. . . . . . . . . . . . . . . 48
3.5 Illustration of e vs. r when 1 +P > 0 and C > 0. . . . . . . . 50
ss 0 C0 0 0
3.6 Diagnostic chart for tracking performance. . . . . . . . . . . . . . . 60
3.7 SRS of systems in Example III.1 . . . . . . . . . . . . . . . . . . . . 61
3.8 Time traces of the output for systems in Example III.1. . . . . . . . 62
3.9 Example of Section 3.2. . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.10 The standard deviations, means, and square root of second moments
of y and yˆ for the disturbance rejection problem of Section 3.2. . . . 64
3.11 System with noise-induced tracking error. . . . . . . . . . . . . . . . 66
3.12 Simulation results for the noise-induced tracking error. . . . . . . . 68
3.13 Demonstration of the noise induced tracking error as a function of
K and σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
AW n
4.1 A-LPNI system and equivalent quasilinear system. . . . . . . . . . . 71
4.2 AS-root locus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3 TE loci of system (4.15) for three cases. . . . . . . . . . . . . . . . . 80
4.4 A sketch of the TE locus for system (4.15) with µ = 1, α = 0.5,
r
−
β = 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
ix