Robust Linear Filter Design via LMIs and Controller Design with Actuator Saturation via SOS ...

132 Pages·2012·0.74 MB·English

Robust Linear Filter Design via LMIs and Controller Design with
Actuator Saturation via SOS Programming
by
Kunpeng Sun
B.S. (Dalian University of Technology, P.R. China) 1996
M.S. (Tsinghua University, P.R. China) 1999
A dissertation submitted in partial satisfaction of the
requirements for the degree of
Doctor of Philosophy
in
Engineering-Mechenical Engineering
in the
GRADUATE DIVISION
of the
UNIVERSITY OF CALIFORNIA, BERKELEY
Committee in charge:
Professor Andrew Packard, Chair
Professor Kameshwar Poolla
Professor Maciej Zworski
Spring 2004
The dissertation of Kunpeng Sun is approved:
Chair Date
Date
Date
University of California, Berkeley
Spring 2004
Robust Linear Filter Design via LMIs and Controller Design with
Actuator Saturation via SOS Programming
Copyright 2004
by
Kunpeng Sun
1
Abstract
Robust Linear Filter Design via LMIs and Controller Design with Actuator
Saturation via SOS Programming
by
Kunpeng Sun
Doctor of Philosophy in Engineering-Mechenical Engineering
University of California, Berkeley
Professor Andrew Packard, Chair
Robust filtering under different assumptions and formulations are considered. Ro-
bust filter design for systems described by time-varying linear fractional transforma-
tion (LFT) uncertain models is reformulated as linear matrix inequalities (LMIs) via
upper bound techniques. The contribution is the treatment of norm bounded (both
structuredandunstructured)LFTuncertaintyusingLMI(ratherthanRiccati)meth-
ods. Furthermore, in the norm bounded unstructured uncertainty case, our results
are less conservative than those by methods based on Riccati equations. Robust filter
design for systems with time-invariant parameter uncertainties in a polytope is also
considered, using parameter dependent Lyapunov functions to solve the problem. In
both cases, we use upper bounds rather than the actual performance objectives.
2
We also exploit that the robust filter design problem (with model uncertainty and
noise)isconvexinthefilterasanoperator. Theimplicationisthatrobustfilterdesign
can be carried out directly, rather than minimizing an upper bound of the objective
function. We show that finite dimensional approximations can be used to obtain sub-
optimal solutions with any degree of accuracy. A design algorithm is proposed, which
is made up of successive finite dimensional approximations. This algorithm requires
a worst case analysis result. A conceptual branch & bound algorithm is outlined, and
a practical algorithm is given.
Polynomial state feedback controller synthesis for systems subject to actuator
saturation is also considered. We are interested in two kinds of problems. The first
one is to design a controller to enlarge a domain of attraction (DOA), and the second
is for disturbance rejection. Sum of squares (SOS) programming is the computational
tool. These synthesis problems are not convex, and ad-hoc algorithms are proposed.
For linear systems with saturation, algorithms here can be used to improve available
results.
Professor Andrew Packard
Dissertation Committee Chair
i
To my parents and Nan,
for their love and support over these years.
ii
Contents
List of Figures iv
List of Tables v
1 Introduction 1
1.1 Filtering problems and robustness . . . . . . . . . . . . . . . . . . . . 2
1.2 Problem formulation issues and problem solving tools . . . . . . . . . 4
1.3 From SDP to SOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Outline and contributions . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Robust filters for time-varying uncertain LFT systems – indirect
method 12
2.1 Problem Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Preliminaries: analysis of uncertain LFT systems . . . . . . . . . . . 18
2.3 Robust H Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2
2.3.1 Robust Filter Synthesis via LMIs . . . . . . . . . . . . . . . . 23
2.3.2 Elimination of Filter Parameters . . . . . . . . . . . . . . . . 29
2.4 Robust H Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
∞
2.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3 Robust filters for time-invariant uncertain systems in polytopes –
indirect method 37
3.1 Problem setup and preliminaries . . . . . . . . . . . . . . . . . . . . . 38
3.2 Filtering result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4 Optimal worst case robust H filter design – direct method 45
∞
4.1 Worst-case analysis via branch & bound . . . . . . . . . . . . . . . . 48
iii
4.1.1 Branch & bound algorithm . . . . . . . . . . . . . . . . . . . . 50
4.2 A practical branch & bound algorithm . . . . . . . . . . . . . . . . . 52
4.2.1 Frequency and scalar complex uncertainty . . . . . . . . . . . 53
4.2.2 Worst-case performance assessment . . . . . . . . . . . . . . . 55
4.3 Filtering problem formulation and
preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4 (cid:178)-suboptimal filters via finite dimensional relaxation . . . . . . . . . . 72
4.5 Design algorithm and error bounds . . . . . . . . . . . . . . . . . . . 76
4.5.1 An algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.5.2 Casting the finite-dimensional problem as an SDP . . . . . . . 78
4.5.3 Bounds via worst case analysis . . . . . . . . . . . . . . . . . . 79
4.5.4 A lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.6 Suboptimal filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.7 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5 Controller Synthesis with Input Saturation – Applications of SOS
Programming 88
5.1 Preliminaries – P-satz and SOS programs . . . . . . . . . . . . . . . . 90
5.1.1 Positivstellensatz and -Procedure . . . . . . . . . . . . . . . 90
S
5.1.2 SOS Programming . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2 State-feedback to enlarge a domain of attraction . . . . . . . . . . . . 93
5.2.1 Controller design via SOS programming . . . . . . . . . . . . 95
5.2.2 Comparisons and a numerical example . . . . . . . . . . . . . 97
5.3 State-feedback for disturbance rejection . . . . . . . . . . . . . . . . . 100
5.3.1 Invariant set enlargement . . . . . . . . . . . . . . . . . . . . 101
5.3.2 Controller design for Problem 4 . . . . . . . . . . . . . . . . . 105
5.3.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . 108
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6 Conclusions 111
Bibliography 114
iv
List of Figures
1.1 Model Based Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 Uncertain LFT systems . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Comparison of Robust and Kalman Filters . . . . . . . . . . . . . . . 35
3.1 Two Robust H Filters . . . . . . . . . . . . . . . . . . . . . . . . . 43
∞
4.1 General Linear Fractional Form . . . . . . . . . . . . . . . . . . . . . 49
4.2 General Robust Design Problem . . . . . . . . . . . . . . . . . . . . . 66
4.3 Uncertain Plant and Filter . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4 Comparison of Various Filters . . . . . . . . . . . . . . . . . . . . . . 87
5.1 saturation and level sets . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2 The invariant ellipsoids and input saturation . . . . . . . . . . . . . . 109
5.3 Smaller invariant set and simulation results . . . . . . . . . . . . . . . 110
v
List of Tables
4.1 Robust FIR Filters with Different Order . . . . . . . . . . . . . . . . 85
4.2 Comparison of Various Filters . . . . . . . . . . . . . . . . . . . . . . 86
5.1 DOA with different parameters . . . . . . . . . . . . . . . . . . . . . 99