Robust Filtering for Uncertain Systems: A Parameter-Dependent Approach

259 Pages·2014·6.239 MB·English

by  Huijun Gao, Xianwei Li (auth.)

Communications and Control Engineering
Huijun Gao
Xianwei Li
Robust Filtering
for Uncertain
Systems
A Parameter-Dependent Approach
Communications and Control Engineering
For furthervolumes:
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Huijun Gao Xianwei Li
•
Robust Filtering
for Uncertain Systems
A Parameter-Dependent Approach
123
Huijun Gao
Xianwei Li
ResearchInstituteofIntelligentControland
Systems
Harbin InstituteofTechnology
Harbin
China
ISSN 0178-5354 ISSN 2197-7119 (electronic)
ISBN 978-3-319-05902-0 ISBN 978-3-319-05903-7 (eBook)
DOI 10.1007/978-3-319-05903-7
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To my parents, Qingxiang Gao
and Yunrong Yan, my wife, Lu Zhao,
and my son, Ying’ao Gao
— Huijun Gao
To my parents, Daiyuan Li
and Helan Chen, and my wife, Yue Gu
— Xianwei Li
Preface
Filtering is one of the basic problems in the fields of systems, control and signal
processing. The goal of filtering is to estimate signals that are unmeasurable
through processing the measured output signals. Since the development of Kal-
man’s optimal filtering theory for stochastic systems and Luenberger’s observer
theoryfordeterministicsystems,manyscholarshave devotedconsiderableefforts
to the problem of filtering. Especially, the well-known Kalman filtering theory
plays a significant role in various social and engineering fields such as aerospace,
astronautics,industrialprocessesandeconomicandfinancialsystems.TheKalman
filtering theory is based on the availability of the precisely-known mathematical
modelofthestudiedplantandtheassumptionofstrictGaussianrandomprocesses
orseries.However,itisusuallydifficulttocharacterizethedynamicsofthestudied
plantexactlybyamathematicalmodel,inevitablyleadingtoanerrorbetweenthe
derived mathematical model and the practical plant; moreover, it is rare for
practicalexternalnoisestocompletelysatisfythestrongGaussianassumption.The
uncertainties existing in systems and signals would greatly degrade the perfor-
mance of a traditional Kalman filter and even cause divergence. Hence, it is of
practical meaning to research filtering theory for uncertain dynamical systems so
as to improve the robustness of a filter against uncertainties.
In systems and control areas, study on uncertain systems has drawn much
attentionfrommanyresearchersforalongtime.Robustcontroltheory,originating
from 1970s, generally solves analysis and synthesis problems of parametric
uncertain systems based on the notion of quadratic stability. Since the late 1980s,
this notion has also been gradually employed to solve robust filtering problems
of uncertain systems, which results in a great number of quadratic approaches to
robust filter design. However, these quadratic approaches have been well recog-
nizedtobeconservativeduetothetheutilizationofacommonquadraticLyapunov
function for the entire uncertainty domain. In recent years, to reduce such con-
servatism and improve the practical applicability of robust filter design methods,
parameter-dependent Lyapunov functions are introduced into robust filtering the-
ory,andparameter-dependentapproachestorobustfilterdesignhavedrawagreat
deal of attention. The essential idea of the most popular parameter-dependent fil-
teringresultsistorelaxLyapunovfunctionstobelinearlyparameter-dependentand
meanwhile to fix some slack matrices, which, though relaxing the quadratic
approaches, still has much limitation.
vii
viii Preface
Inviewofthelimitationoftheexistingparameter-dependentresults,somenew
methods recently have been developed to derive a series of parameter-dependent
approachestorobustfilterdesign,includingours,whichfurtherreleasetheprevious
restrictions on Lyapunov functions, showing great potential in conservatism
reduction. This book systematically summarizes these recent developments
of parameter-dependent filter design methods. Robust H filtering, robust H
2 ?
filteringandrobustenergy-to-peakfilteringarediscussedinthebook,whererobust
H filtering is employed asthe main filtering scheme for various classes of com-
?
plex uncertain dynamical systems including time-delay systems, two-dimensional
systems and networked systems. Moreover, our latest work, some preliminary
resultsonfinitefrequencyH filtering,willalsobeinvolvedattheendofthebook
?
toshowanewfilteringthemeoffutureinterest.Variousexamplesareprovidedin
the book to illustrate the effectiveness of the presented new parameter-dependent
filtering results. All the results are presented in the framework of linear matrix
inequality (LMI).
Fromthebook,readercanfindanoverviewofthelatestadvancesintherobust
filtering area and grasp the state-of-the-art methods on parameter-dependent filter
design.Thisbookcanbeusedasareferencebyresearchersandengineersworking
intheareasrelatedtocontroltheoryandengineering,andsignalprocessing,andit
is especially beneficial for graduate students interested in or focusing on robust
filtering theory and its application. Courses like linear systems, modern control
theory and robust control theory and basic mathematical background are pre-
requisiteforreadingthebook.ThosefamiliarwiththeLMItheorywouldpossibly
read the book more efficiently.
Harbin, China, July 2013 Huijun Gao
Xianwei Li
Acknowledgments
The authors would like to dedicate their gratitude to many colleagues and friends
who have made direct and indirect contribution to this book. First appreciation is
delivered toProf. Tongwen Chen from University of Alberta, Canada, who offers
the first author an opportunity to conduct his postdoctoral research from 2005 to
2007 at University of Alberta, where partial works of the book are completed
underthecooperationwithProf.Chen.SpecialthanksgotoMr.XiangyuMengfor
hiseffortsandcontributiontosomepertinentworkswhenheconductedhisM.Eng.
career under the supervision of the first author. The authors are also grateful to
Prof. James Lam from The University of Hong Kong, Hong Kong, Prof. Zidong
Wang from Brunel University, UK, Professer Peng Shi from The University of
Adelaide, Australia, Prof. Okyay Kaynak from Bogazici University, Turkey, and
Prof.ChanghongWangfromHarbinInstituteofTechnology,China,fortheirlong-
lasting friendship and cooperation that have provided the authors great support,
encouragement and inspiration in study and research. The second author would
liketoexpresshissincerethankstothefirstauthorforhisguidanceandinstruction
in the second author’s study and academic career. Undoubtedly, the authors’
deepest gratitude is forever dedicated to their families for their unreservedly
understanding, love and encouragement.
The writing of the book is financially supported in part by the National 973
ProjectunderGrant2009CB320600973,theNationalNaturalScience Foundation
ofChinaunderGrants61333012,61273201,61329301,60825303and61203035,
and the Key Laboratory of Integrated Automation for the Process Industry,
Northeast University.
ix
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Filtering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Robust Filtering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Uncertainty in Systems . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Uncertainty in Signals. . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.3 Quadratic Robust Filter Design. . . . . . . . . . . . . . . . . . . 5
1.3 Parameter-Dependent Robust Filtering. . . . . . . . . . . . . . . . . . . 7
1.3.1 Significance of Parameter Dependence . . . . . . . . . . . . . 7
1.3.2 Parameter-Dependent Robust Filter Design . . . . . . . . . . 8
1.3.3 Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Organization of the Book. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5.1 Schur Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5.2 Elimination Lemmas. . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5.3 GKYP Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5.4 Jensen Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Part I Quadratic and Parameter-Dependent Filter Design
2 Quadratic Robust Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1 Quadratic Robust H Filter Design . . . . . . . . . . . . . . . . . . . . . 26
2
2.1.1 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.2 H Filtering for Nominal Systems: Continuous-Time. . . . 28
2
2.1.3 Connectiontothe Kalman Filtering: Continuous-Time. . . 35
2.1.4 H Filtering for Nominal Systems: Discrete-Time. . . . . . 39
2
2.1.5 Connection to the Kalman Filtering: Discrete-Time. . . . . 44
2.1.6 Quadratic Robust H Filtering . . . . . . . . . . . . . . . . . . . 47
2
2.2 Quadratic Robust H Filter Design. . . . . . . . . . . . . . . . . . . . . 52
1
2.2.1 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.2.2 H Filtering for Nominal Systems . . . . . . . . . . . . . . . . 54
1
2.2.3 Quadratic Robust H Filtering. . . . . . . . . . . . . . . . . . . 58
1
xi
xii Contents
2.3 Quadratic Robust Energy-to-Peak Filter Design. . . . . . . . . . . . . 61
2.3.1 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.3.2 Energy-to-Peak Filtering for Nominal Systems. . . . . . . . 63
2.3.3 Quadratic Robust Energy-to-Peak Filtering. . . . . . . . . . . 64
2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.5 Summary and Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.5.1 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.5.2 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3 Parameter-Dependent Robust Filter Design . . . . . . . . . . . . . . . . . 83
3.1 Slack Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.1.1 H Performance Criterion. . . . . . . . . . . . . . . . . . . . . . . 84
2
3.1.2 H Performance Criterion. . . . . . . . . . . . . . . . . . . . . . 87
1
3.2 Filter Realization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.2.1 H Filter Realization. . . . . . . . . . . . . . . . . . . . . . . . . . 90
2
3.2.2 H Filter Realization . . . . . . . . . . . . . . . . . . . . . . . . . 94
1
3.3 Linearly Parameter-Dependent Approaches. . . . . . . . . . . . . . . . 96
3.3.1 H Filter Design I. . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
2
3.3.2 H Filter Design I . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
1
3.4 Polynomially Parameter-Dependent Approaches . . . . . . . . . . . . 100
3.4.1 Homogeneous Polynomial . . . . . . . . . . . . . . . . . . . . . . 101
3.4.2 H Filter Design II . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
2
3.4.3 H Filter Design II. . . . . . . . . . . . . . . . . . . . . . . . . . . 109
1
3.5 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.6 Summary and Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.6.1 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.6.2 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Part II Robust Filtering for Time-Delay Systems and 2-D Systems
4 Robust Filtering for Continuous Time-Delay Systems . . . . . . . . . . 125
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.2 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.3 Filter Analysis and Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.3.1 Filter Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.3.2 Filter Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.3.3 Polynomially Parameter-Dependent Approach
to Filter Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.4 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137