Analysis and Synthesis of Dynamic Systems with Positive Characteristics

136 Pages·2017·1.838 MB·English

by  Jun Shen

Springer Theses
Recognizing Outstanding Ph.D. Research
Jun Shen

Analysis and
Synthesis of
Dynamic Systems
with Positive
Characteristics
Springer Theses
Recognizing Outstanding Ph.D. Research
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Jun Shen
Analysis and Synthesis
of Dynamic Systems
with Positive Characteristics
Doctoral Thesis accepted by
The University of Hong Kong, Hong Kong
123
Author Supervisor
Dr. Jun Shen Prof. James Lam
NanjingUniversity ofAeronautics Department ofMechanical Engineering
andAstronautics TheUniversity of HongKong
Nanjing Pokfulam
China Hong Kong
ISSN 2190-5053 ISSN 2190-5061 (electronic)
SpringerTheses
ISBN978-981-10-3879-2 ISBN978-981-10-3880-8 (eBook)
DOI 10.1007/978-981-10-3880-8
LibraryofCongressControlNumber:2017934865
©SpringerNatureSingaporePteLtd.2017
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’
Supervisor s Foreword
Positive systems form a special category of systems possessing the characteristic
that their inputs, states, and outputs are intrinsically nonnegative. They arise nat-
urally as practical systems often involve nonnegative system variables.
Applications of such systems have been found in modeling compartmental net-
works, population evolution, chemical reactors, network congestion control, and
generegulation.Existinglineartheoriesareoftenhandicappedbytheirinflexibility
in handling systems that are defined on cones, rendering most design methods not
directly applicable to positive systems. Therefore, developing new theories on the
analysis and synthesis of positive systems is imperative.
Positive systems have many remarkable properties, one of which is their
robustness against delay in the state variables. This peculiar property reduces the
controllerdesignofdelayedpositivesystemstothatofthecorrespondingdelay-free
systems. In fact, this delay insensitivityis a consequence of cone invariance rather
than positivity. Many classes of structured optimal control problems for positive
delaysystemscanbedirectlycastasconvexoptimization.Linearprogramming,an
effective convex optimization tool with low computational complexity, is particu-
larly suitable for the controller synthesis of positive systems, ranging from
stabilization to robust control with optimal performance indices. Linear-
programming-based controller design algorithms are developed in this book,
which are applicable for large-scale positive systems.
This book is mainly devoted to the analysis and synthesis for several classes of
dynamic systems with positivity or more general cone invariance. In terms of
analysis,theL1-gainofpositivelinearsystemswithunboundeddiscretedelaysand
distributed delays is fully characterized, with applications to multi-agent systems
subjecttocommunicationdelays.Theseresultsarefurtherextendedtolineardelay
systems invariant on a general proper cone. In terms of synthesis, the output-
feedbackstabilizationproblemwithoptimalgainperformanceaswellasthemodel
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vi Supervisor’sForeword
reduction problem for positive systems is investigated. I strongly believe that the
theories developed in this book would notonly be useful for controller design and
implementation for complex systems at a moderate cost, but also help understand
thesystemcharacteristicsinotherrelatedareas,suchascoordinationofmulti-agent
systems, distributed power control of wireless networks, and regulation of genetic
networks.
Pokfulam, Hong Kong Prof. James Lam
March 2017
Abstract
Thisthesisisconcernedwithanalysisandsynthesisproblemsforseveralclassesof
dynamic systems with positivity or a more general cone invariance.
In terms of analysis, stability and performance characterizations are established
for several types of dynamic systems with positivity or cone-invariant property.
Specifically,thefollowingfouraspectsareexploited:(a)Forpositivelinearsystem
with bounded or unbounded time-varying delays, it istheoretically proved that the
‘1=L1-gainisfullydeterminedbythesystemmatrices,whilethetimedelaysplay
norolesinthe‘1=L1-gaincharacterization.Asanapplicationexample,itisshown
that the convergence rate analysis of containment control of multi-agent systems
with diverse communication delays can be cast as stability analysis of a corre-
sponding positive system with multiple delays. (b) For a positive system with
distributeddelays,itisshownthatitsL1-gainisthesameasthatofacorresponding
delay-freepositivesystem.Alongthisline,upperandlowerboundsfortheL1-gain
of a positive system with distributed delays over a bounded time-varying interval
are also given. (c) For a linear delay system which is invariant with respect to a
general proper cone, its asymptotic stability and cone-induced gain turn out to be
insensitive to the magnitude of time delays. (d) For a class of coupled
differential-difference equations, necessary and sufficient conditions on the posi-
tivity and asymptotic stability are presented.
Intermsofsynthesis,severalfundamentalcontrolproblemsarestudied:(a)The
static output-feedback stabilization problem for positive systems is revisited. It is
pointedoutthatforaclassofpositivesystemswhoseoutputmatrixhasaparticular
row echelon form, this problemcan be completely solved via linear programming.
Byduality,thisfactisalsovalidwhenthecolumnechelonformoftheinputmatrix
hasaparticularstructure.Alongthisline,byaugmentingtheoutputmatrixaswell
as the feedback gain matrix, an iterative convex optimization algorithm is devel-
oped for a general multi-input multi-output positive system. (b) The static
output-feedback stabilization problem with optimal L -gain for positive linear
1
systemsisaddressed.Itisshownthatwhenthecontrolinputorthemeasuredoutput
isascalar,thisproblemcanbedirectlysolvedvialinearprogrammingbyaddinga
one-dimensional search. (c) The H1 model reduction problem of discrete-time
vii
viii Abstract
positive linear systems with inhomogeneous initial conditions is investigated.
A necessary and sufficient condition is established for the existence of a desired
reduced-ordermodelsuchthattheoutputerrorbetweentheoriginalsystemandthe
reduced-orderoneisboundedbyaweightedsumofthemagnitudeoftheinputand
that of the initial condition. Moreover, based on congruent transformation and the
dual form of bounded real lemma, several equivalent conditions are derived in
termsoflinearmatrixinequalitiesandaniterativeconvexoptimizationalgorithmis
developed accordingly.
Declaration
I declare that this thesis represents my own work, except where due acknowl-
edgement is made, and that it has not been previously included in a thesis,
dissertation, or report submitted to this university or to any other institution for a
degree, diploma, or other qualifications.
ix