Controller Design for Distributed Parameter Systems

295 Pages·2020·6.269 MB·English

by  Kirsten A. Morris

Communications and Control Engineering
Kirsten A. Morris
Controller Design
for Distributed Parameter
Systems
Communications and Control Engineering
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Kirsten A. Morris
Controller Design
for Distributed Parameter
Systems
123
Kirsten A.Morris
Department ofApplied Mathematics
University of Waterloo
Waterloo, ON,Canada
ISSN 0178-5354 ISSN 2197-7119 (electronic)
Communications andControl Engineering
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Preface
Many systems have dynamics that depend on space as well as on time. Examples
include diffusion, transmission lines, acoustic noise, and structural vibrations.
Because the system state is distributed in space along a continuum, they are
sometimes known as distributed parameter systems (DPS). This is different from
lumpedparametersystems,suchascircuits,wherethereareafinitenumberofstate
variables and the dynamics only depend on time. Lumped parameter systems are
modeled by ordinary differential equations and the state evolves on a
finite-dimensionalspace,usuallyRn.BecausethedynamicsofDPSdependonboth
timeandspace,themodelsarepartialdifferentialequationsandthestateevolveson
an infinite-dimensional state space. For this reason, they are sometimes referred to
as infinite-dimensional systems.
ThepurposeofcontrollerdesignforDPSissimilartothatforanyothersystem.
Every controlled system must of course be stable. Beyond that, the goals are to
improve the response in some well-defined manner, such as driving the system
response to equilibrium optimally in defined sense, or tracking a desired reference
signal. Another common goal is minimization of the system's response to distur-
bances. The related issue of estimation is of great interest for DPS since the dis-
tributed state generally needs to be calculated based on information from a finite
number of sensors.
There is now fairly extensive theory extending the common approaches to
controller and estimation design for lumped systems to DPS. However, for most
practical examples, controller synthesis based directly on the partial differential
equationdescriptionisnotfeasible,sinceaclosed-formexpressionforthesolution
is not available. Instead, a finite-dimensional approximation of the system is first
obtained and controller design is based on this finite-dimensional approximation.
Thehopeisthatthecontrollerhasthedesiredeffectontheoriginalsystem.Thisis
not always the case. However, conditions under which this practical approach to
controllerdesignworkshavebeenobtainedandaredescribedinthisbook.Another
issuewithcontrolandestimatordesignforDPSisthatthecalculationsoftenneedto
bedonewith high-ordersystems,particularlyforproblemsinmorethanonespace
vii
viii Preface
dimension.Algorithmsthatarereliableforlow-ordersystemssometimescannotbe
used and an algorithm suitable for high-order systems is needed.
Another issue for DPS is that because of the distributed nature of the system,
there is often freedom on where to put the control hardware, the actuators and/or
sensors.Thisfreedomcomplicatesthedesignprocess,butthisfreedomcanbeused
to improve performance by careful placement of the hardware.
This book is an introduction to the exciting and still growing subject of control
and estimator design for distributed parameter systems. It is intended to provide a
foundation and techniques for someone wishing to control or estimate DPS. There
are rigorous statements of results, but no proofs. A number of examples are pro-
videdtoillustratethetheory.Toillustratehowtheresultscanbeusedforsystemsin
multiple space dimensions, some examples are in multiple space dimensions. The
notes at the end of each chapter list references where proofs may be found. In
addition, there are references to books for further reading on the topics covered in
each chapter. Ihave made noattempt toprovide anexhaustive bibliography ofthe
extensiveliteratureoncontrolandsystemstheoryfordistributedparametersystems.
Such an attempt would be doomed to failure.
Because this book is an introduction, the class of models is restricted in several
ways. First, all models are linear. Another restriction is to consider only systems
with bounded control and observation. For many situations, such as control on the
boundaryoftheregion,idealizedmodelsleadtoastate-spacerepresentationwhere
the control operator B is unbounded on the state-space. More precisely, it is a
bounded operator into a larger space than the state space. This can sometimes
provide insight into the limitations of control but the analysis is more complicated
than for systems with bounded control and observation. Including a more detailed
model for the actuator often leads to a bounded operator. Similarly, modeling of a
sensor often leads to a bounded observation operator. Reference is provided at the
end of each chapter of generalizations to more general systems, when available.
Systems modeled by delay differential equations also have a solution that
evolvesonaninfinite-dimensionalspace.Althoughthephysicalsituationsarequite
different,theoryandcontroller/estimatordesignarequitesimilartothatofsystems
modeled by partial differential equations. However, the focus of this book is DPS,
and all the examples in this book involve PDE models.
Areviewofthecontrolandestimationtechniquesusedisprovided.Noprevious
knowledge of DPS is assumed. It is also not assumed that the reader has studied
functionalanalysis.TherequiredfunctionalanalysisisinAppendixAorcoveredas
needed.
This book arose from courses taught at the University of Waterloo to students
with varied backgrounds from mathematics and engineering. If students have
previouscoursesinfunctionalanalysisandcontrolsystems,mostofthematerialin
thisbookcanbecoveredinoneterm. Forstudents with noanalysis background,a
slowertreatmentofChaps.2and3isneeded,andperhapsalsosomeofthematerial
intheappendix.Ifthestudentshaveagoodbackgroundinanalysisanddifferential
equations,butnotincontrol,Chaps.2 and3canbe covered more quickly,butthe
later chapters, which cover control and estimator design, will need more attention.
Preface ix
I thank the Canadian National Sciences and Engineering Research Council for
ongoingfinancialsupportofmyresearch,andtheUSAirforceOfficeofScientific
Research for their support since 2010. Travel to learn from colleagues at confer-
ences and at their institutions was funded by these agencies, and some of the
material inthis book arose from research funded by these agencies. I have learnt a
lot from discussions and research with colleagues in both mathematics and engi-
neering.In particular, Iam indebted toM. Vidyasagar and Tom Banks for starting
my education in distributed parameter systems and also to Ruth Curtain. I thank
Amenda Chow, Yanni Guo, Chris Nielsen, Lassi Paunonen, and Hans Zwart for
their very useful detailed comments and suggestions on a draft of this book.
Waterloo, Canada Kirsten A. Morris
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Infinite-Dimensional Systems Theory . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 Riesz-Spectral Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2 Lumer–Phillips Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Sesquilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.5 Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.6 Controllability and Observability. . . . . . . . . . . . . . . . . . . . . . . . . 53
2.7 Input/Output Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.8 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3 Dynamics and Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.1 Sesquilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.2 Spectrum Determined Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.4 External Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4 Optimal Linear-Quadratic Controller Design . . . . . . . . . . . . . . . . . . 103
4.1 Finite-Time Optimal Linear Quadratic Control . . . . . . . . . . . . . . . 104
4.2 Infinite-Time Optimal Linear Quadratic Control . . . . . . . . . . . . . . 107
4.3 Solving the Algebraic Riccati Equation . . . . . . . . . . . . . . . . . . . . 123
4.4 LQ-Optimal Actuator Location . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
xi