Low gain feedback

365 Pages·1999·4.2 MB·English

by  Zongli Lin (auth.)

Lecture Notes
in Control and Information Sciences
240
Editor:M Thoma
Springer
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ZongliLin
Low Gain Feedback
, Springer
SeriesAdvisoryBoard
A.Bensoussan . M.J.Grimble' P.Kokotovic . H.Kwakernaak
J.L.Massey' Y.Z.Tsypkin
Author
ProfessorZongliLin
DepartmentofElectricalEngineering,UniversityofVirginia,
Charlottesville,Virginia22903,USA
ISBN1-85233-081-3 Springer-Verlag London Berlin Heidelberg
BritishlibraryCataloguinginPublicationData
Lin,Zongli
Lowgainfeedback. - (Lecturenotesincontroland
informationsciences ; 240)
1.Feedbackcontrolsystems
I.Title
629.8'312
ISBN1852330813
libraryofCongressCataloging-in-PublicationData
Lin.Zongli,1964-
Lowgainfeedback / ZongliUn
p. cm. -- (Lecturenotesincontrolandinformationsciences
; 240)
Includesbibliographicalreferencesandindex.
ISBN1-85233-081-3 (pbk. : alk.paper)
1.Feedbackcontrolsystems. I.Title. II.Series.
TJ216.L49 1998 98-39948
629.8'3--dc21 CIP
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To Jian, Tony and Vivian
Preface
Comparedtothatofhighgainfeedback, thepowerfulnessoflowgainfeedbackis
lessrecognized. Recently,lowgaindesigntechniqueshavebeenbeingdeveloped
foranincreasingnumberofcontrolproblems,includingcontroloflinearsystems
with saturating actuators, semi-global stabilization of minimum phase input
output linearizable systems, H2- and Hoo-suboptimal control, and nonlinear
Hoo-control.
Thepurposeofthepresentmonographistopresentsomerecentdevelopment
in low gain feedback and its applications in a coherent manner. While most of
the materialsaredrawnfrom recent papers publishedbytheauthorandhisco
authors in the past several years, many design and prooftechniques presented
inthemonographaregreatlyimproved,simplifiedandunified. Themonograph
also contains some new results that have not been submitted for publication.
The intended audience ofthis monograph includes practicing control engi
neers and researchers in areas related to control engineering. An appropriate
background for this monograph would be some first year graduate courses in
linearsystemsandmultivariablecontrol. Somebackgroundinnonlinearcontrol
systems would greatly facilitate the reading ofthe monograph.
Our presentation is organized as follows. In Chapter 1, after a short intro
duction to low gain feedback, in comparison ofhigh gain feedback, we include
a list ofnotations and acronyms that are used throughout the monograph.
.Chapter 2 presents two basic low gain feedback design methodologies, one
based on a direct eigenstructure assignment and the other on the solution ofa
certain parameterized algebraic Riccati equation, and establishes some funda
mentalpropertiesoflowgainfeedback. Bothcontinuous-timeanddiscrete-time
systems are considered.
Chapter 3 contains the first application of low gain feedback: semi-global
asymptotic stabilization oflinear systems with saturating actuators via linear
feedback. Bysemi-globalasymptoticstabilizabilityvialinearfeedback, wemean
that for any a priori given (arbitrarily large) bounded set of the state space,
vii
viii Preface
we find a linear feedback law such that the origin is a locally asymptotically
stable equilibrium point with the given set contained in its basin ofattraction.
Bothstatefeedback andoutputfeedback caseareconsidered. Ourmain results
establish the fact that, if a linear system is asymptotically null controllable
with bounded controls, then, when subject to actuator magnitude saturation,
it is semi-globally asymptotically stabilizable by linear state feedback. If, in
addition, the system is also detectable, then it is semi-globally asymptotically
stabilizablevialinearoutput feedback. Bothcontinuous-timeand discrete-time
systems are considered.
Chapter 4 addresses several fundamental control problems beyond semi
globalasymptoticstabilizationforlinearsystemssubjecttoactuatormagnitude
saturation. Theseproblemsincludeinput-additivedisturbancerejectionandro
bust semi-global asymptotic stabilization in the presenceofmatched nonlinear
uncertainties. To solve these problems, two new design techniques, referred to
as low-and-high gain feedback and low gain based variable structure control,
are developed. Both design techniques are based on the low gain feedback de
signtechnique. Asinthe lowgainfeedback design, therearetwo approachesto
the low-and-highgain feedback design, basingrespectively on the direct eigen
structure assignment and on the solution ofa certain parameterized algebraic
Riccati equation.
Chapter5utilizesthe lowgainfeedback designtechniquestoconstructfeed
back laws that solve semi-global output regulation problems for linear systems
subject to actuator magnitude saturation. Solvability conditions as well as ex
plicit constructionoffeedback lawsthat solvethe problemsareexplicitlygiven.
Our problem formulation follows the classicalformulation ofoutput regulation
forlinearsystemsandisaproblemofcontrollingalinearsystemsubjecttoactu
atorsaturationinordertohaveitsoutputtrack (orreject) a family ofreference
(or disturbance) signals produced by some external generator. The utilization
oflow-and-highgainfeedbackofChapter4inthesolutionofsemi-globaloutput
regulation problems is discussed.
Chapter 6 examines the problem ofsemi-global almost disturbance decou
pling with internal stability for linear systems subject to actuator magnitude
saturation and input additive disturbance. Here by semi-global we mean that
the disturbances are bounded either in magnitude orin energy by any apriori
given (arbitrarilylarge) boundednumber. Thelow-and-highgainfeedbacklaws
as developed in Chapter 4 is utilized to establish that, semi-global almost dis
turbancedecouplingwithlocalasymptoticstabilityisalwayssolvablevialinear
state feedback as long as the system in the absence of actuator saturation is
stabilizable, nomatter where the polesoftheopen loopsystem are, and the10-
Preface ix
cationsofthesepolesplayaroleonlywhensemi-globalasymptoticstabilization
is required.
Chapter 7 develops some scheduling techniques that upgrade the low-and
high gain feedback laws to elevate some ofthe semi-global results into global
ones. Byschedulingthehighgaincomponentofthelow-and-highgainfeedback
law, we show that, in a sharp contrast to global asymptotic stabilization prob
lem as discussed in Chapter 3where open loop poles are required to be in the
closedleft-halfplane, globalfinite gain L,-stabilization can always be achieved
via state feedback, no matter where the open loop poles are. Moreover, the
L,-gain can be made arbitrarily small. By scheduling both the low gain and
high gain component of the low-and-high gain feedback law as developed in
Chapter 4, the semi-global results ofChapter 4 can be made global.
Chapter 8 studies the problem of semi-global asymptotic stabilization of
linearsystemssubjecttobothactuatormagnitudeand ratesaturation. Byuti
lizingthelowgainfeedback designtechnique,itisshownthat, ifalinearsystem
is asymptotically null controllable with bounded controls, then, when subject
to both actuator magnitude and rate saturation, it is semi-globally asymptot
ically stabilizable by linear state feedback. H, in addition, the system is also
detectable, then it issemi-globally asymptotically stabilizablevialinearoutput
feedback. Both continuous-time and discrete-time systems are considered.
Chapter 9 deals with minimum-phase input-output linearizable systems.
Combining low gain feedback and the classical high gain feedback, we obtain
a new family oflow-and-high gain feedback laws and use it to establish semi
global asymptotic stabilizability and the semi-global practical stabilizability of
such systems under some weak conditions, most of which are necessary. The
main rolelowgainfeedback playshereisthe avoidanceoftheso-calledpeaking
phenomenon.
Chapter 10demonstrateshowlowgainfeedback canbe utilized toexplicitly
construct feedback laws that solve the problem of perfect regulation and H2
suboptimal control problems for linear systems with invariant zeros on the jw
axis (unit circlefor discrete-timesystems). Thecomplexitydueto jw-axis (unit
circle) invariant zeros have been well-understood in the literature and, as a
result, they are alwaysexcludedfrom considerationintheexplicit construction
offeedback laws. Both continuous-time and discrete-time case are presented.
Chapter 11 illustrates how low gain feedback can be utilized to solve the
generalH almostdisturbancedecouplingproblem. Therolelowgainfeedback
oo
plays here is the treatment of jw-axis (unit circle) invariant zeros. As in the
H -suboptimal control problems, the major challenge in explicit construction
2
ofsuboptimal feedback laws comes from the presence of jw (unit circle) zeros.
x Preface
In the literature on the explicit construction offeedback laws, these invariant
zeroshavealwaysbeenexcludedfrom consideration. Bothcontinuous-timeand
discrete-time case are considered. Generalization to nonlinear systems is also
included.
The next two chapters include the applications of the low-and-high gain
feedback design techniques tosomephysical systems. Chapter 12considersthe
problem of balancing an inverted pendulum on a carriage, where the physical
limitations impose a constraint on the maximum allowable motion ofthe car
riage. Usinglowgainfeedback designideas,weproviderobust linearcontrollers
that balance the pendulum without violating the maximum allowable motion
constraint.
In Chapter 13, the low-and-highgain design technique ofChapter4is com
bined with another design technique recently developed for linear systemswith
magnitudesaturatingactuators, thepiecewise-linearLQ control,to yieldanew
design technique for linear systems with rate saturating actuators. The com
bined design takes advantages of both design techniques, while avoiding their
disadvantages. An open loop exponentially unstable F-16 class fighter aircraft
is used to demonstrate the effectiveness ofthe combined design method.
Finally, in the Appendix, we collect some technical tools that we have used
in more than one place in the monograph.
I have been fortunate to have the benefit ofthe collaborationofseveral co
workers,from whom I havelearned a great deal. Manyofthe results presented
inthis monographarethe resultsofourcollaboration. Amongtheseco-workers
areDr. SivaBandaoftheAirForceWright Laboratory,Dr. BenM. ChenofNa
tional UniversityofSingapore, Mr. Ravi Mantri ofUS Robotics, Inc., Dr. Meir
Pachter ofthe Air Force Institute ofTechnology, Dr. Ali Saberi ofWashington
State University, Dr. PeddaSannutiofRutgers University, Dr. YacovShamash
of State University of New York at Stony Brook, and Dr. Anton Stoorvogel
of Eindhoven University of Technology, Dr. Andrew R. Teel of University of
California, Santa Barbara.
IamespeciallygratefultoDr.BenChen,Dr.AliSaberi,Dr.YacovShamash,
and Dr. Gang Tao of University of Virginia for the fine examples they set
and their continual support and friendship. I am also grateful to University
of Virginia, in particular the Department of Electrical Engineering and other
colleaguesin the department, for makinganexcellent environmentin which we
live and work. I am indebted to Professor Petar Kokotovic for suggesting the
inverted pendulum system ofChapter 12 as an example for low gain feedback.
I would also like to acknowledge the Air Force Office of Scientific Research,
Preface xi
whose support enabled me to participate in the 1996 AFOSR Summer Faculty
Research Program and to be exposed to flight control problems.
Last, but not the least, lowea greatdeal tomywife, Jian, and mychildren,
TonyandVivian,foralltheirsacrifice. Itistheirunderstanding,encouragement
and love that encourage me to striveon.
ThismonographwastypesetbytheauthorusingD1EjX. Allsimulationsand
numerical computations were carried out in MATLAB. Diagrams were gener
ated using xfig.
Zongli Lin
Charlottesville, Virginia
June, 1998