Robust Control: The Parameter Space Approach

496 Pages·2002·17.718 MB·English

by  Professor Jürgen Ackermann (auth.)

Communications and Control Engineering
Springer-Verlag London Ltd.
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In cooperation with Paul Blue,Tilman Bunte, Levent Guvenc,
Dieter Kaesbauer, Michael Kordt, Michael Muhler
and Dirk Odenthal
Robust Control
The ParameterSpace Approach
Second edition
With321 Figures
i
Springer
Professor Jürgen Ackermann
Deutsches Zentrum fur Luft-und Raumfahrt, Institut fur Robotik und Mechatronik,
Oberpfaffenhofen 82230 Wessling, Germany
Se ries Editors
E.D. Sontag • M. Thoma
ISSN 0178-5354
ISBN 978-1-4471-1099-6 ISBN 978-1-4471-0207-6 (eBook)
DOI 10.1007/978-1-4471-0207-6
British Library Cataloguing in Publication Data
Ackermann, Jurgen
Robust control : the parameter space approach. -2nd ed. -
(Communications and control engineering)
I.Robust control
I.Titie
629.8'312
ISBN 978-1-4471-1099-6
Library of Congress Cataloging-in-Publicalion Data
Ackermann, J. (Jürgen)
Robust control: the parameter space approach I JÜfgen Ackermann.
p. cm. --(Communications and control engineering)
Includes bibliographical references.
ISBN 978-1-4471-1099-6 (alk. paper)
1. Robust contro!. I. Title. 11. Series.
TJ217.2 .A25 2002
629.8--dc21 2001032821
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Originally published by Springer-Verlag London Berlin Heidelberg in 2002
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Preface
Linear control systems with known parameter values are usually described by state
space or transfer function models. Deviations from the nominal parameter values are
then frequently modelled in an assumed generic structure. Examplesare multiplicative
perturbationswith arbitraryphaseand boundedgain ornorm bounded perturbationsof
statespace models. The advantage ofthese approaches isthat onecan develop generic
design procedures and software without restrictions to specific classes ofplants.
In many applications, however, more specific knowledge is availableon how uncer
tain real physical parameters (e.g. mass, velocity, friction coefficient, geometry etc.)
enter a well-known model structure. Even a large uncertainty of a few such essential
parameters can frequently be tolerated by a fixed-gain or gain-scheduled controller, if
it is tailored to the structured parameters. Ifthey are embedded, however, in a larger
number ofgenerically structured or complex-valued perturbations, then only small ad
missible parameter uncertainties result. In order to get non-conservative results, we
focuson essential parametersentering a knownstructure. Additionally,high-frequency
unstructured model uncertainty and sensor noise are considered by frequency domain
specifications on Bode magnitude plots ofdifferent sensitivity functions.
The approach ofthe book isgraphics-oriented,and takesadvantageofthe fact that
the engineer nowadays has increasingly powerfulcomputer graphicson his or her desk.
Optimization-basedapproachesarenot covered. Webelievethatoptimizationshould be
doneat a higher system levelwhere many trade-offsbetween different specifications for
theoverallsystem are required. At the lowrobust feedbackdesign level, the flexibility
for such trade-offsshould be preserved, e.g. by admissiblesolution sets.
In this second edition of Robust Control, the material is arranged such that the
reader is first introduced to the parameter space approach. Once the limitation to a
few uncertain plant parameters or a fewfree controller parameters in a design step is
accepted, the reader isrewarded by easy-to-interpret figuresthat make design conflicts
transparent, and by non-conservative mapping ofspecifications and parameter ranges.
Analysis methods for many uncertain parameters in specifically restricted structures
(affine, tree-structured) are postponed to later chapters. New results for the design
of PID-controllers and frequency domain specifications are included as well as many
newapplications in the case study chapters. Compared to the first edition of 1993,the
newedition is essentially completely rewritten and augmented by contributions ofnew
coauthors.
vi
Contents ofthe Chapters
Chapter 1introducesan idealized crane example to illustrate analyticalstatespace and
transfer function modelling of a plant with uncertain physical parameters q bounded
by an operating domain Q. Further, it reviews the notions of robust controllability
and observability and shows that for robust observability of the crane a sensor for
the rope angle is necessary in addition to a position sensor. The relations between
open-loop and closed-loop parametric characteristic polynomials via state and output
feedback are established. A parametric Hurwitz-stability analysis forthe crane yields a
robust transparent linear controllerstructurewith only threefreecontroller parameters
k1,k2,k3 forpendulumdampingand crabmotion stabilization. The crane isa represen
tative for two-mass systems like the inverted pendulum, two coupled vehicles, a robot
joint, an aircraft rudder deflection. At theend ofChapter 1,wehaveacontrollerstruc
ture and a characteristic polynomial p(s,q,k) with uncertain plant parameters q and
undeterminedcontrollerparametersk. ForChapters 2to 4and 8to 11,this parametric
polynomial serves as the primary interface between the engineering problem and the
mathematical problem ofroot locations ofparametric polynomials.
Chapter 2introduces the concepts ofcritical stability conditions,fictitiuous bound
aries, active boundaries and non-active boundaries, all arising from the last Hurwitz
determinant: Also, the distinction between real root boundaries (RRB), complex root
boundaries (CRB) and infinite root boundaries (IRB) is introduced. The boundary
crossing concept of Frazer and Duncan then leads to the more transparent parameter
sp~eapproach. The pole placement technique allowsa sequential shifting ofpole pairs
in invariance planes as a strategy element for stepwise improvement ofeigenvalue pat
terns. Next, the case forthe parameter space mapping equations being singular, giving
rise to singular frequencies, is discussed. The singular frequencies provide boundaries
between parameter ranges in which the resulting stability regions are topographically
similar. The properties of singular frequencies are exploited for the design of PID
controllers, where forfixed proportional gain the stability regions are convex polygons.
Chapter 3 introduces the basic ideas of the parameter space approach for design
and analysis ofrobust control systems based on eigenvalue specifications. First, some
relations between pole and zero locations and time responses are reviewed. The de
sirable I'-region for the closed-loop poles is used to define f-stability with a boundary
af(a),a E [a-;o"]. Thecraneexample illustrates that griddingofadmissible parame
ter regions may be useful-althoughcomputationally inefficient-foranalysis ofa given
polynomial family. It does not give directions, however, for design steps that improve
the system. Design for two controller parameters at a time can be done by mapping
af for some representative operating conditions, e.g. the vertices of the operating do
main Q. The set ofall simultaneously stabilizingregions for the representatives isthen
the intersection of the stabilizing regions for the representatives. A controller from
this intersection ischosen and the resulting closed-loopsystem isanalyzed by mapping
I' into a plant parameter plane (e.g. load mass and rope length of the crane). This
analysis shows ifthe entire operating domain Qis f-stabilized, not only the represen
tatives. Also, a gain-scheduling approach for measured rope length is illustrated. The
migrationofeigenvaluesunder continuous controller parameterchanges isdiscussed. It
yields a one-to-one correspondence ofopen-loop and closed-loop eigenvaluesas longas
Preface vii
nobranching points have beencrossed. Finally,a simpleexampleshows that controlla
bilityand observabilityofeachplant representative doesnot guaranteethat there exists
a simultaneously stabilizing controller. This is a fundamental difference to controller
design fora nominal plant.
Whilethesecond and third chaptersgivethe motivationforstudying the parameter
space approach, Chapter 4 deals with the mathematical generation of the mapping
equations. First, the boundary of the f-stable region is parameterized as ar(O'),0' E
[0'-;0'+],wherethe scalar 0'plays the same roleasthe frequencyw forHurwitz-stability
boundaries. The polynomial p(s,q) has a root at 0' = 0'*on ar = a(O')+jw(O') if
and only if both the real and the imaginary parts, ofp(a(O'*) +jw(O'*),q) vanish for
some admissible q E Q. Instead of real and imaginary part two linear combinations
thereof are used that can be generated by a simple recursion formula. The resulting
equation has the form D(O')a(q) = 0, where D(O') is a 2 x (n+1) matrix and a(q)
is the coefficient vector ofthe closed-loop characteristic polynomial. If0'is eliminated
from these two equations, then a Hurwitz-type algebraic criterion for r-stability is
obtained. It is very complicated, however. For the parameter space mapping, the two
equations are solved, e.g. for ql(0'),q2(0') that generate the boundary in the (ql'q2)
plane for a sweep over O'. The concept of singular frequencies is generalized to the
boundary ar and isused forthe design ofI'-stabilizing PID-controllers yieldingclosed
loop poles in a shifted left half plane or in a circle. An example with bilinear coefficient
function a(q) introduces us to the treatment ofnon-linear parameter dependencies by
the parameter space approach. An example with complicated polynomial coefficient
functionsshows this particular strength ofthe parameter space approach. The bilinear
example illustrates that for non-linear coefficient functions in a(q), the worst case
operating condition is not necessarily on the boundary of the operating domain, but
may be an interior point. A Jacobian condition is introduced to identify the interior
candidates for the worst case. The resultant method that is used here is described in
Appendix A. A short discussion of extensions to higher dimensional parameter spaces
concludes the chapter.
Chapter 5 now extends the scope beyond the characteristic polynomial as the in
terface between engineering systems and their mathematical robustness analysis and
design. An introductory example illustrates that r-stability does not always imply
good gain and phase margins. Therefore, e-stability is defined to guarantee safety
margins for Nyquist plots from the critical point - 1and from negative-inverse describ
ing functions. Tangent and point conditions yield the boundaries in q-space for which
8-boundaries are crossed. Similar conditions arise when a Popov plot does not admit
a desired sector for the non-linearity. B-stability refers to bounds on Bode magnitude
plots. Again,boundariesoccur fortangentand pointconditions. Mathematically,these
boundary conditions lead to two equations PI(w,ql,q2) = 0,P2(W,ql,q2) = 0 that are
similar to the realand imaginary part ofa polynomial. Finally,in Chapter 5the multi
input, multi-output (MIMO) case is reduced to the same type of mapping condition.
In particular, the H and H conditions are treated.
2 oo
Chapters 6and 7illustrate the combined useofthe tools in casestudies. Chapter 6
introduces us to the lateral, yaw and roll dynamics of car steering. The essential un
certainty is the lateral forcebetween tire and road. The structure ofthe mathematical
viii
modelallowsa robust unilateral decoupling such that the lateral acceleration becomes
independent of the yawrate. In driver assistance systems, control ofthe lateral accel
eration for track following is left to the driver, while the yaw motion is automatically
controlled. Automatic car steering systems take overboth tasks. The ideal concept of
skidding avoidance is based on additional rear wheelsteering. Ifthat is not available,
then several trade-offs must be made. In rollover avoidance, the uncertain height of
the center ofgravity posesa robustness problem. Chapter 7contains three casestudies
in flight control. The first one directly transfers the above robust decoupling concept
to an aircraft in an engine fault situation. Common to both problems is that the yaw
disturbance causeslargeyawmotions followed byoverreaction ofthe driver orpilot. In
the caseofthe aircraft, the robust automatic control system allowsa significantweight
reduction ofthe vertical fin. The other twocasestudies deal with aircraft stabilization
and control ofthe short-periodlongitudinalmode. First, r-specificationsare used,then
B-stability isspecifiedand achievedovera large operating domain.
In Chapter 8, the valueset approach is presented that is particularly suited to the
robustnessanalysisofpolynomialswith manyuncertainparameters. However, there are
restrictions on the kind ofcoefficientfunctions that can be handled. A common basis
is the Mikhailov plot pOw) for Hurwitz-stability, which blows up to a family of plots
for uncertain q E Q,wherethe operating domain Q isa hyperrectangle (Q-box). This
Mikhailovset must contain onestable polynomial and it must not contain the originof
the complexpOw)-plane. The zero exclusioncondition is used in Chapter8 for proofs
of the Kharitonov and edge theorems. The Kharitonov theorem applies to interval
polynomials and requires stability of, at most, four extremal polynomials. The edge
theorem applies to polynomials with affinecoefficientfunctions and requires stability
tests for edges ofthe Q-box. An edge may be checkedfor stability by the Bialas test.
Also,the singularityofvaluesetsisanalyzed. The constructionofMikhailovsetsisdone
for fixed frequencies, i.e. only the scalar w is gridded and not the higher dimensional
Q-box. Forlinear coefficient functions, the valueset forfixedfrequency isa parpolygon
(polygon with pairwise parallel edges) and the origincan enter only through the edges.
The edgeresult alsoholdsforr-stability. In this case, it ismoreefficientto plot a root
locus foreach edge rather than generalizing the Bialas test.
In Chapter9,the valueset approach isapplied to non-linear coefficientfunctions. A
warning example at the beginning shows that unstable islands, even isolated unstable
points, can occur due to a bilinear term in oneofthe coefficients. Thus,simplegeneral
results, as inthe linearcase,cannotbe expected inthe non-linear case. However,useful
tools for the computer-aided rendering ofvaluesets are developed. Formultilinear co
efficientfunctions, the mapping theorem byDesoerprovidesasimplesufficientstability
condition. It says that the convexhull of the value set is generated by the images of
the vertices of the Q-box. It restricts the frequency band in which the actual value
set must be constructed. Necessary and sufficient stability tests for multilinear (and
some polynomial) coefficient functions can be performed if the uncertain parameters
enter in form of a tree-structure into the characteristic polynomial p(s,q), i.e. p(s,q)
may be expressed as a sum ofsubpolynomials, which in turn may be products ofsub
subpolynomials, etc. until basic polynomials are reached. The sequential construction
of the value set becomes possible, even for many uncertain parameters, ifeach uncer-
Preface ix
tain parameter enters only into one of the basic polynomials. Ifsuch a tree-structure
exists, then it can be exploited both for Hurwitz-stability and for f-stability analysis.
Tree-structures arise naturally, for example, in modelling of mass-spring-damper sys
tems and theyare preserved under feedback. Also,given polynomials may beanalyzed
fortree-structures. Ausefulapplication isthe calculationofthe "stability profile",that
is, the right hand boundary ofthe root set.
Chapter 10introducesthe stability radius asa scalar measure ofvicinity toinstabil
ity. Essentially, a ball or box around a test point in parameter space isblown up until
it hits the stability boundary. Tsypkin and Polyak use a frequency-dependent distance
function. Also, an algebraic formulation isgiven for polynomial coefficient functions.
In Chapter 11, the previous results are adapted as far as possible to sampled-data
systems. A problem in the exact treatment here arises, because physical parameters
enter exponentially into the polynomial coefficients. A good approximation can be
achieved, however, by a Poisson series approach.
Appendix Acovers some mathematical background material on polynomials, poly
nomial equations and conic sections. Appendix B gives a short introduction to the
software tool PARADISE (PArametric Robustness Analysis and DesignInteractive Soft
ware Environment). The latest release of the software can be downloaded from
www.robotic.dlr.de/control/paradise.
The bibliography shows that many foundations of the methods of robust control
havebeen laid over a longperiod oftime byscientistsin Russia and othereastern coun
tries (Bialas, Gantmakher, Kharitonov, Mikhailov, Mitrovic, Neimark, Polyak, Popov,
Siljak, Tsypkin, Vishnegradski and many more). Only in the last decades have these
contributions been recognized and further developed in the western literature (Ander
son,Barmish,Bartlett,Bhattacharyya,Dasgupta,Desoer,Fam,Hollot,Horowitz,Jury,
Mansour, Meditch, Tempo and many more).
General Remarks
The prerequisite for the reader isan undergraduatecourse in feedbackcontrolsystems.
We try to keep the mathematics simple. The book is suited for an advanced under
graduatelevelor fora first graduatelevelcoursein robust control. In fact,thematerial
was selected and used for such courses at the University of California, Irvine, at the
Technische Universitat Miinchen and in several short courses. It was also used in con
tinuing education courses at the Carl-Cranz Gesellschaft (CCG), Oberpfaffenhofen,for
participants from industry.
Forthe purposeofsuchcourses, a very restrictiveselection had to be made fromthe
large and rapidly growingliteratureon robust control. Therefore,manyimportantcon
tributions and alternative approaches could not be mentioned. Some cross-references
are given in the form ofremarks. Remarks also indicate possible generalizations, open
problems, and other supplements that are not prerequisites for understanding the fol
lowing sections. The beginner should ignore all remarks, they are intended for the
advanced reader.
x
In the examples with physical parameters, units are given in brackets, e.g. [m]
for meter (to be distinguished from the symbol m for mass) or Is] for seconds (to be
distinguished from the complex variable s of Laplace transforms). In calculations, the
units are omitted. Weuse the followingunits:
Physical Variable Symbol Unit
e
Length meter [m]
Time t second Is]
Mass m kilogram [kg]
Moment ofinertia J [kg m"]
v
Force F newton [N] = [kg-m/s2]
Torque M newton-meter [Nm]
Velocity v [m/s]
Acceleration a [m/s2]
Angles 0:,{3,... radian [rad]
Frequency w [rad/s]
Flight altitude h foot [ft]= 0.3048meter
Mach number M ratio ofv and velocity ofsound,
dimensionless
Acknowledgements
The authors liketo thank the coauthorsofthe first edition ofthis book (1993),whoare
workinginotherfieldsnow. Material written by Andrew Bartlett, WolfgangSieneland
Reinhold Steinhauser has been integrated in the new context. Thanks also to Jessica
Laskeyfortyping partofthe manuscriptand toStefan vonDombrowskiforhistechnical
support.
oberpfaffenhofen Jiirgen Ackermann
June 2002 Paul Blue
Tilman Bunte
Levent Giivenc
Dieter Kaesbauer
Michael Kordt
Michael Muhler
Dirk Odenthal