Optimal control of piecewise affine systems A multi-parametric approach

250 Pages·2008·14.69 MB·English

ETH Library
Optimal control of piecewise affine
systems
A multi-parametric approach
Doctoral Thesis
Author(s):
Baotic, Mato
Publication date:
2005
Permanent link:
https://doi.org/10.3929/ethz-a-005055166
Rights / license:
In Copyright - Non-Commercial Use Permitted
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Diss. ETH No. 15957
Optimal Control of Piecewise Affine Systems
Multi-parametric Approach
— a —
A dissertation submitted to the
SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH
for the degree of
Doctor of Sciences
presented by
MATO BAOTIC
mr. sc, dipl. ing., University of Zagreb, Croatia
born 23.12.1973
citizen of Croatia
accepted on the recommendation of
Prof. Dr. Manfred Morari, examiner
Prof. Dr. Diethard Klatte, co-examiner
Prof. Dr. Nedjeljko Peric, co-examiner
2005
Acknowledgement
Every burden is easier to carry if it can be shared (or at least discussed) with someone.
Every joy is multiplied by the number of persons participating in it. I was happy enough
to make my thesis at such a splendid place as Automatic Control Lab in Zürich. Yes, the
weather in Zürich could have been better in those 4 years, but people could hardly be any
better than the ones at IfA. Professors, secretaries, and, by my account, the best group of
researchers in the world made every working and non-working hour thoroughly enjoyable.
First I would like to thank to Francesco Borrelli and Tobias Geyer. Without Francesco I
don't think I would have started anything of my research. His patience and endurance in
explaining all the research stuff as well as some of the more obscure things (not least one of
which is the position of Bellevue!) were most helpful. Tobias and I embarked on the quest
of getting our PhD's almost simultaneously and, by a stroke of good fortune, we've got it
on the same day. The world is a magnificent place and to be able to discuss any given topic
with him was a real pleasure.
My roommates Valentina Sartori and Michal Kvasnicadeserve aspecial thanks. Our room
rules! Among other things, Michal's dedication to the MPT beyond my wildest dreams made
it what it is - the best hybrid systems toolbox in the world, while Vale never gave up on me
learning Italian by casual listening. Mille baci.
IfFrancesco was responsible for setting me up on the path ofobtaining a PhD, then Frank
J. Christophersen was responsible for getting me there. Without him I would have never
finished many of the papers that are an essential part of this thesis. Yet the most amazing
aspect was not related to work, nor to Frank's vast knowledge of DTgK tricks, but to the
fact that we share a similar taste in so many things in life. Thanks for existing my friend.
It was great to be thought the tricks of the trade by the likes of Alberto Bemporad,
Giancarlo Ferrari-Trecate and Pablo Parrilo.
But the master ofthem all was by far my supervisor Professor Manfred Morari. Having a
supervisor with such a vast scientific knowledge, clarity in explaining the ideas and wisdom
in guiding PhD students through the valleys of ignorance and despair to the holy grail of
knowledge was, for the lack ofa better word, amazing. He was sending us around the globe
and bringing the world to IfA for the same reason - so that we always have a full access to
the vital research information while staying in touch with the reality. Overall, I think that
the proverbial IfA saying "it is always good to talk to MM" says it all. From the bottom of
my heart I thank him for the privilege of adding one more leaf (for the time being at least)
to that "tree" of his PhD students.
Thanks to Fabio D. Torrisi for showing me that the clear thinking is essential for getting
the job done and to Pascal Grieder for bringing a new vigor in our Lab and spurring a
plethora of research activity. Thanks to Alice, Martine, Danielle and Esther for helping
i
Acknowledgement ii
me find the path through the maze of the ETH administration. Andrea B., Andrea G.,
Domenico, Dominik, Eleonora B., Eleonora Z., Gabi, George, Gültekin, Helfried, Ioannis,
Jan, Johan, Kristian, Marc, Milos, Rainer, Raphael, Saso, Urban - you make IfA the greatest
working environment in the world. One has to experience it to believe it.
I would also like to thank Prof. Nedjeljko Peric, Prof. Ivan Petrovic, Miroslav Baric and
Mario Vasak from University of Zagreb for their encouragement and help in all the phases of
my PhD. Mario was instrumental in getting the electronic throttle to behave as the control
theory said it should.
Last but not least, this work would never have been possible if I was not set to life by my
parents Pavo and Manda. They and my brother Marinko have thought me by example the
most valuable lesson: with a little bit of patience and willingness to work one can indeed
move mountains with his own hands. Hvala vam za sve!
Mato Baotic
Zagreb, July 2005
Abstract
This thesis addresses the problem of constrained optimal control of discrete-time linear
hybrid systems.
In their most general form hybrid systems are characterized by the interaction of contin¬
uous-valued components (governed by differential or difference equations) and logic rules.
Among the variety ofequivalent descriptions ofdiscrete-time hybrid systems reported in the
literature we focus on the class of constrained piecewise affine (PWA) systems. Discrete-
time PWA models candescribe alarge number ofprocesses. Moreover, they can approximate
nonlinear discrete-time dynamics via multiple linearizations at different operating points.
Even though PWA systems are a special class of nonlinear systems most of the nonlinear
system andcontroltheorydoesnot applybecauseit requires certainsmoothness assumptions.
For the same reason we also cannot simply use linear control theory in some approximate
manner to design controllers for PWA systems. In the past most tools for the analysis and
control of hybrid systems were ad hoc supported by extensive simulation.
The aim ofthis thesis is to further advance systematicprocedures and develop algorithmic
implementations that give the exact solution to the optimal control problems. A recurring
theme in the thesis is the construction of efficient algorithms for solving various instances
of optimal control problems. An instrumental tool in the development of such algorithms
is the concept of multi-parametric programming, where a quadratic (or linear) optimization
problem is solved off-line for a range of parameters.
Specifically, an efficient implementation of a general multi-parametric quadratic program
is described together with the in-depth analysis of the properties of the solution. The op¬
timal control problem of a constrained linear discrete-time system can now be formulated
as a multi-parametric quadratic program by treating the state vector as a parameter. The
optimal solution is a piecewise affine state-feedback control law that is defined over a poly¬
hedral partition of the feasible state-space. This allows users to carry out most of the
time-consuming/complex computation off-line, while on-line implementation (control action
computation) reduces to a simple set-membership test.
By exploiting the properties of the value function and the optimal control law, new algo¬
rithms are developed that avoid storing the polyhedral regions. The new algorithms signifi¬
cantly reduce the on-line storage demands and computational complexity during evaluation
ofthe PWA feedback control law.
Next, the finite time optimal control problem for constrained discrete-time linear hybrid
systems based on quadratic or linear performancecriteriais tackled. Basic theoretical results
on the structure of the optimal state-feedback solution and of the value function are given.
An algorithm for construction of the solution - a piecewise affine state-feedback control
law defined over possibly non-convex regions - combines multi-parametric programming,
iii
Abstract iv
dynamic programming and basic polyhedral manipulation.
Similar ideas are extended to the infinite time optimal control problems with linear per¬
formance index. A novel algorithm solves the Hamilton-Jacobi-Bellman equation by using
the multi-parametric linear programming solver in a dynamic programming fashion. The
resulting solution when applied in a receding horizon fashion guarantees stability of the
closed-loop system. The important issue of stability guarantees is also addressed with the
introduction of sub-optimal control strategies that generate solutions of lower complexity.
Most of the developed algorithms were tested in two real-life automotive applications:
electronic throttle control and adaptive cruise control. The electronic throttle is used in
automotive applications to control the inflow of air to the vehicle engine by positioning a
throttle plate. The nonlinearities present in the throttle body make the control of the plate
position a challenging task. The electronic throttle is firstly modelled as a PWA system
and then the control strategies developed in this thesis are applied to it. In a multi-object
adaptive cruise control problem the optimal acceleration ofthe car is to be found respecting
traffic rules, safety distances and driver intentions. The hybrid nature of the problem arises
from the multiple objectives that introduce integer variables.
Finally, the MPT toolbox is presented. The MPT toolbox for MATLAB contains all of
the algorithms presented in this thesis as well as a wide range of additional algorithms and
tools developed by the academic community.
Zusammenfassung
Diese Dissertationbehandelt dieThematikoptimaler Regelungsprobleme mit Beschrnkungen
fuer die Klasse der zeitdiskreten, linearen Hybriden Systeme.
In ihrer allgemeinsten Form werden Hybride Systeme durch das Zusammenwirken von
wertekontinuierlichen Systemen, beschrieben durch Differential- oder Differenzengleichun¬
gen, in Verbindungmit logischen Regeln charakterisiert. Aus der Vielfalt, derin derLiteratur
erwähnten und äquivalenten Darstellungsformen Hybrider Systeme, konzentrieren wir uns
hier aufdie Klasse der zeitdiskreten, abschnittsweise affinen Systeme (engl.: piecewise affine
systems, PWA). Diese beschreiben eine grosse Zahl verschiedenster Prozesse. Darüberhinaus
können abschnittsweise affine Systeme nichtlineare Systeme mittels mehrfacher Linearisie¬
rungen an verschiedenen Arbeitspunkten approximieren.
Abschnittsweise affine Systemesind eine spezielle Klasse der nichtlinearen Systeme; jedoch
kann ein Grossteilder nichtlinearen System- und Regelungstheorie bei diesen Systemen nicht
angewendet werden, da zumeist gewisse Stetigkeits- und Glätteannahmen die Voraussetzung
hierfür sind. Aus demselben Grund ist es nicht möglich, die lineare Regelungstheorie in ap¬
proximativer Weise anzuwenden, um einen Regler für abschnittsweise affine Systeme auszu¬
legen. In der Vergangenheit waren die meisten Konzepte für die Analyse und Reglersynthese
von Hybriden Systemen ad hoc und deren Gültigkeit wurde hauptsächlich über extensive
Simulationen begründet.
Das Ziel dieser Dissertation ist sowohl die Weiterentwicklung von systematischen Vorge¬
hen als auch die Entwicklung von Algorithmen zum exakten Lösen von optimalen Rege¬
lungsproblemen. Ein Leitmotiv der Dissertation ist die Entwicklung effizienter Algorithmen
zum Lösen verschiedenster optimaler Regelungsprobleme. Ein wichtiges Instrument in der
EntwicklungdieserAlgorithmen ist das Konzept der multi-parametrischenProgrammierung,
bei der ein quadratisches (oder lineares) Optimierungsproblem für einen Wertebereich von
Parametern offline gelöst wird.
Im besonderen wird eine effiziente Implementierung eines allgemeinen multi-parametri¬
schen, quadratischen Programmes zusammen mit einer detaillierten Eigenschaftsanalyse der
Lösung beschrieben. Das optimale Regelungsproblem für ein beschränktes, lineares, zeit¬
diskretes System kann somit als multi-parametrisches, quadratisches Programm behandelt
werden, bei dem der Zustandsvektor als Parameter betrachtet wird. Die Optimallösung ist
ein abschnittsweise affines Zustandsrückführungsgesetz, das über eine polyhedrische Parti¬
tion des zulässigen Zustandsraumes definiert ist. Dies erlaubt dem Anwender, den grössten
Teil der zeitintensiven und komplexen Berechnung offline auszuführen. Hierdurch verringert
sich die online Bestimmung der Stellgrösse auf einen einfachen Mengenzugehörigkeitstest.
Durch die Nutzung der Eigenschaften des optimalen Kostenfunktionais und des optimalen
Regelgesetzes konnten neue Algorithmen entwickelt werden, die die Speicherung der polyhe-
v
Zusammenfassung vi
drischen Regionen vermeidet. Hierdurch wird eine signifikante Verringern der Speicherungs¬
anforderungen und der Komplexität der Berechnung während der Online-Evaluierung des
abschnittsweise affinen Regelgesetzes ermöglicht.
Danach wird das optimale Regelungsproblem mit endlichem Zeithorizont, basierend auf
quadratischen oder linearen Gütekriterien, für beschränkte, zeitdiskrete, lineare Hybride Sy¬
steme behandelt; es werden grundlegende theoretische Ergebnisse bezüglich der Form und
Struktur des optimalen Regelgesetzes und des optimalen Kostenfunktionais aufgezeigt. Der
Algorithmus zur Bestimmung der optimalen Lösung, die die Form eines abschnittsweise
affinen Regelgesetzes über (möglicherweise) nichtkonvexe Regionen besitzt, basiert auf ei¬
ner Kombination von multi-parametrischer Programmierung, DynamischerProgrammierung
und elementarer Polyhedramanipulation.
Diese Lösungsideen werden auf optimale Regelungsprobleme mit unendlichem Zeithori¬
zont und linearem Gütekriterien ausgeweitet. Ein neuartiger Algorithmus löst die Hamilton-
Jacobi-Bellman Gleichung in Form einer Dynamischen Programmierung durch Anwendung
eines Lösers für multi-parametrische, lineare Programme. Falls eine 'Receding Horizon' Re¬
gelungsstrategie angewendet wird, garantiert die resultierende Lösung die Stabilität des ge¬
schlossenen Regelkreises. Die Wichtigkeit der Stabilitätsgarantie wird darüberhinaus auch
bei der Einführung von suboptimalen Regelungsstrategien mit niedriger Komplexität behan¬
delt.
Die Mehrzahl derentwickelten Algorithmenwurde anzwei verschiedenen Anwendungen im
Automobilbereich getestet: zum einen auf die Regelung einer elektronischen Drosselklappe
und zum anderen auf die adaptive Fahrtregelung. Die elektronische Drosselklappenpositio¬
nierung wird in der Automobiltechnik zur Regulierung der Luftzufuhr zum Fahrzeugmotor
angewendet. Die auftretenden Nichtlinearitäten in der Drossel machen die Regelung der
Klappenposition zu einer herausfordernden Aufgabe. Die elektronische Drosselklappe wur¬
de zuerst als abschnittsweise affines System modelliert und dann die in dieser Dissertation
entwickelten Regelungsstrategien hierauf angewendet. Bei dem adaptiven Fahrtregelungs¬
problem mit multiplen Gütekriterien wird diejenige optimale Beschleunigung des Fahrzeugs
gesucht, die sowohl die Verkehrsregeln, als auch Sicherheitsabstände und die Absichten des
Fahrers, respektiert. Die hybride Natur des Problems wird durch multiple Gütekriterien, die
in das Problem ganzzahlige Variablen einführen, induziert.
Abschliessend wird die MPT Toolbox präsentiert. Diese Toolbox für Matlab beinhaltet
sämtliche Algorithmen welche in dieser Dissertation vorgestellt wurden sowie etliche Stan¬
dardfunktionen der Forschungsgebiete Regelungstechnik und Computational Geometry.
Contents
Acknowledgement i
Abstract iii
Zusammenfassung v
1 Introduction 1
1 MATHEMATICAL PROGRAMMING 5
2 Background 6
2.1 Basic Terminology and Definitions 6
2.2 Convex Optimization 10
3 Polytopes 13
3.1 Definitions 13
3.2 Basic Operations on Polytopes 16
3.2.1 Minimal Representation 16
3.2.2 Chebychev Ball 17
3.2.3 Projection 17
3.2.4 Set-Difference 18
3.2.5 Convex Hull 19
3.2.6 Envelope 19
3.2.7 Vertex Enumeration 20
3.2.8 Pontryagin Difference 20
3.2.9 Minkowski Sum 21
3.3 Polycover and Regiondiff 22
3.3.1 Polycover: MILP formulation 23
3.3.2 Polycover: Branch & Bound algorithm 25
4 Multi-Parametric Programming 29
4.1 Basics of Multiparametric programming 29
4.2 Multi-parametric Quadratic Programming 31
4.2.1 Translation into the standard mp-QP form 32
vn
Contents VÜi
4.2.2 Solution of the mp-QP 32
4.2.3 Direct exploration of the parameter space 37
4.3 Multi-parametric Linear Programming 42
II OPTIMAL CONTROL 43
5 Constrained Finite Time Optimal Control of Linear Systems 44
5.1 Introduction 44
5.2 CFTOC, RHC and their State-feedback PWA Solution 45
5.2.1 CFTOC problem formulation 45
5.2.2 RHC strategy 46
5.2.3 Solution of CFTOC, linear cost Case 46
5.2.4 Solution of CFTOC, quadratic cost Case 48
5.3 Efficient on-line Algorithms 49
5.3.1 Efficient Implementation, linear cost Case 50
5.3.2 Efficient Implementation, quadratic cost Case 51
5.4 Example 59
5.4.1 CFTOC, linear cost Case 60
5.4.2 CFTOC, quadratic cost Case 60
5.5 Conclusion 61
6 Constrained Finite Time Optimal Control of PWA Systems 62
6.1 Introduction 62
6.2 Piecewise Affine Systems 63
6.3 Problem Formulation 64
6.4 Solution Properties 65
6.5 Computation of the Optimal Control Law via Dynamic Programming 71
....
6.5.1 Preliminaries and Basic Steps 72
6.5.2 Multiparametric Programming with Multiple Quadratic Functions 74
. .
6.5.3 Algorithmic Solution of the HJB Equations 77
6.6 Discontinuous PWA systems 80
6.7 Conclusion 81
7 CFTOC and CITOC of PWA Systems with a Linear Performance Index 82
7.1 Introduction 82
7.2 Linear Hybrid Systems 83
7.3 Constrained Finite Time Optimal Control 85
7.3.1 The CFTOC Solution via mp-MILP 86
7.3.2 The CFTOC Solution via Dynamic Programming 87
7.3.3 An Efficient Algorithm for the CFTOC Solution 88
7.3.4 Comments on the Dynamic Programming Approach for the CFTOC
Problem 89
7.3.5 Receding Horizon Control 90