Table Of ContentSPRINGER BRIEFS IN ELECTRICAL AND COMPUTER
ENGINEERING CONTROL, AUTOMATION AND ROBOTICS
Ionela Prodan
Florin Stoican
Sorin Olaru
Silviu-Iulian Niculescu
Mixed-Integer
Representations
in Control Design
Mathematical
Foundations and
Applications
123
SpringerBriefs in Electrical and Computer
Engineering
Control, Automation and Robotics
Series editors
Tamer Başar
Antonio Bicchi
Miroslav Krstic
More information about this series at http://www.springer.com/series/10198
Ionela Prodan Florin Stoican
(cid:129)
Sorin Olaru Silviu-Iulian Niculescu
(cid:129)
Mixed-Integer
Representations in Control
Design
Mathematical Foundations and Applications
123
Ionela Prodan Sorin Olaru
Laboratory of ConceptionandIntegration Laboratory of Signals andSystems
ofSystems CentraleSupélec - CNRS- Université
UniversitéGrenoble Alpes Paris-Sud, UniversitéParis-Saclay
Valence Gif-sur-Yvette
France France
Florin Stoican Silviu-Iulian Niculescu
Department ofAutomatic Control Laboratory of Signals andSystems
andSystemsEngineering CNRS- CentraleSupélec- Université
Politehnica University of Bucharest Paris-Sud, UniversitéParis-Saclay
Bucharest Gif-sur-Yvette
Romania France
ISSN 2191-8112 ISSN 2191-8120 (electronic)
SpringerBriefs inElectrical andComputer Engineering
ISSN 2192-6786 ISSN 2192-6794 (electronic)
SpringerBriefs in Control, Automation andRobotics
ISBN978-3-319-26993-1 ISBN978-3-319-26995-5 (eBook)
DOI 10.1007/978-3-319-26995-5
LibraryofCongressControlNumber:2015955893
©TheAuthor(s)2016
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Preface
“Tobeornottobe?”
(Hamlet,Shakespeare,1601)
While the scope of a book is to present the problem discussed as esoterically as
possible we also have to acknowledge its roots in “real life.” This is the case here
with “control problems with contradictory conditions” which are readily found in
everyday life. They may not be stated so, but each time we balance between
mutually exclusive goals or have to choose a direction in the detriment of another
weactuallysolve(optimallyornot)suchacontrolproblem.An“either/or”decision
isalwaysdifficultforhumanbeings,thesamegoesforoptimizationalgorithmswith
discretevariables.Wehavethusstrongreasonstopursuesolutionsforthisclassof
problems.Asisusuallythecase,thesolutionisbetterorfasterifweunderstandthe
problem’s underlying structure. Therefore, the scope of this book is to provide
efficient constructions which are subsequently put under a mixed-integer form
which can be solved by a computer in a “reasonable” time.
Thisbookrepresentstheculminationofover5yearsofcollaborationworkofthe
authors. The main contributions are the result of work started during the Ph.D.
theses of the first two authors and by subsequent advancements in the following
years.
This book was inspired by our desire to bring to light the importance of the
analysis and control of dynamical systems with conflicting objectives and the
effectiveusageoftheassociatedmixed-integerformulations.Itisworthmentioning
that the topic is not new and monographs exists covering mixed-integer opti-
mization.However,mostofthemassumeaspecificbackgroundinmathematicsand
optimization. The present book is dedicated to a generic class of constraints and
their use in optimization problems and, in this respect, goes deeper into the details
of their construction, representation and computational complexity. It is important
to mention that the present manuscript is mainly dedicated to the problem
description and not to the numerical optimization routines. It focuses on the
mixed-integer aspectsoftheconstraints formulationandtheirrelationshipwith the
optimization-based control design. To our knowledge, another textbook is not
v
vi Preface
currently available that covers a compact treatment of the non-convex feasible set
representation via mixed-integer representations, gathering the recent research
advancements in the literature and illustrating the potential impact on
optimization-based design, as for example in control design.
One of the most important features of the book is that it provides all along the
manuscript the tools for easy reconstruction of the illustrative examples. The
applicationsencompassimportantissuesfromcontroltheory,rangingfrommotion
planning with obstacle and collision avoidance and up to fault tolerant control
schemes.
The book will hopefully not only serve the purpose of disseminating research
results but also of raising the awareness for these challenging, timely and relevant
research topics on optimization and control design. Moreover, we hope that this
book will find attention in the diverse control engineering, computational mathe-
matics and optimization communities and thus will contribute to the development
of mixed-integer representations as a well-defined research field.
Valence Ionela Prodan
Bucharest Florin Stoican
Gif-sur-Yvette Sorin Olaru
Gif-sur-Yvette Silviu-Iulian Niculescu
September 2015
Acknowledgment The work of Florin Stoican is funded by the Sectorial
Operational Programme Human Resources Development 2007–2013 of the
Ministry of European Funds through the Financial Agreement [grant number
POSDRU/159/1.5/S/132395].
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Non-convex Region Description by Hyperplane Arrangements. . . . . 11
2.1 Hyperplane Arrangements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Region Counting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.2 Parametrized Hyperplane Arrangements . . . . . . . . . . . . . . 15
2.2 Polyhedral and Zonotopic Sets . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.1 Polyhedral Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.2 Zonotopic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Non-convex Region Description . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.1 Cell Merging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.2 Numerical Considerations. . . . . . . . . . . . . . . . . . . . . . . . 29
2.4 Notes and Comments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 Mixed-Integer Representations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1 Classical Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Analysis of the Unallocated Tuples . . . . . . . . . . . . . . . . . . . . . . 37
3.2.1 Geometric Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.2 Algebraic Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 The Complement of a Union of Convex Sets . . . . . . . . . . . . . . . 44
3.4 Description of the Feasible Region as Union of Feasible Cells . . . 47
3.5 Description of the Feasible Region Directly Through the
Arrangement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.6 Numerical Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.7 Notes and Comments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4 Control Problems Involving Mixed-Integer Decision Making . . . . . . 57
4.1 Multi-agent Collision Avoidance Problems . . . . . . . . . . . . . . . . . 58
4.2 Extensions for Multi-agent Motion Planning. . . . . . . . . . . . . . . . 64
vii
viii Contents
4.2.1 Shadow Region Description . . . . . . . . . . . . . . . . . . . . . . 65
4.2.2 Mixed-Integer Representations . . . . . . . . . . . . . . . . . . . . 68
4.2.3 Coverage Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2.4 Corner Cutting Problem. . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3 Reference Governor Mechanism for Guaranteed
Fault Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3.1 Output-Based Residual Case. . . . . . . . . . . . . . . . . . . . . . 79
4.3.2 Finite Window Residual Case. . . . . . . . . . . . . . . . . . . . . 82
4.4 Notes and Comments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Appendix: Numerical Data for Illustrative Examples . . . . . . . . . . . . . . 95
Series Editor’s Biographies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Abbreviations
FDI Fault Detection and Isolation
FTC Fault Tolerant Control
KKT Karas–Kuhn–Tucker
LMI Linear Matrix Inequalities
LP Linear Programming
LQ Linear Quadratic
LTI Linear Time Invariant
MILCP Mixed-Integer Linear Constrained Programming
MILP Mixed-Integer Linear Programming
MINLP Mixed-Integer Non-Linear Programming
MIP Mixed-Integer Programming
MIQCP Mixed-Integer Quadratically Constrained Programming
MIQP Mixed-Integer Quadratic Programming
MPC Model Predictive Control
NP-hard Non-deterministic Polynomial-time hard
PWA Piecewise Affine
QP Quadratic Programming
RC Reconfiguration Control
RPI Robust Positive Invariance
UAV Unmanned Aerial Vehicle
ix
