Applied Interval Analysis

Transcription

1 Luc Jaulin, Michel Kieffer, Olivier Didrit and Eric Walter Applied Interval Analysis With Examples in Parameter and State Estimation, Robust Control and Robotics With 125 Figures

2 Contents Preface Notation v xiii Part I. Introduction 1. Introduction What Are the Key Concepts? How Did the Story Start? What About Complexity? How is the Book Organized? 6 Part II. Tools 2. Interval Analysis Introduction Operations on Sets Purely set-theoretic Operations Extended Operations Properties of set Operators Wrappers Interval Analysis Intervals Interval computation Closed intervals Interval vectors Interval matrices Inclusion Functions Definitions Natural inclusion functions Centred inclusion functions Mixed centred inclusion functions 34

3 viii Contents Taylor inclusion functions Comparison Inclusion Tests Interval Booleans Tests Inclusion tests for sets Conclusions Subpavings Introduction Set Topology Distances between compact sets Enclosure of compact sets between subpavings Regulär Subpavings Pavings and subpavings Representing a regulär subpaving as a binary tree Basic Operations on regulär subpavings Implementation of Set Computation Set inversion Image evaluation Conclusions Contractors Introduction Basic Contractors Finite subsolvers Intervalization of finite subsolvers Fixed-point methods Forward-backward propagation Linear programming approach External Approximation Principle Preconditioning Newton contractor Parallel linearization Using formal transformations Collaboration Between Contractors Principle Contractors and inclusion functions Contractors for Sets Defmitions Sets defined by equality and inequality constraints Improving contractors using local search Conclusions 100

4 Contents ix 5. Solvers Introduction Solving Square Systems of Non-linear Equations Characterizing Sets Defined by Inequalities Interval Hüll of a Set Defined by Inequalities First approach Second approach Global Optimization The Moore-Skelboe algorithm Hansen's algorithm Using interval constraint propagation Minimax Optimization Unconstrained case Constrained case Dealing with quantifiers Cost Contours Conclusions 136 Part III. Applications 6. Estimation Introduction Parameter Estimation Via Optimization Least-square parameter estimation in compartmental modelling Minimax parameter estimation Parameter Bounding Introduction The values of the independent variables are known Robustification against outliers The values of the independent variables are uncertain Computation of the interval hüll of the posterior feasible set State Bounding Introduction Bounding the initial State Bounding all variables Bounding by constraint propagation Conclusions Robust Control Introduction Stability of Deterministic Linear Systems Characteristic polynomial 189

5 x Contents Routh criterion Stability degree Basic Tests for Robust Stability Interval polynomials Polytope polynomials Image-set polynomials Conclusion Robust Stability Analysis Stability domains Stability degree Value-set approach Robust stability margins Stability radius Controller Design Conclusions Robotics Introduction Forward Kinematics Problem for Stewart-Gough Platforms Stewart-Gough platforms From the frame of the mobile plate to that of the base Equations to be solved Solution Path Planning Graph discretization of configuration space Algorithms for finding a feasible path Test case Localization and Tracking of a Mobile Robot Formulation of the static localization problem Model of the measurement process Set inversion Dealing with outliers Static localization example Tracking Example Conclusions 267 Part IV. Implementation 9. Automatic Differentiation Introduction Forward and Backward Differentiations Forward differentiation Backward differentiation 273

6 Contents xi 9.3 Differentiation of Algorithms First assumption Second assumption Third assumption Examples Example Example Conclusions Guaranteed Computation with Floating-point Numbers Introduction Floating-point Numbers and IEEE Representation Rounding Special quantities Intervals and IEEE Machine intervals Closed interval arithmetic Handling elementary functions Improvements Interval Resources Conclusions Do It Yourself Introduction Notions of C Program structure Standard types Pointers Passing parameters to a function INTERVAL Class Constructors and destructor Other member functions Mathematical functions Intervals with PROFIL/BIAS BIAS PROFIL Getting started Exercises on Intervals Interval Vectors INTERVAL_VECT0R class Constructors, assignment and function call Operators Friend functions Utilities Vectors with PROFIL/BIAS 326

7 xii Contents 11.8 Exercises on Interval Vectors Interval Matrices Matrices with PROFIL/BIAS Exercises on Interval Matrices Regulär Subpavings with PROFIL/BIAS NODE class Set Inversion with subpavings Image evaluation with subpavings System Simulation and State estimation with subpavings Error Handling Using exit Exception handling Mathematical errors 351 References 353 Index 373