Stability and Stabilization of Time-Delay Systems
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1 Stability and Stabilization of Time-Delay Systems An Eigenvalue-Based Approach Wim Michiels Katholieke Universiteit Leuven Leuven, Belgium Silviu-Iulian Niculescu Laboratoire des Signaux et Systemes Gif-sur-Yvette, France SIHJTL Society for Industrial and Applied Mathematics Philadelphia
2 Preface Symbols Acronyms xiii xix xxi I Stability analysis of linear time-delay Systems 1 1 Spectral properties of linear time-delay Systems Time-delay Systems of retarded type Initial value problem Spectrum: definitions Asymptotic growth rate of Solutions and stability Spectrum: qualitative properties Spectrum: continuity properties Computation of characteristic roots Time-delay Systems of neutral type Initial value problem Spectrum: definitions Asymptotic growth rate of Solutions and stability Spectrum: qualitative properties Spectrum: continuity properties Computation of characteristic roots Notes and references 31 2 Pseudospectra and robust stability analysis Introduction Pseudospectra for nonlinear eigenvalue problems Definition and expressions Connection with stability radii Computational issues Application to time-delay Systems Structured pseudospectra for nonlinear eigenvalue problems 43 v
3 vi Exploiting the system's structure Definition and expressions Computational issues and special cases Application to time-delay Systems Illustrative examples Second-order System Feedback controlled semiconductor laser Notes and references 54 3 Computation of stability regions in parameter Spaces Introduction Basic notions and definitions From D-subdivision to numerical continuation D-subdivision and stability crossing boundaries T-decomposition and delay stability intervals Numerical continuation Computing the crossing direction of characteristic roots Simple crossing characteristic roots Semisimple characteristic roots Further analysis: basic ideas Delay interdependence and crossing direction evaluation Notes and references 81 4 Stability regions in delay-parameter Spaces Introduction Invariance properties Delay shifts and characteristic roots Crossing direction invariance Algebraic methods Elimination principle: basic ideas Matrix pencil approach and crossing characterization Particular cases and other elimination techniques Geometrie methods Identification of crossing points Stability crossing curves Tangents, smoothness, and crossing direction Notes and references Stability of delay rays and delay-interference Introduction Preliminary results Definitions and assumptions Introductory example Properties of some associated matrix-valued funetions Delay-independent stability and delay-interference phenomena Delay-independent stability characterization 121
4 VII Delay-interferencecharacterization Illustrative examples Interterence in parameterized scalar delay Systems Delay rays and second-order delay Systems Notes and references Stability of linear periodic Systems with delays Introduction Systems with fast varying coefficients Averaging periodic Systems Computational tools Analytical tools General case Collocation scheme Computation of stability determining eigenvalues Computation of stability regions Special cases Comparison with the averaging based approach Illustrative examples Variable spindle speed cutting machine Forced elastic column Notes and references 145 II Stabilization and robust stabilization The continuous pole placement method Introduction Motivation Afinite-dimensional Controller for an infinite-dimensional problem Methods based on prediction Scalar example Continuous pole placement algorithm Description of the algorithm Theoretical properties Optimization point of view Illustrative examples Model problem: stabilizing a third-order system General stabilization problems Extensions of State feedback Multiple input, multiple outpul Systems Observer based Controllers Finite-dimensional dynamic State feedback Systems of neutral type Algorithm 173
5 viii Illustrative example Notes and references Stabilizability with delayed feedback: a numerical case study Introduction Characterizationof stabilizable Systems System representation Class of stabilizable Systems for the unit delay Class of stabilizable Systems for arbitrary delay values Discussion Noncyclic System matrix Simultaneous stabilization over a delay interval Notes and references The robust stabilization problein Introduction Stability radii as robustness measures Stabilization versus robust stabilization Robust stabilization procedure Continuity properties Algorithm Illustrative example Notes and references Stabilization using a direct eigenvalue optimization approach Introduction Eigenvalue optimization approach Smoothness properties of spectral abscissa function Gradient sampling algorithm Application to linear time-delay Systems Extension to nonlinear time-delay Systems Illustrative examples Model problem: a third-order System Semiconductor laser Notes and references 216 III Applications Stabilization by delayed Output feedback: single delay case Introduction Characterization of all stabilizable second-order Systems Necessary conditions for stabilizability Controller construction Prerequisites Stabilization using the delay parameter 229
6 ix Stabilization using the delay and gain parameter Geometry of stability regions Identification of crossing points Classification of stability crossing curves, smoothness, and crossing directions Illustrative examples Second-order System Sixth-order System Notes and references Stabilization by delayed Output feedback: multiple delay case Introduction Necessary conditions for stabilizability Stabilization of multiple integrators Control laws based on numerical differentiation with backward differences Control laws based on exact pole placement and low-gain design Illustrative example Notes and references Congestion control algorithms in networks Algorithms for Single connection modeis with two delays Model and related remarks Linear stability analysis Interpretations and discussions TCP/AQM congestion avoidance modeis with one delay Model and related remarks Transformation Stability analysis Notes and references Smith predictor for stable Systems: delay sensitivity analysis Introduction Sensitivity of stability w.r.t. infinitesimal delay mismatches Instability mechanism Conditions for practical stability Stability analysis and critical delay mismatches Geometry of stability regions Identification of crossing points Stability crossing curves: smoothness and crossing directions Illustrative example Multivariable case Practical stability condition Stability domain 289
7 X 14.7 Notes and referenees Controlling unstable Systems using finite spectrum assignment Introduction Preliminaries Implementation of the integral Instability mechanism Stability conditions Removing restrictions Delay mismatch Output feedback Static output feedback Dynamic output feedback and relations with Smith Predictors Notes and referenees Consensus problems in traffic flow applications Introduction Extension of stability Iheory to Systems with distributed delays Conditions for the realizalion ofa consensus Prerequisites Computation of stability regions Examples Other modeis Notes and referenees Stability analysis of delay modeis in biosciences Introduction Delay effects on stability in some human respiration modeis Delay model and its linearization Stability analysis and delay intervals Discussions and interpretations Delays in immune dynamics model of leukemia Delay model and its linearization Stability analysis in the delay-parameter space Illustrative example and discussions Notes and referenees 342 Appendix 345 A. 1 Rouche's theorem 345 A.2 Structured Singular value (ssv) 345 A.3 Continuity properties 347 A.4 Interdependency of numbers 347 A.5 Software 348
8 XI Bibliography 351 Index 375
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