Nonlinear Systems Tracking

PREFACE Systems, control, and computers Dynamical systems Dynamical systems and computers Dynamical systems and control Control goal Tracking control tasks On the book Goals of the book The book structure and composition In gratitude SYSTEMS AND CONTROL BASIS Introduction Time and systems Time, physical variables, and systems Notational preliminaries Classes of the systems Sets Set basis Neighborhood Continuity of sets Set contraction Systems General dynamical systems Plants and control systems Existence and solvability Fundamental control principle Desired regime Introduction Concept and definitions Origins of time-varying models Introduction Deviations and mathematical models TRACKABILITY Trackability concept On system and control concepts Controllability and observability Disturbance rejection or compensation New control concepts Time and control Perfect trackability concepts Perfect trackability Perfect natural trackability Perfect trackability criteria Output space criteria State space criteria Both spaces criteria Perfect natural trackability criteria Output space criteria State space criteria Both spaces criteria Imperfect trackability concepts Introduction to (imperfect) trackability Trackability Natural trackability Elementwise trackability Elementwise natural trackability Imperfect trackability criteria Output space criteria State space criteria Both spaces criteria Imperfect natural trackability criteria Output space criteria State space criteria Both spaces criteria PERFECT TRACKING Tracking generally Primary control goal Tracking versus stability Tracking concepts Tracking characterization and space Various tracking concepts Perfect tracking concept On perfect tracking generally Definitions IMPERFECT TRACKING: STABLE TRACKING Output space definitions Introduction Definitions of L-tracking properties State space definitions Introduction Definitions of L-tracking properties Set tracking Introduction Definitions of set tracking properties CRITERIA FOR STABLE TRACKING Introduction Lyapunov methods and methodologies Tracking accuracy and definitions Suitable mathematical models Comparison and (semi)definite functions Comparison functions Semidefinite functions Definite functions Decrescent functions Time-invariant vector definite functions Time-varying vector definite functions Sets and functions Positive definite function induces sets Set invariance relative to a function Semidefinite functions and time-varying sets Definite functions relative to time-varying sets Decrescent functions and time-varying sets Families F of time-varying sets Outline of the Lyapunov method Physical origin of the Lyapunov method Lyapunov method Lyapunov theorems for nonlinear systems Lyapunov original methodologies Lyapunov method extended to tracking Criteria: Asymptotically contractive sets Criteria: Noncontractive time-varying sets CLM: Motion and set tracking Introduction Systems smooth properties Systems and generating functions Criteria: Systems with continuous motions Criteria: Systems with differentiable motions Time-varying set tracking Time-varying set and motion tracking Conditions for stable tracking Conditions for exponential tracking FINITE REACHABILITY TIME TRACKING Output space definitions Finite scalar reachability time Finite vector reachability time State space definitions Finite scalar reachability time tracking Elementwise state FVRT tracking Criteria on contractive sets Introduction Stable tracking with FSRT Stable tracking with FVRT Criteria on noncontractive sets Stable tracking with FSRT Stable tracking with FVRT FRT tracking control synthesis Internal dynamics space Output space REQUIRED TRACKING QUALITY AND CONTROL SYNTHESIS Natural tracking control concept Tracking quality: Output space Output space tracking operator Tracking operator properties Reference output Tracking algorithm and initial conditions Tracking algorithms: Output space Matrix notation meaning Examples of tracking algorithms NTC synthesis: Output space General NTC theorem: Output space NTC synthesis: Output space Tracking quality: State space State space tracking operator Tracking operator properties Reference state vector RR Tracking algorithm and initial conditions Tracking algorithms: State space Matrix notation meaning Examples of tracking algorithms NTC synthesis: State space General NTC theorem: State space NTC synthesis in the state space Tracking quality: Both spaces Both spaces (BS) tracking operator Tracking operator properties The reference BS vector Tracking algorithm and initial conditions Tracking algorithms: Both spaces Matrix notation meaning Examples of tracking algorithms NTC synthesis: Both spaces General NTC theorem: Both spaces (BS) NTC synthesis in both spaces CONCLUSION Systems, control, tracking, trackability Perturbed systems Control goal: Tracking Tracking demands trackability Lyapunov theory and tracking Lyapunov theory extended to tracking Consistent Lyapunov methodology: Tracking Lyapunov control synthesis High-quality tracking: Control synthesis Tracking with finite reachability time Demanded tracking quality Trackability theory and tracking theory importance APPENDICES Notation Abbreviations Indexes Letters Names Symbols Matrix and vector notation Sets Units Dini derivatives Definitions of derivatives Properties Proofs for Part III Proof of Theorem 89 Proof of Theorem 152 Proof of Theorem 157 Proof of Theorem 160 Proof of Theorem 162 Proof of Theorem 164 Proof of Theorem 165 Proof of Theorem 167 Proofs for Part VII Lemma 1 Lemma 2 Proof of Theorem 363 Proof of Theorem 364 Proof of Theorem 368 Lemma 415 USED LITERATURE INDEXES Author index Subject index

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