502 Pages
    by CRC Press

    502 Pages 16 B/W Illustrations
    by CRC Press

    Tracking is the goal of control of any object, plant, process, or vehicle. From vehicles and missiles to power plants, tracking is essential to guarantee high-quality behavior.

    Nonlinear Systems Tracking establishes the tracking theory, trackability theory, and tracking control synthesis for time-varying nonlinear plants and their control systems as parts of control theory. Treating general dynamical and control systems, including subclasses of input-output and state-space nonlinear systems, the book:

    • Describes the crucial tracking control concepts that comprise effective tracking control algorithms
    • Defines the main tracking and trackability properties involved, identifying properties both perfect and imperfect
    • Details the corresponding conditions needed for the controlled plant to exhibit each property
    • Discusses various algorithms for tracking control synthesis, attacking the tracking control synthesis problems themselves
    • Depicts the effective synthesis of the tracking control, under the action of which, the plant behavior satisfies all the imposed tracking requirements resulting from its purpose

    With clarity and precision, Nonlinear Systems Tracking provides original coverage, presenting discovery and proofs of new tracking criteria and control algorithms. Thus, the book creates new directions for research in control theory, enabling fruitful new control engineering applications.


    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


    Time and systems
    Time, physical variables, and systems
    Notational preliminaries
    Classes of the systems

    Set basis
    Continuity of sets
    Set contraction

    General dynamical systems
    Plants and control systems
    Existence and solvability
    Fundamental control principle

    Desired regime
    Concept and definitions

    Origins of time-varying models
    Deviations and mathematical models


    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
    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


    Tracking generally
    Primary control goal
    Tracking versus stability

    Tracking concepts
    Tracking characterization and space
    Various tracking concepts

    Perfect tracking concept
    On perfect tracking generally


    Output space definitions
    Definitions of L-tracking properties

    State space definitions
    Definitions of L-tracking properties

    Set tracking
    Definitions of set tracking properties


    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
    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


    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
    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


    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


    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


    Matrix and vector notation

    Dini derivatives
    Definitions of derivatives

    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



    Author index

    Subject index


    Lyubomir T. Gruyitch has very rich international academic and research experience. Now retired, he was a professor at the École Nationale d'Ingénieurs, which integrated with the Institut Polytechnique de Sévenans into the University of Technology of Belfort–Montbéliard, in France; the AECI professor of control in the Department of Electrical Engineering at the University of Natal, Durban, South Africa; and a professor of automatic control in the Faculty of Mechanical Engineering at the University of Belgrade, Serbia. He has also been a visiting professor at Louisiana State University, Baton Rouge, USA; the University of Notre Dame, Indiana, USA; and the University of Santa Clara, California, USA. He continues to teach and participate at conferences on an invited basis. Dr. Gruyitch is the author of several published books and many scientific papers on dynamical systems, control systems, and time and its relativity. He has participated at many scientific conferences throughout the world. Republic of France promoted Professor Gruyitch Doctor Honoris Causa at the University of Science and Technology, Lille. He has been honored with several awards, including the highest award by the Faculty of Mechanical Engineering, University of Belgrade, for teaching and scientific contributions to the faculty, 1964–1992; and an award from the Yugoslav Air Force Academy for teaching achievements in the undergraduate course foundations of automatic control. Dr. Gruyitch earned his Certified Mechanical Engineer (Dipl. M. Eng.), Master of Electrical Engineering Sciences (M. E. E. Sc.), and Doctor of Engineering Sciences (D.Sc.) degrees from the University of Belgrade.

    "Numerous publications and books present various aspects of tracking, but I do not know of another book only devoted to tracking and its various aspects. ... I used to teach tracking in my courses of process control and stability analysis of complex nonlinear systems, and this book could be very interesting to improve my courses. ... This book gives a complete presentation of the various aspects of tracking of nonlinear and/or time varying systems, including the determination of the maximum error for ill-defined and/or perturbed systems."
    —Pierre Borne, École Centrale de Lille, France