Nonlinear Systems Tracking: 1st Edition (Paperback) book cover

Nonlinear Systems Tracking

1st Edition

By Lyubomir T. Gruyitch

CRC Press

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Paperback: 9781138749511
pub: 2017-07-26
Currently out of stock
Hardback: 9781498753258
pub: 2016-01-06

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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.


"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

Table of Contents


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

About the Author

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.

Subject Categories

BISAC Subject Codes/Headings:
TECHNOLOGY & ENGINEERING / Engineering (General)
TECHNOLOGY & ENGINEERING / Power Resources / Electrical