Nonlinear Systems Tracking  book cover
1st Edition

Nonlinear Systems Tracking

ISBN 9781138749511
Published July 26, 2017 by CRC Press

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Book Description

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.

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

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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