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.
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
Biography
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
