Finite-Time Stability: An Input-Output Approach

By Francesco Amato, Gianmaria De Tommasi and Alfredo Pironti

Systematically presents the input-output finite-time stability (IO-FTS) analysis of dynamical systems, covering issues of analysis, design and robustness

The interest in finite-time control has continuously grown in the last fifteen years. This book systematically presents the input-output finite-time stability (IO-FTS) analysis of dynamical systems, with specific reference to linear time-varying systems and hybrid systems. It discusses analysis, design and robustness issues, and includes applications to real world engineering problems.

While classical FTS has an important theoretical significance, IO-FTS is a more practical concept, which is more suitable for real engineering applications, the goal of the research on this topic in the coming years.

Key features:

• Includes applications to real world engineering problems.
• Input-output finite-time stability (IO-FTS) is a practical concept, useful to study the behavior of a dynamical system within a finite interval of time.
• Computationally tractable conditions are provided that render the technique applicable to time-invariant as well as time varying and impulsive (i.e. switching) systems.
• The LMIs formulation allows mixing the IO-FTS approach with existing control techniques (e. g. H∞ control, optimal control, pole placement, etc.).

This book is essential reading for university researchers as well as post-graduate engineers practicing in the field of robust process control in research centers and industries. Topics dealt with in the book could also be taught at the level of advanced control courses for graduate students in the department of electrical and computer engineering, mechanical engineering, aeronautics and astronautics, and applied mathematics.

• Language: English
• Publisher: Wiley
• Release date: Aug 1, 2018
• ISBN: 9781119140559

Book preview

Dedication

To my mother

F. A.

To my family, for all the time I've subtracted to their love

G. D. T.

To Teresa and Andrea

A. P.

Preface

The concept of finite‐time stability (FTS) is useful to study the behavior of dynamical systems within a finite‐time horizon. This concept permits to specify bounds on the state and/or the output of a dynamical system, given a bound on its initial state, and/or to constrain the input to belong to a specific class of signals. It follows that finite‐time stability is an attractive concept from the engineering point of view, since it gives the possibility to quantitatively specify the transient response of a dynamical system to exogenous inputs (disturbances).

FTS was first introduced in the Russian literature more than sixty years ago [1-3]. The original definition dealt with the state response of autonomous systems: a system is said to be finite‐time stable if, given a bound on the initial condition, its state does not exceed a certain threshold during a specified time interval. During the sixties and seventies, FTS appeared also in the Western literature [4-6], together with the related concept of practical stability. This pioneering works, although developing a nice theoretical framework, did not provide computationally tractable conditions for checking the FTS of a given dynamical system, unless simple cases were considered. Therefore, for a long period, this field of research was neglected by control scientists.

At the end of the last century, the development of the Linear Matrix Inequality theory (LMI, [7]) has fueled new interest in the field of finite‐time control. In particular, starting from the beginning of the twenty‐first century, FTS and finite‐time stabilization have been investigated in the context of linear systems (e.g., [8-14]). According to this modern approach to FTS, conditions for analysis and design are provided in terms of feasibility problems involving both LMIs [7] and Differential Linear Matrix Inequalities (DLMIs, [15]), or in terms of solutions of Differential Lyapunov Equations (DLEs, [16]).

As far as state FTS is concerned, an effort has been made in order to extend the results obtained for linear systems to the context of nonlinear systems (e.g., [12, 17, 18]), hybrid systems ([19-23]), and stochastic systems ([18, 23-29] among others).

In order to extend the finite‐time stability concept to the input‐output case, the definition of input‐output finite‐time stability (IO‐FTS) was originally given by the authors in [30, 31]. A dynamical system is said to be input‐output finite‐time stable if, given a class of input signals bounded over a specified time horizon, the output of the system does not exceed an assigned threshold during the considered time interval. IO‐FTS extends the finite‐time stability framework to the case of non‐autonomous dynamical systems, giving the possibility to set quantitative constraints on the transient response to disturbances.

Just as state FTS is an independent concept with respect to Lyapunov stability, also IO‐FTS is not related to classic IO‐stability [32]. The main differences between classic IO‐stability and IO‐FTS are that the latter involves signals defined over a finite‐time interval, does not necessarily require the input and output to belong to the same class, and that quantitative bounds on both input and output must be specified.

The material presented in this book collects and extends the results published by the authors since 2010 on the major control system journals. Besides presenting the main theoretical results to solve both the IO‐FTS analysis and synthesis problems for different classes of dynamical systems, a number of case studies are presented as examples of practical applications of finite‐time control techniques. Numerical issues related to the solution of DLMIs feasibility problems that arise in the proposed finite‐time theory are also discussed, in order to give some guidance to their practical solution.

Chapter 1 introduces the considered finite‐time stability framework and presents some preliminary background results that are exploited throughout this monograph. Necessary and sufficient conditions to check IO‐FTS for linear systems are provided in Chapter 2, while Chapter 3 deals with the solution of the stabilization (i.e., synthesis) problem. IO‐FTS of linear system with nonzero initial conditions is considered in Chapter 4, while the case of IO‐FTS with additional constraints on the control input is discussed in Chapter 5. Robust and mixed finite‐time/ control is presented in Chapter 6, which concludes the discussion concerning the case of linear dynamical systems. The extension of the IO‐FTS concepts to a special class of hybrid systems, namely the impulsive dynamical linear systems, is addressed in Chapters 7 and 8; the case of uncertain resetting times for this type of discontinuous dynamical systems is also considered in Chapter 9.

It is important to remark that the IO‐FTS approach is useful to refine the system behavior during the transient phase, while classical IO (Lyapunov) stability is a fundamental requirement to guarantee the correct behavior at steady state; therefore, it is a good practice to satisfy both requirements when designing a control system. To this end, in Chapter 10, we illustrate a hybrid architecture, where the controller is implemented by both finite‐time control techniques and the classical robust control approach.

The book is completed by five appendices. Appendices A and B provide some preliminary results on LTV systems and matrix algebra; Appendix C illustrates some numerical techniques to solve optimization problems with D/DLMIs constraints, while some MATLAB scripts that solve this type of optimization problems are presented in Appendix D. Appendix E discusses some real‐world examples where the IO‐FTS approach can be exploited.

There are some issues that are not presented in this book, in particular those ones that are currently in progress. For example, we do not discuss the extension of the IO‐FTS theory to nonlinear, as well as stochastic systems and systems with delays. Here, impulsive systems are only considered from the deterministic point of view, while there is a growing interest for impulsive and switched systems regulated by stochastic phenomena; for such topics the interested reader is referred to the specific literature; see also Section 1.5 of the book.

Catanzaro & Naples, November 2017

Francesco Amato,

Gianmaria De Tommasi,

Alfredo Pironti

List of Acronyms

Abbreviations

Mathematical Symbols

Set Theory

Numerical Sets

Vector and Matrix Operators

Special Matrices

Norms

Function Spaces

Miscellaneous

1

Introduction

This first chapter has the twofold objective of introducing the framework of input‐output finite‐time stability (IO‐FTS), together with the notation that will be used throughout the book, and providing some useful background on the analysis of the behavior of dynamical systems.

In order to introduce the topics dealt with in this monograph, we first recall the concept of state FTS, and then we will extend it to the input‐output case, both with zero and nonzero initial conditions. The former extension correspond to the concept of IO‐FTS, while the latter represents a generalization of the finite‐time boundedness (FTB) concept, namely IO‐FTS with nonzero initial conditions (IO‐FTS‐NZIC).

Roughly speaking, FTS involves the behavior of the system state for an autonomous dynamical system with nonzero initial conditions, while IO‐FTS looks at the input‐output behavior of the system, with zero initial conditions. IO‐FTS‐NZIC mixes the two concepts, considering the input‐output finite‐time control problem with a nonzero initial condition. The common points to these definitions is that they are defined over a finite‐time interval and that quantitative bounds are given for the admissible signals during this interval.

1.1 Finite‐Time Stability (FTS)

The concept of finite‐time stability (FTS) dates back to the fifties, when it was introduced in the Russian literature ([1–3]); later, during the sixties, this concept appeared for the first time in Western journals [4–6].

Given the dynamical system

1.1

where , we can give the following formal definition, which restates the original definition in a way consistent with the notation adopted in this monograph; in the following we consider the finite‐time interval , with .

Definition 1.1 (FTS, [2, 4, 8])

Given the time interval , a set , and a family of sets , system (1.1) is said to be finite‐time stable with respect to (wrt) if

1.2

where, with a slight abuse of notation, denotes the solution of (1.1) starting from at time .⋄

Note that, in general, the set , called outer (or trajectory) set, possibly depends on time; obviously must contain theinner (or initial) set , for well‐posedness of Definition 1.1.

An issue that is important to clarify is why the property expressed by (1.2) is called FTS.

In order to answer this question, we recall the classical definition of Lyapunov stability (LS, [32, Ch. 4]; see also Appendix A.3). Let be an equilibrium point for system (1.1), i.e., for all . The equilibrium point is said to be stable in the sense of Lyapunov if for each , there exists a positive scalar , possibly depending on and , such that , implies

and this holds for all .

The key points in the above definition are: an equilibrium point is stable if, once an arbitrary value for has been fixed, which defines a ball centered in , then it must be possible to build an inner ball (of radius ) such that, whenever the initial condition is inside such ball, the trajectory of the system starting from does not exit the outer ball (of radius ). Moreover this property holds for an infinite time horizon, that is, for all between and infinity.

Note that LS is a qualitative concept, that is, both the inner and the outer ball are not quantified; therefore, LS can be regarded as a structural property: a given equilibrium point is either stable or it is