Finite-Time Stability: An Input-Output Approach E-Book

Francesco Amato

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: Wiley Series in Dynamics and Control of Electromechanical Systems
• Sprache: Englisch

Seitenzahl: 237

Veröffentlichungsjahr: 2018

Beschreibung

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.

Leseprobe

Table of Contents

Cover

Dedication

Preface

List of Acronyms

Chapter 1: Introduction

1.1 Finite‐Time Stability (FTS)

1.2 Input‐Output Finite‐Time Stability

1.3 FTS and Finite‐Time Convergence

1.4 Background

1.5 Book Organization

Chapter 2: Linear Time‐Varying Systems: IO‐FTS Analysis

2.1 Problem Statement

2.2 IO‐FTS for

Exogenous Inputs

2.3 A Sufficient Condition for IO‐FTS for

Exogenous Inputs

2.4 Summary

Chapter 3: Linear Time‐Varying Systems: Design of IO Finite‐Time Stabilizing Controllers

3.1 IO Finite‐Time Stabilization via State Feedback

3.2 IO‐Finite‐Time Stabilization via Output Feedback

3.3 Summary

Chapter 4: IO‐FTS with Nonzero Initial Conditions

4.1 Preliminaries

4.2 Interpretation of the Norm of the Operator

4.3 Sufficient Conditions for IO‐FTS‐NZIC

4.4 Design of IO Finite‐Time Stabilizing Controllers NZIC

4.5 Summary

Chapter 5: IO‐FTS with Constrained Control Inputs

5.1 Structured IO‐FTS and Problem Statement

5.2 Structured IO‐FTS Analysis

5.3 State Feedback Design

5.4 Design of an Active Suspension Control System Using Structured IO‐FTS

5.5 Summary

Chapter 6: Robustness Issues and the Mixed

/FTS Control Problem

6.1 Preliminaries

6.2 Robust and Quadratic IO‐FTS with an

Bound

6.3 State Feedback Design

6.4 Case study: Quadratic IO‐FTS with an

Bound of the Inverted Pendulum

6.5 Summary

Chapter 7: Impulsive Dynamical Linear Systems: IO‐FTS Analysis

7.1 Background

7.2 Main Results: Necessary and Sufficient Conditions for IO‐FTS in Presence of

Signals

7.3 Example and Computational Issues

7.4 Main Result: A Sufficient Condition for IO‐FTS in Presence of

Signals

7.5 Summary

Chapter 8: Impulsive Dynamical Linear Systems: IO Finite‐Time Stabilization via Dynamical Controllers

8.1 Problem Statement

8.2 IO Finite‐Time Stabilization of IDLSs:

Signals

8.3 IO Finite‐Time Stabilization of IDLSs:

Signals

8.4 Summary

Chapter 9: Impulsive Dynamical Linear Systems with Uncertain Resetting Times

9.1 Arbitrary Switching

9.2 Uncertain Switching

9.3 Numerical Example

9.4 Summary

Chapter 10: Hybrid Architecture for Deployment of Finite‐Time Control Systems

10.1 Controller Architecture

10.2 Examples

10.3 Summary

Appendix A: Fundamentals on Linear Time‐Varying Systems

A.1 Existence and Uniqueness

A.2 The State Transition Matrix

A.3 Lyapunov Stability of Linear Time‐Varying Systems

A.4 Input to State and Input to Output Response

Appendix B: Schur Complements

Appendix C: Computation of Feasible Solutions to Optimizations Problems Involving DLMIs

C.1 Numerical Solution to a Feasibility Problem Constrained by a DLMI Coupled with LMIs

C.2 Numerical Solution to a Feasibility Problem Constrained by a D/DLMI Coupled with LMIs

Appendix D: Solving Optimization Problems Involving DLMIs using MATLAB®

D.1 MATLAB® Script for the Solution of the Optimization Problem with DLMI/LMI Constraints Presented in Example 2.2

D.2 MATLAB® Script for the Solution of the D/DLMI/LMI Feasibility Problem Presented in Section 8.3.1

Appendix E: Example s of Applications of IO‐FTS Control Design to Real‐World Systems

E.1 Building Subject to Earthquakes

E.2 Quarter Car Suspension Model

E.3 Inverted Pendulum

E.4 Yaw and Lateral Motions of a Two‐Wheel Vehicle

References

Index

End User License Agreement

List of Tables

Chapter 05

Table E.1 Model parameters for the considered six story building (N=6).

Table E.2 Parameters of the inverted pendulum.

Table E.3 Two‐wheel model parameters.

Chapter 02

Table 2.1 Maximum values of

satisfying Theorem 2.3 for the LTV system (2.36).

Chapter 07

Table 7.1 Values of

obtained exploiting condition

ii)

in Theorem 7.1 for the IDLS system (7.27).

Table 7.2 Values of

obtained exploiting condition

iii)

in Theorem 7.1 for the IDLS system (7.27).

List of Illustrations

Chapter 5

Figure E.1 Lumped parameters model of an N‐story building.

Figure E.2 Ground acceleration, velocity, and displacement of El Centro earthquake.

Figure E.3 Schematic representation of the active suspension system.

Figure E.4 Scheme of the inverted pendulum.

Figure E.5 Schematic representation of the bicycle, along with the various symbols adopted for its description, and the ground reference frame.

Chapter 1

Figure 1.1 Given a time interval

, and the two ellipsoidal domains delimited by

and by the constant matrix

, a second‐order system is finite‐time stable if all the trajectories over the considered time interval are like the one reported in light gray. Furthermore, in dark gray are reported two examples of trajectories that are not finite‐time stable.

Figure 1.2 Free response of the LS stable LTI system 1.5 when the initial state is set equal to

. Although the considered LTI system is Lyapunov stable, the same system can be either FTS or not, depending on the FTS parameters.

Figure 1.3 Free response of the LS unstable LTI system 1.6 when the initial state is set equal to

. Even when Lyapunov unstable systems are considered, the finite‐time stability depends on the chosen parameters.

Figure 1.4 Time response of system (1.2) to the unitary step function. When the weighting matrix

is considered, then the weighted output exceeds

; hence, the system is not IO‐FTS. On the other hand, it can be proved that for all the exogenous inputs

belonging to the class of bounded signals in the time interval

, if the weighting matrix

is considered, then the weighted output never exceeds

.

Chapter 2

Figure 2.1 Time evolution of the output of system (2.46) when the exogenous input is set equal to

.

Chapter 3

Figure 3.1 Uncontrolled base floor velocity and displacement.

Figure 3.2 Controlled base floor velocity and displacement.

Figure 3.3 Control force applied to the base floor.

Figure 3.4 Time evolution of the output

, and the weighted output

of system (3.21), when the exogenous input is set equal to

.

Figure 3.5 Time evolution of the weighted output

, and of the control input

, when the exogenous input is set equal to

, and when system (3.21) is IO finite‐time stabilized by means of an output feedback controller.

Figure 3.6 Time evolution of the weighted output

and of

, when the exogenous input is set equal to

, and when the output feedback controller is designed including the additional constraints (3.22) in order to limit the control input.

Chapter 4

Figure 4.1 Relationships between the operators

,

and their duals.

Figure 4.2 Time evolution of the weighted output

of system (4.39) for different choices of the initial state, when the exogenous input is set equal to

.

Figure 4.3 Time evolution of the weighted output

of the closed loop system of Example 4.2, when the initial state

is taken equal to

and the exogenous input

is equal to

in the interval

.

Chapter 5

Figure 5.1 Ground asperity considered for the design of the structured IO‐FTS controller for the active suspension system.

Figure 5.2 Bump response: structured IO‐FTS time‐varying controller (–), constrained

controller (‐ ‐).

Figure 5.3 Bump response: time behavior of the weighted output

and

when the structured IO‐FTS time‐varying controller is considered.

Chapter 6

Figure 6.1 The considered state feedback control configuration.

Figure 6.2 Weighted output

for 100 random realizations of the open loop uncertain system considered in Section 6.2.2.

Figure 6.3 Weighted output

for 100 random realizations of the closed loop uncertain system considered in Section 6.3.1.

Figure 6.4 Disturbance force

considered in the nonlinear simulation of the inverted pendulum.

Figure 6.5 Time traces of the cart position

, of the pendulum angle

, and of the control input

, when the disturbance shown in Figure 6.4 is applied to the inverted pendulum.

Chapter 7

Figure 7.1 Example of the time behavior of the impulsive system (7.27), when an input in

is considered.

Figure 7.2 Weighted output

of system (7.27) when the input shown in Figure 7.1 is applied to the impulsive system (7.27), and

is taken equal to 2.

Figure 7.3 Time evolution of the exogenous input

, of the output

, and of the weighted output

for the IDLS considered in Section 7.4.1.

Chapter 8

Figure 8.1 Weighted output

of the IDLS (8.10) when the exogenous input is set equal to

in the time interval

.

Figure 8.2 Control inputs

and weighted output

for the IDLS (8.10) when it is IO finite‐time stabilized by a controller designed exploiting Theorem 8.1.

Figure 8.3 State‐feedback controller gains obtained by solving the feasibility problem (8.9) for IDLS considered in Section 8.3.1.

Figure 8.4 Control action

and weighted output

for the closed‐loop system considered in Section 8.3.1.

Chapter 9

Figure 9.1 Switching signal

for SLS considered in Section 9.3.

Figure 9.2 Weighted output

for the two linear systems considered in Section 9.3, when

in

, and

.

Figure 9.3 Weighted output

for the IDLS considered in Section 9.3, when

in

, and

.

Figure 9.4 Worst case weighted output

for the IDLS considered in Section 9.3, in the case of uncertain switching, when

, and the exogenous input is taken equal to 1 in the time interval

.

Chapter 10

Figure 10.1 Block diagram of the proposed hybrid architecture for the implementation of a controller based on an IO‐FTS design approach.

Figure 10.2 Hybrid automaton implementing the supervisor block provided in the architecture reported in Figure 10.1.

Figure 10.3 Hybrid automaton for the possible implementation of the active suspension control system based on structured IO finite‐time stabilization.

Figure 10.4 Ground asperity considered to prove the effectiveness of the hybrid controller for the active suspension system.

Figure 10.5 Response to the ground asperity reported in Figure 10.4. Hybrid controller (–), constrained

controller (‐ ‐). The circles denote the time instants when the discrete state

of the hybrid automaton is activated.

Figure 10.6 Side wind velocity profile considered as exogenous input for the simulation described in Section 10.2.2.

Figure 10.7 Behavior of the hybrid control architecture for vehicle collision avoidance. The circles denote the time instants when the state

of the hybrid automaton is activated, that is, the time instants when the event

occurs, causing the activation of the IO finite‐time stabilizing controller for a time window of length

.

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Wiley Series in Dynamics and Control of Electromechanical Systems

Finite-Time Stability

Amato, De Tommasi and Pironti

August 2018

An Input-Output Approach

Process Control System Fault Diagnosis

Gonzalez, Qi and Huang

September 2016

A Bayesian Approach

Variance-Constrained Multi-Objective

Ma, Wang and Bo

April 2015

Stochastic Control and Filtering

Sliding Mode Control of Uncertain

Wu, Shi and Su

July 2014

Parameter-Switching Hybrid Systems

Algebraic Identification and Estimation

Sira-Ramírez, García Rodríguez,

May 2014

Methods in Feedback Control Systems Cortes

Romero and Luviano Juárez

Finite-Time Stability

An Input-Output Approach

Francesco Amato

University of Catanzaro Magna Græcia Italy

Gianmaria De Tommasi

University of Naples Federico II Italy

Alfredo Pironti

University of Naples Federico II Italy

Copyright

This edition first published 2018

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The right of Francesco Amato, Gianmaria De Tommasi, and Alfredo Pironti to be identified as the authors of this work has been asserted in accordance with law.

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Library of Congress Cataloging-in-Publication Data:

Names: Amato, Francesco, author. | De Tommasi, Gianmaria, author. | Pironti, Alfredo, author.

Title: Finite-time stability : an input-output approach / Francesco Amato, University of Catanzaro Magna Græcia, IT, Gianmaria De Tommasi University of Naples Federico II, IT, Alfredo Pironti, University of Naples Federico II, Italy.

Description: First edition. | Hoboken, NJ : John Wiley & Sons, Inc., 2018. | Series: Wiley series in dynamics and control of electromechanical systems | Includes bibliographical references and index. |

Identifiers: LCCN 2018016200 (print) | LCCN 2018017626 (ebook) | ISBN 9781119140566 (pdf) | ISBN 9781119140559 (epub) | ISBN 9781119140528 (cloth)

Subjects: LCSH: Stability. | System design.

Classification: LCC QA871 (ebook) | LCC QA871 .A43 2018 (print) | DDC 515/.392-dc23

LC record available at https://lccn.loc.gov/2018016200

Cover design: Wiley

Cover image: © agsandrew/GettyImages

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

DLE

differential Lyapunov equation

DLMI

differential linear matrix inequality

D/DLE

differential/difference Lyapunov equation

D/DLMI

differential/difference linear matrix inequality

FTB

finite‐time boundedness

FTS

finite‐time stability

IDLS

impulsive dynamical linear system

SLS

switching linear system

IO‐FTS

input‐output finite‐time system

LMI

linear matrix inequality

LTI

linear time‐invariant

LTV

linear time‐varying

LS

Lyapunov stability

AS

Arbitrary switching

US

Uncertain switching

Mathematical Symbols

such that

for all

there exists

equal by definition

is equivalent to

implies

Set Theory

the element

belongs to the set

the union of the sets

and

the set

is a subset of the set

the set

is a

strict

subset of the set

closed interval

open interval

the set composed of the elements of the set

which do not belong to the set

Numerical Sets

(

)

nonnegative (positive) integer numbers

field of real numbers

nonnegative real numbers

set of the

‐tuple of real numbers

real matrices with

rows and

columns

Vector and Matrix Operators

the

‐th element of the vector

the

‐th element of the matrix

determinant of the square matrix

inverse of the square matrix

transpose of matrix

block diagonal matrix with

,

,

,

on the diagonal

maximum eigenvalue of the positive definite matrix

minimum eigenvalue of the positive definite matrix

is (symmetric) positive definite

is (symmetric) positive semidefinite

is (symmetric) negative definite

is (symmetric) negative semidefinite

is (symmetric) positive definite

is (symmetric) positive semidefinite

the rank of matrix

Kronecker product of matrices

and

:=

where

,

and

. When

defines the internal product in

, between

and

.

Special Matrices

identity matrix; the dimension will be clear from the context

0

zero matrix; the dimension will be clear from the context

Norms

1Introduction

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).