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Finite-time stability (FTS) is a more practical concept than classical Lyapunov stability, useful for checking whether the state trajectories of a system remain within pre-specified bounds over a finite time interval. In a linear systems framework, FTS problems can be cast as convex optimization problems and solved by the use of effective off-the-shelf computational tools such as LMI solvers. Finite-time Stability and Control exploits this benefit to present the practical applications of FTS and finite-time control-theoretical results to various engineering fields. The text is divided into two parts:

· linear systems; and

· hybrid systems.

The building of practical motivating examples helps the reader to understand the methods presented.

Finite-time Stability and Control is addressed to academic researchers and to engineers working in the field of robust process control. Instructors teaching graduate courses in advanced control will also find parts of this book useful for their courses.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
In this chapter, we discuss the organization of the book.
Francesco Amato, Roberto Ambrosino, Marco Ariola, Carlo Cosentino, Gianmaria De Tommasi

Linear Systems

Frontmatter

Chapter 2. FTS Analysis of Continuous-Time Linear Systems

Abstract
This chapter deals with the finite-time analysis and design problems for CT-LTV systems. Necessary and sufficient conditions for FTS are provided. One condition is based on the state transition matrix and is of poor usefulness in practice; the other conditions involve either DLMIs or DLEs. It is shown that the DLE-based condition is more efficient from the computational point of view; however, DLMIs are useful in the context of the design problem. Indeed, by the latter approach, we solve both the state and output feedback finite-time stabilization problems. The model of the car suspension system is used as a test case to show the effectiveness of the proposed techniques.
Francesco Amato, Roberto Ambrosino, Marco Ariola, Carlo Cosentino, Gianmaria De Tommasi

Chapter 3. Controller Design for the Finite-Time Stabilization of Continuous-Time Linear Systems

Abstract
In this chapter, we consider the finite-time stabilization problem. The starting point is condition (v) in Theorem 2.1, which allows the derivation of a necessary and sufficient condition for the finite-time stabilization via state and output feedback. The technique is then illustrated through the application to the car suspension system example introduced in Sect. 2.​5.
Francesco Amato, Roberto Ambrosino, Marco Ariola, Carlo Cosentino, Gianmaria De Tommasi

Chapter 4. Robustness Issues

Abstract
In this chapter, we deal with CT-LTV systems subject to uncertainties; it is assumed that the uncertain part can be modeled according to the classical linear fractional transformation (LFT) form, which covers many cases of practical interest. The concept of quadratic FTS (QFTS) is introduced; QFTS implies the existence of a quadratic Lyapunov function that allows us to prove the FTS of the given system for all admissible uncertainties. The main result of the chapter is a necessary and sufficient condition for QFTS in terms of either DLMIs or DLEs; then synthesis conditions are derived.
Francesco Amato, Roberto Ambrosino, Marco Ariola, Carlo Cosentino, Gianmaria De Tommasi

Chapter 5. FTS of Discrete-Time Linear Systems

Abstract
In this chapter, we deal with some finite-time control problems for discrete-time linear time-varying (DT-LTV) systems. First, we provide necessary and sufficient conditions for finite-time stability; such conditions can be regarded as the discrete-time counterparts of the results stated in Theorem 2.1. Indeed, one condition involves the computation of the state transition matrix; the other condition requires the solvability of a difference Lyapunov equality (DLE) or the feasibility of a difference linear matrix inequality (DLMI), respectively. The DLMI condition is the basis for the solution of the state and output feedback problems, which, again, are expressed in terms of optimization problems involving DLMIs.
Francesco Amato, Roberto Ambrosino, Marco Ariola, Carlo Cosentino, Gianmaria De Tommasi

Chapter 6. FTS Analysis Via PQLFs

Abstract
This chapter deals with the FTS analysis for CT-LTV systems when the initial and trajectory domains are piecewise quadratic. The approaches presented in the previous chapters of the book make use of quadratic Lyapunov functions to perform the FTS analysis and control of a given system. This is consistent with the fact that the initial and trajectory domains have been assumed to be ellipsoidal. The main contribution of this chapter is to consider a more general class of Lyapunov functions, namely the family of Piecewise Quadratic Lyapunov Functions (PQLFs). In particular, a novel sufficient condition for the FTS of CT-LTV systems, based on the PQLFs approach, is provided; then a procedure is proposed to convert such a condition into a computationally tractable problem. The numerical examples included at the end of the chapter are divided into three parts: first, we introduce a comparison with the other results presented in this book when ellipsoidal initial and trajectory domains are considered; then we present an example in which the domains have different structures, that is, a polytopic initial domain and an ellipsoidal trajectory domain; finally, the general case in which both the domains are polytopic is discussed in the third example. In the last two examples, the conditions developed in Chap. 2 can only be applied with added conservativeness due to the necessity of approximating the polytopic domains by ellipsoidal domains.
Francesco Amato, Roberto Ambrosino, Marco Ariola, Carlo Cosentino, Gianmaria De Tommasi

Hybrid Systems

Frontmatter

Chapter 7. FTS of IDLSs

Abstract
In this chapter, we consider the class of CT-LTV systems with finite state jumps, which are linear continuous-time systems whose states undergo finite jump discontinuities at discrete instants of time. Such systems, namely Impulsive Dynamical Linear Systems (IDLSs) (Haddad et al., Impulsive and Hybrid Dynamical Systems, Princeton University Press, Princeton, 2006), can be regarded as a special class of hybrid systems, and they can be either time-dependent (TD-IDLSs) if the state jumps are time-driven or state-dependent (SD-IDLSs) if the state jumps occur when the trajectory reaches an assigned subset of the state space, the so-called resetting set. TD-IDLSs can also be seen as a special case of switching linear systems (Liberzon, Switching in Systems and Control, Springer, Berlin, 2003). Lyapunov stability and stabilization of hybrid systems have been thoroughly discussed in the literature (see, for instance, the monographs Liberzon, Switching in Systems and Control, Springer, Berlin, 2003; Haddad et al., Impulsive and Hybrid Dynamical Systems, Princeton University Press, Princeton, 2006; Pettersson, Analysis and Design of Hybrid Systems. Ph.D. Thesis, 1999, and references therein). In this chapter, we propose some necessary and sufficient conditions for the FTS of TD-IDLSs, while only a sufficient condition will be provided for SD-IDLSs.
Francesco Amato, Roberto Ambrosino, Marco Ariola, Carlo Cosentino, Gianmaria De Tommasi

Chapter 8. Controller Design for the Finite-Time Stabilization of IDLSs

Abstract
Finite-time stabilization of IDLSs is tackled in this chapter. The D/DLMI feasibility problems introduced in Chap. 7 will be here exploited to derive necessary and sufficient conditions for finite-time stabilization of TD-IDLSs, while only sufficient conditions will be given in the case of SD-IDLSs. In the conclusion section, we will show, by means of some examples, how to extend the presented results to the more general case of time-dependent SLSs.
Francesco Amato, Roberto Ambrosino, Marco Ariola, Carlo Cosentino, Gianmaria De Tommasi

Chapter 9. Robustness Issues for IDLSs

Abstract
This chapter deals with the FTS of uncertain hybrid systems. First, we extend the definition of QFTS given in Chap. 4 to the class of IDLSs subject to norm-bounded uncertainties. Necessary and sufficient conditions to check QFTS are provided for the case of uncertain TD-IDLSs, while only a sufficient condition can be given for SD-IDLSs. The case of switching systems subject to uncertain resetting times is then considered; sufficient conditions to check FTS for the class of SLSs are given.
Francesco Amato, Roberto Ambrosino, Marco Ariola, Carlo Cosentino, Gianmaria De Tommasi

Backmatter

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