Analysis of switched and hybrid systems-beyond piecewise quadratic methods (original) (raw)
Piecewise polynomial Lyapunov functions for a class of switched nonlinear systems
42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475), 2003
This paper proposes sufficient conditions to the regional stability analysis of switched nonlinear systems with time-varying parameters. The nonlinear sub-modes of operation are described by means of differential-algebraic equations involving the state and an auxiliary nonlinear vector. We then use piecewise polynomial Lyapunov functions and a relaxation technique that lead to a convex characterization of the problem in terms of linear matrix inequalities.
A polynomial approach for stability analysis of switched systems
Systems & Control Letters, 2010
A polynomial approach to deal with the stability analysis of switched non-linear systems under arbitrary switching using dissipation inequalities is presented. It is shown that a representation of the original switched problem into a continuous polynomial system allows us to use dissipation inequalities for the stability analysis of polynomial systems. With this method and from a theoretical point of view, we provide an alternative way to search for a common Lyapunov function for switched non-linear systems.
Robust stability analysis of nonlinear hybrid systems
2009
Abstract We present a methodology for robust stability analysis of nonlinear hybrid systems, through the algorithmic construction of polynomial and piecewise polynomial Lyapunov-like functions using convex optimization and in particular the sum of squares decomposition of multivariate polynomials. Several improvements compared to previous approaches are discussed, such as treating in a unified way polynomial switching surfaces and robust stability analysis for nonlinear hybrid systems.
Hybridization for Stability Analysis of Switched Linear Systems
Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control, 2016
In this paper, we present a hybridization method for stability analysis of switched linear hybrid system (LHS), that constructs a switched system with polyhedral inclusion dynamics (PHS) using a state-space partition that is specific to stability analysis. We use a previous result based on quantitative predicate abstraction to analyse the stability of PHS. We show completeness of the hybridization based verification technique for the class of asymptotically stable linear system and a subclass of switched linear systems whose dynamics are pairwise Lipschitz continuous on the state-space and uniformly converging in time. For this class of systems, we show that by increasing the granularity of the region partition, we eventually reach an abstract switched system with polyhedral inclusion dynamics that is asymptotically stable. On the practical side, we implemented our approach in the tool Averist, and experimentally compared our approach with a state-of-the-art tool for stability analysis of hybrid systems based on Lyapunov functions. Our experimental results illustrate that our method is less prone to numerical errors and scales better than the traditional approaches. In addition, our tool returns a counterexample in the event that it fails to prove stability, providing feedback regarding the potential reason for instability. We also examined heuristics for the choice of state-space partition during refinement.
Stability Criteria for Switched and Hybrid Systems
2005
The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving these problems in a number of diverse communities , and to review some problems that remain open. An important contribution of our work is to bring together material from several areas of research and to present results in a unified manner. We begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions. Closely related to the concept of stability are the notions of exponential growth rates and converse Lyapunov theorems, both of which are discussed in detail. In particular, results on common quadratic Lyapunov functions and piecewise linear Lyapunov functions are presented, as they represent constructive methods for proving stability, and also represent problems in which significant progress has been made. We also comment on the inherent difficulty of determining stability of switched systems in general which is exemplified by NP-hardness and undecidability results. We then proceed by considering the stability of switched systems in which there are constraints on the switching rules; be it through dwell time requirements or state dependent switching laws. Also in this case the theory of Lyapunov functions and the existence of converse theorems is reviewed. We briefly comment on the classical Lure' problem and on the theory of stability radii, both of which encapture many of the features of switched systems and are rich sources of practical results on the topic. Finally, both as an application, and an introduction to stochastic positive switched systems, a switched linear model of TCP dynamics is derived and several results presented.
Stability of switched and hybrid systems
Proceedings of 1994 33rd IEEE Conference on Decision and Control
This paper outlines work on the stability analysis of hybrid systems. Particularly, we concentrate on the continuous dynamics and model the finite dynamics as switching among finitely many continuous systems. We introduce multiple Lyapunov functions as a tool for analyzing Lyapunov stability. We use IFS theory as a tool for Lagrange stability. By enforcing the conditions of our theorems, one can also synthesize hybrid systems with desired stability properties.
A review of stability results for switched and hybrid systems
2001
Abstract--Hybrid and switched dynamic systems are of major research interest nowadays due to their use as models in many applications in computer science and systems control. One of the major problems in hybrid and switched dynamic systems is establishing their key property of stability, which also is important in controller design. Stability may prove also critical for real-time systems, embedded systems, and hybrid systems in general that arise in computer science problems where verification tests are undecidable.
Stabilizing Supervisory Control of Hybrid Systems Based on Piecewise Linear Lyapunov Functions 1
2000
In this paper, the stability of discrete-time piecewise linear hybrid systems is in- vestigated using piecewise linear Lyapunov functions. In particular, we consider switched discrete-time linear systems and we identify classes of switching sequences that result in stable trajectories. Given a switched linear system, we present a systematic methodology for computing switching laws that guarantee stability based on the matrices of the system. In the proposed approach, we assume that each individual subsystem is stable and admits a piece- wise linear Lyapunov function. Based on these Lyapunov functions, we compose "global" Lyapunov functions that guarantee stability of the switched linear system. A large class of stabilizing switching sequences for switched linear systems is characterized by computing conic partitions of the state space.
Multiple Lyapunov functions and other analysis tools for switched and hybrid systems
IEEE Transactions on Automatic Control, 1998
In this paper, we introduce some analysis tools for switched and hybrid systems. We first present work on stability analysis. We introduce multiple Lyapunov functions as a tool for analyzing Lyapunov stability and use iterated function systems (IFS) theory as a tool for Lagrange stability. We also discuss the case where the switched systems are indexed by an arbitrary compact set. Finally, we extend Bendixson's theorem to the case of Lipschitz continuous vector fields, allowing limit cycle analysis of a class of "continuous switched" systems.
International Journal of Control, 2002
In this paper, the stability of switched linear systems is investigated using piecewise linear Lyapunov functions. In particular, we identify classes of switching sequences that result in stable trajectories. Given a switched linear system, we present a systematic methodology for computing switching laws that guarantee stability based on the matrices of the system. In the proposed approach, we assume that each individual subsystem is stable and admits a piecewise linear Lyapunov function. Based on these Lyapunov functions, we compose "global" Lyapunov functions that guarantee stability of the switched linear system. A large class of stabilizing switching sequences for switched linear systems is characterized by computing conic partitions of the state space. The approach is applied to both discrete-time and continuous-time switched linear systems.