Multiple Lyapunov functions and other analysis tools for switched and hybrid systems

  title={Multiple Lyapunov functions and other analysis tools for switched and hybrid systems},
  author={Michael S. Branicky},
  journal={IEEE Trans. Autom. Control.},
  • M. Branicky
  • Published 1 April 1998
  • Computer Science, Mathematics
  • IEEE Trans. Autom. Control.
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 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… 
A Note on Multiple Lyapunov Functions and Stability Condition for Switched and Hybrid Systems
A new Lyapunov stability condition is proposed, which complements the existing stability conditions that evaluate the value of multiple LyAPunov functions at the starting points or the end points, and also applies to the switching control problem for stabilization of nonholonomic systems.
A multiple Lyapunov functions approach for stability of switched systems
  • Jin Lu, L. Brown
  • Mathematics
    Proceedings of the 2010 American Control Conference
  • 2010
In this paper, a multiple Lyapunov functions based approach is presented for the stability analysis for switched systems. Compared with existing Multiple Lyapunov functions approaches in the
Stability analysis of deterministic and stochastic switched systems via a comparison principle and multiple Lyapunov functions
This paper presents a general framework for analyzing stability of nonlinear switched systems, by combining the method of multiple Lyapunov functions with a suitably adapted comparison principle in
New stability results for discrete-time switched systems: a new multiple piecewise convex Lyapunov function approach
In this paper, a new multiple Lyapunov function is proposed to obtain the improved stability conditions for discrete-time switched systems. Firstly, a multiple piecewise convex Lyapunov function
Stability Analysis of Linear Switched Systems with Mixed Delays
This paper addresses the stability of the switched systems with discrete and distributed time delays. By applying Lyapunov functional and function method, we show that, if the norm of system matrices
Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach
The approach followed in this paper looks at the existence of a switched quadratic Lyapunov function to check asymptotic stability of the switched system under consideration and shows that the second condition is, in this case, less conservative.
Stability and stabilization of piecewise affine and hybrid systems: an LMI approach
In this paper we present various algorithms both for stability analysis and state-feedback design for discrete-time piecewise affine systems. Our approach hinges on the use of piecewise quadratic
On formalism and stability of switched systems
In this paper, we formulate a uniform mathematical framework for studying switched systems with piecewise linear partitioned state space and state dependent switching. Based on known results from the
The stability analysis of switched systems
A stability approach of switched systems is presented. We propose the concepts of attractive region and semi-attractive region, and use of them as a tool for analyzing the Lyapunov stability of
On the stability and robustness of switched systems
  • Qing Hui
  • Computer Science, Mathematics
    2009 European Control Conference (ECC)
  • 2009
Converse Lyapunov theorems for a finite family of pairwise commuting semistable systems are developed and some robustness results for asymptotic stability and semistability of switched systems are derived.


Stability of switched and hybrid systems
  • M. Branicky
  • Mathematics
    Proceedings of 1994 33rd IEEE Conference on Decision and Control
  • 1994
This paper outlines some preliminary work on the stability analysis of switched and hybrid systems. The hybrid systems considered are those that combine continuous dynamics, represented by
Construction of piecewise Lyapunov functions for stabilizing switched systems
This paper discusses the problem of stabilizing a pair of switched linear systems. A control law is developed using a Lyapunov function having a piecewise continuous derivative. A Lyapunov function
A Stabilizing Switching Scheme for Multi Controller Systems
The stability of a multi-controller system is analyzed using Lyapunov theory. Stability is guaranteed by a switching strategy determined by a combination of separate Lyapunov functions for the
Analyzing continuous switching systems: theory and examples
  • M. Branicky
  • Mathematics
    Proceedings of 1994 American Control Conference - ACC '94
  • 1994
This paper details work on ordinary differential equations that continuously switch among regimes of operation. In the first part, we develop some tools for analyzing such systems. We prove an
Computation of piecewise quadratic Lyapunov functions for hybrid systems
The search for a piecewise quadratic Lyapunov function is formulated as a convex optimization problem in terms of linear matrix inequalities to demonstrate the flexibility and power of the approach.
Recent results on Lyapunov-theoretic techniques for nonlinear stability
Establishes a converse Lyapunov function theorem useful for studying stability of systems with disturbances. The result applies to global stability with respect to closed, not necessarily compact,
On the Controllability and Observability of Hybrid Systems
This paper considers a special class of hybrid systems, whose state space is a cross-product space of an Euclidean space and a finite-state space. Such models may be used to represent systems subject
Stability theory for hybrid dynamical systems
  • H. Ye, A. Michel, L. Hou
  • Mathematics
    Proceedings of 1995 34th IEEE Conference on Decision and Control
  • 1995
Hybrid systems which are capable of exhibiting simultaneously several kinds of dynamic behavior in different parts of a system are of great current interest. In the present paper we first formulate a
A unified framework for hybrid control
We propose a very general framework for hybrid control problems that encompasses several types of hybrid phenomena considered in the literature. A specific control problem is studied in this
Nonlinear Systems Analysis
Non-linear Differential Equations with Unique Solutions, Proof of the Kalman-Yacubovitch Lemma and proof of the Frobenius Theorem.