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

@article{Branicky1998MultipleLF, title={Multiple Lyapunov functions and other analysis tools for switched and hybrid systems}, author={Michael S. Branicky}, journal={IEEE Trans. Autom. Control.}, year={1998}, volume={43}, pages={475-482} }

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…

## 3,205 Citations

A Note on Multiple Lyapunov Functions and Stability Condition for Switched and Hybrid Systems

- Mathematics, Computer Science2007 IEEE International Conference on Control Applications
- 2007

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

- MathematicsProceedings 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

- Mathematics, Computer ScienceSIAM J. Control. Optim.
- 2006

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

- Mathematics2019 Chinese Control Conference (CCC)
- 2019

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

- Mathematics
- 2010

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

- Mathematics, Computer ScienceIEEE Trans. Autom. Control.
- 2002

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

- MathematicsProceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187)
- 2000

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

- Mathematics
- 2012

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

- 2001 International Conferences on Info-Tech and Info-Net. Proceedings (Cat. No.01EX479)
- 2001

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

- Computer Science, Mathematics2009 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.

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