# 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…

## 3,331 Citations

### 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

- MathematicsSIAM 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

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- 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

- MathematicsIEEE 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

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- 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…

### On formalism and stability of switched systems

- MathematicsJournal of Control Theory and Applications
- 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

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- 2001

The concepts of attractive region and semi-attractive region are proposed and used as a tool for analyzing the Lyapunov stability of switched systems and are more practical than the other presented ones.

### On the stability and robustness of switched systems

- 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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