# 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…

## 3,258 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

- 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

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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.

### Analysis of Switched and Hybrid Systems

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- 2004

This paper presents a method for stability analysis of switched and hybrid systems using polynomial and piecewise polynomial Lyapunov functions. Computation of subh functions can be performed using…

## References

SHOWING 1-10 OF 36 REFERENCES

### Stability of switched and hybrid systems

- MathematicsProceedings 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

- MathematicsProceedings of 1994 33rd IEEE Conference on Decision and Control
- 1994

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…

### Analyzing continuous switching systems: theory and examples

- MathematicsProceedings 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

- Mathematics1997 European Control Conference (ECC)
- 1997

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

- MathematicsProceedings of 1994 American Control Conference - ACC '94
- 1994

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

- Mathematics1988 American Control Conference
- 1988

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

- MathematicsProceedings 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

- MathematicsProceedings of 1994 33rd IEEE Conference on Decision and Control
- 1994

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

- MathematicsIEEE Transactions on Systems, Man, and Cybernetics
- 1980

Non-linear Differential Equations with Unique Solutions, Proof of the Kalman-Yacubovitch Lemma and proof of the Frobenius Theorem.