Quadratic Lyapunov functions for systems with state-dependent switching

In this paper, we consider the existence of quadratic Lyapunov functions for certain types of switched linear systems. Given a partition of the state-space, a set of matrices (linear dynamics), and a matrix-valued function A ( x ) constructed by associating these matrices with regions of the state-s...

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Bibliographic Details
Published inLinear algebra and its applications Vol. 433; no. 1; pp. 52 - 63
Main Authors Griggs, Wynita M., King, Christopher K., Shorten, Robert N., Mason, Oliver, Wulff, Kai
Format Journal Article
LanguageEnglish
Published Amsterdam Elsevier Inc 15.07.2010
Elsevier
Subjects
Algebra
Exact sciences and technology
Hybrid systems
Linear and multilinear algebra, matrix theory
Lyapunov functions
Mathematics
Quadratic stability
Sciences and techniques of general use
52A10
Hybrid systems
Lyapunov functions
93D05
15A18
Quadratic stability
Partition
State space
Discrete system
State space method
Quadratic system
Matrix function
Hybrid system
Linear system
Positive definite matrix
Eigenvalue problem
Non linear effect
Lyapunov function
Quadratic function
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Summary:In this paper, we consider the existence of quadratic Lyapunov functions for certain types of switched linear systems. Given a partition of the state-space, a set of matrices (linear dynamics), and a matrix-valued function A ( x ) constructed by associating these matrices with regions of the state-space in a manner governed by the partition, we ask whether there exists a positive definite symmetric matrix P such that A ( x ) T P + PA ( x ) is negative definite for all x ( t ) . For planar systems, necessary and sufficient conditions are given. Extensions for higher order systems are also presented.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2010.02.011