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Stability of Linear Parameter Varying and Linear Switching Systems
Author(s) -
Wirth Fabian
Publication year - 2003
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200310348
Subject(s) - irreducibility , mathematics , linear system , lyapunov function , classification of discontinuities , stability (learning theory) , class (philosophy) , set (abstract data type) , exponential stability , nonlinear system , pure mathematics , mathematical analysis , computer science , physics , quantum mechanics , machine learning , artificial intelligence , programming language
Abstract We consider stability of families of linear time‐varying systems, that are determined by a set of time‐varying parameters which adhere to certain rules. The conditions are general enough to encompass on the one hand stability questions for systems that are frequently called linear parameter varying systems in the literature and on the other hand also linear switching systems, in which parameter variations are allowed to have discontinuities. Combinations of these two sets of assumptions are also possible within the framework studied here. Under the assumption of irreducibility of the sets of system matrices, we show how to construct parameter dependent Lyapunov functions for the systems under consideration that exactly characterize the exponential growth rate. It is clear that such Lyapunov functions do not exist in general. But every system of our class can be reduced to a finite number of subsystems for which irreducibility holds.