A new methodology for the stability analysis of pairwise triangularizable and related switching systems |
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Authors: | Shorten Robert; Cairbre Fiacre O |
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Institution: |
1 The Hamilton Institute, National University of Ireland, Maynooth, Co. Kildare, Ireland 2 Department of Mathematics, National University of Ireland, Maynooth, Co. Kildare, Ireland
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Abstract: | We consider the asymptotic stability of the time-varying dynamicsystem : = A(t)x, A(t) Rn x n, A(t) A= {A1, ..., Am}, where Ai is Hurwitz and where a set of non-singularmatrices Ti j exist such that any pair of matrices {Ti j AiTi j1, Ti j Aj Ti j1}, i, j {1, ..., m}, areupper triangular. Switching systems of this form are referredto as pairwise triangularizable switching systems. It can beestablished that (a) pairwise triangularizability is not sufficientto guarantee the existence of a common quadratic Lyapunov functionfor the linear time-invariant dynamic systems Ai : = Ai x; (b) additional conditions can be specified which guaranteeasymptotic stability of the switching system . In this paperwe also show that pairwise triangularizability is not even sufficientto guarantee asymptotic stability of the switching system .We also show that the method of proof of stability in (b), whichdoes not assume the existence of a common quadratic Lyapunovfunction, can be used to prove the asymptotic stability of moregeneral switching systems (systems that are not pairwise triangularizable).Finally, we show that our results can be used as the basis forthe design of practical control systems; namely, for the designof an automobile speed switched controller with guaranteed stabilityproperties. |
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Keywords: | hybrid systems non-quadratic Lyapunov stability switched linear systems |
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