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Lie- and chronologico-algebraic tools for studying stability of time-varying systems

✍ Scribed by Andrey V. Sarychev

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
180 KB
Volume
43
Category
Article
ISSN
0167-6911

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✦ Synopsis

We will study stability and asymptotic stability for time-varying systems described by ODEs of the form αΊ‹ = f( -1 t; x), where f(t; x) is 1-periodic with respect to t and ΒΏ0 is a small parameter. Since the discovery of stabilizing e ect of vibration in the reverse pendulum example, there have been a lot of study regarding stability of such systems and design of fast-oscillating stabilizing feedback laws. In this paper we suggest an approach which is kind of high-order averaging procedure based on Lie algebraic formalism and the formalism of chronological calculus. This latter is a method of asymptotic analysis for ows generated by time-variant ODE. We apply the approach to study stability issues for linear and nonlinear systems. In particular, we derive conditions of stability for the second-and third-order linear di erential equations with periodic fast-oscillating coe cients, we study output-feedback stabilization of bilinear systems and consider high-order averaging procedure for nonlinear systems under homogeneity assumptions. At the end we study the problem of stabilization of nonholonomic (control-linear) systems by means of time-varying feedbacks.

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