Stability of Planar Switched Systems: The Linear Single Input Case

SIAM Journal on Control and Optimization (Impact Factor: 1.46). 01/2002; 41(1):89-112. DOI: 10.1137/S0363012900382837
Source: DBLP


We study the stability of the origin for the dynamical system ˙ x(t )= u(t)Ax(t )+( 1− u(t))Bx(t), where A and B are two 2 × 2 real matrices with eigenvalues having strictly negative real part, x ∈ R,. In the space of these parameters, the shape and the convexity of the region in which there is stability are studied. This bidimensional problem assumes particular interest since linear systems of higher dimensions can be reduced to our situation. Key words. stability, planar, random switching function, switched systems AMS subject classifications. 93D20, 37N35 PII. S0363012900382837

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    • "For n = 2 this approach led to a deep qualitative understanding of the problem in terms of the value function corresponding to the optimal control problem [21] (see also [22]). A dynamic programming approach [13] provided the first explicit expression for the value function for the case n = 2 [14] (see also [9] and the closely related work [5]). These issues are described in detail in the recent survey paper [12]. "
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    ABSTRACT: The absolute stability problem (ASP) entails determining a critical parameter value for which the stability of a feedback system, composed of an nth-order linear system and a sector-bounded nonlinear function, loses its stability. The ASP is one of the oldest open problems in the theory of stability and control. Recently, it is attracting consider-able interest, as solving it is equivalent to providing a necessary and sufficient condition for the stability of linear switched systems under arbitrary switching. Pyatnitsky pioneered the most promising approach for addressing the ASP. His approach is based on using optimal control techniques for characterizing the "most destabilizing" sector-bounded nonlinear function. This is equivalent to characterizing the "most destabilizing" switching law for a linear switched system under arbitrary switching. In this paper, we develop a new approach to the ASP which is based on a Lie-algebraic analysis of the switching function that determines the optimal control. We show that the finiteness of the associated Lie algebra implies that the switching function itself is the solution of a switched linear system of order at most n 2 . Furthermore, the switching function has a special and symmetric structure. This makes it possible to obtain an explicit analytic expression for the switching function for low orders of n. We demonstrate this using two examples.
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    • "Note that the matrix A in the previous example is singular. It is actually possible to see, by using for instance the results of [2] [8], that for n = 2, whenever M := conv{A, B}, A, B are non-singular and ρ(M) = 0, the supremum is always reached. This justifies the following question. "
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    ABSTRACT: Consider continuous-time linear switched systems on R^n associated with compact convex sets of matrices. When the system is irreducible and the largest Lyapunov exponent is equal to zero, there always exists a Barabanov norm (i.e. a norm which is non increasing along trajectories of the linear switched system together with extremal trajectories starting at every point, that is trajectories of the linear switched system with constant norm). This paper deals with two sets of issues: (a) properties of Barabanov norms such as uniqueness up to homogeneity and strict convexity; (b) asymptotic behaviour of the extremal solutions of the linear switched system. Regarding Issue (a), we provide partial answers and propose four open problems motivated by appropriate examples. As for Issue (b), we establish, when n = 3, a Poincar\'e-Bendixson theorem under a regularity assumption on the set of matrices defining the system. Moreover, we revisit the noteworthy result of N.E. Barabanov [5] dealing with the linear switched system on R^3 associated with a pair of Hurwitz matrices {A, A + bcT }. We first point out a fatal gap in Barabanov's argument in connection with geometric features associated with a Barabanov norm. We then provide partial answers relative to the asymptotic behavior of this linear switched system.
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    • "Optimal control of hybrid systems has been widely studied (Cassandras, Pepyne, & Wardi, 2001; Dmitruk & Kaganovich, 2008, 2011; Spinelli, Bolzern, & Colaneri, 2006). The problem studied in this paper is closely related to the variational approach to the stability of switched systems previous developed by Boscain (2002), Margaliot (2006) and Rapoport (1996). We provide an analytic solution for a particular class of switched systems using sufficient conditions via the necessary conditions based on the Pontryagin principle. "
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    Automatica 09/2013; 49(9):2874-2880. DOI:10.1016/j.automatica.2013.06.001 · 3.02 Impact Factor
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