期刊
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
卷 15, 期 1, 页码 84-105出版社
SIAM PUBLICATIONS
DOI: 10.1137/140959031
关键词
hybrid system; piecewise smooth system; dimension reduction; noninvertibility; Lyapunov exponents; saltation matrix; discontinuity mapping
资金
- Italian Ministry for University and Research [FIRB RBFR08TIA4]
In discontinuous systems, the fundamental solution matrix of the linearized dynamics about a reference trajectory can be noninvertible. This feature can be exploited, for instance, to design robust control algorithms, to synchronize a network, or to stabilize otherwise unstable or chaotic dynamics. In this paper we classify all the phenomena that cause rank defect in the fundamental solution matrix of a generic discontinuous system. We relate these phenomena to simple geometric conditions at a point of vector field switching, sliding, or impact, and we derive necessary and sufficient conditions for the rank defect. This constitutes a valuable tool to detect flow noninvertibility or to purposefully include it in the design of a system. In terms of Lyapunov exponents, the singularity of the fundamental solution matrix means that an infinitesimal sphere of initial perturbations is mapped onto a lower-dimensional ellipsoid. This consequently reduces the number of finite exponents and, most remarkably, makes them depend on the full history of the reference trajectory used for the computation. We introduce a numerical procedure which allows the computation of the Lyapunov exponents by resorting to a properly reduced variational system. The validity of the approach is verified in two examples.
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