1. Necessary and Sufficient Conditions for the Noninvertibility of Fundamental Solution Matrices of a Discontinuous System.
- Author
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Bizzarri, Federico, Colombo, Alessandro, Dercole, Fabio, and Gajani, Giancarlo Storti
- Subjects
MATRICES (Mathematics) ,LYAPUNOV exponents ,MATRIX exponential ,ROBUST control ,HYBRID systems ,DYNAMICAL systems - Abstract
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. [ABSTRACT FROM AUTHOR]
- Published
- 2016
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