Name
Title | ||
|---|---|---|
Necessary and Sufficient Conditions for the Noninvertibility of Fundamental Solution Matrices of a Discontinuous System. |
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. |
Year | DOI | Venue |
|---|---|---|
2016 | 10.1137/140959031 | SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS |
Keywords | Field | DocType |
hybrid system,piecewise smooth system,dimension reduction,noninvertibility,Lyapunov exponents,saltation matrix,discontinuity mapping | Ellipsoid,Vector field,Control theory,Mathematical analysis,Matrix (mathematics),Singularity,Fundamental solution,Chaotic,Robust control,Mathematics,Lyapunov exponent | Journal |
Volume | Issue | ISSN |
15 | 1 | 1536-0040 |
Citations | PageRank | References |
0 | 0.34 | 0 |
Authors | ||
4 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Federico Bizzarri | 1 | 131 | 31.78 |
| Alessandro Colombo | 2 | 7 | 4.22 |
| Fabio Dercole | 3 | 47 | 14.32 |
| Giancarlo Storti Gajani | 4 | 96 | 13.09 |