A Filippov sliding vector field on an attracting co-dimension 2 discontinuity surface, and a limited loss-of-attractivity analysis

Abstract

We consider sliding motion, in the sense of Filippov, on a discontinuity\ud surface Σ of co-dimension 2. We characterize, and restrict\ud to, the case of Σ being attractive through sliding. In this situation,\ud we show that a certain Filippov sliding vector field f_F (suggested\ud in Alexander and Seidman, 1998 [2], di Bernardo et al., 2008 [6],\ud Dieci and Lopez, 2011 [10]) exists and is unique. We also propose\ud a characterization of first order exit conditions, clarify its relation to\ud generic co-dimension 1 losses of attractivity for Σ, and examine\ud what happens to the dynamics on Σ for the aforementioned vector\ud field f_F . Examples illustrate our results

Exit from sliding in piecewise-smooth flows: Deterministic vs. determinacy-breaking

“…Exit from high codimension sliding (figure 1(iii-iv)) has hardly been studied as yet, though substantial steps in this direction are starting to be made, for example in [13] where the problem of computability of solutions at exit points is raised in particular. Our aim here is to open up this problem by demonstrating basic but non-trivial behaviours induced by exit from sliding.…”

Dynamics at a Switching Intersection: Hierarchy, Isonomy, and Multiple Sliding

“…In [9], a convex canopy of the form (7) is taken over field values f µ 1 ,µ 2 ,... , but the result above shows how easily this is extended to a combination over input values g µ 1 ,µ 2 ,... . This reveals some ambiguity in the argument of [8] that one particular combination can be justified over another by smoothing out the discontinuity. The alternatives of parametric or field combination imply that different choices of smoothing may in fact be chosen to justify any such convex combinations, and the ambiguity of such smoothing is discussed in [16].…”

“…In [8,9], Filippov's method is refined by reducing the dimension of F as suggested above. The method is motivated by certain attractivity assumptions and restrictions on the number of intersecting manifolds, but the approach can actually be applied far more generally, extending straightforwardly to the intersection of m manifolds in at least m dimensions for any finite (positive) integer m. The more general approach opens a new world of dummy dynamics inside the intersection, and leads to multiple or nonlinear sliding modes, which will be introduced here.…”

“…One must solve the system and investigate how many valid vectors there are within the convex canopy F that are tangent to the discontinuity surface D. In the special case when all of the local vector fields are directed towards the intersection (i.e. it is attractive), then unique sliding vectors do exist, and this case is considered in [8,9]. In slightly weaker conditions, when the intersection attracts nearby dynamics and is reachable through sliding motion outside the intersection, uniqueness is possible as shown in [8].…”

“…it is attractive), then unique sliding vectors do exist, and this case is considered in [8,9]. In slightly weaker conditions, when the intersection attracts nearby dynamics and is reachable through sliding motion outside the intersection, uniqueness is possible as shown in [8].…”