Hidden Degeneracies in Piecewise Smooth Dynamical Systems

When a flow suffers a discontinuity in its vector field at some switching surface, the flow can cross through or slide along the surface. Sliding along the switching surface can be understood as the flow along an invariant manifold inside a switching layer. It turns out that the usual method for finding sliding modes — the Filippov convex combination or Utkin equivalent control — results in a degeneracy in the switching layer whenever the flow is tangent to the switching surface from both sides. We derive the general result and analyze the simplest case here, where the flow curves parabolically on either side of the switching surface (the so-called fold–fold or two-fold singularities). The result is a set of zeros of the fast switching flow inside the layer, which is structurally unstable to perturbation by terms nonlinear in the switching parameter, terms such as [Formula: see text] [where the superscript does mean “squared”]. We provide structurally stable forms, and show that in this form the layer system is equivalent to a generic singularity of a two timescale system. Finally we show that the same degeneracy arises when a discontinuity is smoothed using standard regularization methods.

Switching Control Blending Methodology with Single Parameter Dependent and its Application to Tiltrotor

Because direct switching between controllers will generate abrupt change of control signal causing severe actuators workload, a switching control blending methodology is presented in this paper. By the blending of weighted multi-controllers output signals in which the weight value depends on a single switching parameter, the control signals can be switched smoothly and the closed-loop dynamic acts continuously. In this method, the design of multi-controllers can use any available methods while the closed-loop switching stability is insured by rational division of closed-loop state-space along the switching parameter. The closed-loop switching stability under control blending is presented, and a state-space division algorithm is developed. With the application to the tiltrotor, the simulation results show that the presented control methodology can provide all modes control ability and make control signals switched smoothly.

Numerical Investigation of Rough Surfaces: Coupling Approach

A numerical analysis of the flow field in rough microchannel is carried out decomposing the computational physical domain into kinetic and continuum sub-domains. Each domain size is determined by the value of a proper threshold parameter, based on the local Knudsen number and local gradients of macro-parameters. This switching parameter is computed from a preliminary Navier–Stokes solution throughout the whole physical domain. The solution is then advanced in time simultaneously in both kinetic and continuum domains: the coupling is achieved by matching half fluxes at the interface of the kinetic and Navier–Stokes domains, taking care of the conservation of momentum, energy and mass through the interface. The roughness geometry is modeled as a series of triangular obstructions with a relative roughness up to a maximum of 5% of the channel height. A wide range of Mach numbers is considered, from nearly incompressible to chocked flow conditions and a Reynolds number up to 100. Accuracy and discrepancies between full Navier Stokes, kinetic and coupled solutions are discussed, assessing the range of applicability of first order slip condition in rough geometries. The effect of the roughness is discussed via Poiseuille number as a function of local Knudsen and Mach numbers.