Jitter in Piecewise-Smooth Dynamical Systems with Intersecting Discontinuity Surfaces

Abstract

Mathematical models involving switches -in the form of differential equations with discontinuities -can accomodate real-world non-idealities through perturbations by hysteresis, time-delay, discretization, and noise. These are used to model the processes associated with switching in electronic control, mechanical contact, predator-prey preferences, and genetic or cellular regulation. The effect of such perturbations on rapid switching dynamics about a single switch are somewhat benign: in the limit that the size of the perturbation goes to zero the dynamics is given by Filippov's sliding solution. When multiple switches are involved, however, perturbations can have complicated effects, as shown in this paper. In the zero-perturbation limit, hysteresis, time-delay, and discretization cause erratic variation or 'jitter' for stable sliding motion, whilst noise generates a relatively regular sliding solution similar to the canopy solution (an extension of Filippov's solution to multiple switches). We illustrate the results with a model of a switched power circuit, and showcase a variety of complex phenomena that perturbations can generate, including chaotic dynamics, exit selection, and coexisting sliding solutions.

A unified approach to explain contrary effects of hysteresis and smoothing in nonsmooth systems

“…The surface consists of two half-planes, u = +1 for y + α > 0 and u = −1 for y − α < 0, which are consistent with (2)-( 3) when α = 0. These half-planes are connected by a plane segment on which u = −y/α and |u| < 1, which is consistent with the condition u ∈ [−1, 1] from (13). Hysteresis manifests as a relaxation between the half-planes u = 1, y ≥ −α and u = −1, y ≤ α.…”

“…for ε = 0 the system (19) is formally equivalent to the system (17), and hence to the system (2) with (13), and moreover is formally equivalent to the system (2)-(3) in the limit α = 0. A proper justification of these statements if given in the following sections.…”

“…The results here apply to a single attracting switching surface. The situation for two or more switches turns out to be even richer and more intriguing, see [13].…”

A compendium of Hopf-like bifurcations in piecewise-smooth dynamical systems

“…In phase space, two switching manifolds generically intersect on a codimension-two surface. Determining an appropriate sliding vector field on such a surface is surprisingly difficult in a general setting, see [49,98,102] for recent developments, however here we are only concerned with two-dimensional systems for which two switching manifolds can be expected to intersect at a point. The dynamics near two transversally intersecting switching manifolds, depends on the direction of the vector field in each of the four regions bounded by the switching manifolds [81,99].…”

Acceleration of the Measurement Time of Thermopiles Using Sigma-Delta Control

“…In our case the control is implemented as a double-loop circuit based on thermal sigma-delta modulation [17,18]. This modulation provides an implicit analog-to-digital conversion, that makes it suitable for many sensing applications, even in the case of heterogeneous systems [19,20]. Moreover, this approach also inherits other benefits of sigma-delta converters, such as quantization noise shaping [21].…”