Nonlinear adaptive stabilization of a class of planar slow-fast systems at a non-hyperbolic point

Abstract

Non-hyperbolic points of slow-fast systems (also known as singularly perturbed ordinary differential equations) are responsible for many interesting behavior such as relaxation oscillations, canards, mixed-mode oscillations, etc. Recently, the authors have proposed a control strategy to stabilize nonhyperbolic points of planar slow-fast systems. Such strategy is based on geometric desingularization, which is a well suited technique to analyze the dynamics of slow-fast systems near non-hyperbolic points. This technique transforms the singular perturbation problem to an equivalent regular perturbation problem. This papers treats the nonlinear adaptive stabilization problem of slow-fast systems. The novelty is that the point to be stabilized is non-hyperbolic. The controller is designed by combining geometric desingularization and Lyapunov based techniques. Through the action of the controller, we basically inject a normally hyperbolic behavior to the fast variable. Our results are exemplified on the van der Pol oscillator.

Stabilization of a class of slow–fast control systems at non-hyperbolic points

“…In the context of slow-fast systems it was first introduced in [4], and has been further developed afterwards, see e.g. [10,11,[19][20][21] and [12][13][14][15] for some applications in control systems.…”

“…However, with the increased interest in shaping the behavior of complex multi-timescale systems, we require control techniques that can tackle problems where the timescale separation is not global. Preliminary steps in this regard have been developed for regulation purposes in [12,14] in the planar case, and [15] for the case of singularities of quadratic degeneracy.…”

“…The idea is to show that Geometric Desingularization [4,21] can be used in combination with control strategies to stabilize degenerate points of slow-fast systems, c.f. [12,14] for the planar case. In few words, the Geometric Desingularization technique is a suitable change of coordinates, well-defined around singularities, that allows a simpler analysis of the involved behavior of slow-fast systems without global timescale separation.…”

Improving the Region of Attraction of a Non-Hyperbolic Point in Slow-Fast Systems With One Fast Direction

“…Controllers that locally stabilize the origin of systems like (1) have been recently proposed in e.g. [14], [15]. These controllers deal with a class of singular perturbation problems for which the common regularity condition ∂ g ∂ z (x, z, 0) = 0 does not hold.…”

Model Order Reduction and Composite Control for a Class of Slow-Fast Systems Around a Non-Hyperbolic Point

“…This is not the only way in which a slow-fast system may lose normal hyperbolicity. Another one is, e.g., due to degenerate singularities induced by nonlinear terms in the layer equation, compare with [14], [15], [16].…”