Teixeira singularities in 3D switched feedback control systems

Abstract

This paper is concerned with the analysis of a singularity that can occur in threedimensional discontinuous feedback control systems. The singularity is the two-fold -a tangency of orbits to both sides of a switching manifold. Particular attention is placed on the Teixeira singularity, which previous literature suggests as a mechanism for dynamical transitions in this class of systems. We show that such a singularity cannot occur in classical single-input single-output systems in the Lur'e form. It is, however, a potentially dangerous phenomenon in multiple-input multiple-output switched control systems. The theoretical derivation is illustrated by means of a representative example.

Sliding bifurcations and non-determinism in systems with switching

“…The determinant (24), which in terms of Q is simply SCB det[SC∂ x P, ∂ x Q, SC], must not vanish, hence SCB = 0, and therefore Q = 0. This is a general proof of a result found for a specific example in Colombo et al (2010), that continuity between f + and f − is achieved at generic two-folds in SIMO systems. Note, however, that the derivatives J + and J − are still nonequal.…”

“…Following Colombo et al (2010), we begin with the classical Lur'e problem with p outputs and m inputs, in which a state x ∈ R 3 evolves according tȯ…”

“…This means a pair of folds can exist, and if CBE ± = 0 they intersect, but do so nontransversally since (26) vanishes (Colombo et al (2010)). …”

“…In (ii) the orbit is attracted to a stable periodic orbit (bold). Simulation from Colombo et al (2010). and µ = −0.2, corresponding to w + w − = 1.022 and w + w − = 0.64 respectively, both cases involving faux canards.…”

“…The meeting between attracting and repelling regions of sliding has also been shown to produce repetitive yet unpredictable dynamics, constituting a nondeterministic form of chaos Colombo and Jeffrey (2011). The conditions for all these behaviours have been derived across a number of papers by Colombo et al (2010); Colombo and Jeffrey (2011); Jeffrey and Hogan (2011), but are presented here in a general form applicable to any piecewise-smooth (Filippov) system, and then applied to a typical model of switched feedback control.…”