Second-order Filippov systems: sliding dynamics without sliding regions

Abstract

This paper develops fundamental mathematical theory for second-order Filippov systems. These are discontinuous ordinary differential equations with solutions defined in the sense of Filippov, and whose first Lie derivatives vary continuously across discontinuity surfaces. Unlike generic Filippov systems, discontinuity surfaces consist only of crossing regions and their boundaries where both adjacent vector fields are tangent to the discontinuity surface. Crossing orbits spiral around invisible-invisible tangency surfaces, and we derive a formula for the attractive or repulsive strength of these surfaces. We prove crossing orbits cannot converge to tangency surfaces in finite time (no Zeno), and that the limiting dynamics consists of Filippov solutions on the tangency surfaces (second-order sliding motion). We derive a vector field that governs this motion, and characterise the stability of equilibria on tangency surfaces. The methodology is applied to a model of a mechanical oscillator with compliant impacts, and a model of ant colony migration. We also relate second-order Filippov systems to second-order sliding mode control, and show that for two-dimensional systems the results reduce to known theory.

Figures (9)

• Figure 1: Phase portraits of Filippov systems of the form \ref{['eq:f']}. The discontinuity surface $\Sigma$ is coloured green for crossing regions and grey for attracting sliding regions. Throughout this paper sliding orbits on sliding regions are coloured red, as in panel (a), while sliding orbits on tangency surfaces are coloured orange, as in panel (b).
• Figure 2: A phase portrait of \ref{['eq:f']} with \ref{['eq:B']}. The blue trajectory (virtual) is the orbit of $\dot{x} = f^L(x)$ passing through the invisible fold of $f^L$.
• Figure 3: A phase portrait of a three-dimensional second-order Filippov system illustrating the three types of regions defined in Definition \ref{['df:region']}. This figure was drawn using \ref{['eq:C']}; the coordinate axes are omitted for clarity. The blue trajectories are the virtual parts of the cubically tangent orbits of $\dot{x} = f^L(x)$ and $\dot{x} = f^R(x)$.
• Figure 4: A sketch illustrating the first return maps $P_L$ and $P_R$.
• Figure 5: A sketch illustrating the orbits $\phi(t)$ and $\xi(t)$ of Theorem \ref{['th:consistency']}.

Theorems & Definitions (30)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Example 2.7
• Definition 2.8
• Theorem 2.1
• proof