An efficient algorithm for solving piecewise-smooth dynamical systems

Abstract

This article considers the numerical treatment of piecewise-smooth dynamical systems. Classical solutions as well as sliding modes up to codimension-2 are treated. An algorithm is presented that, in the case of non-uniqueness, selects a solution that is the formal limit solution of a regularized problem. The numerical solution of a regularized differential equation, which creates stiffness and often also high oscillations, is avoided.

Keywords: Codimension-2 manifold; Filippov solution; Hidden dynamics; Piecewise-smooth systems; Regularization; Scaling invariance.

Fig. 1

Transition function

Fig. 2

Flowchart of possible switchings from a classical solution

Fig. 3

Flowchart of switchings from a codimension-1 sliding mode exiting Σ₁

Fig. 4

Entering the codimension-2 manifold

Fig. 5

Flowchart of switchings from a codimension-1 entering Σ₁ ∩Σ₂. In the case of multiple solutions of (4.6), λ₂ is the value that is closest to − 1. Here, and in the following, the term “Filippov solution” means a solution according to Definition 2.2

Fig. 6

Flowchart of switchings from a codimension-1 entering Σ₁ ∩Σ₂ (cont.). Condition (4.4) is assumed in addition to the assumption (4.3) of the flowchart of Fig. 5

Fig. 7

Flowchart for exiting a codimension-2 sliding of type (a); c.f., case (A) of [18]

Fig. 8

Flowchart for exiting a codimension-2 sliding of type (b); for brevity we use the notation $f_i^{++}$ and $f_i^{--}$ for $f_i^{1,1}$ and $f_i^{-1,-1}$, respectively. All functions are evaluated at the collapsing stationary point

Fig. 9

Flowchart of possible exits from a codimension-2 sliding under the assumption that ${\partial }_1{g}_1(y^{\ast },{\lambda }_1^{\ast },{\lambda }_2^{\ast })=0$ and ${\partial }_2{g}_2(y^{\ast },{\lambda }_1^{\ast },{\lambda }_2^{\ast })>0$. The vertical asymptote of g₁ = 0 is outside the unit square

Fig. 10

Flowchart of possible exits from a codimension-2 sliding under the assumption that ${\partial }_1{g}_1(y^{\ast },{\lambda }_1^{\ast },{\lambda }_2^{\ast })=0$ and ${\partial }_2{g}_2(y^{\ast },{\lambda }_1^{\ast },{\lambda }_2^{\ast })>0$. The vertical asymptote of g₁ = 0 is inside the unit square to the right of ${\lambda }_1^{\ast }$

Fig. 11

Vector field of the hidden dynamics in the situation where both stationary points coalesce. The first seven pictures correspond to the colored boxes of Fig. 8 in the same order. Red arrows represent the solution after the switching

Fig. 12

Vector field of the hidden dynamics for the situations discussed in Fig. 9 (upper pictures) and in Fig. 10 (lower pictures). Twenty solutions corresponding to initial values that are random perturbations of the equilibrium $(u_1^{\ast },u_2^{\ast })$ are plotted in red

Fig. 13

Basin of attraction for the two classical solutions