Finite Elements with Switch Detection for Direct Optimal Control of Nonsmooth Systems

TL;DR

Convergence results for the FESD method are derived, local uniqueness of the solution and convergence of numerical sensitivities are proven, and in an optimal control problem benchmark with FESd the authors achieve up to five orders of magnitude more accurate solutions than a standard approach for the same computational time.

Abstract

This paper introduces Finite Elements with Switch Detection (FESD), a numerical discretization method for nonsmooth differential equations. We consider the Filippov convexification of these systems and a transformation into dynamic complementarity systems introduced by [Stewart, 1990]. FESD is based on solving nonlinear complementarity problems and can automatically detect nonsmooth events in time. If standard time-stepping Runge-Kutta (RK) methods are naively applied to a nonsmooth ODE, the accuracy is at best of order one. In FESD, we let the integrator step size be a degree of freedom. Additional complementarity conditions, which we call cross complementarities, enable exact switch detection, hence FESD can recover the high order accuracy that the RK methods enjoy for smooth ODE. Additional conditions called step equilibration allow the step size to change only when switches occur and thus avoid spurious degrees of freedom. Convergence results for the FESD method are derived, local uniqueness of the solution and convergence of numerical sensitivities are proven. The efficacy of FESD is demonstrated in several simulation and optimal control examples. In an optimal control problem benchmark with FESD, we achieve up to five orders of magnitude more accurate solutions than a standard time-stepping approach for the same computational time.

Figures (12)

• Figure 1: Illustration of active sets at different points. It can be seen that ${\mathcal{I}}(x(t_1)) = {\mathcal{I}}_0 = \{1\}$. At $x(t_{\mathrm{s},1})$ the trajectory crosses the surface of discontinuity between $R_1$ and $R_2$, hence ${\mathcal{I}}(x(t_{\mathrm{s},1})) = {\mathcal{I}}_1^0 = \{1,2\}$ and later ${\mathcal{I}}_1 = \{2\}$. The segment between $x(t_{\mathrm{s},2})$ and $x(t_{\mathrm{s},3})$ is a sliding mode and we have ${\mathcal{I}}_2^0 = \{2,3\}$ and ${\mathcal{I}}_2 = \{2,3\}$. Finally we have at $x(t_{\mathrm{s},3})$ that ${\mathcal{I}}_3^0 = \{1,2,3,4\}$.
• Figure 2: Illustration of the arguments of Lemma \ref{['lem:lambda_cont']} and Remark \ref{['rem:lambda_at_switch']} on the Example \ref{['ex:switching_cases']} (a) and its corresponding DCS \ref{['eq:dcs_example']} for $t\in[0,2]$ and $x(0) = -1$ with ${t_{\mathrm{s}}} = -\frac{1}{3}$. The functions $\mu = \min(-x,x)$, $\lambda_1 = x-\mu$ and $\lambda_2 =- x-\mu$ are continuous. At the switching point ${t_{\mathrm{s}}}$ we have $\dot{\lambda}_1({t_{\mathrm{s}}}^-)=0$, $\dot{\lambda}_1({t_{\mathrm{s}}}^-)>0$ and $\dot{\lambda}_2({t_{\mathrm{s}}}^-)<0$, $\dot{\lambda}_1({t_{\mathrm{s}}}^-)=0$.
• Figure 3: Illustration of the analytic solution and a polynomial solution approximation to a PSS via an IRK Radau-IIA method of order 7. The left plot shows an approximation with a fixed step size where an active-set change happens on a stage point. The right plot shows an approximation obtained with FESD (based on the same IRK method) where the switch happens on the boundary. The circles represent the stage values, the vertical dotted lines the finite elements boundaries, and the vertical dashed line the switching time ${t_{\mathrm{s}}}$.
• Figure 4: An illustration of the standard complementarity conditions $\Psi(\mathbf{\Theta},\mathbf{\Lambda}) =0$ (left plot) and the standard complementarity conditions augmented by $0=G_{\mathrm{cross}}(\mathbf{\Theta},\mathbf{\Lambda})$ (right plot). The dots represent the stage values. The vertical dotted line marks the finite element boundaries, and the vertical dashed line marks the switching time ${t_{\mathrm{s}}}$. In the standard case (left plot), an active-set change can happen at any complementarity pair. With the cross complementarities \ref{['eq:cross_comp']} (right plot) an active-set change can only happen on the boundaries of a finite element.
• Figure 5: Illustration of the discontinuity of the solution map of \ref{['eq:fesd_equation']} for the PSS $\dot{x} \in 2-\mathrm{sign}(x) +x^2$ for $T = 0.2$ and ${N_\mathrm{FE}} =2$.

Theorems & Definitions (15)

• Proposition 4
• Lemma 5
• Remark 6
• Example 3
• Theorem 7: Theorem 3.2 Stewart1990b
• Example 4
• Proposition 9
• Lemma 10
• Lemma 14: Corollary 6.1 in Matsaglia1974
• Theorem 15