Switched Optimal Control with Dwell Time Constraints

TL;DR

This work tackles switched optimal control problems with dwell-time constraints by embedding the discrete switching into a continuous variable $v\in[0,1]$, solving an embedded problem (EOCP) and a modified version (MEOCP) that uses a concave auxiliary cost $L_v(v)$ to drive solutions to bang-bang form. Since embedding alone cannot guarantee dwell-time satisfaction, the authors introduce a dwell-time filtering layer that post-processes the MEOCP solution using mode-insertion gradients $D_{\sigma,s,\alpha}$ to remove violations with minimal cost impact. The filter deterministically removes offending switches and selects replacement modes over $[\tau_i,\tau_i+T)$, with optional re-solving to recover near-optimality on the remaining trajectory. A mass-spring-damper example demonstrates feasibility for small dwell-time bounds and reveals the trade-off between higher switching frequence and performance, validating the method’s practicality while acknowledging limitations in optimality after filtering and its current restriction to two subsystems.

Abstract

This paper presents an embedding-based approach for solving switched optimal control problems (SOCPs) with dwell time constraints. At first, an embedded optimal control problem (EOCP) is defined by replacing the discrete switching signal with a continuous embedded variable that can take intermediate values between the discrete modes. While embedding enables solutions of SOCPs via conventional techniques, optimal solutions of EOCPs often involve nonexistent modes and thus may not be feasible for the SOCP. In the modified EOCP (MEOCP), a concave function is added to the cost function to enforce a bang-bang solution in the embedded variable, which results in feasible solutions for the SOCP. However, the MEOCP cannot guarantee the satisfaction of dwell-time constraints. In this paper, a MEOCP is combined with a filter layer to remove switching times that violate the dwell time constraint. Insertion gradients are used to minimize the effect of the filter on the optimal cost.

Figures (6)

• Figure 1: The typical switching mode signal in order from top to bottom: original switching mode signal, replace mode $1$ during interval $[\tau_i,\tau_i+T)$, replace mode $0$ during interval $[\tau_i,\tau_i+T)$
• Figure 2: Mass-Spring-Damper System
• Figure 3: The solution of MEOCP (without a dwell-time constraint). The plot on top shows the switching mode signal as a function of time and the plot at the bottom shows the states of the systems as a function of time.
• Figure 4: The results after adding the filter (for a dwell-time constraint $T=0.1 s$) to the solution of MEOCP. The plot on top shows the switching mode signal as a function of time and the plot at the bottom shows the states of the systems as a function of time.
• Figure 5: The results after adding the filter (for a dwell-time constraint $T=0.2 s$) to the solution of MEOCP. The plot on top shows the switching mode signal as a function of time and the plot at the bottom shows the states of the systems as a function of time.