Tracking-in-range Formulations for Numerical Optimal Control

TL;DR

This work presents novel optimal control formulations to solve tracking-in-range problems, for both problems requiring the tracker to be always in range, and problems allowing the tracker to go out of range to yield overall better outcomes.

Abstract

In contrast to set-point tracking which aims to reduce the tracking error between the tracker and the reference, tracking-in-range problems only focus on whether the tracker is within a given range around the reference, making it more suitable for the mission specifications of many practical applications. In this work, we present novel optimal control formulations to solve tracking-in-range problems, for both problems requiring the tracker to be always in range, and problems allowing the tracker to go out of range to yield overall better outcomes. As the problem naturally involves discontinuous functions, we present alternative formulations and regularisation strategies to improve the performance of numerical solvers. The extension to in-range tracking with multiple trackers and in-range tracking in high dimensional space are also discussed and illustrated with numerical examples, demonstrating substantial increases in mission duration in comparison to traditional set-point tracking.

Figures (7)

• Figure 1: Solutions to a 1-D tracking problem. All solutions have equal energy consumption. 'a.i.r' stands for always-in-range and 'n.a.i.r' stands for not always-in-range.
• Figure 2: Lagrange cost comparison between different formulations with $x_r = 1.5$ m and $\delta = 1.5$ m
• Figure 3: Effect of constants $k_1$ and $k_2$ on the stage cost approximation
• Figure 4: Changes of the Lagrange cost as $k_1$, $k_2$ and $\rho$ increase iteratively
• Figure 5: Solutions to multi-tracker 1D in-range tracking problems