Optimal Control of Reduced Left-Invariant Hybrid Control Systems

Abstract

Optimal control is ubiquitous in many fields of engineering. A common technique to find candidate solutions is via Pontryagin's maximum principle. An unfortunate aspect of this method is that the dimension of system doubles. When the system evolves on a Lie group and the system is invariant under left (or right) translations, Lie-Poisson reduction can be applied to eliminate half of the dimensions (and returning the dimension of the problem to the back to the original number). Hybrid control systems are an extension of (continuous) control systems by allowing for sudden changes to the state. Examples of such systems include the bouncing ball - the velocity instantaneously jumps during a bounce, the thermostat - controls switch to on or off, and a sailboat undergoing tacking. The goal of this work is to extend the idea of Lie-Poisson reduction to the optimal control of these systems. If $n$ is the dimension of the original system, $2n$ is the dimension of the system produced by the maximum principle. In the case of classical Lie-Poisson reduction, the dimension drops back down to $n$. This, unfortunately, is impossible in hybrid systems as there must be an auxiliary variable encoding whether or not an event occurs. As such, the analogous hybrid Lie-Poisson reduction results in a $n+1$ dimensional system. The purpose of this work is to develop and present this technique.

Figures (3)

• Figure 1: A schematic of Lie-Poisson reduction as applied to hybrid systems. An event occurs at $g_1\in\Sigma$ and the arc gets mapped to $\Delta(g_1)$. The map $J_R:T^*G\to\mathfrak{g}^*$ allows for the dimension reduction, and the corresponding reduced momentum jumps by $\delta \mu$
• Figure 2: A trajectory with initial conditions $(x_0,y_0,\theta_0) = (0,0,0)$. The auxiliary parameters are $C=D=1$ and $\mu_\theta(0) = -1$. The black arc is the initial trajectory which maps to the red trajectories after the first reset. The two red arcs indicate following both lifted reset maps \ref{['eq:decon_recon_reset']}. Notice that the next reset (red to blue) is independent on which reset map was chosen.
• Figure 3: A plot of the running cost against time corresponding to the trajectories in Fig. \ref{['fig:state_trajectory']}. This cost is multi-valued as there are two choices for \ref{['eq:decon_recon_reset']}. Notice that at resets, both options result in the same total cost.

Theorems & Definitions (14)

• Definition 1: Tangent Preserving oprea2023study
• Definition 2: Left-Invariant Hybrid Control System
• Proposition 1
• proof
• Definition 3: Normal LIHCS
• Remark 1
• Theorem 1: Lie-Poisson Reduction mech_symmetry
• Remark 2
• Remark 3
• Theorem 2: Impact Lie-Poisson Reduction oprea2023study