Stable Near-Optimal Control of Nonlinear Switched Discrete-Time Systems: An Optimistic Planning-Based Approach

Originating in the artificial intelligence literature, optimistic planning (OP) is an algorithm that generates near-optimal control inputs for generic nonlinear discrete-time systems whose input set is finite. This technique is, therefore, relevant for the near-optimal control of nonlinear switched systems for which the switching signal is the control, and no continuous input is present. However, OP exhibits several limitations, which prevent its desired application in a standard control engineering context, as it requires, for instance, that the stage cost takes values in <inline-formula><tex-math notation="LaTeX">$[0, 1]$</tex-math></inline-formula>, an unnatural prerequisite, and that the cost function is discounted. In this article, we modify OP to overcome these limitations, and we call the new algorithm <inline-formula><tex-math notation="LaTeX">${\rm OP}_{\text{min}}$</tex-math></inline-formula>. We then analyze <inline-formula><tex-math notation="LaTeX">${\rm OP}_{\text{min}}$</tex-math></inline-formula> under general stabilizability and detectability assumptions on the system and the stage cost. New near-optimality and performance guarantees for <inline-formula><tex-math notation="LaTeX">${\rm OP}_{\text{min}}$</tex-math></inline-formula> are derived, which have major advantages compared to those originally given for OP. We also prove that a system whose inputs are generated by <inline-formula><tex-math notation="LaTeX">${\rm OP}_{\text{min}}$</tex-math></inline-formula> in a receding-horizon fashion exhibits stability properties. As a result, <inline-formula><tex-math notation="LaTeX">${\rm OP}_{\text{min}}$</tex-math></inline-formula> provides a new tool for the near-optimal, stable control of nonlinear switched discrete-time systems for generic cost functions.