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
$[0, 1]$
, 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
${\rm OP}_{\text{min}}$
. We then analyze
${\rm OP}_{\text{min}}$
under general stabilizability and detectability assumptions on the system and the stage cost. New near-optimality and performance guarantees for
${\rm OP}_{\text{min}}$
are derived, which have major advantages compared to those originally given for OP. We also prove that a system whose inputs are generated by
${\rm OP}_{\text{min}}$
in a receding-horizon fashion exhibits stability properties. As a result,
${\rm OP}_{\text{min}}$
provides a new tool for the near-optimal, stable control of nonlinear switched discrete-time systems for generic cost functions.