On merging constraint and optimal control-Lyapunov functions

TL;DR

This paper presents a convex optimization-based method to merge control Lyapunov functions shaped by constraints and optimality criteria, ensuring control-sharing in active constraint regions for constrained stabilization.

Contribution

It introduces a novel partial control-sharing approach that guarantees merging of CLFs with convex optimization, enabling constrained LQ stabilization with bounded complexity.

Findings

The partial control-sharing condition is formulated as a convex optimization problem.

The method successfully merges constraint-shaped and optimal control Lyapunov functions.

The approach solves the constrained linear-quadratic stabilization problem with local optimality.

Abstract

Merging two Control Lyapunov Functions (CLFs) means creating a single "new-born" CLF by starting from two parents functions. Specifically, given a "father" function, shaped by the state constraints, and a "mother" function, designed with some optimality criterion, the merging CLF should be similar to the father close to the constraints and similar to the mother close to the origin. To successfully merge two CLFs, the control-sharing condition is crucial: the two functions must have a common control law that makes both Lyapunov derivatives simultaneously negative. Unfortunately, it is difficult to guarantee this property a-priori, i.e., while computing the two parents functions. In this paper, we propose a technique to create a constraint-shaped "father" function that has the control-sharing property with the "mother" function. To this end, we introduce a partial control-sharing, namely, the control-sharing only in the regions where the constraints are active. We show that imposing partial control-sharing is a convex optimization problem. Finally, we show how to apply the partial control-sharing for merging constraint-shaped functions and the Riccati-optimal functions, thus generating a CLF with bounded complexity that solves the constrained linear-quadratic stabilization problem with local optimality.