Parameter-Dependent Control Lyapunov Functions for Stabilizing Nonlinear Parameter-Varying Systems

TL;DR

This paper develops parameter-dependent control Lyapunov functions for stabilizing nonlinear parameter-varying systems, enabling robust, gain-scheduled control synthesis that fully captures nonlinearities, with practical convex conditions and input constraints.

Contribution

It introduces a novel framework for PD-CLFs that directly handles nonlinearities in NPV systems, using sum of squares programming for joint synthesis of controllers and Lyapunov functions.

Findings

Convex conditions for PD-CLF synthesis via sum of squares programming

Maximized stabilization region for polynomial NPV systems

Validated approach through numerical simulations including rocket landing case

Abstract

This paper introduces the concept of parameter-dependent (PD) control Lyapunov functions (CLFs) for gain-scheduled stabilization of nonlinear parameter-varying (NPV) systems. It shows that given a PD-CLF, a min-norm control law can be constructed by solving a robust quadratic program. For polynomial control-affine NPV systems, it provides convex conditions, based on the sum of squares programming, to jointly synthesize a PD-CLF and a PD controller while maximizing the PD region of stabilization. Input constraints can be straightforwardly incorporated into the synthesis procedure. Unlike traditional linear parameter-varying (LPV) methods that rely on linearization or over-approximation to get an LPV model, the proposed framework fully captures the nonlinearities of the system dynamics. The theoretical results are validated through numerical simulations, including a 2D rocket landing case study under varying mass and inertia.