Robust Local Stabilization of Nonlinear Systems with Controller-Dependent Norm Bounds: A Convex Approach with Input-Output Sampling

TL;DR

This paper introduces a convex, sampling-based method for designing robust controllers for nonlinear systems with unknown dynamics, addressing input-dependent nonlinearities through iterative SDP relaxations.

Contribution

It proposes a novel convex approach using input-output sampling and iterative SDPs to synthesize controllers for nonlinear systems with input-dependent uncertainties.

Findings

Effective control synthesis demonstrated on numerical examples

Convex relaxations enable handling of non-convex control design problems

Iterative algorithm improves robustness and stability guarantees

Abstract

This letter presents a framework for synthesizing a robust full-state feedback controller for systems with unknown nonlinearities. Our approach characterizes input-output behavior of the nonlinearities in terms of local norm bounds using available sampled data corresponding to a known region about an equilibrium point. A challenge in this approach is that if the nonlinearities have explicit dependence on the control inputs, an a priori selection of the control input sampling region is required to determine the local norm bounds. This leads to a "chicken and egg" problem, where the local norm bounds are required for controller synthesis, but the region of control inputs needed to be characterized cannot be known prior to synthesis of the controller. To tackle this issue, we constrain the closed-loop control inputs within the sampling region while synthesizing the controller. As the resulting synthesis problem is non-convex, three semi-definite programs (SDPs) are obtained through convex relaxations of the main problem, and an iterative algorithm is constructed using these SDPs for control synthesis. Two numerical examples are included to demonstrate the effectiveness of the proposed algorithm.