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

Sze Kwan Cheah, Diganta Bhattacharjee, Maziar S. Hemati, Ryan J. Caverly

TL;DR

The paper addresses stabilizing nonlinear systems with unknown, controller-dependent nonlinearities by learning local input-output bounds from sampled data and formulating a robust, data-driven controller synthesis in LFT form. It converts the nonconvex synthesis into three convex SDP relaxations solved iteratively, while constraining the closed-loop inputs to stay within the data-supported region. The method yields local asymptotic stabilization within an ellipsoidal region and demonstrates reduced conservatism over purely data-driven methods via two 2D examples, including an inverted pendulum. The approach bridges data-driven I/O characterization with model-based LFT-based synthesis to enable practical stabilization under limited information. Future work will extend to parametric uncertainties, exogenous signals, and output-feedback design.

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.

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

TL;DR

The paper addresses stabilizing nonlinear systems with unknown, controller-dependent nonlinearities by learning local input-output bounds from sampled data and formulating a robust, data-driven controller synthesis in LFT form. It converts the nonconvex synthesis into three convex SDP relaxations solved iteratively, while constraining the closed-loop inputs to stay within the data-supported region. The method yields local asymptotic stabilization within an ellipsoidal region and demonstrates reduced conservatism over purely data-driven methods via two 2D examples, including an inverted pendulum. The approach bridges data-driven I/O characterization with model-based LFT-based synthesis to enable practical stabilization under limited information. Future work will extend to parametric uncertainties, exogenous signals, and output-feedback design.

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.
Paper Structure (5 sections, 4 theorems, 32 equations, 2 figures, 1 algorithm)

This paper contains 5 sections, 4 theorems, 32 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Control Synthesis
  4. Numerical Examples
  5. Conclusions and Future Work

Key Result

Lemma II.1

(Young's Relation CaverlyLMI): Consider $\mbf{X} \in \mathbb{R}^{m \times n}$ and $\mbf{Y} \in \mathbb{R}^{m \times n}$. For any $\mbf{S} >0$, it holds that

Figures (2)

  • Figure 1: Simulation results of the closed-loop system \ref{['eq:closed_loop_system_example_1']} with $r=0.5$ and $\mbf{K}=-0.7151-0.6762$.
  • Figure 2: Simulation results of the closed-loop system \ref{['eq:closed_loop_system_example_2']} with $\mbf{K}=-13.4283-13.5242$.

Theorems & Definitions (9)

  • Lemma II.1
  • Theorem III.1
  • proof
  • Remark III.2
  • Remark III.3
  • Lemma III.1
  • proof
  • Lemma III.2
  • proof