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Stabilization of Nonlinear Systems with State-Dependent Representation: From Model-Based to Direct Data-Driven Control

Lidong Li, Rui Huang, Lin Zhao

TL;DR

The paper addresses stabilizing nonlinear systems represented in state-dependent form using both model-based and direct data-driven control. The main method is offline LMIs-based design for the pair $(A(x),B(x))$ and polytope bounds, extended to data-driven settings via Petersen's lemma to enforce stability for all data-consistent models. Key contributions include local exponential stabilization with an estimated ROA, robustness to disturbances, and ROA under input saturation, all derived from finite data; and validation through numerical and physical experiments including a quadrotor attitude system. The work delivers end-to-end stability and safety guarantees directly from data without explicit system identification, advancing nonlinear data-driven control beyond first-order approximations or online SDRE schemes.

Abstract

This paper presents a novel framework for stabilizing nonlinear systems represented in state-dependent form. We first reformulate the nonlinear dynamics as a state-dependent parameter-varying model and synthesize a stabilizing controller offline via tractable linear matrix inequalities (LMIs). The resulting controller guarantees local exponential stability, maintains robustness against disturbances, and provides an estimate of the region of attraction under input saturation. We then extend the formulation to the direct data-driven setting, where a known library of basis functions represents the dynamics with unknown coefficients consistent with noisy experimental data. By leveraging Petersen's lemma, we derive data-dependent LMIs that ensure stability and robustness for all systems compatible with the data. Numerical and physical experimental results validate that our approach achieves rigorous end-to-end guarantees on stability, robustness, and safety directly from finite data without explicit model identification.

Stabilization of Nonlinear Systems with State-Dependent Representation: From Model-Based to Direct Data-Driven Control

TL;DR

The paper addresses stabilizing nonlinear systems represented in state-dependent form using both model-based and direct data-driven control. The main method is offline LMIs-based design for the pair and polytope bounds, extended to data-driven settings via Petersen's lemma to enforce stability for all data-consistent models. Key contributions include local exponential stabilization with an estimated ROA, robustness to disturbances, and ROA under input saturation, all derived from finite data; and validation through numerical and physical experiments including a quadrotor attitude system. The work delivers end-to-end stability and safety guarantees directly from data without explicit system identification, advancing nonlinear data-driven control beyond first-order approximations or online SDRE schemes.

Abstract

This paper presents a novel framework for stabilizing nonlinear systems represented in state-dependent form. We first reformulate the nonlinear dynamics as a state-dependent parameter-varying model and synthesize a stabilizing controller offline via tractable linear matrix inequalities (LMIs). The resulting controller guarantees local exponential stability, maintains robustness against disturbances, and provides an estimate of the region of attraction under input saturation. We then extend the formulation to the direct data-driven setting, where a known library of basis functions represents the dynamics with unknown coefficients consistent with noisy experimental data. By leveraging Petersen's lemma, we derive data-dependent LMIs that ensure stability and robustness for all systems compatible with the data. Numerical and physical experimental results validate that our approach achieves rigorous end-to-end guarantees on stability, robustness, and safety directly from finite data without explicit model identification.
Paper Structure (17 sections, 11 theorems, 131 equations, 12 figures)

This paper contains 17 sections, 11 theorems, 131 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Model-Based Controller
  3. Problem Formulation
  4. Model-Based Controller Design
  5. Robustness Against Disturbances
  6. System Subject to Input Saturation
  7. Data-Driven Controller
  8. Problem Formulation
  9. Data-Driven Controller Design
  10. Robustness Against Disturbances
  11. System Subject to Input Saturation
  12. Discussion on Region of Attraction
  13. Comparisons with Existing Methods
  14. Experiment
  15. Numerical Experiment
  16. ...and 2 more sections

Key Result

Lemma 1

Given a positive real number $r$, if there exist $P \succ 0$, $\epsilon > 0$ and $K$ such that then there exists a region of attraction and the closed-loop system $x^{+} = ( A(x) + B(x)K ) x$ starting from any $x(0) \in \mathcal{B}_{r_0}$ will exponentially converge to the origin.

Figures (12)

  • Figure 1: Phase portrait of the closed-loop system with model-based controller.
  • Figure 2: State response of the closed-loop system with model-based controller under disturbance.
  • Figure 3: Estimate of ROA, $\mathcal{B}_{r_0} \cap \mathcal{E}_{P,1}$, for different saturation levels. The red solid ball is $\mathcal{B}_{r_0}$; the black dashed ellipsoid is $\mathcal{E}_{P,1}$; the background is the phase portrait regarding the closed-loop system $x^{+} = A(x)x + B(x) \mathop{\mathrm{sat}}\nolimits(Kx)$.
  • Figure 4: Phase portrait of the closed-loop system with data-driven controller.
  • Figure 5: State response of the closed-loop system with data-driven controller under disturbance.
  • ...and 7 more figures

Theorems & Definitions (22)

  • Definition 1
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 2
  • Lemma 3
  • proof
  • ...and 12 more