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Data-driven stabilization of nonlinear systems via descriptor embedding

Mohammad Alsalti, Claudio De Persis, Victor G. Lopez, Matthias A. Müller

TL;DR

The paper introduces descriptor embedding to stabilize nonlinear systems from data by lifting dynamics into a linear-parameter-varying descriptor form. It develops data-dependent LMIs that yield a state-dependent, polytope-constrained gain $u=K(x)Z(x)$, enabling either local or global stabilization depending on the domain and the existence of a global $L$. Extensions cover inexact basis expansions and robust stabilization under noise, and the framework is broadened to general input-affine nonlinear systems with corresponding data-driven LMIs. Simulations show competitive performance and larger estimated regions of attraction compared with non-cancellation approaches, especially when leveraging the system's inherent nonlinearities. Overall, descriptor embedding provides a flexible, data-driven pathway to stability guarantees beyond traditional nonlinear cancellation strategies.

Abstract

We introduce the notion of descriptor embedding for nonlinear systems and use it for the data-driven design of stabilizing controllers. Specifically, we provide sufficient data-dependent LMI conditions which, if feasible, return a stabilizing nonlinear controller of the form $u=K(x)Z(x)$ where $K(x)$ belongs to a polytope and $Z$ is a user-defined function. The proposed method is then extended to account for the presence of uncertainties and noisy data. Furthermore, a method to estimate the resulting region of attraction is given using only data. Simulation examples are used to illustrate the results and compare them to existing methods from the literature.

Data-driven stabilization of nonlinear systems via descriptor embedding

TL;DR

The paper introduces descriptor embedding to stabilize nonlinear systems from data by lifting dynamics into a linear-parameter-varying descriptor form. It develops data-dependent LMIs that yield a state-dependent, polytope-constrained gain , enabling either local or global stabilization depending on the domain and the existence of a global . Extensions cover inexact basis expansions and robust stabilization under noise, and the framework is broadened to general input-affine nonlinear systems with corresponding data-driven LMIs. Simulations show competitive performance and larger estimated regions of attraction compared with non-cancellation approaches, especially when leveraging the system's inherent nonlinearities. Overall, descriptor embedding provides a flexible, data-driven pathway to stability guarantees beyond traditional nonlinear cancellation strategies.

Abstract

We introduce the notion of descriptor embedding for nonlinear systems and use it for the data-driven design of stabilizing controllers. Specifically, we provide sufficient data-dependent LMI conditions which, if feasible, return a stabilizing nonlinear controller of the form where belongs to a polytope and is a user-defined function. The proposed method is then extended to account for the presence of uncertainties and noisy data. Furthermore, a method to estimate the resulting region of attraction is given using only data. Simulation examples are used to illustrate the results and compare them to existing methods from the literature.

Paper Structure

This paper contains 12 sections, 10 theorems, 104 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Descriptor embedding for nonlinear systems
  4. Data-based stabilization of nonlinear systems
  5. Main result
  6. Global vs local stabilization
  7. Extensions
  8. Inexact basis function expansion and robust stabilization
  9. More general class of systems
  10. Discussion and conclusions
  11. On PDLF structure for poly-quadratically admissible LPVd systems
  12. Example of a function L satisfying Assumption \ref{['asmp:L']}

Key Result

Theorem 1

System eqn:polytopic_singularLPV (or equivalently eqn:polytopic_singularLPV_SVDform) is poly-quadratically admissible if and only if $A_{22}(\rho_t)$ is invertible for all $\rho_t\in\mathcal{D}\subseteq\mathbb{R}^{p}$, and is poly-quadratically stable.$\square$

Figures (4)

  • Figure 1: ROA estimates using all three methods (NLmin refers to the method from DePersis2023). For each method, we show the largest sub level set $\mathcal{R}_{c_{\max}}$ contained in its corresponding set $\mathcal{V}$ (the latter shown only for the methods proposed in the paper and not for the NLmin approach). The areas of each region are as follows: NLmin: 14.6960 (in black), Theorem \ref{['thm:Kx']}: 15.4224 (in green) representing approximately 5% increase over NLmin, and Corollary \ref{['cor:Kx']}: 15.4224 (in blue) also representing approximately 5% increase over NLmin.
  • Figure 2: Numerical estimates of the true ROA for all three methods. The areas of each region are as follows: NLmin: 31.8450, Theorem \ref{['thm:Kx']}: 40.2670 representing approximately 26.4% increase over NLmin, and Corollary \ref{['cor:Kx']}: 40.1383, approximately a 26% increase over NLmin.
  • Figure 3: Numerical estimates of the true ROA of the closed-loop systems using (i) a polytopic controller obtained using Theorem \ref{['thm:Kx_robust']} (in green) and (ii) a controller obtained from DePersis2023 (in black). The areas of each region are as follows: 213.6048 for the nonlinearity minimization approach and 272.7167 for the proposed approach corresponding to an increase of approximately 27.7% in the size of the ROA.
  • Figure 4: (Top): Data-dependent ROA estimate (in green) for the closed-loop system whose control law is designed using Theorem \ref{['thm:Kx_general']}. In contrast, the method from guo2023data returns an empty estimate of the ROA. (Bottom): Numerical estimates of the true ROA of the closed-loop systems using (i) a controller obtained using Theorem \ref{['thm:Kx_general']} (in green) and (ii) a controller obtained from guo2023data (in black). The areas of each region are as follows: 11.8030 for the nonlinearity minimization approach and 17.6380 for the proposed approach corresponding to an increase of approximately 49.4% in the size of the ROA.

Theorems & Definitions (26)

  • Definition 1
  • Theorem 1: bara2011dilatedbarbosa2018
  • Theorem 2: bara2011dilated
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Remark 3
  • Theorem 3
  • proof
  • ...and 16 more