Second-Order Sampling-Based Stability Guarantee for Data-Driven Control Systems

TL;DR

This paper tackles robust stability guarantees for nonlinear data-driven control systems with uncertainty by introducing a sampling-based framework that discretizes infinite-state stability conditions using margins. The core innovation is second-order margins, which shrink quadratically with the discretization interval $\tau$, enabling more precise stability evaluations across broad function classes ${\mathcal{F}(N)}$ and ${\mathcal{F}^{+}(N)}$ including GP means, GP uncertainties, SE kernels, polynomials, and DNNs. The authors develop recursive algorithms ${\mathcal{A}}$ and ${\mathcal{A}^{+}}$ to compute tight $\underline{m}$ and $\overline{m}$ margins with $O'(\tau^2)$ accuracy, and integrate these margins into an optimization-based design of Lyapunov functions and controllers to guarantee a stability region $\mathbb{X}_{\mathrm{S}}$ while balancing control performance. A numerical demonstration on a partially unknown pendulum using GP-based nominal dynamics confirms that the method yields enlarged stability regions and verifies the improved margins as $\tau$ decreases, illustrating practical applicability to data-driven control with GP, kernel-based, and NN models.

Abstract

This study presents a sampling-based method to guarantee robust stability of general control systems with uncertainty. The method allows the system dynamics and controllers to be represented by various data-driven models, such as Gaussian processes and deep neural networks. For nonlinear systems, stability conditions involve inequalities over an infinite number of states in a state space. Sampling-based approaches can simplify these hard conditions into inequalities discretized over a finite number of states. However, this simplification requires margins to compensate for discretization residuals. Large margins degrade the accuracy of stability evaluation, and obtaining appropriate margins for various systems is challenging. This study addresses this challenge by deriving second-order margins for various nonlinear systems containing data-driven models. Because the size of the derived margins decrease quadratically as the discretization interval decreases, the stability evaluation is more accurate than with first-order margins. Furthermore, this study designs feedback controllers by integrating the sampling-based approach with an optimization problem. As a result, the controllers can guarantee stability while simultaneously considering control performance.

Figures (8)

• Figure 1: Concept of sampling-based approaches for guaranteeing the stability condition $\Phi(\boldsymbol{x})<0$ for all states $\boldsymbol{x}$.
• Figure 2: Illustration of a stability region $\mathbb{X}_{\mathrm{S}}$, region of attraction $\mathbb{X}_{\mathrm{A}}$, target region $\mathbb{X}_{\mathrm{T}}$, and level sets of $V(\boldsymbol{x})$.
• Figure 3: Two-dimensional illustration of simplexes $\mathbb{X}_{i}$ in $\mathbb{X}$.
• Figure 4: One-dimensional illustration of $\xi(\boldsymbol{x})$, its linear interpolation ${\xi}^{\mathrm{I}}(\boldsymbol{x})$, lower margin ${\underline{m}(\xi)}$, and upper margin ${\overline{m}(\xi)}$.
• Figure 5: With initial $\boldsymbol{u}$ and $V$

Theorems & Definitions (65)

• Remark 1: Model set
• Definition 1: Practical stability
• Definition 2: Stability region
• Proposition 1: Region of attraction and target region
• proof
• Remark 2: Application to stability analysis
• Definition 3: Simplexes (Fig. \ref{['fig:simplexes']})
• Definition 4: Maximum sampling interval (Fig. \ref{['fig:simplexes']})
• Definition 5: Linear interpolations (Fig. \ref{['fig:LI_margins']})
• Definition 6: Lower and upper margins (Fig. \ref{['fig:LI_margins']})