Robust Data-Driven Control for Nonlinear Systems Using their Digital Twins and Quadratic Funnels

TL;DR

This work addresses safe data-driven control for nonlinear plants using imperfect digital twins by modeling plant–twin deviations as a locally time-varying system and constructing data-consistent uncertainty sets from finite trajectories. It then synthesizes time-varying quadratic funnels around the twin trajectory via LMIs and the Matrix S-lemma, guaranteeing constraint satisfaction and robust invariance. The method operates in online segments, updating the funnel and feedback gain with new data while maintaining stability guarantees. The approach yields practical safety certificates and demonstrates improved tracking over twin-based open-loop deployment, with potential extensions to output feedback and tighter uncertainty representations.

Abstract

This paper examines a robust data-driven approach for the safe deployment of systems with nonlinear dynamics using their imperfect digital twins. Our contribution involves proposing a method that fuses the digital twin's nominal trajectory with online, data-driven uncertainty quantification to synthesize robust tracking controllers. Specifically, we derive data-driven bounds to capture the deviations of the actual system from its prescribed nominal trajectory informed via its digital twin. Subsequently, the dataset is used in the synthesis of quadratic funnels -- robust positive invariant tubes around the nominal trajectory -- via linear matrix inequalities built on the time-series data. The resulting controller guarantees constraint satisfaction while adapting to the true system behavior through a segmented learning strategy, where each segment's controller is synthesized using uncertainty information from the previous segment. This work establishes a systematic framework for obtaining safety certificates in learning-based control of nonlinear systems with imperfect models.

Figures (4)

• Figure 1: Interconnection and data flow between the physical plant and its digital twin.
• Figure 2: Segmented data collection and synthesis. During $\mathcal{T}^{D}_{i-1}$ we collect $(H_{i-1},\,\Xi_{i-1},\,H^{+}_{i-1})$. At $k_i$, these are passed to a convex synthesis step that returns $(P_i, K_i)$ for segment $\mathcal{T}_i$.
• Figure 3: Schematic of per-segment quadratic funnels in input (left) and state (right) spaces. At time $k$, the 1-level set $\{\eta:\eta^\top P_i \eta \le 1\}$ is centered at $\hat{x}_{\text{nom}}(k)$. Under $\xi=K_i\eta$, any trajectory starting inside remains inside; the induced input ellipsoid $\mathcal{E}_u(R_i)$ respects input limits.
• Figure 4: Closed-loop tracking on the 2-DoF arm. The proposed data-driven controller (red) tracks the nominal trajectory computed on the digital twin (black) while remaining within the certified state funnels (gray) and respecting state constraints (green). For comparison, applying the twin controller directly to the plant (dashed gray) leads to larger excursions and constraint violations.

Theorems & Definitions (17)

• Definition 1: Funnel
• Lemma 1: Disturbance Bound
• Lemma 2: Variation in Linearization
• Lemma 3: Bound on Total Disturbance
• Lemma 4: Bound on Jacobian Variation
• Definition 2: Admissible Variation Set
• Definition 3: Data-Consistent System Set
• Remark 1
• Definition 4: Quadratic Funnel
• Remark 2: Computing $P_{\min}(k)$ and $R_{\max}(k)$