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Kernel-based Learning for Safe Control of Discrete-Time Unknown Systems under Incomplete Observations

Zewen Yang, Xiaobing Dai, Weijie Yang, Bahar İlgen, Aleksandar Anžel, Georges Hattab

TL;DR

This work tackles safe control for discrete-time high-order systems with unknown dynamics and partial state observability. It develops a kernel ridge regression (KRR) based model for the unknown dynamics within an RKHS, coupled with a state observer and a data-acquisition strategy that enables learning from limited measurements. A Lyapunov-based analysis yields explicit conditions for ultimate boundedness of both tracking and observation errors, integrating the KRR prediction error bound into the control design. Simulations demonstrate substantial improvements in tracking accuracy and estimation performance compared to controllers without learning, highlighting the practical significance for safety-critical applications.

Abstract

Safe control for dynamical systems is critical, yet the presence of unknown dynamics poses significant challenges. In this paper, we present a learning-based control approach for tracking control of a class of high-order systems, operating under the constraint of partially observable states. The uncertainties inherent within the systems are modeled by kernel ridge regression, leveraging the proposed strategic data acquisition approach with limited state measurements. To achieve accurate trajectory tracking, a state observer that seamlessly integrates with the control law is devised. The analysis of the guaranteed control performance is conducted using Lyapunov theory due to the deterministic prediction error bound of kernel ridge regression, ensuring the adaptability of the approach in safety-critical scenarios. To demonstrate the effectiveness of our proposed approach, numerical simulations are performed, underscoring its contributions to the advancement of control strategies.

Kernel-based Learning for Safe Control of Discrete-Time Unknown Systems under Incomplete Observations

TL;DR

This work tackles safe control for discrete-time high-order systems with unknown dynamics and partial state observability. It develops a kernel ridge regression (KRR) based model for the unknown dynamics within an RKHS, coupled with a state observer and a data-acquisition strategy that enables learning from limited measurements. A Lyapunov-based analysis yields explicit conditions for ultimate boundedness of both tracking and observation errors, integrating the KRR prediction error bound into the control design. Simulations demonstrate substantial improvements in tracking accuracy and estimation performance compared to controllers without learning, highlighting the practical significance for safety-critical applications.

Abstract

Safe control for dynamical systems is critical, yet the presence of unknown dynamics poses significant challenges. In this paper, we present a learning-based control approach for tracking control of a class of high-order systems, operating under the constraint of partially observable states. The uncertainties inherent within the systems are modeled by kernel ridge regression, leveraging the proposed strategic data acquisition approach with limited state measurements. To achieve accurate trajectory tracking, a state observer that seamlessly integrates with the control law is devised. The analysis of the guaranteed control performance is conducted using Lyapunov theory due to the deterministic prediction error bound of kernel ridge regression, ensuring the adaptability of the approach in safety-critical scenarios. To demonstrate the effectiveness of our proposed approach, numerical simulations are performed, underscoring its contributions to the advancement of control strategies.
Paper Structure (9 sections, 5 theorems, 30 equations, 3 figures, 1 algorithm)

This paper contains 9 sections, 5 theorems, 30 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. System Description and Objective
  4. Kernel Ridge Regression
  5. Learning-based Safe Control with KRR
  6. Data Acquisition and Analysis
  7. Tracking Control with Performance Guarantee
  8. Simulation
  9. Conclusion

Key Result

Lemma 1

Given a kernel $\kappa(\cdot,\cdot)$ with corresponding RKHS $\mathcal{H}_{\kappa}$ and a data set $\mathbb{D} = \{ \bm{x}^{(\iota)}, z^{(\iota)} \}_{\iota = 1, \cdots, N}$, the solution of eqn_KRR_optimization is presented as where $\bm{k}(\cdot) \!\!=\!\! [\kappa(\!\bm{x}^{(1)\!}, \cdot\!), \!\cdots\!, \kappa(\!\bm{x}^{(N)\!}, \cdot\!)]^T$, $\bm{z} \!=\! [z^{(1)}, \!\cdots\!, z^{(N)}]^T$ and $\

Figures (3)

  • Figure 1: The manifold of the unknown function $f(\cdot)$, the states of the collected training data set $\mathbb{D}$, and the state trajectory during the data acquisition procedure.
  • Figure 2: The plots of states $\bm{x}$ with KRR and its error bound (orange), without KRR (blue), and the desired trajectory (black dashed). The partial magnification plots (right).
  • Figure 3: The curves of $\| \bm{e}(t) \|$ and $\| \hat{\bm{e}}(t) \|$ over time $t$.

Theorems & Definitions (10)

  • Lemma 1: Representer theorem
  • Definition 1
  • Lemma 2
  • proof
  • Remark 1
  • Lemma 3: hashimoto2022learning
  • Lemma 4
  • proof
  • Theorem 1
  • proof