Kernel-based Learning for Safe Tracking Control of Discrete-Time Unknown Systems with Incomplete State Observations

Core Concepts

A kernel-based learning approach for safe tracking control of discrete-time unknown systems with partially observable states, integrating a state observer to achieve accurate trajectory tracking.

Abstract

The paper presents a kernel-based learning approach for safe tracking control of discrete-time nonlinear dynamical systems with unknown dynamics and partially observable states.
Key highlights:

The system is described by a class of high-order discrete-time nonlinear dynamics, where the nonlinear function f(·) is unknown but modeled using kernel ridge regression (KRR).
A tailored data acquisition strategy is introduced to collect training data for KRR, leveraging auxiliary state variables to handle the limitation of partial state measurements.
The learning-based control law is designed by integrating KRR with a state observer, ensuring the adaptability and safety of the approach in the presence of system uncertainties.
Theoretical analysis is provided to derive deterministic bounds for both tracking error and observation error, guaranteeing the control performance.
Numerical simulations demonstrate the effectiveness of the proposed method, showcasing a significant improvement in tracking performance compared to the controller without learning.

The core contribution is the development of a kernel-based learning framework for safe tracking control of unknown discrete-time systems with incomplete state observations, which addresses the challenges of system uncertainties and partial state measurements through the integration of KRR and a state observer.

Stats

The measurement noise v(tk) is bounded by |v(tk)| ≤ v̄, where v̄ ∈ R≥0.
The measurement noise w(ι) in the data set D is bounded by |w(ι)| ≤ w̄, where w̄ ∈ R≥0.
The Lipschitz constant of the unknown function f(·) is Lf = √2LκB, where Lκ is the Lipschitz constant of the kernel κ(·, ·) and B is the RKHS norm bound of f(·).
The upper bound of the measurement noise w in the data set D is derived as w̄ = ((2/T)^(n-1) + Lf√(1-(2/T)^(2n))/(1-(2/T)^2)) v̄.

Quotes

"Safe control for dynamical systems is critical, yet the presence of unknown dynamics poses significant challenges."
"Machine learning techniques facilitate the discernment of latent patterns and the derivation of models from data, thereby augmenting the control performance in the face of incomplete system knowledge."
"Kernel methods offer a distinctive advantage by providing theoretical error bounds making significant contributions to the domain of safety-critical control scenarios."

Key Insights Distilled From

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

by Zewe... at arxiv.org 05-03-2024

https://arxiv.org/pdf/2405.00822.pdf

Deeper Inquiries

How can the proposed approach be extended to handle time-varying or stochastic uncertainties in the system dynamics?

The proposed approach can be extended to handle time-varying or stochastic uncertainties by incorporating adaptive learning mechanisms. One way to address time-varying uncertainties is to update the kernel-based model periodically using new data acquired over time. This continuous learning process allows the model to adapt to changes in the system dynamics. Additionally, introducing stochastic kernels or incorporating probabilistic models such as Gaussian processes can enable the framework to account for stochastic uncertainties. By modeling the uncertainties as random variables with known distributions, the control strategy can be designed to be robust to stochastic variations in the system.

What are the potential limitations of the kernel-based learning approach, and how can they be addressed to further improve the control performance?

One potential limitation of the kernel-based learning approach is the computational complexity associated with training the model, especially for high-dimensional systems. This can lead to increased training times and memory requirements, making real-time implementation challenging. To address this limitation, techniques such as sparse kernel methods or online learning algorithms can be employed to reduce the computational burden while maintaining accuracy. Additionally, optimizing the hyperparameters of the kernel function and regularization parameters can enhance the model's performance and generalization capabilities.
Another limitation is the sensitivity of kernel methods to the choice of kernel function and its parameters. Inaccurate selection of the kernel can lead to suboptimal performance. To mitigate this, a systematic approach to kernel selection, such as cross-validation or Bayesian optimization, can be utilized to identify the most suitable kernel for the system dynamics. Regularization techniques can also be applied to prevent overfitting and improve the model's robustness.

What are the implications of the kernel-based learning framework for the broader field of safe and adaptive control of complex dynamical systems?

The kernel-based learning framework offers several implications for the field of safe and adaptive control of complex dynamical systems. Firstly, by leveraging non-parametric models and theoretical error bounds, the framework provides a principled approach to modeling uncertainties and designing control strategies for safety-critical applications. The ability to capture complex system dynamics with limited data makes kernel methods well-suited for adaptive control in dynamic environments.
Furthermore, the integration of kernel-based learning with state observers enhances the system's observability and controllability, enabling accurate state estimation and trajectory tracking. This combination of learning-based control and observation facilitates the development of robust and adaptive control systems that can operate effectively under partial observability and unknown dynamics.
Overall, the kernel-based learning framework contributes to advancing the state-of-the-art in safe and adaptive control by providing a solid foundation for data-driven control strategies, enabling improved performance, robustness, and adaptability in complex dynamical systems.