Safe Learning of Regions of Attraction for Uncertain, Nonlinear Systems with Gaussian Processes (1603.04915v3)

Abstract: Control theory can provide useful insights into the properties of controlled, dynamic systems. One important property of nonlinear systems is the region of attraction (ROA), a safe subset of the state space in which a given controller renders an equilibrium point asymptotically stable. The ROA is typically estimated based on a model of the system. However, since models are only an approximation of the real world, the resulting estimated safe region can contain states outside the ROA of the real system. This is not acceptable in safety-critical applications. In this paper, we consider an approach that learns the ROA from experiments on a real system, without ever leaving the true ROA and, thus, without risking safety-critical failures. Based on regularity assumptions on the model errors in terms of a Gaussian process prior, we use an underlying Lyapunov function in order to determine a region in which an equilibrium point is asymptotically stable with high probability. Moreover, we provide an algorithm to actively and safely explore the state space in order to expand the ROA estimate. We demonstrate the effectiveness of this method in simulation.

• The paper proposes a method to safely learn and expand regions of attraction for uncertain nonlinear systems by modeling unknown dynamics with Gaussian Processes and using experimental data.
• Gaussian Processes model unknown dynamics, enabling safe data collection within the known safe region to refine the region of attraction.
• This method offers significant potential for safety-critical systems where exact modeling is challenging, providing an adaptive way to ensure robust and safe system operation.

Analyzing Safe Learning Techniques for Regions of Attraction in Nonlinear Systems using Gaussian Processes

The paper under discussion presents an intricate paper of nonlinear control systems, specifically focusing on the safe identification and expansion of regions of attraction (ROA) through Gaussian Processes (GPs). The authors, Berkenkamp, Moriconi, Schoellig, and Krause, propose a method of determining ROAs safely by utilizing experimental data taken from real systems, rather than relying entirely on potentially flawed model-based estimates.

Problem Formulation and Approach

The problem of determining the ROA is pivotal in control theory, especially for safety-critical systems where the operational boundaries must ensure safety under various conditions. Traditional approaches usually estimate ROAs based on system models but face significant issues when model inaccuracies arise, jeopardizing safety. The authors address this by proposing a novel methodology that actively learns the ROA through experimentation without exiting the true ROA, thereby ensuring operational safety throughout the learning process.

The paper's approach hinges on a few critical assumptions about the system modeled as

$\dot{x}(t) = f(x(t), u(t)) + g(x(t), u(t)),$

where $f$ is a known a priori model, and $g$ encapsulates unknown dynamics and model errors. The innovation lies in implementing Gaussian Processes to model $g$, allowing for the estimation of these unknown dynamics by leveraging observed data.

Methodology

The authors employ Gaussian Processes due to their ability to naturally incorporate uncertainty into the modeling of system dynamics. This facilitates the use of the GP's inherent probabilistic properties to derive high-probability safety guarantees, integral to controlling a system without encroaching upon unsafe regions.

A Lyapunov function, a scalar function often used in control systems to prove stability, is used within this GP framework to ascertain and expand the ROA iteratively. Suppose the Lyapunov function satisfies a particular negativity condition across the ROA. In that case, this is used to judiciously sample new system states, permitting safe exploration and refinement of the ROA.

In practice, the method incrementally updates the model based on evaluations within currently known safe sets and extends the ROA by actively probing the state-space boundary of the existing ROA, thereby engendering a conservative yet systematically expanding ROA.

Simulation and Results

Simulation experiments demonstrate that this method can effectively and robustly learn and expand the ROA over time. In particular, the given experimental results on an inverted pendulum, despite unaligned model parameters, validate that the learning process can refine the ROA to more closely resemble the true dynamical constraints of the system.

The methodology highlights both the efficacy of utilizing GPs in a control environment and the pragmatic potential of integrating learning processes to refine operational safety zones actively.

Theoretical and Practical Implications

The theoretical advancements showcased include leveraging GPs for system dynamics modeling and integrating these with Lyapunov theory for safety-critical applications. Practically, this translates into significant potential benefits for systems where exact modeling is challenging or impossible, offering an adaptive method to ensure robust and safe systems operation.

Future studies may explore refining model accuracy through more sophisticated kernel functions or optimization algorithms in GPs, tailored for specific domains within nonlinear control systems, and extend this work to multidimensional control contexts.

In summary, this paper makes a profound contribution to safely learning and expanding ROAs for uncertain nonlinear systems, signifying an important step towards more adaptive, reliable control strategies in safety-critical applications.