Computation-Aware Learning for Stable Control with Gaussian Process

TL;DR

The paper presents a computation-aware framework for Gaussian process dynamical models in robotics, distinguishing mathematical uncertainty from computational uncertainty arising from limited resources. By treating the GP inverse covariance as a random variable and tracking combined uncertainty, it derives a stability analysis via Lyapunov functions and robust ROA estimation under compute constraints. It then formulates a CLF-based optimization (CLF-SOCP) that incorporates both uncertainty sources, with an explicit stabilizing policy extending Sontag's formula to uncertain models. The framework is validated through 1D nonlinear and inverted pendulum simulations, as well as a quadrotor tracking experiment, demonstrating improved safety and performance under computational limits and offering a practical path to stable online learning on resource-constrained platforms.

Abstract

In Gaussian Process (GP) dynamical model learning for robot control, particularly for systems constrained by computational resources like small quadrotors equipped with low-end processors, analyzing stability and designing a stable controller present significant challenges. This paper distinguishes between two types of uncertainty within the posteriors of GP dynamical models: the well-documented mathematical uncertainty stemming from limited data and computational uncertainty arising from constrained computational capabilities, which has been largely overlooked in prior research. Our work demonstrates that computational uncertainty, quantified through a probabilistic approximation of the inverse covariance matrix in GP dynamical models, is essential for stable control under computational constraints. We show that incorporating computational uncertainty can prevent overestimating the region of attraction, a safe subset of the state space with asymptotic stability, thus improving system safety. Building on these insights, we propose an innovative controller design methodology that integrates computational uncertainty within a second-order cone programming framework. Simulations of canonical stable control tasks and experiments of quadrotor tracking exhibit the effectiveness of our method under computational constraints.

Figures (7)

• Figure 1: Quadrotor tracking control under computational constraints. The green line denotes the proposed computation-aware learning control method in this paper, showing improved tracking accuracy due to adaptive, conservative stability constraints compared to the agnostic one (blue line).
• Figure 2: Workflow of computation-aware learning for stable control. This figure illustrates the three main stages of developing a control system with computational constraints. Initially, in Section \ref{['sec.model learning']} (computation-aware dynamical model learning), the model's uncertainty is broken down into mathematical and computational parts. Following that, in Section \ref{['sec.stability analysis']} (computation-aware stability analysis), the system's stability is analyzed by estimating its Region of Attraction (ROA) under various uncertainties. In Section \ref{['sec.controller design']} (computation-aware controller design), a controller is designed to ensure system stability with minimal control effort, using the control lyapunov function-second order cone programming (CLF-SOCP) method. The flowchart concludes with a decision-making process that evaluates whether the system's computational capacity needs upgrading before deployment.
• Figure 3: Comprehensive evaluation of model learning and stable control in a nonlinear 1D System \ref{['eq.1D system model']}. During the GP model training, we systematically gather 25 data points per online update and vary the CG iterations for model learning ($i=4$, $i=12$, $i=25$). The combined uncertainty of the learned dynamics, Lyapunov function $V(x)$ and its derivative $\dot{V}(x)$ in \ref{['eq.asymptotic stability']} respectively, decompose into the mathematical uncertainty ($$∎) and computational uncertainty ($$∎). (a) Computation-aware model learning. The computational uncertainty diminishes with increasing $i$, while mathematical uncertainty remains constant. (b) Computation-aware ROA estimation. The estimated ROA grows with each iteration, converging towards the true ROA using a given control policy $\pi(x)=-2.5 x$. (c) Computation-aware controller design. Our method \ref{['eq.clf-SOCP']} always ensures a negative $\dot{V}(x)$, signifying stable control, which is impressive even when the model is learned with only 4 iterations. (d) Computation-agnostic controller design. This design (setting $\sigma_i^{\textup{comp}} \equiv 0$ in \ref{['eq.clf-SOCP']}), which omits computational uncertainty, yields a much smaller ROA compared to the counterparts in (c), potentially leading to unsafety in regions where $\dot{V}(x)$ is positive.
• Figure 4: Computation-aware ROA estimation for inverted pendulum system given the linear quadratic controller. The ROA estimate using the prior model exceeds the true ROA. In contrast, the ROA estimates using computation-aware GP models do not exceed the true ROA and increase with the number of CG iterations.
• Figure 5: Comparison of computation-aware (Our method) and computation-agnostic berkenkamp2016safe ROA estimation for inverted pendulum system under computation constraints. Neglecting computational uncertainty can lead to overly optimistic estimates of a system’s stability, potentially misclassifying unstable regions ($$∎) as stable.

Theorems & Definitions (23)

• Remark 1
• Remark 2
• Remark 3
• Lemma 1: Pointwise Convergence of the Posterior Mean
• Remark 4
• Definition 1: Asymptotic Stability
• Definition 2: Region of Attraction
• Lemma 2: Level Set Estimates of ROA khalil2002nonlinear
• Theorem 1: ROA Estimation using Computation-Aware GP Model
• proof