[Paper Review] On the Robustness of Data-Driven Controllers for Linear Systems

Rajasekhar Anguluri, Abed AlRahman Al Makdah+2 more|arXiv (Cornell University)|Dec 21, 2019

Fault Detection and Control Systems17 references4 citations

TL;DR

This paper introduces a probabilistic framework to quantify the robustness of data-driven state-feedback controllers for linear systems under stochastic perturbations in training data. By leveraging a first-order approximation of the data-driven controller design map, it derives tight upper and lower bounds on the probability of spectral radius exceeding one, enabling stability guarantees that depend on perturbation statistics, data size, algorithm sensitivity, and nominal system properties.

ABSTRACT

This paper proposes a new framework and several results to quantify the performance of data-driven state-feedback controllers for linear systems against targeted perturbations of the training data. We focus on the case where subsets of the training data are randomly corrupted by an adversary, and derive lower and upper bounds for the stability of the closed-loop system with compromised controller as a function of the perturbation statistics, size of the training data, sensitivity of the data-driven algorithm to perturbation of the training data, and properties of the nominal closed-loop system. Our stability and convergence bounds are probabilistic in nature, and rely on a first-order approximation of the data-driven procedure that designs the state-feedback controller, which can be computed directly using the training data. We illustrate our findings via multiple numerical studies.

Motivation & Objective

• To address the lack of robustness guarantees in data-driven control under adversarial or stochastic training data corruption.
• To quantify how targeted perturbations in training data affect the stability of closed-loop systems with data-driven controllers.
• To develop a framework that links controller robustness to perturbation statistics, data size, algorithm sensitivity, and spectral properties of the nominal system.
• To provide probabilistic bounds on instability probability that are computationally tractable and directly based on training data.
• To enable comparative analysis of different data-driven control algorithms based on their sensitivity to data perturbations.

Proposed method

• Models the data-driven controller design as a Fréchet-differentiable map from training data to controller gain matrix.
• Uses a first-order Taylor expansion of the controller map to approximate its sensitivity to input data perturbations.
• Derives a probabilistic bound on the spectral radius of the closed-loop system under perturbed data using the Jacobian of the controller map.
• Computes the Jacobian matrix $ J_X $ numerically from training data using techniques analogous to automatic differentiation.
• Applies concentration inequalities (e.g., Gaussian tail bounds) to derive upper and lower bounds on the probability of instability.
• Validates the framework via Monte Carlo simulations on a vehicle dynamics model with controlled data corruption.

Experimental results

Research questions

• RQ1 How does the probability of instability in a data-driven controller vary with the variance and sparsity of perturbations in the training data?
• RQ2 What is the dependence of controller robustness on the size of the training dataset and the sensitivity of the data-driven algorithm?
• RQ3 How do the spectral properties of the nominal closed-loop system influence the stability of the perturbed system?
• RQ4 Can first-order approximation of the controller design map yield tight and meaningful probabilistic bounds on instability?
• RQ5 How does the sensitivity of different data-driven control algorithms to data perturbations affect their robustness in practice?

Key findings

• The upper bound on instability probability decreases as the perturbation variance decreases, and converges to zero when variance is sufficiently small.
• The lower bound on instability probability increases with perturbation variance and approaches one as variance grows, indicating worst-case instability likelihood.
• The Jacobian norm $ J_{\text{max}} $, which measures algorithm sensitivity, decreases with increasing training data dimension, implying improved robustness with larger datasets.
• The rate at which instability probability increases with the number of corrupted data points is independent of the system dimension, a key insight for scalability.
• The theoretical bounds tightly capture the Monte Carlo estimate of instability probability across varying perturbation variances, validating the framework’s accuracy.
• The upper bound becomes loose for high-dimensional systems, indicating a need for tighter analytical bounds in future work.