[Paper Review] Robust Data-Driven Control for Systems with Noisy Data

Chin-Yao Chang, Andrey Bernstein|arXiv (Cornell University)|Jul 19, 2022

Control Systems and Identification4 citations

TL;DR

This paper proposes a robust data-driven control framework for linear and switched linear systems using noisy input-output data, without assuming statistical properties of the noise. It derives an upper bound on modeling error due to data noise, applies diagonal scaling as a preconditioning method to tighten this bound, and uses the bound to design robust feedback controllers that ensure stability and performance despite data uncertainty.

ABSTRACT

This paper presents a robust data-driven controller design based on the noisy input-output data without assumptions on the statistical properties of the noises. We start with the direct data-representation of system models that take elements from behavioral system theory, followed by analyses of the upper bound of the "modeling" error with the data representation with presence of noises. Some pre-conditioning methods are put into the context based on how the derived bound is structured. We lastly leverage the upper bound to develop robust controllers that ride through the data noises.

Motivation & Objective

• Address the challenge of designing robust controllers when system data is corrupted by noise without prior assumptions on noise distribution.
• Characterize the upper bound of modeling errors arising from noisy data in data-driven system identification.
• Improve the tightness of the error bound through preconditioning techniques, particularly diagonal scaling.
• Develop robust feedback controllers for linear and switched linear systems using the derived error bounds.
• Demonstrate numerically that preconditioning reduces both condition numbers and actual modeling errors, enhancing controller robustness.

Proposed method

• Use behavioral system theory to represent system dynamics directly from noisy input-output data matrices without explicit system identification.
• Derive an analytical upper bound on the norm of the modeling error (δBA) in terms of the data matrix condition number and noise level, without assuming noise statistics.
• Apply diagonal scaling (Ruiz algorithm) as a preconditioning method to reduce the condition number of the data matrix, thereby tightening the error bound.
• Formulate a semidefinite programming (SDP) problem to compute robust feedback gain matrices that account for the worst-case error bound.
• Extend the framework to switched linear systems by modeling each mode separately and applying the same error-bound and controller design procedure per mode.
• Use the Frobenius norm and pseudo-inverse of data matrices to compute the error bound and feedback gains in a numerically stable way.

Experimental results

Research questions

• RQ1 What is the tightest possible analytical upper bound on data-driven modeling error when noise is present and its statistical properties are unknown?
• RQ2 How can preconditioning techniques such as diagonal scaling reduce the condition number of the data matrix and thereby improve the robustness of data-driven controllers?
• RQ3 To what extent does reducing the condition number of the data matrix reduce actual modeling errors in practice?
• RQ4 Can the derived error bound be effectively used to design robust feedback controllers that guarantee stability and performance under data noise?
• RQ5 How does the proposed method perform on switched linear systems with multiple modes, and can the error bound be applied per mode?

Key findings

• Diagonal scaling reduced the condition number of the data matrix by a factor of approximately 10 across all five modes in the numerical example.
• The upper bound on the normalized modeling error (‖δBA‖ / ‖[B A]‖) was reduced by 3% to 17% across modes after preconditioning, with the largest improvement in Mode 2 (17%).
• The actual modeling error (‖δBA‖ / ‖[B A]‖) decreased by 3% to 15% after preconditioning, indicating that reduced condition number improves actual estimation accuracy.
• The robust feedback controller designed using the preconditioned error bound successfully stabilized the system, driving the state to the origin within a moderate number of steps.
• Column selection preconditioning was ineffective in this case due to the randomized nature of the algorithm and the imposed lower bound on the number of columns.
• The proposed method enables robust controller design without requiring prior knowledge of noise statistics, relying only on a norm-bound on the noise.