Data-Driven Superstabilizing Control under Quadratically-Bounded Errors-in-Variables Noise

Core Concepts

The author presents a method for data-driven super stabilizing control under quadratically-bounded errors-in-variables noise, utilizing a sum-of-squares hierarchy of semidefinite programs to generate controllers. The approach aims to eliminate input and measurement noise processes, enhancing tractability.

Abstract

This content delves into the intricacies of data-driven super stabilizing control under quadratically-bounded errors-in-variables noise. The paper introduces a framework for full-state-feedback stabilizing control using sum-of-squares hierarchies of semidefinite programs. It focuses on eliminating input and measurement noise processes to improve effectiveness and tractability. The study proposes methods for robust superstabilization in challenging EIV noise scenarios, emphasizing computational complexity and practical applications.
The paper discusses the Error-in-Variables model of system identification/control, addressing nonconvex optimization problems due to corrupted observed data. It highlights the generation of super stabilizing controllers through solving semidefinite programs based on quadratic bounds on input and measurement noise. The study employs a theorem of alternatives to enhance tractability by removing noise processes from the equations.
Furthermore, it explores set-membership direct Data Driven Control (DDC) frameworks, focusing on data-consistent plants and stabilized plants by designed controllers. The Matrix S-Lemma is utilized for proofs of robust control when the noise model is defined by a matrix ellipsoid. Various methodologies like Farkas-based certificates and Sum of Squares (SOS) certificates are discussed for stabilization in different scenarios.
The content also addresses the computational complexity involved in discretizing infinite-dimensional linear programs into finite-dimensional convex optimization problems using SOS-matrix truncations. It provides insights into incorporating process noise into the framework and presents a numerical example demonstrating the application of these methods in practice.

Stats

Instances of such quadratic bounds include elementwise norm bounds (at each time sample), energy bounds (across the entire signal), and chance constraints.
A controller u = Kx certifiably stabilizes all plants consistent with the data will be able to stabilize the true system with probability Pjoint.
For this specific example, algorithms from Theorems 1 and 2 both fail to find a common quadratically stabilizing controller.
The per-noise probability is chosen as δx = δu = (0.95)1/(2T −1) = 0.9981.
In this work, we will ensure superstabilization under quadratically bounded noise.
Computational complexity grows linearly with 'Count' scaling while polynomial growth occurs with increasing 'Size'.
Superstabilizing control is performed under a decay-bound objective λ∗ = minλ,K λ ≥ ∥A+BK∥∞ ∀(A, B) ∈ P.

Quotes

"The Error-in-Variables model involves nontrivial input and measurement corruption resulting in generically nonconvex optimization problems."
"Superstabilizing controllers are generated through the solution of a sum-of-squares hierarchy of semidefinite programs."
"Our goal is to find a gain matrix K such that full-state-feedback control policy ut = Kxt can simultaneously stabilize all plants consistent with observed data."

Key Insights Distilled From

Data-Driven Superstabilizing Control under Quadratically-Bounded Errors-in-Variables Noise

by Jared Miller... at arxiv.org 03-07-2024

https://arxiv.org/pdf/2403.03624.pdf

Deeper Inquiries

How can these methods be extended beyond linear systems to nonlinear systems

To extend these methods beyond linear systems to nonlinear systems, one can explore techniques like polynomial chaos expansions, neural network approximations, or kernel-based methods. Polynomial chaos expansions allow for the representation of uncertainties in a system through orthogonal polynomials, enabling the analysis and control of nonlinear dynamics with uncertain parameters. Neural network approximations leverage deep learning models to approximate complex nonlinear functions and capture system dynamics effectively. Kernel-based methods use kernel functions to map data into high-dimensional feature spaces where linear techniques can be applied effectively.

What are potential drawbacks or limitations when applying these techniques in real-world scenarios

When applying these data-driven control techniques in real-world scenarios, several drawbacks or limitations may arise. One limitation is the reliance on accurate data for model identification and controller synthesis; any errors or biases in the collected data can lead to suboptimal control performance. Another drawback is the computational complexity associated with solving large-scale optimization problems arising from these methods, which can limit their real-time applicability in fast-paced systems. Additionally, ensuring robustness against unforeseen disturbances or changes in system behavior poses a challenge as these approaches are often designed based on historical data and assumptions that may not hold under all conditions.

How might advancements in machine learning impact or complement these data-driven control strategies

Advancements in machine learning have the potential to significantly impact and complement data-driven control strategies. Machine learning algorithms such as reinforcement learning can be used to adapt controllers online based on feedback from the system's performance, allowing for continuous improvement without explicit modeling of system dynamics. Deep learning models can aid in capturing complex relationships within large datasets for improved prediction accuracy and controller design. Furthermore, advancements in interpretable AI techniques enable better understanding of black-box models generated by machine learning algorithms, enhancing trust and usability of data-driven control strategies across various applications.