[Paper Review] Combining Prior Knowledge and Data for Robust Controller Design

Julian Berberich, Carsten W. Scherer+1 more|arXiv (Cornell University)|Sep 11, 2020

Control Systems and Identification21 citations

TL;DR

This paper presents a robust control framework that unifies prior knowledge and noisy data for linear time-invariant systems using linear matrix inequality (LMI)-based design. By combining structured multipliers from prior knowledge with data-driven multipliers derived from noisy input-state trajectories, the method ensures robust stability and performance guarantees, significantly reducing conservatism compared to black-box data-driven approaches.

ABSTRACT

We present a framework for systematically combining data of an unknown linear time-invariant system with prior knowledge on the system matrices or on the uncertainty for robust controller design. Our approach leads to linear matrix inequality (LMI) based feasibility criteria which guarantee stability and performance robustly for all closed-loop systems consistent with the prior knowledge and the available data. The design procedures rely on a combination of multipliers inferred via prior knowledge and learnt from measured data, where for the latter a novel and unifying disturbance description is employed. While large parts of the paper focus on linear systems and input-state measurements, we also provide extensions to robust output-feedback design based on noisy input-output data and against nonlinear uncertainties. We illustrate through numerical examples that our approach provides a flexible framework for simultaneously leveraging prior knowledge and data, thereby reducing conservatism and improving performance significantly if compared to black-box approaches to data-driven control.

Motivation & Objective

• Address the limitation of existing data-driven control methods that ignore available prior knowledge about system structure or parameters.
• Develop a systematic approach to integrate prior knowledge (e.g., known system matrices, uncertainty structure) with finite, noisy data for robust controller synthesis.
• Provide strong theoretical guarantees—specifically robust stability and performance—using a unified multiplier-based uncertainty description.
• Extend the framework to output-feedback control using noisy input-output data and to handle nonlinear uncertainties.
• Demonstrate that combining prior knowledge with data reduces conservatism and improves performance over purely data-driven or model-based methods.

Proposed method

• Formulate a full-block uncertainty representation that captures both prior knowledge and data-derived uncertainty using structured multipliers.
• Transform prior knowledge into equivalent multipliers for a full-block uncertainty structure, preserving structural information without loss.
• Learn data-based multipliers from finite, noisy input-state trajectories using a novel, unifying disturbance description that bounds noise energy.
• Combine prior and data-derived multipliers into a single uncertainty description for robust controller design via LMI optimization.
• Design static state-feedback and output-feedback controllers using LMI feasibility conditions that guarantee robust H∞ performance.
• Extend the framework to handle nonlinear uncertainties and input-output data by leveraging the same multiplier-based uncertainty modeling.

Experimental results

Research questions

• RQ1 How can prior knowledge about system matrices or uncertainty structure be systematically combined with noisy data for robust controller design?
• RQ2 What is a unifying and general disturbance description that can represent both prior knowledge and data-based uncertainty bounds?
• RQ3 Can the integration of prior knowledge and data reduce conservatism and improve performance compared to purely data-driven or model-based approaches?
• RQ4 How can the proposed framework be extended to output-feedback control using noisy input-output data?
• RQ5 What is the impact of the proposed multiplier-based uncertainty description on the robustness and performance of the resulting controller?

Key findings

• The proposed framework achieves an H∞-norm bound of γ = 0.22 on the disturbance-to-error channel in the numerical example, meeting design specifications.
• The data-driven controller design closely matches the performance of a nominal controller based on exact model knowledge, demonstrating strong theoretical guarantees.
• The integration of prior knowledge and data reduces conservatism compared to black-box data-driven methods, leading to improved performance.
• The framework successfully handles both input-state and input-output data, enabling robust output-feedback controller design.
• The use of a unified multiplier-based disturbance description generalizes and extends existing noise bounding techniques in the literature.
• The approach is extendable to nonlinear uncertainties and maintains robust stability and performance under structured uncertainty.