Combining Prior Knowledge and Data for Robust Controller Design

TL;DR

The paper addresses robust controller design for uncertain LTI systems by merging partial prior knowledge with finite noisy data, modeled via a linear fractional transformation and multiplier LMIs. It introduces a unifying framework that first translates prior information into transformed multipliers for a full-block uncertainty $\tilde{\Delta}=B_w\Delta$, then learns additional multipliers from data to form a combined set $\bm{\tilde{\Delta}}_{\rm com}$ that exactly captures all information available about the uncertainty. Synthesis conditions are LMIs guaranteeing stability and robust $\mathcal{H}_2$ performance (and, in extensions, robust quadratic performance for nonlinear uncertainties) for all $\Delta\in\bm{\Delta}_{\rm com}$, with practical realizations via $L=K\mathcal{X}$ and standard Schur complement reformulations. Numerical examples, including an aerial/satellite-like system, demonstrate substantial conservatism reduction and performance gains when exploiting both prior structure and data, compared to purely data-driven or purely model-based approaches, and show the framework's extension to data-driven loop-shaping through $\mathcal{H}_{\infty}$ objectives.

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.

Figures (6)

• Figure 1: Uncertain plant generating data for robust control.
• Figure 2: Illustration of the satellite system in Example \ref{['ex:introduction']}. This figure as well as the example are adapted from franklin2019feedback.
• Figure 3: Guaranteed closed-loop $\mathcal{H}_2$-norm according to the four scenarios in Section \ref{['subsec:ex_H2']}, depending on the noise level $\bar{d}$. The different scenarios are 1) using only prior knowledge, 2) using only available data (equivalent to waarde2022from), 3) using prior knowledge and data, and 4) using exact model knowledge.
• Figure 4: Guaranteed closed-loop $\mathcal{H}_2$-norm when using prior knowledge and data (scenario 3) in Section \ref{['subsec:ex_H2']}), depending on the noise level $\bar{d}$ and for different disturbance multipliers in Assumption \ref{['ass:noise_multipliers']}.
• Figure 5: Guaranteed closed-loop $\mathcal{H}_2$-norm when using prior knowledge and data (scenario 3) in Section \ref{['subsec:ex_H2']}), depending on the data length $N$ and for different disturbance multipliers in Assumption \ref{['ass:noise_multipliers']}.

Theorems & Definitions (3)

• proof
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