Direct Data-Driven $H_{\infty}$ Control for Linear Discrete-Time Delay Systems

This paper introduces a direct data-driven approach to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$H_{\infty}$</tex> control for linear discrete-time input delay systems. It has been previously demonstrated that the input and state trajectories of a linear system can be represented using persistently exciting input data. Based on this, the paper presents a data-driven representation of the system using a Hankel matrix derived from input-state data collected through experiments. Stability conditions are then formulated in terms of linear matrix inequalities that rely solely on this data representation, thereby eliminating the need for explicit system matrices. These conditions are further extended to address <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$H_{\infty}$</tex> control problems, which constitute the primary objective of this paper. A numerical example illustrates the effectiveness of the proposed data-driven approach.