A TBLMI Framework for Harmonic Robust Control

Abstract

The primary objective of this article is to demonstrate that problems related to stability and robust control in the harmonic context can be effectively addressed by formulating them as semidefinite optimization problems, invoking the concept of infinite-dimensional Toeplitz block linear matrix inequalities (TBLMIs). One of the central challenges tackled in this study pertains to the efficient resolution of these infinite-dimensional TBLMIs. Exploiting the structured nature of such problems, we introduce a consistent truncation method that effectively reduces the problem to a finite-dimensional convex optimization problem. By consistent, we mean that the solution to this finite-dimensional problem allows to closely approximate the infinite-dimensional solution with arbitrary precision. Furthermore, we establish a link between the harmonic framework and the time domain setting, emphasizing the advantages over periodic differential LMIs. We illustrate that our proposed framework is not only theoretically sound but also practically applicable to solving $H_{2}$ and $H_\infty$ harmonic control design problems. To enable this, we extend the definitions of $H_{2}$ and $H_\infty$ norms into the harmonic space, leveraging the concepts of the harmonic transfer function and the average trace operator for Toeplitz block operators. Throughout this article, we support our theoretical contributions with a range of illustrative examples that demonstrate the effectiveness of our approach.