Robust Control of General Linear Delay Systems under Dissipativity: Part I -- A KSD based Framework

This paper introduces an effective framework for designing memoryless dissipative full-state feedbacks for general linear delay systems via the Krasovski\u{i} functional (KF) approach, where an unlimited number of pointwise and general distributed delays (DDs) exists in the state, input and output. To handle the infinite dimensionality of DDs, we employ the Kronecker-Seuret Decomposition (KSD) which we recently proposed for analyzing matrix-valued functions in the context of delay systems. The KSD enables factorization or least-squares approximation of any number of $\mathcal{L}^2$ DD kernels from any number of DDs without introducing conservatism. This also facilitates the construction of a complete-type KF with flexible integral kernels, following from an application of a novel integral inequality derived from the least-squares principle. Our solution includes two theorems and an iterative algorithm to compute controller gains without relying on nonlinear solvers. A challenging numerical example, intractable for existing methods, underscores the efficacy of this approach.