Dissipative Control of General Linear Time-Delay Systems: Applications of the Kronecker-Seuret Decomposition

TL;DR

This paper introduces a unified framework using Kronecker-Seuret decomposition for stabilizing complex linear time-delay systems with multiple delays, overcoming previous theoretical and numerical limitations.

Contribution

The paper develops a novel approach employing KSD for analyzing and controlling systems with unlimited pointwise and distributed delays, enabling controller synthesis without nonlinear solvers.

Findings

Successfully stabilizes systems with complex delay structures.

Addresses problems that previous methods could not solve.

Provides an iterative algorithm for controller gain computation.

Abstract

Stabilizing autonomous linear time delay systems, particularly when addressing an unlimited number of pointwise and distributed delays (DDs) under dissipative constraints, poses a significant challenge. Existing solutions are often hindered by theoretical limitations, numerical obstacles, or an inability to address the complexities of the delay integral kernels. In this paper, we propose a unified framework to tackle the above problem by employing the concept of the Kronecker-Seuret decomposition (KSD) for matrix-valued functions, which we recently have developed for the analysis of complex delay structures in coordination with the Krasovski\u{\i} functional approach. Our strategy can simultaneously address two distinct control problems, where the matrix kernels of DDs can contain an unlimited number of square-integrable functions. We show in detail how the KSD can factorize and approximate different kernel functions simultaneously without introducing conservatism. Furthermore, the use of KSD also enables us to construct complete-type functionals, whose integral kernels can include any number of weakly differentiable and linearly independent functions, underpinned by the utilization of novel integral inequalities derived from the least-squares principle. The solution to each synthesis problem comprises two theorems accompanied by an iterative algorithm, which can be utilized as a single package to compute controller gains, thus eliminating the need for nonlinear solvers. We present the testing results of two challenging examples, which could not be addressed by existing methods, to demonstrate the effectiveness of our methodology. Additionally, the paper reviews recent advancements in the research of time-delay systems, providing a valuable reference for both emerging and established researchers.