Dissipative Stabilization of Linear Systems with Time-Varying General Distributed Delays (Complete Version)

TL;DR

This paper introduces new methods for stabilizing linear systems with general, time-varying distributed delays, using dissipativity-based conditions and integral inequalities, without relying on common approximation techniques.

Contribution

It develops novel stabilization techniques for systems with measurable, time-varying distributed delays, handling arbitrary square-integrable kernels directly without approximations.

Findings

Derived sufficient conditions for dissipative stabilization via matrix inequalities.

Proposed a new integral inequality-based Krasovskii functional.

Validated methods through numerical examples.

Abstract

New methods are developed for the stabilization of a linear system with general time-varying distributed delays existing at the system's states, inputs and outputs. In contrast to most existing literature where the function of time-varying delay is continuous and bounded, we assume it to be bounded and measurable. Furthermore, the distributed delay kernels can be any square-integrable function over a bounded interval, where the kernels are handled directly by using a decomposition scenario without using approximations. By constructing a Krasovski\u{i} functional via the application of a novel integral inequality, sufficient conditions for the existence of a dissipative state feedback controller are derived in terms of matrix inequalities without utilizing the existing reciprocally convex combination lemmas. The proposed synthesis (stability) conditions, which take dissipativity into account, can be either solved directly by a standard numerical solver of semidefinite programming if they are convex, or reshaped into linear matrix inequalities, or solved via a proposed iterative algorithm. To the best of our knowledge, no existing methods can handle the synthesis problem investigated in this paper. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed methodologies.