Inversion of Separable Kernel Operators in Coupled Differential-Functional Equations and Application to Controller Synthesis

TL;DR

This paper develops a direct algebraic method to invert kernel operators in coupled differential-functional equations, facilitating control synthesis without power series expansions, and demonstrates its application with a numerical example.

Contribution

It introduces a novel algebraic approach for inverting separable kernel operators, enhancing control synthesis techniques for coupled differential-functional systems.

Findings

Inverse kernel operator can be explicitly constructed algebraically.

The domain of the infinitesimal generator is an invariant subspace.

Numerical example demonstrates the method's effectiveness.

Abstract

This article presents the inverse of the kernel operator associated with the complete quadratic Lyapunov-Krasovskii functional for coupled differential-functional equations when the kernel operator is separable. Similar to the case of time-delay systems of retarded type, the inverse operator is instrumental in control synthesis. Unlike the power series expansion approach used in the previous literature, a direct algebraic method is used here. It is shown that the domain of definition of the infinitesimal generator is an invariant subspace of the inverse operator if it is an invariant subspace of the kernel operator. The process of control synthesis using the inverse operator is described, and a numerical example is presented using the sum-of-square formulation.