State Estimator Design: Addressing General Delay Structures with Dissipative Constraints

TL;DR

This paper presents a novel dissipative estimator design for continuous-time delay systems with unlimited pointwise and distributed delays, utilizing Krasovski
2f functionals and a new matrix function decomposition to avoid conservatism.

Contribution

It introduces the Kronecker-Seuret Decomposition for matrix-valued functions, enabling delay kernel factorization without conservatism, and formulates the estimator design as sequential convex SDP problems.

Findings

Successfully stabilizes systems with complex delay structures

Eliminates the need for nonlinear solvers in design process

Demonstrates effectiveness through numerical experiments

Abstract

Dissipative estimator (observer) design for continuous time-delay systems poses a significant challenge when an unlimited number of pointwise and general distributed delays (DDs) are concerned. We propose an effective solution to this semi-open problem using the Krasovski\u{\i} functional (KF) framework in conjunction with a quadratic supply rate function, where both the plant and the estimator can accommodate an unlimited number of pointwise and general distributed delays with an unlimited number of square-integrable kernels. A key contribution is the introduction of a control concept called Kronecker-Seuret Decomposition (KSD) for matrix-valued functions, which allows for the factorizations or approximations of any DD integral kernel without introducing conservatism. Moreover, using KSD facilitates the construction of complete-type KFs with integral kernels that can contain any number of weakly differentiable and linearly independent functions. Our proposed solution is formulated as sequential convex SDP problems and is set out in two theorems along with an off-line iterative algorithm, which eliminates the need for nonlinear numerical solvers. We show the effectiveness of our method using two challenging numerical experiments, including a system stabilized by a non-smooth controller.