Decay Rate Assignment through Multiple Spectral Values in Delay Systems

• Published in: IEEE transactions on automatic control 2024-08, p.1-15
• Main Authors: Boussaada, Islam, Mazanti, Guilherme, Niculescu, Silviu-Iulian, Michiels, Wim
• Format: Article
• Language: English
• Subjects: Asymptotic stability; Automatic; characteristic function; delay; Delays; Dynamical Systems; Engineering Sciences; exponential stability; Green–Hille transformation; hypergeometric functions; Kummer functions; Linear systems; Mathematics; Numerical stability; Optimization and Control; partial pole placement; Polynomials; Systematics
• ISSN: 0018-9286
• DOI: 10.1109/TAC.2024.3447117

This paper focuses on a spectral property for linear time-invariant (LTI) dynamical systems represented by delay-differential equations (DDEs) entitled multiplicity-induced-dominancy (MID), which consists, roughly speaking, in the spectral abscissa of the system being defined by a multiple spectral value. More precisely, we focus on the MID property for spectral values with over-order multiplicity, i.e., a multiplicity larger than the order of the DDE. We highlight the fact that a root of over-order multiplicity is necessarily a root of a particular polynomial, called the elimination-produced polynomial , and we address the MID property using a suitable factorization of the corresponding characteristic function involving special functions of Kummer type. Additional results and discussion are provided in the case of the n-th order integrator, in particular on the local optimality of a multiple root. The derived results show how the delay can be further exploited as a control parameter and are applied to some problems of stabilization of standard benchmarks with prescribed exponential decay.