On the Lyapunov Matrix of Linear Delay Difference Equations in Continuous Time

The fundamental matrix and the delay Lyapunov matrix of linear delay difference equations are introduced. Some properties of the Lyapunov matrix, and the jump discontinuities of its derivative are proven, leading to its construction in the case of single delay or commensurate delays. An approximation is proposed for the non-commensurate case

Necessary and sufficient stability conditions for integral delay systems

A Lyapunov-Krasovskii functional with prescribed derivative whose construction does not require the stability of the system is introduced. It leads to the presentation of stability/instability theorems. By evaluating the functional at initial conditions depending on the fundamental matrix we are able to present necessary and sufficient stability conditions expressed exclusively in terms of the delay Lyapunov matrix for integral delay systems. Some examples illustrate and validate the stability conditions.Comment: This paper has been submitted to International Journal of Robust and Nonlinear Contro

Stability analysis and estimate of the region of attraction of a human respiratory model

International audienceIn this paper, we complete the stability analysis of various human respiratory non-linear time delay models introduced in. More precisely, we present a detailed mathematical analysis of the stability of the nonlinear model trivial equilibrium , an estimate of its region of attraction and exponential estimates of the solutions starting in this region. The proposed approach is constructive and it is based on the use of Lyapunov-Krasovskii functionals of complete type for time-delay systems with a cross term in the time derivative

Linear time-delay systems: the complete type functionals approach

[EN] Recent results on Lyapunov-Krasovskii functionals of complete type for linear time-delay systems are presented. The main concepts and results are introduced for the single delay system case, and necessary and sufficient stability conditions expressed in terms of the Lyapunov delay matrix are explained. The use of complete type functionals in analysis and controller design is discussed. The contribution focuses mainly at results of researchers in Mexico.[ES] Se introducen resultados recientes del enfoque de funcionales de Lyapunov-Krasovski de tipo completo para sistemas lineales con retardos. Se explican brevemente los principales conceptos y resultados para el caso de sistemas con un retardo así como las condiciones necesarias y suficientes de estabilidad expresadas en terminos del análogo de la matriz de Lyapunov. Las extensiones  de este tipo de condiciones de estabilidad a otras clases de sistemas con retardos son expuestas brevemente. Tambien se presentan aplicaciones existentes del efoque de funcionales de tipo completo a problemas de analisis y de diseño de controladores. El trabajo se enfoca a contribuciones de investigadores de Mexico a este tema de estudio.Este trabajo ha sido realizado parcialmente gracias al apoyo del Conacyt, México, Proyecto A1-S-24796.Mondié, S.; Gomez, M. (2022). Contribuciones al estudio de sistemas lineales con retardos: el enfoque de funcionales de tipo completo. Revista Iberoamericana de Automática e Informática industrial. 19(4):381-393. https://doi.org/10.4995/riai.2022.16828OJS38139319

Assigning the Kronecker invariants of a matrix pencil by row or column completions

AbstractThe challenge consists in describing the relationships between the Kronecker invariants of a matrix pencil and one of its subpencils. For a given subpencil, an algorithm for constructing a matrix pencil with prescribed Kronecker invariants should also be proposed

Exponential Stabilization of a Class of Nonlinear Neutral Type Time-Delay Systems, an Oilwell Drilling Model Example

International audienceThis paper deals with exponential stabilization of the class of nonlinear neutral type time-delay systems that can be transformed into a multi-model system. The approach is based on Lyapunov-Krasovskii techniques and uses a descriptor representation. The exponential stability properties are proved using an appropriate change of variables associated with a polytopic representation. The results are given in terms of LMIs. As an application example, we determine an e ective stabilizing controller for an oilwell drilling system

Backstepping for Uncertain Nonlinear Systems with a Delay in the Control

International audienceThe recent new backstepping control design strategy based on the introduction of artificial delays and/or dynamic extensions is adapted to a family of systems. That way, globally asymptotically stabilizing control laws for fundamental systems which cannot be handled by other techniques are determined

Finite Spectrum Assignment of Unstable Time-Delay Systems with a Safe Implementation

Abstract—The instability mechanisms, related to the implementation of distributed delay controllers in the context of finite spectrum assignment, were studied in detail in the past few years. In this note we introduce a distributed delay control law that assigns a finite closed-loop spectrum and whose implementation with a sum of point-wise delays is safe. This property is obtained by implicitly including a low-pass filter in the control loop. This leads to a closed-loop characteristic quasipolynomial of retarded type, and not one of neutral type, which was shown to be a cause of instability in previous schemes. Index Terms—Delay equations, finite spectrum assignment. I