Editorial for the special issue ‘Time‐delay systems: Recent trends and applications’

Differential-difference equations, differential integral equations, and functional differential equations have been subjects of recurrent study with origins in geometry and number theory. Picard, in 1908, emphasized the importance of considering hereditary effects in physical systems, suggesting the need for understanding the behaviour of complex functional equations that account for past influences*. Volterra's works in the early 20th century further delved into integro-differential equations, particularly in modeling viscoelasticity and species interaction. Driven by the necessity of modeling engineering systems and control, the field gained in popularity after the Second World War, specially in Soviet Union and United States. Despite engineers' awareness of hereditary effects, the lack of theoretical groundwork limited detailed discussions. Over the past five decades, extensive development in the theory of functional differential equations has occurred, becoming integral in various applications like viscoelasticity, mechanics, and biology. The 1950s witnessed notable activity in the field, with key publications from researchers such as Myshkis, Krasovskii, Razumikhin, Bellman, Cooke, Halanay, Yoshizawa and Hale shedding light on the subject until the early 1960s. Most research primarily focused on linear equations, especially concerning stability analysis under nonlinear perturbations. The use of Laplace transforms and Lyapunov's second method aided in understanding stability properties and understanding the ways the parameters affect the qualitative behavior of the dynamics. Presently, functional differential equations are considered as well developed as ordinary differential equations. However, the journey from ODE ideas to FDE expressions was lengthy, with the need for a new approach to tackle complex problems that were challenging through traditional means. Despite initial resistance, the modern analytic and geometric theory of FDEs has flourished, offering insights into various scientific disciplines. Year 2024 marks the 100th anniversary of the birth of two prominent figures in the field of time-delay systems: Nikolay Nikolayevich Krasovskii and Aristide Halanay, whose monographs,1, 2 and most notably their subsequent English translations3, 4 served as first and foremost references on the field–along with other major books dating from the 1960s. More than 60 years after the seminal monograph by Krasovskii (1959) and the one by Halanay (1963), time-delay systems can be considered as an established area, with research flourishing in many heterogeneous directions. The special issue collects 21 works by some of the leading researchers on the topic, focusing on fundamental problems, recent trends, and applications. We divided the contributions in five main sections, each of them devoted to a major active area of research in the broad field of time-delay systems. We start with four contributions on problems concerning stability and stabilization of time-delay systems. Then, we move to results focusing on uncertainty and robustness properties, listing six papers dealing with these aspects. Observer theory and observer-based controllers are the main subject of other four contributions. Three works on predictors and their application to nonlinear systems with delays are then presented, before closing the issue with four contributions which focus on delay-based approaches of partial differential equations (PDE) models. The next Section briefly presents each contribution. Alexander Aleksandrov, Denis Efimov, Emilia Fridman In their paper, A. Aleksandrov et al.5 deal with the local input-to-state stability property of a second-order system with vector position, affected by either constant or time-varying delays, with a power nonlinearity of the degree higher than one, and which does not contain a velocity-proportional damping term. By a Lyapunov-Razumikhin approach, the conditions for the local input-to-state stability are provided. New time estimates on decay of solutions are obtained. The approach is extended to attitude stabilization of a rigid body, and it is illustrated by simulations. Vittorio De Iuliis and Costanzo Manes The paper by V. De Iuliis and C. Manes6 addresses delay-independent stability for linear time-varying delay differential systems. It initially focuses on positive systems, presenting two delay-independent conditions for exponential input-to-state stability with explicit ISS gains. The extension to systems without sign constraints follows, utilizing a state-bounding approach that exploits the properties of positive systems. Due to the time-varying nature of the considered systems, the obtained stability conditions are formulated as an infinite number of inequalities. Implementation challenges are addressed, highlighting special cases where conditions can be represented as a finite-dimensional linear programming problem. Epiphane Loko, Antoine Chaillet, Iasson Karafyllis The article by E. Loko et al.7 outlines a method for constructing a Lyapunov-Krasovskii functional for time-delay systems whose stability analysis can be conducted via the Razumikhin or Halanay approaches. The proposed approach accounts for the presence of input disturbances, emphasizing input-to-state stability and exponential ISS. The constructed functional is shown to be coercive and dissipative in terms of the state history norm. An application to the study of the coupled ODE-PDE model of a chemical reactor illustrates the novel approach and shows how it allows to ensure ISS in terms of the supremum norm of the state. Andrey Polyakov and Miroslav Krstic A. Polyakov and M. Krstic8 address the problem of fixed-time stabilization by static feedback for delay-free and input delay systems. Their work presents a static nonlinear homogeneous feedback design for linear time-invariant systems, ensuring a constant settling time for all nonzero initial conditions through feedback gain dependence on the initial state. An ISS-based analysis of the robustness of the closed-loop system against measurement noise and exogenous perturbations is assessed for both delay-free and input delay systems. Marco A. Gomez, Christopher D. Cruz-Ancona In their paper, M. A. Gomez and C. D. Cruz-Ancona9 consider a robust control problem for uncertain linear time-delay systems. Assuming the exponential stabilizability of the nominal system, an innovative sliding mode control is presented, based on a new class of sliding manifolds that can be derived from the Lyapunov-Krasovskii functional of the complete type constructed for the nominal system. An important consequence is that every time-delay system with matched perturbations whose associated nominal system is exponentially stabilizable is robustly asymptotically stabilizable. César-Fernando Méndez-Barrios, Juan-Diego Torres-García, Silviu-Iulian Niculescu In the paper by C. Mendéz-Barrios et al.,10 the authors consider the control problem of a class of strictly proper linear time-invariant single-input/single-output systems by a PD controller, for which the derivative action has been implemented by means of a delay-difference approximation scheme using multiple delays. They study the behavior of the characteristic roots of the closed-loop system and characterize the improperly-posed case, that is, the situations where "small delays" induce instability in the closed-loop system. Hui Peng, Yanling Ding, Tian Qi, Jie Chen H. Peng et al.11 propose a stability analysis of a standard consensus protocol over a network of N N continuous-time one dimensional linear agents, with constant uniform communication delays and additive measurement noise. In particular, the case of unstable and non-minimum phase agents is considered, and both cases of directed and undirected communication graphs is investigated. The authors define a robustness parameter, the Delay Consensus Margin, which is the largest delay for which the considered protocol can robustly achieve consensus, and an H 2 H₂ performance criterion with respect to the measurement noise. These robustness/performance parameters are evaluated as functions of the unstable pole/zero pair of the agents and of the network topology, characterized by the eigenvalues of the Laplacian. Huan Phan-Van, Keqin Gu The paper by H. Phan-Van and K. Gu12 considers linear systems with a given partition structure, and for this class of systems they develop a recursive method that finds and computes underlying structured invariant subspaces, thus unveiling a blocky multichannel feedback structure for the system, where the feedback can include delays and uncertainties. An interesting aspect is that such invariant subspaces allow the dimensionality reduction of the delay channels, thus paving the way for efficient use of general analysis tools, such as the Lyapunov-Krasovskii functional approach for robust stability analysis. Jin Zhang, Emilia Fridman J. Zhang and E. Fridman13 investigate the stabilization of linear uncertain systems with unknown control directions. They employ a bounded extremum seeking controller accounting for a small, time-varying measurement delay. This study extends existing literature by considering disturbances that not only possess a constant component but also include small discontinuous variations arising from quantization. Two types of measurements, that is, the state measurements and the state quadratic norm ones, are examined. For the ISS analysis the authors present the system as an ODE with delayed perturbations and employ variation of constants formula leading to explicit conditions in terms of simple inequalities. The method allow for larger parameter uncertainties compared to existing results. Congran R. Zhao, Wei Lin The article by C.R. Zhao and W. Lin14 deals with sampled-data control for nonlinear systems featuring both state and input delays, implemented through memoryless sampled-data feedback. The class of uncertain systems under consideration contains uncontrollable/unobservable linearization and is not stabilizable, even locally, by any linear or smooth feedback. The author develop both state and output feedback control schemes under sample and hold conditions, by the emulation method. Utilizing the Lyapunov-Krasovskii functional theorem in conjunction with robust control principles, the study establishes the global asymptotic stability of the proposed sampled-data state and output feedback controllers for hybrid closed-loop systems with delays and uncertainty. This stability holds provided that both the input delay and sampling period remain within certain limits. Stefano Battilotti The paper by S. Battilotti15 presents a general observer theory for differential delay systems, employing various types of symmetries to construct semi-global and global observers with bounded or unbounded solutions. It introduces symmetry notions inspired by those in ordinary and partial differential systems, highlighting the role of symmetry in system detectability and observer design. Symmetries, whether exact or asymptotic, facilitate system mapping and approximation, leading to effective observer design, with parameters that may be constant or updated online for improved performance. Mario Di Ferdinando, Giordano Pola, Stefano Di Gennaro, Pierdomenico Pepe, Alessandro Borri In the paper by M. Di Ferdinando et al.16 a continuous-time tracking controller for a class of nonlinear time-delay systems, based on Germani's nonlinear observer, is firstly designed. Then, sufficient conditions are provided for the existence of a suitably fast sampling and of an accurate quantization of the input/output channels such that the digital implementation of the proposed continuous-time observer-based tracking controller ensures the semi-global practical stability property of the related sampled-data quantized closed-loop tracking error system, with arbitrarily small final target ball of the origin. Marcello Guarro, Francesco Ferrante, Ricardo G. Sanfelice M. Guarro et al.17 propose a hybrid observer for state estimation over networks under the assumption that the delayed measurements of the output of the plant at time instants, not necessarily periodic, are accompanied by timestamps provided by a clock that synchronizes with the clock of the observer in finite time. In this framework, convergence properties of the estimation error are addressed. Illustrated examples complete the presentation showing the effectiveness of the proposed hybrid observer. Haifang Li, Bin Zhou, Wim Michiels In their paper, H. Li et al.18 deal with sensor fault estimation of linear systems in the presence of unknown inputs in input and output channels. Relying on matrix equation theory, the unknown inputs are first removed from the output channel, and then an augmented descriptor system is constructed by treating both system state and sensor fault as (pseudo)-state variables. The augmented descriptor system is then transformed into a regular one, followed by the elimination of the unknown input from the state equation. Finally, a reduced-order prescribed-time sensor fault estimator is obtained by using periodic delayed observers. Imoleayo Abel, Mrdjan Janković, Miroslav Krstic In the work by I. Abel et al.19 the authors deal with the problem of enforcing safety for nonlinear systems with multiple inputs affected by distinct time delays, via the use of robust control barrier functions. Two control approaches are introduced, which enforce safety before all input time delays have been compensated by state predictors, whenever it is possible to do so, treating longer delayed inputs as known disturbances. So, whenever possible, a subset of input channels with shorter delays will be utilized for keeping the system in the admissible safe set before longer input delays have been compensated. Nikolaos Bekiaris-Liberis In his work, N. Bekiaris-Liberis20 investigates the Cooperative Adaptive Cruise Control design for vehicular platoons in the presence of delays. The case of homogeneous vehicle's dynamics, described by a third-order nonlinear system with input delay, is considered, and a new control scheme based on a nonlinear predictor-feedback is proposed. In addition, a novel approach for stability and ℒ ∞ L∞ string stability analysis is also provided. The design's robustness to delay uncertainty is discussed. Bryan Rojas-Ricca, Fernando Castanos, Sabine Mondié The article by B. Rojas-Ricca et al.21 analyzes the high-gain prediction approach for a class of nonlinear systems affected by constant input delay. The problem is addressed in the framework of weighted homogeneity and input-to-state stability. The canonical form for uniformly observable nonlinear systems allows tuning the linear-part spectrum by multiplicity-induced dominancy and ensuring closed-loop system input-to-state stability using the descriptor method of the Lyapunov–Krasovskii approach. Due to the trade-off between delay and gain margin, the limitation of high gain is overcome using an efficient cascade of subpredictors. Kaïs Ammari, Islam Boussaada, Silviu-Iulian Niculescu, Sami Tliba K. Ammari et al.22 address the boundary control problem of the transport equation. Inspired by the pole placement approach, the authors propose a control method based on assigning an appropriate exponential decay rate to the solution of the closed-loop system. The corresponding controller consists of an autoregressive relation connecting the input and output of the transport equation. The obtained results provide an analytical lower bound for the exponential decay rate of the solution. Paul-Erik Haacker, Iasson Karafyllis, Miroslav Krstic, Mamadou Diagne By exploiting the relation between first-order hyperbolic partial differential equations (PDEs) and an equivalent delay-based (dynamical) system representation, P.-E. Haacker et al.23 propose several feedback laws for population systems modeled by age-structured hyperbolic PDEs under realistic assumptions such as dilution is governed by actuation dynamics and population density is positive. More precisely, the static output backstepping control with unrestricted dilution, global stabilization with dilution constrained to a finite positive interval and full-state stabilization under state constraints are explicitly discussed. Vladimir Rasvan V. Rasvan24 addresses critical cases in difference operator stability for neutral functional differential equations arising from 1D hyperbolic partial differential equations dynamics in mechanical and hydraulic engineering, showing a connection between nonasymptotic stability and energy losses. The paper shows that by appropriately considering energy losses, critical stability properties can be mitigated, leading to asymptotic stability for the system's dynamics and also ensuring other asymptotic properties. Shanshan Wang, Jie Qi, Miroslav Krstic In their paper, S. Wang et al.25 introduce a novel delay-adaptive control approach for a class of first-order hyperbolic partial integro-differential equations with an unknown input delay. Given the inherent challenge in accurately determining the exact delay value, it emerges as a significant source of uncertainty in the process under consideration. The proposed delay-adaptive control method combines the infinite-dimensional backstepping technique with a Lyapunov argument to design unknown parameter's update law and ensure the closed-loop global stability. Confident that this special issue will shed light on the hot research topic of time-delay systems, we would like to thank the authoritative researchers for providing so interesting contributions, in various directions. We would like to thank the so many involved reviewers for their invaluable help in the revision process. Finally, our thanks go to the editorial board of the International Journal of Robust and Nonlinear Control, and in particular to the editor-in-chief Mike Grimble, for the encouragement and the advise during the project and the realization of this special issue. We believe that this special issue will be helpful to both well trained researchers as well as to beginners PhD students, who can find here an up-to-date picture of the latest achievements in this important research field.