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The numerical minimization of the spectral abscissa function of linear time-invariant time-delay systems, an established approach to compute stabilizing controllers with a fixed structure, often gives rise to minima characterized by active characteristic roots with multiplicity higher than one. At the same time, recent theoretical results reveal situations where the so-called multiplicity induced dominancy property holds, i.e. a sufficiently high multiplicity implies that the root is dominant. Using an integrative approach, combining analytical characterizations, computation of characteristic roots and numerical optimization, a complete characterization of the stabilizability of second-order systems with input delays is provided, for both state feedback and delayed output feedback. The level sets of the minimal achievable spectral abscissa are also characterized. These results shed light on the complex relations between (configurations involving) multiple roots, the property of being dominant roots and the property of corresponding to (local/global) minimizers of the spectral abscissa function. [ABSTRACT FROM AUTHOR]

Qiao et al.21 developed adaptive PID controllers for trajectory tracking of robotic manipulators with uncertainties, which consist of a standard PID controller augmented by an adaptation law. Özbay and Gündes4 addressed the problems of pole placement and delay margin by the design of low-order controllers, including specifically PID controllers. Shi et al.11 considered a so-called dynamic equational PID controller, which augments the derivative control in the PID controller to high-order differentials. For well over a century, from the enabling of early governors to a standard tool in process control, PID (proportional-integral-derivative) control stood out as the most favored method for its unparalleled simplicity, ease of implementation, and cost-effectiveness, and indeed has been an enduring gift to feedback control. [Extracted from the article]

This article focuses on the root behavior of the characteristic function of some LTI SISO systems controlled by a PD‐control when a delay‐difference operator is used to approximate the derivative action. We will focus on the cases when the corresponding stability problem is improperly posed for "small" delay values. In this context, we will express the solutions of the corresponding characteristic function as power series and exploit their structure in deriving the asymptotic behavior of the characteristic roots. This analysis will allow detecting and explicitly characterizing the cases when delay‐difference approximations lead to improperly posed stability problems. Furthermore, in the case when the derivative approximation guarantees the properly posedness of the closed‐loop system, the robustness of the scheme with respect to the delay margin of the delay‐difference approximation and control gains parameters are explicitly addressed. More precisely, upper bounds on the delays as well as the robustness of the corresponding controller are computed. Some illustrative examples complete the presentation. [ABSTRACT FROM AUTHOR]

This paper presents the analysis of the stability properties of proportional-integralderivative (PID) controllers for dynamical systems with multiple state delays, focusing on the mathematical characterization of the potential sensitivity of stability with respect to infinitesimal parametric perturbations. These perturbations originate, for instance, from neglecting feedback delay, a finite-difference approximation of the derivative action, or neglecting fast dynamics. It is also shown that adding a low-pass filter, which might be necessary to reduce sensitivity to high-frequency noise, under certain conditions on the derivative feedback gain may destabilize the closed loop even for arbitrarily large cutoff frequencies. The analysis of these potential sensitivity problems leads us to the introduction of a "robustified" notion of stability called strong stability, inspired by the corresponding notion for neutral functional differential equations. We prove that strong stability can be achieved by adding a low-pass filter with a sufficiently large cutoff frequency to the control loop on the condition that the filter itself does not destabilize the nominal closed loop system. Throughout the paper, the theoretical results are illustrated by examples that can be analyzed analytically, including, among others, a third-order unstable system for which both proportional and derivative control action are necessary for achieving stability, while the regions in the gain parameter-space for stability and strong stability are not identical. Besides the analysis of strong stability, a computational procedure is provided for designing strongly stabilizing PID controllers. Computational case studies illustrating this design procedure complete the presentation. [ABSTRACT FROM AUTHOR]

This article addresses the boundary control problem of the transport equation. Namely, we propose a control method, which is merely a delayed output feedback relying on a partial pole placement idea, that consists in assigning an appropriate exponential decay rate to the closed‐loop system's solution. The proposed control structure appearing in the transport boundary, which has proven its effectiveness in controlling finite dimensional systems, consists of an autoregressive relation linking the transport equation's input and output. The obtained result provides an analytical lower bound for the solution's exponential decay. The contribution is concluded with a numeric discussion on the concrete implementation of the method with a special emphasis on the delay seen as a control parameter. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we study the behavior of characteristic roots of a class of strictly proper LTI SISO systems that are subject to a PD controller for which the derivative action has been implemented by means of a delay‐difference approximation scheme that uses multiple delays. To such an end, the improperly‐posed case is discussed in depth, that is, the situations where "small delays" induce instability in the closed‐loop system. The effect of adding more delays on the improperly‐posed case is analyzed and a methodology for determining the set of parameters that produces such a situation for a given number of delays is derived. Some illustrative examples complete the presentation. [ABSTRACT FROM AUTHOR]

This work revisits recent results on maximal multiplicity induced-dominancy for spectral values in reduced-order time-delay systems and extends it to the general class of second-order retarded differential equations. A parametric multiplicity-induced-dominancy property is characterized, allowing to a delayed stabilizing design with reduced complexity. As a matter of fact, the approach is merely a delayed-output-feedback where the candidates' delays and gains result from the manifold defining the maximal multiplicity of a real spectral value, then, the dominancy is shown using the argument principle. Sensitivity of the control design with respect to the parameters uncertainties/variation is discussed. Various reduced order examples illustrate the applicative perspectives of the approach. [ABSTRACT FROM AUTHOR]

• We construct a chain of two interconnected chemostats, assuming competitive exclusion. • The first chemostat cultivates the unadvantaged competitor. The second cultivates two ones. • The output of the first chemostat is the input of the second one. • We carry out a stability analysis by constructing a Lyapunov functional. We study a nonlinear system of differential delay equations describing a model of a chain of two chemostats, where one contains two microbial species in competition for a single limiting nutrient and receives an external input of the less advantaged competitor, which is cultivated in an external chemostat. We obtain sufficient conditions ensuring coexistence of all the species in competition which consist in upper delay bounds. [ABSTRACT FROM AUTHOR]

This paper aims at establishing a simple yet efficient solution to the problem of trajectory tracking with input constraint of a non-linear longitudinal vehicle model. We make use of differential flatness by embedding the constraint into the reference trajectory design. [ABSTRACT FROM AUTHOR]

This paper deals with some problems concerning the zero behaviour of a class of sampled-data SISO systems. More precisely, given a continuous-time system, we derive all the sampling intervals guaranteeing the invariance of the number of unstable zeros in each interval. To perform such an analysis, we adopt an eigenvalue perturbation-based approach. Various illustrative numerical examples complete the presentation. [ABSTRACT FROM AUTHOR]

The analysis of time-delay systems mainly relies on detecting and understanding the spectral values bifurcations when crossing the imaginary axis. This paper deals with the zero singularity, essentially when the zero spectral value is multiple. The simplest case in such a configuration is characterized by an algebraic multiplicity two and a geometric multiplicity one, known as the Bogdanov-Takens singularity. Moreover, in some cases the codimension of the zero spectral value exceeds the number of the coupled scalar-differential equations. Nevertheless, to the best of the author's knowledge, the bounds of such a multiplicity have not been deeply investigated in the literature. It is worth mentioning that the knowledge of such an information is crucial for nonlinear analysis purposes since the dimension of the projected state on the center manifold is none other than the sum of the dimensions of the generalized eigenspaces associated with spectral values with zero real parts. Motivated by a control-oriented problems, this paper provides an answer to this question for time-delay systems, taking into account the parameters' algebraic constraints that may occur in applications. We emphasize the link between such a problem and the incidence matrices associated with the Birkhoff interpolation problem. In this context, symbolic algorithms for LU-factorization for functional confluent Vandermonde as well as some classes of bivariate functional Birkhoff matrices are also proposed. [ABSTRACT FROM AUTHOR]

Abstract: When a time-delay system involves multiple imaginary roots (MIRs), the stability analysis will become much more complicated than that in the case with only simple imaginary roots (SIRs). An MIR may exhibit different splitting behaviors and, to the best of the authors’ knowledge, their properties have not been fully investigated. In this paper, we focus on characterizing the invariance properties for MIRs with any multiplicity. Furthermore, the proposed methodology makes it possible to also cover some degenerate cases already encountered and discussed in the literature. In addition, we propose an easily implemented frequency-sweeping method, making it possible to derive the asymptotic behavior without invoking the Puiseux series. [Copyright &y& Elsevier]

This work addresses the output feedback stabilisation problem for a class of linear single-input single-output systems subject to I/O network delays. More precisely, we are interested in the characterisation of the set of delay and gain parameters guaranteeing the stability of the closed-loop system. To perform such an analysis, we adopt aneigenvalue perturbationbased approach. Various illustrative numerical examples complete the presentation. [ABSTRACT FROM PUBLISHER]

Abstract: For linear systems with pointwise or distributed delay in the inputs which are stabilized through the reduction approach, we propose a new technique of construction of Lyapunov–Krasovskii functionals. These functionals allow us to establish the ISS property of the closed-loop systems relative to additive disturbances. [Copyright &y& Elsevier]

Abstract: For time-varying forward-complete nonlinear systems with delay in the input, a new reduction model approach is proposed. It presents three advantages. First, the corresponding control laws do not include distributed terms. Second, it yields closed-loop systems with positive solutions that can be easily derived. Finally, the stabilized systems possess some robustness properties that can be estimated. [Copyright &y& Elsevier]

This article focuses on the stabilisation problem of fluid-flow delay models of transmission control protocol/active queue management (TCP/AQM) networks by using a proportional-integral (PI) controller as AQM strategy. More precisely, the complete set of PI controllers that exponentially stabilises the corresponding linear time-delay system is derived. Using the particular geometric properties of this set of the controller parameters, the issues of robustness to uncertainty in the network parameters and to perturbation in the controller coefficients are addressed. Then, a methodology to compute a non-fragile PI AQM controller is provided. Finally, exponential estimates for the closed-loop system solutions, allowing to evaluate the performance of the corresponding PI-controlled closed-loop system, are proposed by using a Lyapunov-Krasovskii functional approach. An illustrative example completes the presentation. [ABSTRACT FROM AUTHOR]

This paper addresses a new control strategy for synchronizing two or more distributed and interconnected dynamic systems having communication time delays. The proposed strategy that uses the Smith predictor principle and delay information not only achieves synchronization but also preserves the natural local dynamics of each subsystem without being affected by the feedback nature of control. The proposed synchronization scheme is generalized to cases that deal with an arbitrary number of heterogeneous interconnected systems through dynamic scaling of input under a ring-type network configuration. In addition, possibility of applying the proposed scheme to nonlinear systems is discussed. Simulation and experimental tests are conducted to validate theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose various estimates of the attraction region for a class of nonlinear time-delay systems of the form The approach is constructive and makes use of a Lyapunov–Krasovskii functional associated to the linear part. Several illustrative examples (delayed logistic equation, stabilizing nonlinear oscillations by delayed output feedback, congestion control in high-performance networks and hereditary phenomena in physics) complete the presentation. [ABSTRACT FROM PUBLISHER]

In this paper, we present a geometric method for describing the effects of the ‘delay-induced uncertainty’ on the stability of a standard Smith predictor control scheme. The method consists of deriving the ‘stability crossing curves’ in the parameter space defined by the ‘nominal delay’ and ‘delay uncertainty’, respectively. More precisely, we start by computing the ‘crossing set’, which consists of all frequencies corresponding to all points on the stability crossing curve, and next we give their ‘complete classification’, including also the explicit characterization of the ‘directions’ in which the zeros cross the imaginary axis. This approach complements existing algebraic stability tests, and it allows some new insights in the stability analysis of such control schemes. Several illustrative examples are also included. [ABSTRACT FROM PUBLISHER]

In this paper, we propose a synchronization scheme to achieve a high level of consistency in peer-to-peer-based shared virtual environments (SVEs), as well as to display natural and realistic motions of virtual objects, in collaborative haptic tasks. The synchronization scheme utilizes an advanced feedback controller to compensate for the state error between geographically separated sites with a significant amount of time delay. It is designed using the mathematical model of a two-user SVE manipulating a freely moving object represented as a mass with damping resistance, with a haptic interface. Thanks to feedback control theory of time delay systems, the controller is shown to result in closed-loop stability and is be robust to perturbations in the time delay. Together with the synchronization control, a recovery filter is also designed and integrated so as to preserve the natural behavior of the synchronized object, which is, otherwise, affected by the feedback control action. In addition to verifying the theoretical results, two experiments using real Internet and local area network communications are carried out. These tests clearly support the validity of the analyses and demonstrate the applicability of the synchronization scheme. [ABSTRACT FROM AUTHOR]

This paper characterizes the stability crossing curves of a class of linear systems with gamma-distributed delay with a gap. First, we describe the crossing set, i.e., the set of frequencies where the characteristic roots may cross the imaginary axis as the parameters change. Then, we describe the corresponding stability crossing curves, i.e., the set of parameters such that there is at least one pair of characteristic roots on the imaginary axis. Such stability crossing curves divide the parameter space ℝ+2 defined by the mean delay and the gap into different regions. Within each such region, the number of characteristic roots on the right half complex plane is fixed. This naturally describes the regions of parameters where the system is stable. The classification of the stability crossing curves is also discussed. Some illustrative examples (Cushing equation in biology, traffic flow models in transportation systems, and control over networks of a simplified helicopter model) are also presented. [ABSTRACT FROM AUTHOR]

The problem of the asymptotic stability independent of delays for a class of linear systems including multiple delays is addressed. Both cases where the delays are allowed to vary independently of each other and where they are restricted to a one-dimensional subspace of the delay-parameter space are considered. It the latter case it turns out that the resulting dependency between the delays (rationally independent, rationally dependent, commensurate) plays an important role. The stability conditions are expressed in terms of the spectral properties of some appropriate complex matrices. As a consequence of the stability study, a complete characterization of the delay interference phenomenon is given. Furthermore, a connection is established with the stability theory for continuous-time delay-difference equations, subjected to delay perturbations. Various illustrative examples complete the paper. [ABSTRACT FROM AUTHOR]

Abstract: For the general linear scalar time-delay systems of arbitrary order with two delays, this article provides a detailed study on the stability crossing curves consisting of all the delays such that the characteristic quasipolynomial has at least one imaginary zero. The crossing set, consisting of all the frequencies corresponding to all the points in the stability crossing curves, are expressed in terms of simple inequality constraints and can be easily identified from the gain response curves of the coefficient transfer functions of the delay terms. This crossing set forms a finite number of intervals of finite length. The corresponding stability crossing curves form a series of smooth curves except at the points corresponding to multiple zeros and a number of other degenerate cases. These curves may be closed curves, open ended curves, and spiral-like curves oriented horizontally, vertically, or diagonally. The category of curves are determined by which constraints are violated at the two ends of the corresponding intervals of the crossing set. The directions in which the zeros cross the imaginary axis are explicitly expressed. An algorithm may be devised to calculate the maximum delay deviation without changing the number of right half plane zeros of the characteristic quasipolynomial (and preservation of stability as a special case). [Copyright &y& Elsevier]

This note focuses on the stability analysis of some classes of nonlinear time-delay models, encountered as fluid models for TCP/AQM network. By combining analytical and numerical tools, the attractors of these models, as well as the local and global behaviors of the solutions are studied. Among others, the presence of a chaotic attractor is shown, which supports the proposition that TCP itself as a deterministic process can cause or contribute to chaotic behavior in a network. The main goals of the paper are firstly to provide qualitative and quantitative information on the dynamics of the models under consideration, and secondly to illustrate the capabilities of computational tools for stability and bifurcation analysis of delay differential equations to analyze fluid flow models. [ABSTRACT FROM AUTHOR]

We address the output feedback stabilization problem of linear finite-dimensional SISO systems. Limitations of static time-invariant output feedback on stabilizability are well known. We investigate and compare the possibilities of two recently proposed simple modifications/generalizations of static time-invariant output feedback to remove such limitations. The first approach consists of introducing a time-delay in the control law, which can be treated as an additional control parameter. The second approach consists of making the gain time-varying. We show that both approaches are complementary. Existing theoretical results are brought together in a unifying framework. Numerical procedures for the construction of the controllers are provided. Robustness w.r.t. both parametric and delay uncertainty are dealt with. As an illustration the stabilizability of all second-order systems is completely determined. [ABSTRACT FROM PUBLISHER]

This paper addresses the control of time-delay systems whose relative degree does not exceed two. An adaptive Smith controller together with an adaptive law similar to the delay-free case is proposed. By using Lyapunov–Krasovskii functionals for an appropriate model transformation of the original system, semiglobal stability of the closed-loop system and asymptotic convergence of the output error is established. Strict positive realness together with the low relative degree of the plant is exploited to establish the stability properties. Robustness properties of the adaptive controller are briefly discussed. [Copyright &y& Elsevier]

Provides an overview of the stability and control of time-delay systems. Examples of time-delay systems; Stability criteria and performance measure; Feedback control.

This paper focuses on the problem of uniform asymptotic stability of a class of linear neutral systems. Sufficient conditions for delay-dependent stability are given in terms of the existence of solutions of some linear matrix inequalities. Furthermore, the proposed technique extends to neutral systems the results obtained for delay-difference equations using model transformations. Illustrative examples are included. [Copyright &y& Elsevier]

This paper focuses on the robustness analysis of a simple (second-order) control algorithm for data transfer in high-speed networks. The model under consideration (derived using a fluid approximation technique) can be found in Izmailov (SIAM J. Contr. Optimiz. 34 (1996) 1767). The novelty of the approach lies in characterizing the stability of the scheme in the delay-parameter space (where the delays correspond to the control-time interval, and to the round-trip time, respectively). Thus, we shall compute some delay-insensitive measures for the algorithm which give upper and lower bounds for the uncertainty in the knowledge of the round-trip times. [Copyright &y& Elsevier]

The article discusses various reports published within the issue, including one on identifiability and identification of delay systems and another on Lie algebras and Lyapunov transformations in the stability and control of some class nonlinear delay systems.

This paper focuses on Forced oscillations (periodic and almost periodic) in some dynamical models described by coupled differential and difference equations. The existence and stability result is based on a theorem on invariant manifolds on Banach spaces due to Kurzweil and Halanay. The state‐space (based on Liapunov–Krasovskii functionals combined with the S‐procedure) approach is considered. An illustrative example is presented and some conclusions are drawn concerning sharpness and ease in applications of the method. [ABSTRACT FROM PUBLISHER]

This paper focuses on the stability analysis of a class of neutral linear systems. Some sufficient delay-dependent stability conditions are derived in terms of an appropriate model transformation of the original system, transformation similar to the one used in the retarded case. The analysis makes use of an appropriate construction of a degenerate Lyapunov-Krasovskii functional. Several computational schemes and frequency-domain interpretations, as well as an example are also presented. [ABSTRACT FROM AUTHOR]

This note focuses on the problem of asymptotic stability and hyperbolicity of a class of linear systems described by delay-differential equations including commensurable delays. An unitary approach for the considered problems is proposed via a matrix-pencil technique. Necessary and sufficient conditions, delay-independent or delay-dependent, are given in terms of the generalized eigenvalue distribution of two constant and regular matrix pencils: one associated with finite time delays and the other one associated with infinite delay. Furthermore, the proposed results are easy to check in numerical examples. An example from the neural-network field has also been considered. [ABSTRACT FROM PUBLISHER]

This paper focuses on 'mixed' delay-independent/delay-dependent asymptotic stability problems of a class of linear systems described by delay-differential equations involving several constant but unknown delays. We give some sufficient conditions for characterizing unbounded stability regions in the delays parameter space. The proposed approach makes use of some appropriate Liapunov-Krasovskii functionals, and the results obtained are expressed in terms of matrix inequalities. We also discuss several ways to construct such analytic functionals. These results allow us to recover (or to improve) as limit cases previous delay-independent or/and delay-dependent conditions from the control literature. [ABSTRACT FROM AUTHOR]

Abstract: The structure of positive polynomials on a torus is derived from recent results of real algebraic geometry. As an application, we propose some simple conditions for testing the hyperbolicity/stability of a generic class of linear systems of retarded type. [Copyright &y& Elsevier]

In this work, we show that the coexistence of the maximal number of real spectral values of generic single-delay retarded second-order differential equations guarantees the realness of the rightmost spectral value. From a control theory standpoint, this entails that a delayed proportional-derivative (PD) controller can stabilize a delayed second-order differential equation. By assigning the maximum number of negative roots to the corresponding characteristic function (a quasipolynomial), we establish the conditions for asymptotic stability. If the assigned real spectral values are uniformly distributed, we specify a necessary and sufficient condition for the rightmost root to be negative, thus guaranteeing the exponential decay rate of the system's solutions. We illustrate the proposed design methodology in the delayed PD control of the damped harmonic oscillator. It is worth mentioning that this work represents a natural continuation of Amrane et al. (2018) and Bedouhene et al. (2020), addressing the problem of coexisting real spectral values for linear dynamical systems including delays in their models. [ABSTRACT FROM AUTHOR]

In this paper, which is a direct continuation and generalization of the recent works by the authors [17, 35], we show the validity of the generic multiplicity-induced-dominancy property for a general class of linear functional differential equations with a single delay, including the retarded as well as the neutral cases. The result is based on an appropriate integral representation of the corresponding characteristic quasipolynomial functions involving some appropriate degenerate hypergeometric functions. [ABSTRACT FROM AUTHOR]

An important question of ongoing interest for linear time-delay systems is to provide conditions on its parameters guaranteeing exponential stability of solutions. Recent works have explored spectral techniques to show that, for some low-order delay-differential equations of retarded type, spectral values of maximal multiplicity are dominant, and hence determine the asymptotic behavior of the system, a property known as multiplicity-induced-dominancy. This work further explores such a property and shows its validity for general linear delay-differential equations of retarded type of arbitrary order including a single delay in the system's representation. More precisely, an interesting link between characteristic functions with a real root of maximal multiplicity and Kummer's confluent hypergeometric functions is exploited. We also provide examples illustrating our main result. [ABSTRACT FROM AUTHOR]

This work exploits structural properties of a class of functional Vandermonde matrices to emphasize some qualitative properties of a class of linear autonomous nth order differential equation with forcing term consisting in the delayed dependent-variable. More precisely, it deals with the stabilizing effect of delay parameter coupled with the coexistence of the maximal number of real spectral values. The derived conditions are necessary and sufficient and, to the best of the authors’ knowledge, represent a novelty in the literature. Under appropriate conditions, such a configuration characterizes the spectral abscissa corresponding to the studied equation. A new stability criterion is proposed. This criterion extends recent results in factorizing quasipolynomial functions. The applicative potential of the proposed method is illustrated through the stabilization of coupled oscillators. [ABSTRACT FROM AUTHOR]

The article discusses several papers published within the issue including one by A. M. Perdon and M. Anderlucci about disturbance decoupling problem, another by K. Liu, V. Suplin and E. Fridman regarding stability analysis of linear systems and a paper by E. Witrant and colleagues on a model-based feedback control strategy for indoor temperature regulation.

A question of ongoing interest for linear Time-delay systems is to determine conditions on the equation parameters that guarantee the exponential stability of solutions. In recent works a new interesting property of time-delay systems was emphasized. As a matter of fact, the multiple spectral values for time-delay systems was characterized by using a Birkhoff/Vandermonde-based approach. Then, a multiplicity induced stability criteria were exhibited for reduced order systems; scalar delay-equations and a special class of second order systems. This work, further explores such a criteria and shows their applicability to the control of a mechanical system. [ABSTRACT FROM AUTHOR]

The event-driven model-free control is proposed in this article, which deals with the 'trade-off' between computational cost and system performance. Model-free controllers demand low computational resources and have high robustness, which is especially suitable for embedded systems with complex dynamics and/or affected by disturbances. The proposed method is implemented in two distinct nonlinear Multiple Input, Multiple Output (MIMO) motion models: a quadrotor model hovering in different weather circumstances and a vehicular friction control model operating in variable road surface conditions. Under the time- and event-driven schemes, the model-free control is compared with standard control strategies in various realistic scenarios. [ABSTRACT FROM AUTHOR]

This paper concerns some spectral properties of the scalar dynamical system defined by a linear delay-differential equation with two positive delays. More precisely, the existing links between the delays and the maximal multiplicity of the characteristic roots are explored, as well as the dominance of such roots compared with the spectrum localization. As a by-product of the analysis, the pole placement issue is revisited with more emphasis on the role of the delays as control parameters in defining a partial pole placement guaranteeing the closed-loop stability with an appropriate decay rate of the corresponding dynamical system. [ABSTRACT FROM AUTHOR]

We contribute to the perturbation theory of nonlinear eigenvalue problems in three ways. First, we extend the formula for the sensitivity of a simple eigenvalue with respect to a variation of a parameter to the case of multiple nonsemisimple eigenvalues, thereby providing an explicit expression for the leading coefficients of the Puiseux series of the emanating branches of eigenvalues. Second, for a broad class of delay eigenvalue problems, the connection between the finite-dimensional nonlinear eigenvalue problem and an associated infinite-dimensional linear eigenvalue problem is emphasized in the developed perturbation theory. Finally, in contrast to existing work on analyzing multiple eigenvalues of delay systems, we develop all theory in a matrix framework, i.e., without reduction of a problem to the analysis of a scalar characteristic quasi-polynomial. [ABSTRACT FROM AUTHOR]

A recent work by Mazenc and Malisoff provides a trajectory-based approach for proving stability of time-varying systems with time-varying delays. Here, we provide several significant applications of their approach. In two results, we use a Lyapunov function for a corresponding undelayed system to provide a new method for proving stability of linear continuous-time time-varying systems with bounded time-varying delays. We allow uncertainties in the coefficient matrices of the systems. Our main results use upper bounds on an integral average involving the delay. The results establish input-to-state stability with respect to disturbances. We also provide a novel reduction model approach that ensures global exponential stabilization of linear systems with a time-varying pointwise delay in the input, which allows the delay to be discontinuous and uncertain. Finally, we provide an alternative to the reduction model method, based on a different dynamic extension. Our examples demonstrate the usefulness of our findings in several settings. [ABSTRACT FROM AUTHOR]

We study a class of continuous-time nonlinear systems with discrete measurements, model uncertainty, and sensor noise. We provide an estimator of the state for which the observation error enjoys a variant of the exponential input-to-state stability property with respect to the model uncertainty and sensor noise. A valuable novel feature is that the overshoot term in this stability estimate only involves a recent history of uncertainty values. Also, the rate of exponential convergence can be made arbitrarily large by reducing the supremum of the sampling intervals. Our proof uses a recently developed trajectory based approach. We illustrate our work using a model for a pendulum whose suspension point is subjected to an unknown time-varying bounded horizontal oscillation. [ABSTRACT FROM AUTHOR]

This paper addresses set invariance properties for linear time-delay systems. More precisely, the first goal of the article is to review known necessary and/or sufficient conditions for the existence of invariant sets with respect to dynamical systems described by linear discrete time-delay difference equations (dDDEs). Secondly, we address the construction of invariant sets in the original state space (also called D -invariant sets) by exploiting the forward mappings. The notion of D -invariance is appealing since it provides a region of attraction, which is difficult to obtain for delay systems without taking into account the delayed states in some appropriate extended state space model. The present paper contains a sufficient condition for the existence of ellipsoidal D -contractive sets for dDDEs, and a necessary and sufficient condition for the existence of D -invariant sets in relation to linear time-varying dDDE stability. Another contribution is the clarification of the relationship between convexity (convex hull operation) and D - invariance of linear dDDEs. In short, it is shown that the convex hull of the union of two or more D - invariant sets is not necessarily D - invariant , while the convex hull of a non-convex D - invariant set is D - invariant . [ABSTRACT FROM AUTHOR]

The paper considers the problem of stabilization of systems possessing a multiple zero eigenvalue at the origin. The controller that we propose, uses multiple delayed measurements instead of derivative terms. Doing so, we increase the performances of the closed loop in presence of system uncertainties and/or noisy measurements. The problem formulation and the analysis is presented through a classical engineering problem which is the stabilization of an inverted pendulum on a cart moving horizontally. On one hand, we perform a nonlinear analysis of the center dynamics described by a three dimensional system of ordinary differential equations with a codimension-three triple zero bifurcation. On the other hand, we present the complementary stability analysis of the corresponding linear time invariant system with two delays describing the behavior around the equilibrium. The aim of this analysis is to characterize the possible local bifurcations. Finally, the proposed control scheme is numerically illustrated and discussed. [ABSTRACT FROM AUTHOR]

SUMMARY Interval observers are constructed for discrete-time systems. First, time-invariant interval observers are proposed for a family of nonlinear systems. Second, it is shown that, for any time-invariant exponentially stable discrete-time linear system with additive disturbances, time-varying exponentially stable discrete-time interval observers can be constructed. The latter result relies on the design of time-varying changes of coordinates, which transform a linear system into a nonnegative one. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]