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• 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]

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]

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 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]

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]

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]

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]

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]

We studied the existence of positively invariant sets for linear delay-difference equations. In particular, we regarded two strong stability notions: robust (with respect to delay parameter) asymptotic stability for the discrete-time case and delay-independent stability for the continuous-time case. The correlation between these stability concepts is also considered. Furthermore, for the delay-difference equations with two delay parameters, we provided a computationally efficient numerical routine which is necessary to guarantee the existence of contractive sets of Lyapunov-Razumikhin type. This condition also appears to be necessary and sufficient for the delay-independent stability and sufficient for the robust asymptotic stability. [ABSTRACT FROM AUTHOR]

This paper focuses on the location of the non-asymptotic zeros of Whittaker and Kummer confluent hypergeometric functions. Based on a technique proposed by E. Hille for the analysis of solutions of some second-order ordinary differential equations, we characterize the sign of the real part of zeros of Whittaker and Kummer functions and provide estimates on the regions of the complex plane where those zeros can be located. Our main result is a correction of a previous statement by G. E. Tsvetkov whose propagation has induced mistakes in the literature. In particular, we review some results of E. B. Saff and R. S. Varga on the error of Padé's rational approximation of the exponential function, which are based on the latter. [ABSTRACT FROM AUTHOR]

Abstract: For a family of nonlinear discrete-time systems with input, output and uncertain terms, a new interval observer is designed. Its main feature is that it is composed of two copies of classical observers. This interval observer applies in the presence of unknown bounded nonlinear terms and additive disturbances and is used to achieve asymptotic stability through an appropriate choice of dynamic output feedback. An illustrative example completes the presentation. [Copyright &y& Elsevier]

Abstract: Congestion control is a fundamental building block in packet switching networks such as the Internet due to the fact that communication resources are shared. It has been shown that the plant dynamics is essentially made up of an integrator plus time delay and that a proportional controller plus a Smith predictor defines a simple and effective controller. It has also been shown that the TCP congestion control can be modelled using a Smith predictor plus a proportional controller. Due to the importance of this control structure in the field of data network congestion control, we analyse the robust stability of the closed-loop system in the face of delay uncertainties that are present in data networks due to queuing. In particular, by applying a geometric approach, we derive a bound on the proportional controller gain which is necessary and sufficient to guarantee the closed-loop stability for a given bound on the delay uncertainty. [Copyright &y& Elsevier]

Abstract: This paper presents a dynamic model of the helium flow in the cryogenic distribution line (QRL) used in the Large Hadron Collider (LHC) at CERN. The study is focused on the return pumping line, which transports gaseous helium at low pressure and temperature over . Our aim is to propose a new real-time model of the QRL while taking into account the non-homogeneous transport phenomena. The flow model is based on 1D Euler equations and considers convection heat transfers, hydrostatic pressure and friction pressure drops. These equations are discretized using a finite difference method based on an upwind scheme. A specific model for the interconnection cells is also proposed. The corresponding simulation results are compared with experimental measurements of a heat wave along the line that results from a quench of a superconducting magnet. Different hypotheses are presented and the influence of specific parameters is discussed. [Copyright &y& Elsevier]

Abstract: This paper presents a process and control simulator for industrial helium cryogenic plants controlled by Programmable Logic Controllers (PLC). This simulator can be used for different purposes such as operator training, test of the PLC programs or the optimization of the plant. The different component models used in the simulator are detailed and explained. Various large scale cryogenic plants used for the particle accelerator LHC (Large Hadron Collider) at CERN have been modeled and simulated. The good agreement between the simulation results and the dynamic behavior of real plants is demonstrated with experimental results. Various discussions complete the presentation. [Copyright &y& Elsevier]

In this paper, the output feedback stabilization of an underactuated chain of scalar hyperbolic systems coupled at one end with a finite-dimensional system is addressed. As encountered in practical applications, only the first PDE subsystem is actuated. The measurements are collocated with the actuation. The proposed approach uses a recursive dynamics interconnection framework. More precisely, we first estimate (delayed) values of the states at each subsystem's boundaries. Then, we design a state-predictor that gives access to the boundary states' present and future values. It becomes finally possible to design a state-observer for the entire system. Combining this observer with a stabilizing state-feedback law, we can recursively design an output-feedback control law that stabilizes the whole chain. Some illustrative examples complete the presentation. [ABSTRACT FROM AUTHOR]

Abstract: This paper studies the stability of 2-D dynamic systems. We consider systems characterized by 2-D polynomials and 2-D state–space descriptions. For each description, we derive necessary and sufficient stability conditions, which all require only the computation of a constant matrix pencil. The stability of the underlying system can then be determined by inspecting the generalized eigenvalues of the matrix pencil. The results consequently yield 2-D stability tests that can be checked both efficiently and with high precision. Additionally, frequency-sweeping tests are also obtained which complement the matrix-pencil tests and are likely to be more advantageous analytically. [Copyright &y& Elsevier]

The problem of globally asymptotically stabilizing by bounded feedback of an oscillator with an arbitrary large delay in the input is solved. A first solution follows from a general result on the global stabilization of null controllable linear systems with delay in the input by bounded control laws with a distributed term. Next, it is shown through a Lyapunov analysis that the stabilization can be achieved as well when neglecting the distributed terms. It turns out that this main result is intimately related to the output feedback stabilization problem. [Copyright &y& Elsevier]

This paper presents a review of past and present results and approaches in the area of motion planning using MIP (Mixed-integer Programming). Although in the early 2000s MIP was still seen with reluctance as method for solving motion planning-related problems, nowadays, due to increases in computational power and theoretical advances, its extensive modeling capabilities and versatility are coming to the fore and enjoy increased application and appreciation. This class of control problems involves, essentially, either a selection from a limited number of alternatives or a constrained optimization problem over a non-convex domain. In both situations, MIP has proven to be an efficient modeling technique as it will be shown in the present review paper. Furthermore, an emphasis is laid on the existing alternatives for implementation and on various experimental validations documented in the literature. [ABSTRACT FROM AUTHOR]

Abstract In this paper we derive explicit lower bounds on the delay margin of second-order unstable delay systems achievable by PID control, which provide a priori a guaranteed range of delay values over which a second-order delay plant can be stabilized by a PID, and more generally, a finite-dimensional LTI controller. This is done by deriving lower bounds on the delay margin achievable by PD controllers, whereas this margin constitutes a lower bound on that achievable by PID controllers. Analysis shows that in most cases our results can be significantly less conservative than the lower bounds obtained elsewhere using more sophisticated, general LTI controllers. [ABSTRACT FROM AUTHOR]

This paper studies the migration of double imaginary roots of the systems’ characteristic equation when two parameters are subjected to small deviations. The proposed approach covers a wide range of models. Under the least degeneracy assumptions, we found that the local stability crossing curve has a cusp at the point that corresponds to the double root, and it divides the neighborhood of this point into an S-sector and a G-sector. When the parameters move into the G-sector, one of the roots moves to the right half-plane, and the other moves to the left half-plane. When the parameters move into the S-sector, both roots move either to the left half-plane or the right half-plane depending on the sign of a quantity that depends on the characteristic function and its derivatives up to the third order. [ABSTRACT FROM AUTHOR]

This article presents basic concepts and recent research directions about the stability of sampled-data systems with aperiodic sampling. We focus mainly on the stability problem for systems with arbitrary time-varying sampling intervals which has been addressed in several areas of research in Control Theory. Systems with aperiodic sampling can be seen as time-delay systems, hybrid systems, Input/Output interconnections, discrete-time systems with time-varying parameters, etc. The goal of the article is to provide a structural overview of the progress made on the stability analysis problem. Without being exhaustive, which would be neither possible nor useful, we try to bring together results from diverse communities and present them in a unified manner. For each of the existing approaches, the basic concepts, fundamental results, converse stability theorems (when available), and relations with the other approaches are discussed in detail. Results concerning extensions of Lyapunov and frequency domain methods for systems with aperiodic sampling are recalled, as they allow to derive constructive stability conditions. Furthermore, numerical criteria are presented while indicating the sources of conservatism, the problems that remain open and the possible directions of improvement. At last, some emerging research directions, such as the design of stabilizing sampling sequences, are briefly discussed. [ABSTRACT FROM AUTHOR]

The PID control is favored in controlling industrial processes for its ease of implementation. In this paper we present an analytical study, with an aim at stability robustness, tracking performance, and pole placement of first-order unstable plants under PID control. We employ a multiplicity-induced dominancy (MID) strategy to design and tune PID controller gains, which is shown to possess desirable properties that ensure specified decay rate of the closed-loop response and the stability robustness of the closed-loop system in spite of variations in the delay parameter. We also study the tradeoff between delay robustness and tracking performance of PID controllers. We show that under the constraint of steady-state tracking, the maximal delay robustness range can be computed by solving a unimodal pseudo-concave optimization problem. [ABSTRACT FROM AUTHOR]

Abstract: This paper addresses the implementation of a predictive control strategy for Unmanned Air Vehicles in the presence of bounded disturbances. The goal is to prove the feasibility of such a real-time optimization-based control design and to demonstrate its tracking capabilities for the nonlinear dynamics with respect to a reference trajectory which is pre-specified via differential flatness. In order to benefit from the computational advantages of the linear predictive control formulations, an off-line linearization strategy of the nonlinear model of the vehicle along the flat trajectory is employed. The proposed method exhibits effective performance validated through software-in-the-loop simulations and real flight tests on different Unmanned Aerial Vehicles (UAVs). [Copyright &y& Elsevier]

Abstract: In this paper, a PD controller design for haptic systems under delayed feedback is considered. More precisely, a complete stability analysis of a haptic system where local dynamics are described by some second-order mechanical dynamics is presented. Next, using two optimization techniques ( and stability, margin optimization) an optimal choice for the controller gains is proposed. The derived results are tested on a three degree-of-freedom real-time experimental platform to illustrate the theoretical results. [Copyright &y& Elsevier]

Abstract: This paper aims to provide a numerical algorithm able to locate all unstable poles, and therefore the characterization of the stability as a function of the delay, for a class of linear fractional-order neutral systems with multiple commensurate delays. We start by giving the asymptotic position of the chains of poles and the conditions for their stability for a small delay. When these conditions are met, the root continuity argument and some simple substitutions allow us to determine the locations where some roots cross the imaginary axis, providing therefore the complete characterization of the stability windows. The same method can be extended to provide the position of all unstable poles as a function of the delay. [Copyright &y& Elsevier]

Abstract: This technical communique introduces a new concept of set invariance with respect to linear discrete time dynamics affected by delay. We are interested in the definition and characterization of sequences of cyclically invariant subsets in the state space. The algebraic conditions established in the late ’80s for linear dynamics are generalized to invariance analysis in the presence of delays for given sequences of polyhedral sets. [Copyright &y& Elsevier]

Abstract: Definitions for pseudospectra and stability radii of an analytic matrix function are given, where the structure of the function is exploited. Various perturbation measures are considered and computationally tractable formulae are derived. The results are applied to a class of retarded delay differential equations. Special properties of the pseudospectra of such equations are determined and illustrated. [Copyright &y& Elsevier]

In most of the numerical examples of time-delay systems proposed in the literature, the number of unstable characteristic roots remains positive before and after a multiple critical imaginary root (CIR) appears (as the delay, seen as a parameter, increases). This fact may lead to some misunderstandings: (i) A multiple CIR may at most affect the instability degree; (ii) It cannot cause any stability reversals (stability transitions from instability to stability). As far as we know, whether the appearance of a multiple CIR can induce stability is still unclear (in fact, when a CIR generates a stability reversal has not been specifically investigated). In this paper, we provide a finer analysis of stability reversals and some new insights into the classification: the link between the multiplicity of a CIR and the asymptotic behavior with the stabilizing effect. Based on these results, we present an example illustrating that a multiple CIR's asymptotic behavior is able to cause a stability reversal. To the best of the authors' knowledge, such an example is a novelty in the literature on time-delay systems. [ABSTRACT FROM AUTHOR]