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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]

This work considers a class of delay eigenvalue problems that admit a spectrum similar to that of a Hamiltonian matrix, in the sense that the spectrum is symmetric with respect to both the real and imaginary axes. We propose a method to iteratively approximate the eigenvalues closest to a given purely real or imaginary shift, while preserving the symmetries of the spectrum. To this end, our method exploits the equivalence between the considered delay eigenvalue problem and the eigenvalue problem associated with a linear but infinite-dimensional operator. To compute the eigenvalues closest to the given shift, we apply a specifically chosen shift-invert transformation to this linear operator and compute the eigenvalues with the largest modulus of the new shifted and inverted operator using an (infinite) Arnoldi procedure. The advantage of the chosen shift-invert transformation is that the spectrum of the transformed operator has a "real skew-Hamiltonian"-like structure. Furthermore, it is proven that the Krylov subspace constructed by applying this operator satisfies an orthogonality property in terms of a specifically chosen bilinear form. By taking this property into account during the orthogonalization process, it is ensured that, even in the presence of rounding errors, the obtained approximation for, e.g., a simple, purely imaginary eigenvalue is simple and purely imaginary. The presented work can thus be seen as an extension of [V. Mehrmann and D. Watkins, SIAM J. Sci. Comput., 22 (2001), pp. 1905-1925] to the considered class of delay eigenvalue problems. Although the presented method is initially defined on function spaces, it can be implemented using finite-dimensional linear algebra operations. The performance of the resulting numerical algorithm is verified for two example problems: the first example illustrates the advantage of the proposed approach in preserving purely imaginary eigenvalues when working in finite precision, while the second one demonstrates its applicability to a large scale problem. [ABSTRACT FROM AUTHOR]

The fundamental notion of relative degree is studied for a class of linear timedelay systems of retarded type, where the standard assumption of commensurate delays is dropped. Algebraic conditions are provided that fully exploit the delay dependence structure. Even though the analyzed systems do not have a nontrivial feedthrough from input to output, it is shown that the relative degree may be sensitive to infinitesimal delay perturbations, which is at the basis of a novel notion of relative degree, called strong relative degree. This notion is characterized algebraically and computationally in both a SISO and a MIMO setting. By means of the obtained characterizations and a benchmark problem, which illustrates that invariant zeros may be characterized as zeros of quasi-polynomials of retarded, neutral, or advanced type, new light is shed on existence conditions of a normal form. The novel concepts and theoretical results play a key role in the design and analysis of extended PD controllers, for which an algorithm and numerical experiments are presented. [ABSTRACT FROM AUTHOR]

We present an active control method for achieving non-collocated vibration absorption, implemented on a lumped mass system. The task of suppressing vibrations at a number of frequencies, at a target mass is recast into a problem of assigning zeros of the transfer function from the external excitation force to the position of the target mass. The controller design involves computing the controller parameters by utilizing the available output feedback to directly assign a set of zeros on the imaginary axis while simultaneously preserving the overall stability with a sufficient margin. The design also takes into account delays in the feedback path. The presence of delays leads to an infinite dimensional system for which, there exists in general, infinitely many characteristic roots. The requirement of achieving sufficient damping in the resulting closed-loop system together with suppressing vibrations of the desired frequency leads to a constrained optimization problem of minimizing the spectral abscissa function subject to zero-location constraints. These constraints exhibit polynomial dependence on the controller parameters. We present two approaches for controller parameterization for the single and multiple input case. Both approaches are based on constraint elimination and exploit the property that the constraints are affine in a selection of the gain parameters. Finally, the methodology is verified by simulating on a spring–mass–damper system following which, it is then validated experimentally on a laboratory setup. [ABSTRACT FROM AUTHOR]

In this article, we are concerned with the output synchronization of a group of heterogeneous agents with nonlinear dynamics. It is shown that via dynamic coupling, the output synchronization problem is transformed into a family of output homogenization and internal-model-based synchronization problems. In contrast to conventional approaches, the adopted approach does not rely on the generation of a specific reference trajectory, to be tracked by all agents. Based on this premise, we propose a distributed state-feedback protocol driving all the heterogeneous multiagent individual systems onto a copy of a virtual reference signal. Invoking the results from the theory of output regulation and cooperative control, the state-feedback controller synthesis is conducted to achieve the output synchronization of heterogeneous multiagent networks. Immediate extensions of the obtained results to observer-based control are also discussed. The effectiveness of the proposed output synchronization algorithm is verified through theoretical analysis and case studies. [ABSTRACT FROM AUTHOR]

Summary: This paper studies the problem of sensor fault estimation for linear systems in the presence of unknown inputs in both input and output channels. Relying on matrix equation theory, the unknown inputs are first removed from the output channel. Second, an augmented descriptor system is constructed by treating both system state and sensor fault as (pseudo)‐state variables. Third, the augmented descriptor system is 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. This matrix equation based approach is then complemented by an alternative one, which relies on successive transformations of state, input and outputs. This approach sheds light on the adopted solvability conditions throughout the paper and on the role of different directions in the output space in handling unknown inputs, sensor faults and the core state estimation problem. Finally, two numerical examples are presented to illustrate the effectiveness of the proposed methodologies. [ABSTRACT FROM AUTHOR]

AbstractThis article deals with the analysis and design of an active spatial mechanical vibration absorber and verification of its performance in collocated and non-collocated vibration absorption of a spatial flexible primary structure. The considered active absorber with six degrees of freedom has six identical natural frequencies. It is a so-called unifrequency absorber. The unifrequency property is achieved by delayed feedback, according to the concept of a delayed resonator. A proper tuning of the delayed feedback compensates the damping and variations in stiffness of mechanical parts, resulting in the behavior of an ideal absorber. The absorber was attached to a three-dimensional flexible frame with many degrees of freedom transformed to a reduced order model in modal coordinates. The use of a unifrequency absorber for two types of tasks is shown in this article. In the collocated case, the absorber is placed at the location where we want to suppress vibration. Such a placement is often not possible for various reasons. Therefore, the non-collocated absorber is also investigated which is remote from the point of desired vibration suppression. The absorber has considerable potential for use in improving the dynamic behavior of robots, complex rotor systems, and other mechanical systems suffering from unwanted fully spatial vibrations. [ABSTRACT FROM AUTHOR]

This paper provides a mathematical characterization of the robust (strong) H-infinity norm of an uncertain linear time-invariant system with discrete delays in terms of the robust distance to instability of an associated characteristic matrix. The considered class of uncertainties consists of real-valued, structured, Frobenius norm-bounded matrix uncertainties that act on the coefficient matrices. The robust H-infinity norm, defined as the worst-case value of the H-infinity norm over all admissible uncertainty values, is an important measure of robust performance to quantify the worst-case disturbance rejection of an uncertain dynamical system. For the considered system class, this robust H-infinity norm is however a fragile measure, as for a particular instance of the uncertainties, the H-infinity norm might be sensitive to arbitrarily small perturbations on the delays. Therefore, we introduce the robust strong H-infinity norm, inspired by the notion of strong stability of delay differential equations of neutral type, which takes into account both the uncertainties on the system matrices and infinitesimal delay perturbations. This quantity is a continuous function of both the elements of the system matrices and the delays. The main contribution of this work is the introduction of a relation between this robust strong H-infinity norm and the robust distance to instability of an associated characteristic matrix. This relation is subsequently employed in a novel algorithm for computing the robust strong H-infinity norm of uncertain time-delay systems. [ABSTRACT FROM AUTHOR]

Summary: We present novel theoretical concepts for linear time‐periodic systems with multiple delays, which are closely related to the spectral properties and Lyapunov matrices. At the basis of the main results is the associated dual system, constructed by transposition of the systems matrices and affine transformations of their arguments. We introduce, for the first time, the concepts of the ℋ2 norm and the dual Lyapunov matrix of periodic systems with delays. We show that the primal and dual system have the same ℋ2 norm, characterized by primal and dual delay Lyapunov equations, which extend the well‐known results for time‐invariant systems with delays, and periodic systems without delays. Having at hand the pair of primal‐dual Lyapunov matrices, along with some energy interpretations, allow us to generalize the concept of position balancing and explore its potential for model reduction. The obtained results are illustrated by several examples, including the delayed Mathieu equation. [ABSTRACT FROM AUTHOR]

We develop an eigenvalue-based approach for the stability assessment and stabilization of linear systems with multiple delays and periodic coefficient matrices. Delays and period are assumed commensurate numbers, such that the Floquet multipliers can be characterized as eigenvalues of the monodromy operator and by the solutions of a finite-dimensional nonlinear eigenvalue problem, where the evaluation of the characteristic matrix involves solving an initial value problem. We demonstrate that such a dual interpretation can be exploited in a two-stage approach for computing dominant Floquet multipliers, where global approximation is combined with local corrections. Correspondingly, we also propose two novel characterizations of left eigenvectors. Finally, from the nonlinear eigenvalue problem formulation, we derive computationally tractable expressions for derivatives of Floquet multipliers with respect to parameters, which are beneficial in the context of stability optimization. Several numerical examples show the efficacy and applicability of the presented results. [ABSTRACT FROM AUTHOR]

This paper is concerned with polynomial approximations of the spectral abscissa function (defined by the supremum of the real parts of the eigenvalues) of a parameterized eigenvalue problem, which are closely related to polynomial chaos approximations if the parameters correspond to realizations of random variables. Unlike previous work, we highlight the major role of this function smoothness properties. Even if the eigenvalue problem matrices are analytic functions of the parameters, the spectral abscissa function may not be differentiable, and even non-Lipschitz continuous, due to multiple rightmost eigenvalues counted with multiplicity. This analysis demonstrates smoothness properties not only heavily affect the approximation errors of the Galerkin and collocation based polynomial approximations, but also the numerical errors in the evaluation of coefficients in the Galerkin approach with integration methods. A documentation of the experiments, conducted on the benchmark problems through the software Chebfun, is publicly available. [ABSTRACT FROM AUTHOR]

Summary: We introduce a new robust stability measure for systems with multiple pointwise delays, which is called smoothed spectral abscissa, consistently with the existing measure for delay‐free systems. Its main characteristics are that it is smooth with respect to the system parameters and it provides a trade‐off between the decay rate of the system solution and the H2 norm of a transfer matrix related with the system. The smoothed spectral abscissa is implicitly defined in terms of the H2 norm of an auxiliary system, and its computation is based on the so‐called delay Lyapunov matrix. We show that these features make the smoothed spectral abscissa suitable for the design of robust controllers by using standard gradient‐based optimization techniques and exploiting a novel characterization of the derivatives of the delay Lyapunov matrix with respect to the system parameters. [ABSTRACT FROM AUTHOR]

A novel approach for the design of delayed output-feedback controllers that optimize a quadratic cost function is presented. The proposal is based on the formulation of the optimization of the quadratic performance criterion as a minimization problem whose objective function is given in terms of the so-called delay Lyapunov matrix. The computation of the delay Lyapunov matrix sensitivity with respect to the controller gains allows obtaining the gradient of the objective function and solving the minimization problem within a gradient-based optimization framework. The potential of the approach is shown by some examples of practical relevance. [ABSTRACT FROM AUTHOR]

A delay Lyapunov matrix corresponding to an exponentially stable system of linear time-invariant delay differential equations can be characterized as the solution of a boundary value problem involving a matrix valued delay differential equation. This boundary value problem can be seen as a natural generalization of the classical Lyapunov matrix equation. Lyapunov matrices play an important role in constructing Lyapunov functionals and in H2 optimal control. In this paper we present a general approach for computing delay Lyapunov matrices and H2 norms for systems with multiple discrete delays, whose applicability extends toward problems where the matrices are large and sparse, and the associated positive semidefinite matrix has a low rank. The problems addressed are challenging, because the boundary value problem is matrix valued with a structure that is much harder to exploit than in the delay-free case, and its solution is in the generic situation nonsmooth. In contrast to existing methods that are based on solving the boundary value problem directly, our method is grounded in solving standard Lyapunov equations of increased dimensions. It combines several ingredients: (i) a spectral discretization of the system of delay equations, (ii) a targeted similarity transformation which induces a desired structure and sparsity pattern and, at the same time, favors accurate low-rank solutions of the corresponding Lyapunov equation, and (iii) a Krylov method for large-scale matrix Lyapunov equations. The structure of the problem is exploited in such a way that the final algorithm does not involve a preliminary discretization step and provides a fully dynamic construction of approximations of increasing rank. Interpretations are also given in terms of a projection method directly applied to a standard linear infinite-dimensional system, which is equivalent to the original time-delay system. Throughout the paper two didactic examples are used to illustrate the properties of the problem, the challenges and methodological choices, while numerical experiments are reported to illustrate the effectiveness of the algorithms. [ABSTRACT FROM AUTHOR]

This paper presents a novel algorithm to compute the real stability radius for a linear delay system of retarded type with multiple delays. The real stability radius is the distance to instability, measured as the minimal real-valued perturbation that renders the system unstable. Our method is based on characterizing this distance to instability as the inverse of the global maximum of a real structured singular value function. We develop a criss-cross type algorithm that globally converges to this maximum, and whose convergence rate seems to be superlinear and sometimes quadratic in numerical experiments. The algorithm exploits that the intersections of these singular value functions with constant functions can be written as purely imaginary eigenvalues of certain delay eigenvalue problems (DEP) with positive and negative delays. This is an extension of the well-known linear case (without delays) where this results in algebraic eigenvalue problems. In addition, a novel numerical solver to compute all the imaginary eigenvalues of this DEP is also presented. It combines an approximation using spectral discretizations and an automatic procedure to determine the required number of discretization points. Finally, due to the presence of multiple eigenvalues at the maximum, these approximations are corrected with a block-Newton algorithm for nonlinear eigenvalue problems. [ABSTRACT FROM AUTHOR]

A complex methodology for control of flexible multi-body systems is proposed with the objective to achieve a favorable distribution of system motion so that the oscillatory mode of the flexible part is not excited. As the key element, the recently proposed concept of a feedback loop with an inverse distributed delay shaper is adopted. Unlike in existing works, the mutual coupling between the primary (controlled) structure and secondary (flexible) structure of oscillatory nature is explicitly taken into account in the controller design. First, an easy to apply method to isolate the flexible mode to be targeted in the shaper design is proposed. Secondly, the interconnection of the system, shaper and the controller is formulated as a set of delay differential algebraic equations. Then the spectral optimization control design technique is applied to achieve fast dynamics of the infinite dimensional system. The viability of the overall methodology is validated by both simulations and experiments in an extensive case study example. [ABSTRACT FROM AUTHOR]

The authors present an integrated approach for designing low‐order multi‐objective controllers for linear time‐delay systems (TDSs), combining recently developed methods for reduction of delay systems and fixed‐order control design, respectively. As a benchmark problem for process control applications, a model of an experimental heat transfer setup is used, which serves to motivate the adopted combination of model reduction and control technique. The former corresponds to a Krylov based reduction procedure, which is generalised to multiple‐input‐multiple‐output systems with state, input and output delays, and allows the adaptive construction of an accurate low‐order linear time‐invariant approximation of the original linear TDS. Concerning the latter, a fixed‐order controller is synthesised for the delay‐free approximation, exploiting a recently proposed linear matrix inequality based framework for fixed‐order controller design, and validated on the original model. In this way, the systematic overall design procedure, which is grounded in convex optimisation, complements approaches based on directly optimising stability and performance measures as a function of the controller parameters, which may lead to highly nonconvex, even non‐smooth optimisation problems. The successful design of a fixed‐order multi‐ H2 controller is validated on the benchmark problem, confirming the potential of the adopted approach for realistic industrial applications. [ABSTRACT FROM AUTHOR]

We present an algorithm to compute the pseudospectral abscissa for a nonlinear eigenvalue problem. The algorithm relies on global under-estimator and over-estimator functions for the eigenvalue and singular value functions involved. These global models follow from eigenvalue perturbation theory. The algorithm has three particular features. First, it converges to the globally rightmost point of the pseudospectrum, and it is immune to nonsmoothness. The global convergence assertion is under the assumption that a global lower bound is available for the second derivative of a singular value function depending on one parameter. It may not be easy to deduce such a lower bound analytically, but assigning large negative values works robustly in practice. Second, it is applicable to large-scale problems since the dominant cost per iteration stems from computing the smallest singular value and associated singular vectors, for which efficient iterative solvers can be used. Furthermore, a significant increase in computational efficiency can be obtained by subspace acceleration, that is, by restricting the domains of the linear maps associated with the matrices involved to small but suitable subspaces, and solving the resulting reduced problems. Occasional restarts of these subspaces further enhance the efficiency for large-scale problems. Finally, in contrast to existing iterative approaches based on constructing low-rank perturbations and rightmost eigenvalue computations, the algorithm relies on computing only singular values of complex matrices. Hence, the algorithm does not require solutions of nonlinear eigenvalue problems, thereby further increasing efficiency and reliability. This work is accompanied by a robust implementation of the algorithm that is publicly available. [ABSTRACT FROM AUTHOR]

This paper is concerned with an eigenvalue based stabilization method for linear autonomous delay systems with uncertain parameters, modeled by a random vector. The dependence of the characteristic matrix on the uncertain parameters can be nonlinear; also delay values can be subject to uncertainty. Unlike the stability optimization methods for deterministic problems, which minimize the spectral abscissa, our approach shows better robust properties based on a more realistic model, where the uncertainty is taken into account by minimizing an objective function, consisting of the mean of the spectral abscissa with a variance penalty. The properties of the spectrum of delay differential algebraic equations of retarded type are analyzed and an integration method to compute the mean and variance of the spectral abscissa, which describes the stability property of the system, is presented. Subsequently the analysis of the behavior of the gradient of the characteristic roots is addressed, and a suitable explicit formula of the gradient of the objective function is presented. Finally, for designing the controller a suitable optimization routine, requiring the objective function and its gradient, is presented. The efficacy of the method is illustrated with numerical examples, including a model of an experimental heat-exchanger. The algorithms developed are publicly available. [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]

Basins of attraction are instrumental to study the effect of input saturation in control systems, as these sets characterise the initial conditions for which the control strategy induces attraction to the desired state. In this paper, we describe these sets when the open-loop system is exponentially unstable and the system is controlled by actuators with both constant time-delays and saturation. Estimates of the basin of attraction are provided and the allowable time-delay in the control loop is determined with a novel piecewise quadratic Lyapunov–Krasovskii functional that exploits the piecewise affine nature of the system. As this approach leads to sufficient, but not to necessary conditions for attractivity, we present simulations for two examples to show the applicability of the results. [ABSTRACT FROM AUTHOR]

The absorption properties and active actuation of two-dimensional delayed resonators (2D-DR) are studied from various aspects. For a general 2D-DR, the actuation forces and energy supply needed for entire vibration absorption of a platform excited by a periodic force and torque are determined. For this purpose, a model describing the full absorber-platform dynamics is transformed into a form that allows to express the relevant design quantities as a function of the absorber's parameters only. Subsequently, the parameters of the 2D-DR absorber, specifically the stiffness and damping of the connecting links, are optimized to turn the underlying three degree of freedom problem to a one degree of freedom problem. For this configuration, the stability and robustness in vibration absorption are studied. Based on the performed analysis, design rules are specified for the optimized 2D-DR design, including its positioning on the vibrating platform. A numerical case study demonstrates the 2D-DR absorber design and assessment of the operable frequency range for a specific set of voice-coil actuators. [ABSTRACT FROM AUTHOR]

We perform a linear stability analysis of synchronized equilibria in networks of diffusely coupled oscillators, affected by distributed delays in the coupling, and we characterize the structure of the emanating solutions in the bifurcations, under the assumption that the delay kernels are equal to each other. The motivation comes from the fact that valuable quantitative and qualitative information about the occurrence and type of synchronous or partially synchronous solutions can be obtained from this linear analysis. We analyze stability of synchronized equilibria as a function of the parameters of a so-called shifted gamma-distributed delay, which allows to represent or approximate a large class of distributed delay kernels. We also present an asymptotic analysis method, which is particularly suitable for studying the effect of a distribution of delays on stability. We apply the methods to networks of coupled Lorenz systems, where we highlight that stability is favored by a distributed delay compared with a discrete delay with the same average value. Among others, we show that if the coupling is diffusive, the synchronized equilibria become asymptotically stable for large values of the coupling gain, that is, the systems locally asymptotically synchronize at an equilibrium, independently of the network topology. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, a structure-preserving model reduction approach for a class of delay differential equations is proposed. Benefits of this approach are, firstly, the fact that the delay nature of the system is preserved after reduction, secondly, that input–output stability properties are preserved and, thirdly, that a computable error bound reflecting the accuracy of the reduction is provided. These results are applicable to large-scale linear delay differential equations with constant delays, but also extensions to a class of nonlinear delay differential equations with time-varying delays are presented. The effectiveness of the results is evidenced by means of an illustrative example. [ABSTRACT FROM AUTHOR]

909This paper considers interpolating matrix polynomials P(λ) in Lagrange and Hermite bases. A classical approach to investigating the polynomial eigenvalue problem P(λ)x = 0 is linearization, by which the polynomial is converted into a larger matrix pencil with the same eigenvalues. Since the current linearizations of degree n Lagrange polynomials consist of matrix pencils with n + 2 blocks, they introduce additional eigenvalues at infinity. Therefore, we introduce new linearizations which overcome this. Initially, we restrict to Lagrange and barycentric Lagrange matrix polynomials and give two new and more compact linearizations, resulting in matrix pencils of n + 1 and n blocks for polynomials of degree n. For the latter, there is a one-to-one correspondence between the eigenpairs of P(λ) and the eigenpairs of the pencil. We also prove that these linearizations are strong. Moreover, we show how to exploit the structure of the proposed matrix pencils in Krylov-type methods, so that in this case we only have to deal with linear system solves of matrices of the original matrix polynomial dimension. Finally, we generalize for Hermite interpolation and introduce new linearizations for Hermite Lagrange and barycentric Hermite matrix polynomials. Again, we can show that the linearizations are strong and that there is a one-to-one correspondence of the eigenpairs. [ABSTRACT FROM AUTHOR]

This paper revisits a recently developed methodology based on the matrix Lambert W function for the stability analysis of linear time invariant, time delay systems. By studying a particular, yet common, second order system, we show that in general there is no one to one correspondence between the branches of the matrix Lambert W function and the characteristic roots of the system. Furthermore, it is shown that under mild conditions only two branches suffice to find the complete spectrum of the system, and that the principal branch can be used to find several roots, and not the dominant root only, as stated in previous works. The results are first presented analytically, and then verified by numerical experiments. [ABSTRACT FROM AUTHOR]

Abstract: We consider networks of time-delayed diffusively coupled systems and relate conditions for synchronization of the systems in the network to the topology of the network. First we present sufficient conditions for the solutions of the time-delayed coupled systems to be bounded. Next we give conditions for local synchronization and we show that the values of the coupling strength and time-delay for which there is local synchronization in any network can be determined from these conditions. In addition we present results on global synchronization in relation to the network topology for networks of a class of nonlinear systems. We illustrate our results with examples of synchronization in networks with FitzHugh–Nagumo model neurons and Hindmarsh–Rose neurons. [Copyright &y& Elsevier]

The full synchronization of coupled nonlinear oscillators has been widely studied. In this paper we investigate conditions for which partial synchronization of time-delayed diffusively coupled systems arises. The coupling configuration of the systems is described by a directed graph. As a novel quantitative result we first give necessary and sufficient conditions for the presence of forward invariant sets characterized by partially synchronous motion. These conditions can easily be checked from the eigenvalues and eigenvectors of the graph Laplacian. Second, we perform stability analysis of the synchronized equilibria in a (gain,delay) parameter space. For this analysis the coupled nonlinear systems are linearized around the synchronized equilibria and then the resulting characteristic function is factorized. By such a factorization, it is shown that the relation between the behaviour of different agents at the zero of the characteristic function depends on the structure of the eigenvectors of the weighted Laplacian matrix. By determining the structure of the solutions in the unstable manifold, combined with the characterization of invariant sets, we predict which partially synchronous regimes occur and estimate the corresponding coupling gain and delay values. We apply the obtained results to networks of coupled Hindmarsh–Rose neurons and verify the occurrence of the expected partially synchronous regimes by using a numerical simulation. We also make a comparison with an existing approach based on Lyapunov functionals. [ABSTRACT FROM AUTHOR]

Abstract: We present a numerical method to plot the root locus of Single-Input–Single-Output (SISO) dead-time systems with respect to the controller gain or the system delay. We compute the trajectories of characteristic roots of the closed-loop system on a prescribed complex right half-plane. We calculate the starting, branch and boundary crossing roots of root-locus branches inside the region. We compute the root locus of each characteristic root based on a predictor–corrector type continuation method. To avoid the high sensitivity of roots with respect to the locus parameter in the neighborhood of branch points, the continuation method relies on a natural parameterization of the root-locus trajectory in terms of a distance in the (characteristic root, locus parameter)-space. The method is numerically stable for high order SISO dead-time systems. [Copyright &y& Elsevier]

Abstract: Spectral discretization methods are well established methods for the computation of characteristic roots of time-delay systems. In this paper a method is presented for computing all characteristic roots in a given right half plane. In particular, a procedure for the automatic selection of the number of discretization points is described. This procedure is grounded in the connection between a spectral discretization and a rational approximation of exponential functions. First, a region that contains all desired characteristic roots is estimated. Second, the number of discretization points is selected in such a way that in this region the rational approximation of the exponential functions is accurate. Finally, the characteristic roots approximations, obtained from solving the discretized eigenvalue problem, are corrected up to the desired precision by a local method. The effectiveness and robustness of the procedure are illustrated with several examples and compared with DDE-BIFTOOL. [Copyright &y& Elsevier]

We analyze and design H-infinity controllers for general time-delay systems with time-delays in systems' state, inputs, and outputs. We allow the designer to choose the order of the controller and to introduce constant time-delays into the controller. The closed-loop system of the plant and the controller is modeled by a system of delay differential algebraic equations (DDAEs). The advantage of the DDAE modeling framework is that any interconnection of systems and controllers prone to various types of delays can be dealt with in a systematic way, without using any elimination technique. We present a predictor-corrector algorithm for the H-infinity norm computation of systems described by DDAEs. Instrumental to this we analyze the properties of the H-infinity norm. In particular, we illustrate that it may be sensitive with respect to arbitrarily small delay perturbations. Due to this sensitivity, we introduce the strong H-infinity norm, which explicitly takes into account small delay perturbations, inevitable in any practical control application. We present a numerical algorithm to compute the strong H-infinity norm for DDAEs. Using this algorithm and the computation of the gradient of the strong H-infinity norm with respect to the controller parameters, we minimize the strong H-infinity norm of the closed-loop system based on nonsmooth, nonconvex optimization methods. By this approach, we tune the controller parameters and design H-infinity controllers with a prescribed order or structure. [ABSTRACT FROM AUTHOR]

We will establish here a formula for the convergence factor of the method called residual inverse iteration, which is a method for nonlinear eigenvalue problems and a generalization of the well-known inverse iteration. The formula for the convergence factor is explicit and involves quantities associated with the eigenvalue to which the iteration converges, in particular the eigenvalue and eigenvector. Residual inverse iteration allows for some freedom in the choice of a vector w and we can use the formula for the convergence factor to analyze how it depends on the choice of w. We also use the formula to illustrate the convergence when the shift is close to the eigenvalue. Finally, we explain the slow convergence for double eigenvalues by showing that under generic conditions, the convergence factor is one, unless the eigenvalue is semisimple. If the eigenvalue is semisimple, it turns out that we can expect convergence similar to the simple case. [ABSTRACT FROM AUTHOR]

We present a model order reduction method which allows the construction of a reduced, delay-free model of a given dimension for linear time-delay systems, whose characteristic matrix is nonlinear due to the presence of exponential functions. The method builds on the equivalent representation of the time-delay system as an infinite-dimensional linear problem. It combines ideas from a finite-dimensional approximation via a spectral discretization, on the one hand, and a Krylov-Padé model reduction approach, on the other hand. The method exhibits a good spectral approximation of the original model, in the sense that the smallest characteristic roots are well approximated and the nonconverged eigenvalues of the reduced model have a favorable location, and it preserves moments at zero and at infinity. The spectral approximation is due to an underlying Arnoldi process that relies on building an appropriate Krylov space for the linear infinite-dimensional problem. The preservation of moments is guaranteed, because the chosen finite-dimensional approximation preserves moments and, in addition, the space on which one projects is constructed in such a way that the preservation of moments carries over to the reduced model. The implementation of the method is dynamic, since the number of grid points in the spectral discretization does not need to be chosen beforehand and the accuracy of the reduced model can always be improved by doing more iterations. It relies on a reformulation of the problem involving a companion-like system matrix and a highly structured input matrix, whose structure are fully exploited. [ABSTRACT FROM AUTHOR]

The stabilisation and robustification of a time-delay system is the topic of this article. More precisely, we want to minimise the H2 norm of the transfer function corresponding to the class of linear time-invariant input-output systems with fixed time delays in the states. Due to the presence of the delays, the transfer function is a nonrational, nonlinear function, and the classical procedure which involves solving Lyapunov equations is no longer applicable. We therefore propose an approach based on a spectral discretisation applied to a reformulation of the time-delay system as an infinite-dimensional standard linear system. In this way, we obtain a large delay-free system, which serves as an approximation to the original time-delay system, and which allows the application of standard H2 norm optimisation techniques. We give an interpretation of this approach in the frequency domain and relate it to the approximation of the nonlinear terms in the time-delay transfer function by means of a rational function. Using this property, we can provide some insight into the convergence behaviour of the approximation, justifying its use for the purpose of H2 norm computation. Along with this, the easy availability of derivatives with respect to the original matrices allows for an efficient integration into any standard optimisation framework. A few numerical examples finally illustrate how the presented method can be employed to perform optimal H2 norm design using smooth optimisation techniques. [ABSTRACT FROM AUTHOR]

The paper presents an optimization-based algorithm for stabilizing retarded systems using a state-derivative feedback controller. It is shown that an application of such a controller results in neutral dynamics of the closed-loop system if small feedback delays occur. Therefore, the strong stability theory of neutral systems needs to be considered in the controller synthesis problem. The stabilization approach is based on minimizing the spectral abscissa of the closed-loop system over the controller parameter space, subject to a strong stability constraint. The constrained optimization problem is first turned into an unconstrained problem by application of a barrier method. Subsequently, the optimization is performed using both Broyden-Fletcher-Goldfarb-Shanno (BFGS) and hybrid algorithms for non-smooth optimization Matlab utilities. In the application example, the theoretical results are applied to regenerative chatter suppression in cutting process. [ABSTRACT FROM PUBLISHER]

We consider the characterization and computation of H∞ norms for a class of time-delay systems. It is well known that in the finite-dimensional case the H∞ norm of a transfer function can be computed using the connections between the corresponding singular value curves and the imaginary axis eigenvalues of a Hamiltonian matrix, leading to the established level set methods. We show a similar connection between the transfer function of a time-delay system and the imaginary axis eigenvalues of an infinite-dimensional linear operator Lξ. Based on this result, we propose a predictor-corrector algorithm for the computation of the H∞ norm. In the prediction step, a finite-dimensional approximation of the problem, induced by a spectral discretization of the operator Lξ, and an adaptation of the algorithms for finite-dimensional systems, allow us to obtain an approximation of the H∞ norm of the transfer function of the time-delay system. In the next step the approximate results are corrected to the desired accuracy by solving a set of nonlinear equations which are obtained from the reformulation of the eigenvalue problem for the linear infinite-dimensional operator Lξ as a finite-dimensional nonlinear eigenvalue problem. These equations can be interpreted as characterizations of peak values in the singular value plot. The effects of the discretization in the predictor step are fully characterized, and the choice of the number of discretization points is discussed. The paper concludes with a numerical example and the presentation of the results of extensive benchmarking. [ABSTRACT FROM AUTHOR]

Abstract: If is an eigenvalue of a time-delay system for the delay then is also an eigenvalue for the delays , for any . We investigate the sensitivity, periodicity and invariance properties of the root for the case that is a double eigenvalue for some . It turns out that under natural conditions (the condition that the root exhibits the completely regular splitting property if the delay is perturbed), the presence of a double imaginary root for some delay implies that is a simple root for the other delays , . Moreover, we show how to characterize the root locus around . The entire local root locus picture can be completely determined from the square root splitting of the double root. We separate the general picture into two cases depending on the sign of a single scalar constant; the imaginary part of the first coefficient in the square root expansion of the double eigenvalue. [Copyright &y& Elsevier]

The behavior of the rightmost characteristic roots of linear time-delay systems as a function of the delay parameter is studied for small values of a gain parameter. The results explain the qualitative behavior of the stability regions in the delay parameter space of oscillatory systems subjected to large delays, and shed new light on the corresponding stabilization problem. In particular, for a chain of multiple oscillators the determination of a stabilizing value of the delay parameter is interpreted as a phase synchronization problem. The results are illustrated with the output feedback stabilization of an oscillator and the stability analysis of a gyroscopic system. [ABSTRACT FROM AUTHOR]

Abstract: The pseudospectrum of a linear time-invariant system is the set in the complex plane consisting of all the roots of the characteristic equation when the system matrices are subjected to all possible perturbations with a given upper bound. The pseudospectral abscissa is defined as the maximum real part of the characteristic roots in the pseudospectrum and, therefore, it is for instance important from a robust stability point of view. In this paper we present an accurate method for the computation of the pseudospectral abscissa of retarded delay differential equations with discrete pointwise delays. Our approach is based on the connections between the pseudospectrum and the level sets of an appropriately defined complex function. The computation is done in two steps. In the prediction step, an approximation of the pseudospectral is obtained based on a rational approximation of the characteristic matrix and the application of a bisection algorithm. Each step in this bisection algorithm relies on checking the presence of the imaginary axis eigenvalues of a complex matrix, similar to the delay free case. In the corrector step, the approximate pseudospectral abscissa is corrected to any given accuracy, by solving a set of nonlinear equations that characterizes the extreme points in the pseudospectrum contours. [Copyright &y& Elsevier]

Abstract: A novel method for the determination of controller parameters in a broad class of linear control systems affected by time-delays is presented. This method is based on an appropriate shaping of the spectrum of the closed-loop system. Its application follows two steps. First, a number of rightmost poles, smaller than the number of controller parameters, are directly assigned. This leads to constraints on the controller parameters. By using algebraic techniques a complete parametrization of all controllers satisfying these constraints is obtained. In the second step, the remaining degrees of freedom in the parameter space are used to shift the remaining part of the system spectrum as far to the left as possible. This corresponds to an optimization problem involving a nonsmooth, non-convex objective function. An extensive real-plant example is presented to demonstrate the effectiveness and applicability of the procedure. [Copyright &y& Elsevier]

Abstract: A stability analysis of linear systems with stochastically varying delay is performed. It is assumed that the delay function has the form of a sawtooth with switches occurring at the arrival times of a homogeneous Poisson process. Several notions of stochastic stability are considered and corresponding stability criteria are derived. For two examples the different criteria are compared and the effect on stability of various deterministic approximations is examined. [Copyright &y& Elsevier]

We consider the synchronization problem of an arbitrary number of coupled nonlinear oscillators with delays in the interconnections. The network topology is described by a directed graph. Unlike the conventional approach of deriving directly sufficient synchronization conditions, the approach of the paper starts from an exact stability analysis in a (gain, delay) parameter space of a synchronized equilibrium and extracts insights from an analysis of its bifurcations and from the corresponding emerging behavior. Instrumental to this analysis a factorization of the characteristic equation is employed that not only facilitates the analysis and reduces computational cost but also allows to determine the precise role of the individual agents and the topology of the network in the (in)stability mechanisms. The study provides an algorithm to perform a stability and bifurcation analysis of synchronized equilibria. Furthermore, it reveals fundamental limitations to synchronization and it explains under which conditions on the topology of the network and on the characteristics of the coupling the systems are expected to synchronize. In the second part of the paper the results are applied to coupled Lorenz systems. The main results show that for sufficiently large coupling gains, delay-coupled Lorenz systems exhibit a generic behavior that does not depend on the number of systems and the topology of the network, as long as some basic assumptions are satisfied, including the strong connectivity of the graph. Here the linearized stability analysis is strengthened by a nonlinear stability analysis which confirms the predictions based on the linearized stability and bifurcation analysis. This illustrates the usefulness of the exact linearized analysis in a situation where a direct nonlinear stability analysis is not possible or where it yields conservative conditions from which it is hard to get qualitative insights in the synchronization mechanisms and their scaling properties. In the examples several network topologies are considered. [ABSTRACT FROM AUTHOR]

Abstract: This paper points out specific stability problems of the state derivative and proportional–derivative state feedback which may result from the presence of otherwise negligible delays in the control loops. It is shown that the stability problems arise from the neutral character of the closed-loop system with the state derivative action under small delay perturbations. As the main result, a stability criterion is derived for a safe implementation of the state derivative feedback. Furthermore, a scheme including filtered state derivative feedback is proposed to overcome the destabilizing effects of small feedback delays. Two application examples are presented. From a theoretical point of view, the main contribution of the paper lies in the stability theory of linear neutral equations with dependent delays, subjected to delay perturbations. [Copyright &y& Elsevier]

In this paper we consider the stabilizability of dynamical systems via time-delayed state and output feedback control. Based on an eigenvalue optimization approach in combination with a continuation argument and a result characterizing properties of the optimum eigenvalue configurations, we obtain explicit expressions for stabilizability boundaries for two-dimensional systems. As time-delayed feedback can be used to approximate derivative feedback, we also give a comparison of the stabilizability via both types of feedback. [ABSTRACT FROM AUTHOR]

This paper focuses on consensus problems for a class of linear systems with distributed delay that are encountered in modeling traffic flow dynamics. In the application problems the distributed delay, whose kernel is a ?-distribution with a gap, represents the human drivers' behavior in the average. The aim of the paper is to give a characterization of the regions in the corresponding delay parameter space, where a consensus is reached for all initial conditions. The structure and properties of the system are fully exploited, which leads to explicit and computationally tractable expressions. As a by-product a stability theory for distributed delay systems with a ?-distribution kernel is developed. Also explicit expressions for the consensus function(al) of time-delay systems with constant and distributed delays that solve a consensus problem are provided. Several illustrative examples complete the presentation. [ABSTRACT FROM AUTHOR]