Time Delay Operator Research Articles

A generalized Padé approximation of time delay operator

This paper addresses the problem of fractional order Pade approximation for the time delay operator. Based on optimization method, a fitting polynomial with basic order α is constructed to approximate the time delay operator. This makes for two generalized Pade approximations in the form of normal and symmetric approximation, respectively. The advantage of the proposed method lies in the fact that the resulting approximate model is in the standard transform function form of any commensurate order α, making all of the existing stability analysis methods and control design tools for such systems amenable to the fractional order time delay systems. Some numerical examples are given to demonstrate the effectiveness of the proposed method.

Integral quadratic constraints for delayed nonlinear and parameter-varying systems

The stability and performance of nonlinear and linear parameter varying (LPV) time delayed systems are analyzed. First, the input/output behavior of the time delay operator is bounded in the frequency domain by integral quadratic constraints (IQCs). A simple geometric interpretation is used to derive new IQCs for both constant and varying delays. Second, the performance of nonlinear and LPV delayed systems is bounded using dissipation inequalities that incorporate IQCs. The nonlinear or LPV part of the system is treated directly in the analysis and not bounded by IQCs. This step makes use of recent results that show, under mild technical conditions, that an IQC has an equivalent representation as a finite-horizon time-domain constraint. A numerical example with a nonlinear delayed system is provided to demonstrate the effectiveness of the method.

A constructive algebraic analysis approach to Artstein's reduction of linear time-delay systems

Artstein's results show that a first-order linear differential system with delayed inputs is equivalent to a first-order linear differential system without delay under an invertible transformation which includes integral and time-delay operators. Within a constructive algebraic approach, we show how this reduction can be found again, generalized and interpreted as a particular isomorphism between modules defining the two above linear systems. Moreover, we prove that Artstein's reduction can be obtained in an automatic way by means of symbolic computation techniques, and thus can be implemented in dedicated computer algebra systems.

Fuzzy Opinion Networks: A Mathematical Framework for the Propagation of Opinions and Their Uncertainties Across Social Networks

Human opinions are inherently fuzzy, because the carrier of opinions is natural language where fuzzy words and vague expressions are the mainstays rather than exceptions, thus a good mathematical theory for opinion dynamics should consider the opinion and its uncertainty simultaneously.We propose a new mathematical framework for the evolution and propagation of opinions and their uncertainties, called Fuzzy Opinion Network (FON), which is the connection of a number of Gaussian Nodes, possibly through some weighted average, time-delay or logic operators, where a Gaussian Node is a Gaussian fuzzy set with the center and the standard deviation being the node inputs and the fuzzy set itself being the node output. In this framework an opinion is modeled as a Gaussian fuzzy set with the center representing the opinion itself and the standard deviation characterizing the uncertainty about the opinion. We study the basic connections of Fuzzy Opinion Networks, including basic center, basic standard deviation (sdv), basic center-sdv, chain-in-center and chain-in-sdv connections, and we analyze a number of dynamic connections, including self-feedback, compromising husband with persistent wife (or vice versa), compromising with each other, ring connection with state-dependent uncertainty, “smart” student versus “stubborn” student, and bounded confidence connection, to show how opinions and their uncertainties propagate and evolve across different network structures and scenarios.The main insights gained from the mathematical analyses of the FONs include:(a) Opinion leaders are important for a community, in the sense that the anxiety (uncertainty) of all members in the community may easily go to infinity if everyone compromises fully with all others (i.e., if there is no opinion leader in the community); when there is an opinion leader (a stubborn node in the FON), however, the anxieties of all members in the community will converge to a finite number;(b) Speed matters for time-varying FONs; for example, depending on the speed of gaining confidence through communicating with others, the uncertainties of the members in the community may stabilize at a finite number if the speed is fast or go to infinity if the speed is slow; and,(c) For state-dependent connections such as the Ring Connection or the Bounded Confidence Connection studied in this paper, a consensus will be reached if a central agent collects the opinions of all individuals and discloses them back to each individual (this shows that the media is a powerful machine to control people’s opinion), while in the decentralized control scenario (people know only the opinions of neighbors), different communities emerge where people in the same community reach a consensus but these different consensuses remain separated forever (this explains why people in the same country tend to share similar opinions while people at different countries usually have different opinions).

Robust Control Techniques Applied to the Hot-Dip Galvanizing Process

This paper presents new results to compute the time-delay margin of the hot-dip galvanizing control system. Compared to previous works on this issue, this paper considers a mathematical model of the plant with two inputs and one output. The inputs are used to regulate the output, which represents the Zinc mass coating of steel strip. To achieve this objective, a multivariable PI controller is used, this controller is tuned applying the well known Ziegler and Nichols method, and then the maximum time-delay is computed in order to guarantee the stability property of the close loop control system. The work bases its results on a transformation of the time-delay operator and it is performed in order to get a two variable polynomial; after this, to obtain the robust stability property, a result based on the Hurwitz matrix is applied.

SCATTERING THEORY FOR LATTICE OPERATORS IN DIMENSION d ≥ 3

This paper analyzes the scattering theory for periodic tight-binding Hamiltonians perturbed by a finite range impurity. The classical energy gradient flow is used to construct a conjugate (or dilation) operator to the unperturbed Hamiltonian. For dimension d ≥ 3, the wave operator is given by an explicit formula in terms of this dilation operator, the free resolvent and the perturbation. From this formula, the scattering and time delay operators can be read off. Using the index theorem approach, a Levinson theorem is proved which also holds in the presence of embedded eigenvalues and threshold singularities.

Delay Approximation of Fractional Integrals

AbstractThis paper explores the calculation of fractional integrals by means of the time delay operator. The study starts by reviewing the memory properties of fractional operators and their relationship with time delay. Based on the time response of the Mittag‐Leffler function an approximation of fractional integrals consisting of time delayed samples is proposed. The tuning of the approximation is optimized by means of a genetic algorithm. The results demonstrate the feasibility of the new perspective and the limits of their application.

A Dynamic Feedforward Neural Network Based on Gaussian Particle Swarm Optimization and its Application for Predictive Control

A dynamic feedforward neural network (DFNN) is proposed for predictive control, whose adaptive parameters are adjusted by using Gaussian particle swarm optimization (GPSO) in the training process. Adaptive time-delay operators are added in the DFNN to improve its generalization for poorly known nonlinear dynamic systems with long time delays. Furthermore, GPSO adopts a chaotic map with Gaussian function to balance the exploration and exploitation capabilities of particles, which improves the computational efficiency without compromising the performance of the DFNN. The stability of the particle dynamics is analyzed, based on the robust stability theory, without any restrictive assumption. A stability condition for the GPSO+DFNN model is derived, which ensures a satisfactory global search and quick convergence, without the need for gradients. The particle velocity ranges could change adaptively during the optimization process. The results of a comparative study show that the performance of the proposed algorithm can compete with selected algorithms on benchmark problems. Additional simulation results demonstrate the effectiveness and accuracy of the proposed combination algorithm in identifying and controlling nonlinear systems with long time delays.

Time delay in scattering by potentials and by magnetic fields with two supports at large separation

We study the asymptotic behavior of the time delay (defined as the trace of the Eisenbud–Wigner time delay operator) for scattering by potential and by magnetic field with two compact supports as the separation of supports goes to infinity. The emphasis is placed on analyzing how different the asymptotic formulae are in potential and magnetic scattering. The difference is proper to scattering in two dimensions.

Ore algebras: a polynomial approach to nonlinear time-delay systems

In this paper Ore algebra defined over the field of meromorphic functions is introduced to provide a polynomial approach to nonlinear time-delay systems. In comparison to the systems without delays the polynomials related to the nonlinear time-delay systems belong now to the multivariate skew polynomial ring and have the sense of differential time-delay operators. Then the neccessary and sufficient condition for accessibility of nonlinear time-delay systems is presented in terms of common left factors of such polynomilas. The paper also introduces transfer functions of nonlinear time-delay systems and show their basic properties.

Factoring and decomposing a class of linear functional systems

Within a constructive homological algebra approach, we study the factorization and decomposition problems for a class of linear functional (determined, over-determined, under-determined) systems. Using the concept of Ore algebras of functional operators (e.g., ordinary/partial differential operators, shift operators, time-delay operators), we first concentrate on the computation of morphisms from a finitely presented left module M over an Ore algebra to another one M′, where M (resp., M′) is a module intrinsically associated with the linear functional system Ry = 0 (resp., R′ z = 0). These morphisms define applications sending solutions of the system R′ z = 0 to solutions of R y = 0. We explicitly characterize the kernel, image, cokernel and coimage of a general morphism. We then show that the existence of a non-injective endomorphism of the module M is equivalent to the existence of a non-trivial factorization R = R 2 R 1 of the system matrix R. The corresponding system can then be integrated “in cascade”. Under certain conditions, we also show that the system Ry = 0 is equivalent to a system R′ z = 0, where R′ is a block-triangular matrix of the same size as R. We show that the existence of idempotents of the endomorphism ring of the module M allows us to reduce the integration of the system Ry = 0 to the integration of two independent systems R 1 y 1 = 0 and R 2 y 2 = 0. Furthermore, we prove that, under certain conditions, idempotents provide decompositions of the system Ry = 0, i.e., they allow us to compute an equivalent system R′z = 0, where R′ is a block-diagonal matrix of the same size as R. Applications of these results in mathematical physics and control theory are given. Finally, the different algorithms of the paper are implemented in the Maple package M orphisms based on the library oremodules.

On the Consistency of the Blocked Neural Network Estimator in Time Series Analysis

We describe a nonlinear regression problem, where the regression functions have an additive structure and the dependent variable is a one-dimensional time series. Multivariate time series with unknown time delay operators are used as independent variables. By fitting a feedforward neural network with block structure to the data, we estimated the additive regression function and, parallel to this, the time lags. We present the consistency proof of neural network weights estimator and the time lag estimator independently from each other. In the practical part of the article, we present the useful feature of blocked neural networks to estimate the relevance measures of each input variable in a simple way. Furthermore, we propose an approach to solve the well-known variable selection problem for the class of nonlinear multivariate beta-mixing time series models considered here. Finally, we apply the methodology to an artificial example.

Time delay and resonances in potential scattering

We present stationary expressions for the on-shell time delay operator and for its trace in quantum-mechanical potential scattering. The trace of this operator is shown to represent the time delay for a monoenergetic and well-collimated beam of initial states. For exponentially decaying potentials the total scattering cross section and the trace of the on-shell time delay operator have analytic extensions to complex energies, with poles at most at the usual resonance poles (poles of the analytically continued S-matrix) in the lower half plane and at their mirror images with respect to the real axis. At real energies close to such a pole both the scattering cross section and the time delay are approximately of Breit-Wigner type, the sign of the time delay being related to that of the virial. These results corroborate the idea that a resonance should show up through a peak in the scattering cross section and a large (though positive) time delay.

Algebraic approach to time-delay data analysis for orbiting LISA

The laser phase noise is one of the dominant noises for the gravitational wave detector LISA. Since it is impossible to maintain equal distances between the spacecraft, the time-delay interferometry (TDI) techniques are used to eliminate the laser phase noise along with the optical bench noise. It has been shown for stationary case that the TDI can be formulated in a systematic manner using algebraic geometrical methods, which generate all the data combinations canceling laser phase noise. These data combinations form a module over the polynomials, formally known as first module of syzygies, in three time-delay operators corresponding to the arms of LISA. The rotation induces a difference in up and down light travel times between the spacecraft, which results in a residual laser noise. Recently, several TDI data combinations have been given to cancel the laser phase noise by taking these differences into account. In this paper, we present a rigorous formalism similar to that of stationary case, by taking different arm lengths for up and down beams, resulting in a module over polynomials in six variables. We compute the transfer functions for the six beams and obtain the response to the gravitational wave signal for the generating setmore » of above module.« less

Generalized Discrete Event Abstraction of Continuous Systems: Application to an Integrator

In this paper we present our recent work on generalized discrete event modelling. In the general framework of modelling and simulation of linear systems under time-continuous – or not – signals, the description of the integrator operator becomes fundamental as well as the description the time delay operator and the concept of output feedback. Therefore, this paper shall present the description of an integrator under piecewise linear input trajectories in a first step, then under trajectories described by a sequence of general order polynomials, ensuring a user given accuracy. Moreover, its remarkable behaviour in case of input discontinuities strongly contrast with the rather bad one of classical numerical solvers. Furthermore, a thorough comparison with discrete-time simulation techniques, like the classical Euler one, allows the assessment of an important computational gain using the techniques proposed here. Finally, the complete treatment of an hybrid system example not only illustrates the relevance of this approach, but underlines the interest of its application in the more general context of mixed-mode simulation and distributed simulation.

Time delay interferometry with moving spacecraft arrays

Space-borne interferometric gravitational wave detectors, sensitive in the low-frequency (millihertz) band, will fly in the next decade. In these detectors the spacecraft-to-spacecraft light-travel-times will necessarily be unequal, time-varying, and (due to aberration) have different time delays on up- and down-links. Reduction of data from moving interferometric laser arrays in solar orbit will in fact encounter non-symmetric up- and downlink light time differences that are about 100 times larger than has previously been recognized. The time-delay interferometry (TDI) technique uses knowledge of these delays to cancel the otherwise dominant laser phase noise and yields a variety of data combinations sensitive to gravitational waves. Under the assumption that the (different) up- and downlink time delays are constant, we derive the TDI expressions for those combinations that rely only on four inter-spacecraft phase measurements. We then turn to the general problem that encompasses time-dependence of the light-travel times along the laser links. By introducing a set of non-commuting time-delay operators, we show that there exists a quite general procedure for deriving generalized TDI combinations that account for the effects of time-dependence of the arms. By applying our approach we are able to re-derive the ``flex-free'' expression for the unequal-arm Michelson combinations $X_1$, first presented in \cite{STEA}, and obtain the generalized expressions for the TDI combinations called Relay, Beacon, Monitor, and Symmetric Sagnac.

A dissipative dynamical systems approach to stability analysis of time delay systems

In this paper the concepts of dissipativity and the exponential dissipativity are used to provide sufficient conditions for guaranteeing asymptotic stability of a time delay dynamical system. Specifically, representing a time delay dynamical system as a negative feedback interconnection of a finite-dimensional linear dynamical system and an infinite-dimensional time delay operator, we show that the time delay operator is dissipative with respect to a quadratic supply rate and with a storage functional involving an integral term identical to the integral term appearing in standard Lyapunov–Krasovskii functionals. Finally, using stability of feedback interconnection results for dissipative systems, we develop sufficient conditions for asymptotic stability of time delay dynamical systems. The overall approach provides a dissipativity theoretic interpretation of Lyapunov–Krasovskii functionals for asymptotically stable dynamical systems with arbitrary time delay. Copyright © 2004 John Wiley & Sons, Ltd.

A Cyclic Statistics-Based Parametric Approach to Time-Delay Estimation

The problem of estimating the difference in arrival times of a radio wave impinging on two spatially separated antennas is considered. By exploiting the cyclostationary property of the signal of interest, a new parameter estimation algorithm that is highly tolerant to interference and noises is proposed. The new algorithm models the time-delay operator as a finite impulse response (FIR) filter. From a set of linear equations involving cyclic (cross) correlation, the time delay is estimated to be the index of the FIR filter parameter that has the maximum value. When the signal of interest has a known (or measurable) carrier frequency or keying rate that is distinct from those of interference, the new approach exhibits signal selectivity from the temporally, spectrally, or spatially overlapped interference and noises. The computational complexity of the new approach is comparable to that of the conventional method.

Time-delay operators in semiclassical limit. II. Short-range potentials

This work is a continuation of [27]. We prove that quantum time-delay operator localized in a nontrapping energy interval is in fact an h h -pseudodifferential operator with the h h -principal symbol given by the classical time-delay function composed with the incoming wave operator in classical mechanics. The classical limit of time-delay operator is also given.

Decoupling and coefficient assignment for (A, B, C, D) time-delay systems

Using V as the time-delay operator, a linear invariant time-delay system is expressed by a quadruple (A(δ)), B(δ) C(V) D(V)) with the matrices over a polynomial ring in V. This paper studies the decoupling and coefficient assignment of such a system. For a class of time-delay systems non-controllable over the polynomial ring in V but controllable over the field of rational functions in V, we present sufficient conditions under which the system characteristic polynomial may be determined arbitrarily by feasible control laws. Two examples are investigated to illustrate the theory.