Time-delay Matrix Research Articles

Identification of fractional order time delay system with measurement noise using variable period integration operational matrix

This paper proposes a parameter identification method for fractional-order time-delay systems with measurement noise, based on the Legendre wavelet variable-period integration operational matrix. A variable-period integration operational matrix and a variable-period time-delay integration operational matrix are constructed by variable-period sampling principle. For the impulse noise, global Gaussian noise and local Gaussian noise present in the identification data, the variable-period sampling, operational matrix global coefficient decomposition and matrix local coefficient decomposition methods are used to reconstruct the variable-period integration operational matrix to separate the noise from the data; the fractional-order time-delay system is expressed as an integral equation by using the reconstructed variable-period operational matrix, this reduces the influence of measurement noise on system identification accuracy. To further improve the parameter identification accuracy, the integral equation is solved by the multi-innovation least squares algorithm. The proposed method’s effectiveness is verified through several simulations and an experimental study of a constant-temperature system.

State space reconstruction of Markov chains via autocorrelation structure

Understanding the state space of observed Markov processes is essential for advancing causal inference in a wide range of scientific fields. This paper demonstrates how the previously unknown state space can be reconstructed by exploring the spectrum of the time-delay embedding matrix derived from the autocorrelation sequence of the observed series. It also highlights that the eigenvector associated with the smallest eigenvalue can provide valuable insights into the hidden data generation process itself. The presented results provide a deeper understanding of the complex dynamics of Markov chains and hold promise for enhancing various scientific applications.

Contrastive learning and dynamics embedding neural network for label-free interpretable machine fault diagnosis

Industrial machinery often produces vibration signals that can serve as indicators of underlying faults. However, these signals often need to be labeled, presenting a challenge for accurate and interpretable fault diagnosis. While supervised learning methods, such as deep neural networks, have been applied for fault diagnosis, they need help in effectively distinguishing between different vibration-related faults. In response to this issue, our study introduces an innovative approach for automatic fault diagnosis through the application of the Bootstrap Your Own Latent and Dynamical Systems Model Discovery algorithm (BYOLDIS). This method not only addresses the challenge of unlabelled signals but also provides readily interpretable results. The proposed methodology consists of three fundamental steps. First, we derive a matrix of differential equations to capture the dynamic behavior of faulty bearings. Second, we employ a contrastive learning network alongside a time-delay embedding matrix to reconstruct the coordinates of the fault-dynamical system. Lastly, we construct a library of fault machine dynamic polynomial equations, incorporating prior constraints based on physical models. To assess the effectiveness and robustness of our proposed method, we conducted both simulations and experiments. The results of these case studies affirm that BYOLDIS can accurately diagnose bearing faults and offer dynamic explanations for the diagnostic outcomes. This suggests that BYOLDIS holds substantial promise as a diagnostic tool for processing unlabelled vibrational data.

Adaptive time-domain sliding average method for spall size estimation of roller bearing with skidding

The size of the bearing spall fault can be estimated by the time interval between the roller's entry and exit points in the bearing vibration signal. However, the entry vibration response is typically low amplitude and masked in environmental noise. The time-domain average (TDA) method is effective for periodic impulse enhancement. However, bearing skidding is unavoidable during its running, and the cage's speed fluctuations caused by skidding will lead to non-periodicity of the impulses. The performance of TDA seriously deteriorated with such skidding because of the time-varying period. This paper presents the adaptive time-domain sliding average (ATDSA) method to address the bearing skidding problem. The similarity feature between each impulse segment is utilized, and a time delay matrix is constructed to estimate the skidding rate and mitigate the skidding effect. The frequency spectrum of the reconstructed signal shows clear sidebands with BCF interval. And the obtained averaged waveform has high resolution in both the low-frequency entry and high-frequency exit events. Simulation and experiment results show that it can effectively suppress the skidding effect and estimate the spall size correctly under different speeds.

Wigner–Smith Time Delay Matrix for Electromagnetics: Systems With Material Dispersion and Losses

The Wigner-Smith (WS) time delay matrix relates a system’s scattering matrix to its frequency derivative and gives rise to so-called WS modes that experience well-defined group delays when interacting with the system. For systems composed of nondispersive and lossless materials, the WS time delay matrix previously was shown to consist of volume integrals of energy-like densities plus correction terms that account for the guiding, scattering, or radiating characteristics of the system. This study extends the use of the WS time delay matrix to systems composed of dispersive and lossy materials. Specifically, it shows that such systems’ WS time delay matrix can be expressed by augmenting the previously derived expressions with terms that account for the dispersive and lossy nature of the system, followed by a transformation that disentangles effects of losses from time delays. Analytical and numerical examples demonstrate the new formulation once again allows for the construction of frequency stable WS modes that experience well-defined group delays upon interacting with a system.

Wigner-Smith time delay matrix for acoustic scattering: Theory and phenomenology.

The Wigner-Smith (WS) time delay matrix relates a lossless system's scattering matrix to its frequency derivative. First proposed in the realm of quantum mechanics to characterize time delays experienced by particles during a collision, this article extends the use of WS time delay techniques to acoustic scattering problems governed by the Helmholtz equation. Expression for the entries of the WS time delay matrix involving renormalized volume integrals of energy densities are derived, and shown to hold true, independent of the scatterer's geometry, boundary condition (sound-soft or sound-hard), and excitation. Numerical examples show that the eigenmodes of the WS time delay matrix describe distinct scattering phenomena characterized by well-defined time delays.

Wigner–Smith Time-Delay Matrix for Electromagnetics: Guiding and Periodic Systems With Evanescent Modes

The Wigner-Smith (WS) time delay matrix relates an electromagnetic system’s scattering matrix and its frequency derivative. Previous work showed that the entries of WS time delay matrices of systems excited by propagating waves consist of volume integrals of energy-like field quantities. This paper introduces a generalized WS relationship that applies to systems excited by mixtures of propagating and evanescent fields. Just like its predecessor, the generalized WS relationship allows for the identification of so-called WS modes that interact with the system with well-defined time delays. Furthermore, a technique is developed to compute the WS time delay matrix of a composite system from the WS time delay matrices of its subsystems. Numerical examples demonstrate the usefulness of the generalized WS method when characterizing time delays experienced by fields interacting with guiding and periodic structures that have ports supporting evanescent modes.

Multi-TDOA Estimation and Source Direct Position Determination Based on Parallel Factor Analysis

In this paper, source localization exploiting time difference of arrival (TDOA) information is discussed, and a parallel factor (PARAFAC) analysis based method for multi-TDOA estimation and direct position determination (DPD) is proposed. Firstly, signals from the radiation source are received by multiple antennas through synchronous sensing. Then, the data from multiple antennas is fused by extracting the time delay matrix to construct a PARAFAC model. Thereafter, the time delay matrix is obtained by fast iterative decomposition using the trilinear alternating least square (TALS) algorithm. In the case of no multipath or weak multipath, where the typical scenario is the antennas being deployed on the airborne platform, the multi-TDOA estimation can be obtained simultaneously according to the time delay matrix to improve the processing speed. In the case of multipath, such as the antennas being located on the ground, the DPD cost function directly related to the source position can be established based on the time delay matrix, and the source position estimation can be achieved through grid search. Compared with other DPD methods, the proposed DPD method has better positioning performance and lower complexity. The effectiveness and superiority of the proposed methods are verified by both simulations and actual scenario tests.

Geometry estimation for multiple autonomous underwater vehicles using active sound source

AbstractUnderwater acoustic target detection with multiple autonomous underwater vehicles (AUVs) has the advantage of scalable aperture, however, it depends on the accurate estimation of the shape and structure of the swarm. In this study, a method based on an active sound source is proposed for geometry estimation of distributed nodes. Pulses or chirp signals, transmitted by the source, arrive AUV nodes with different time delays and are collected by the hydrophones, from which the formation geometry could be estimated. Firstly, the frequency‐sliding generalised cross‐correlation (FS‐GCC) feature matrix between the signals received by each node and the reference node is extracted. Secondly, an autoencoder network is employed to enhance the time delay line implied in the FS‐GCC matrix, from which a time delay between two nodes is obtained. A time delay vector could be constructed by concatenating the delays between the reference node and all other nodes when the source lies at different orientations. Lastly, the time delay matrix, which is composed of time delay vectors in a time window when AUV circularly moves around the detection nodes, is further decomposed by singular value decomposition to obtain the formation geometry. The feasibility and effectiveness are verified by the simulation dataset, anechoic pool experiment, and lake trial. The root mean square error of the proposed geometry estimation method drops by 9% and 54% for two typical geometries when compared with ref. [1] for the simulation dataset, which becomes 45% and 37% for the anechoic pool experiment. Though the estimation error increases in lake trials, the proposed method achieved a relatively better precision. It is found that the proposed method has better performance when the formation geometry of AUVs approaches a regular polygon and when the source works in a stop‐and‐go mode. Furthermore, when the source distance meets the far‐field assumption, the azimuthal span of source movement relative to AUVs should be larger than 30°.

Semiclassical calculation of time delay statistics in chaotic quantum scattering

We present a semiclassical calculation, based on classical action correlations implemented by means of a matrix integral, of all moments of the Wigner–Smith time delay matrix, Q, in the context of quantum scattering through systems with chaotic dynamics. Our results are valid for broken time reversal symmetry and depend only on the classical dwell time and the number of open channels, M, which is arbitrary. Agreement with corresponding random matrix theory reduces to an identity involving some combinatorial concepts, which can be proved in special cases.

Distributed secure sampled-data control for distributed generators and energy storage systems in microgrids under abnormal deception attacks

In this work, we investigate the active power sharing (APS) and energy balancing (EB) issues between distributed generators (DGs) and energy storage systems (ESSs) in microgrids with abnormal deception attacks. Firstly, the mathematical model of APS and EB issues under abnormal deception attacks is established, in which attack signals can tamper with the transmitted information and change appropriately with the system’s current state. Next, a distributed secure memory sampled-data controller is designed for APS and EB problems, and the corresponding solving algorithm is given. Furthermore, in order to reduce the conservatism of the results, an improved delay-dependent looped Lyapunov–Krasovskii functional is introduced by adding the combination term of time delay and free matrix. Meanwhile, the high dimensional linear matrix inequality (LMI) in the proposed control algorithm is decomposed into several low dimensional LMIs based on the decoupled method. Finally, a simulation experiment in microgrids is employed to demonstrate the effectiveness of the designed control scheme.

Incremental adaptive optimal control for nonlinear systems with disturbance and input time-delay

In this paper, a model-free incremental adaptive optimal control scheme is proposed for nonlinear systems in presence of disturbance and input time-delay. The linear time-varying approximate model of the nonlinear system is obtained by incremental method, in which the relevant matrix parameters are identified by recursive least squares (RLS) estimation. A time-delay matrix function is constructed by neural network (NN) to eliminate the input time-delay, and the external disturbance of nonlinear system is dealt with by [Formula: see text] optimal control; incremental adaptive dynamic programming (IADP) algorithm is used to get the control law by solving Hamilton–Jacobi–Isaacs (HJI) equation. Convergence analysis of the proposed control scheme is provided. Simulations are given to verify the effectiveness of the proposed control scheme.

Time delay statistics for finite number of channels in all symmetry classes

Within a random matrix theory approach, we obtain spectral statistics of the Wigner time delay matrix Q, for arbitrary channels number M and for all symmetry classes, in fact for the general Dyson parameter β. We also put forth two conjectures: one is related to the large-M expansion of joint cumulants of traces of powers of Q, which generalizes and implies a previous conjecture of Cunden, Mezzadri, Vivo and Simm; the other concerns the tail of the distribution of traces of powers of Q.

Active Tendon Control of Stay Cable by a Giant Magnetostrictive Actuator Considering Time-Delay

Active control of a stay cable through changing its tension is an effective way to lower the vibration amplitude. In this paper, active tendon control of a stay cable using a giant magnetostrictive actuator (GMA), which can generate axial deformation in a short period (normally less than 10 ms) under magnetic fields, is studied considering the time-delay. The bilinear controlled state equation of the small-sag cable is established by employing the Lagrange formulation. The linearization method for the bilinear controlled state equation is proposed and validated by numerical simulations. A GMA is designed and manufactured to actively control a stay cable model and the relationship between the output force and the input voltage is obtained by experiments. Based on the phase-shifting method, a multi-mode control algorithm with the time-delay compensation is proposed, and the time-delay compensation matrix based on the state feedback control algorithm is obtained. Based on a 1:20 scaled cable model, simulations and experiments are conducted to investigate the control performance of the proposed method under different external loads including the free vibration, harmonic excitation and random excitation. The results show that the tendon control of cable vibration by a GMA may be invalid and even amplify the vibration amplitude when the compensation of time-delay is not considered. Moreover, the excellent control performance can be achieved by using the proposed time-delay compensation method.

Stability Analysis of Grid-connected Inverter System Containing Virtual Synchronous Generator under Time Delay and Parameter Uncertainty

Virtual synchronous generator (VSG) control is an effective way to increase the equivalent inertia of grid connected inverter system and improve the stability of the power grid. However, the influence of this control on the stability of the whole system with time delay and parameter uncertainty should be researched further. In this paper, the state space model of the whole grid connected inverter system adopting VSG under control time-delay and parameter uncertainty is established. The Lyapunov functional with time-delay is constructed by employing time-delay dependent free weight matrix transformation. Furthermore, Lyapunov stability theorem is used to obtain the more conservative time-delay stability conditions with and without parameter uncertainty, linear matrix inequality is used to calculate the time-delay stability margin of the system under uncertainty, and the influence of the parameters of VSG control on the delay stability margin is analyzed. The validity of the proposed stability analysis and stability margin solution method are verified by simulation examples.

Integer moments of complex Wishart matrices and Hurwitz numbers

We give formulae for the cumulants of complex Wishart (LUE) and inverse Wishart matrices (inverse LUE). Their large- N expansions are generating functions of double (strictly and weakly) monotone Hurwitz numbers which count constrained factorisations in the symmetric group. The two expansions can be compared and combined with a duality relation proved in [F. D. Cunden, F. Mezzadri, N. O’Connell, and N. J. Simm, Moments of random matrices and hypergeometric orthogonal polynomials, Comm. Math. Phys. 369 (2019), no. 3, 1091–1145] to obtain: i) a combinatorial proof of the reflection formula between moments of LUE and inverse LUE at genus zero and, ii) a new functional relation between the generating functions of monotone and strictly monotone Hurwitz numbers. The main result resolves the integrality conjecture formulated in [F. D. Cunden, F. Mezzadri, N. J. Simm, and P. Vivo, Correlators for the Wigner–Smith time-delay matrix of chaotic cavities, J. Phys. A 49 (2016), no. 18, 18LT01, 20 pp] on the time-delay cumulants in quantum chaotic transport. The precise combinatorial description of the cumulants given here may cast new light on the concordance between random matrix and semiclassical theories.

SVM Based on Gaussian and Non-Gaussian Double Subspace for Fault Detection

In industrial production processes, the data usually have high-dimensional characteristics. When a support vector machine (SVM) is used for fault detection, it takes a long time to run. For high-dimensional data, principal component analysis (PCA) and independent component analysis (ICA) are very effective dimensionality reduction algorithms. Since the data follow different distributions in industrial processes, PCA is commonly used to process Gaussian-distributed data, and ICA is used to process non-Gaussian distributed data. Addressing these limitations, a novel fault detection method of SVM based on Gaussian and non-Gaussian double subspace (DSSVM) is proposed. The Kolmogorov-Smirnov (KS) test is used to determine the normal distribution characteristics of the process variables in the original data. The process variables are divided into a Gaussian subspace and a non-Gaussian subspace. Fault detection models of the Gaussian subspace are established based on PCA, and models of the non-Gaussian subspace are based on ICA. To reduce the effect of autocorrelation among process variables on the SVM, the delay and time difference input characteristics are introduced into the principal components obtained in the Gaussian subspace and independent components obtained in the non-Gaussian subspace. Finally, the time delay matrix and time difference matrix are combined, and the SVM model is used for fault detection and monitoring. The DSSVM method reduces the data dimensions and eliminates the effect of autocorrelation among process variables on the detection results. The proposed method is applied to a multivariable numerical simulation and the Tennessee-Eastman industrial process. Comparisons with simulation results for the kernel principal component analysis (KPCA), kernel independent component analysis (KICA), SVM and the statistical process monitoring method based on variable distribution characteristics (VDSPM) further verify the effectiveness of the algorithm.

Wigner–Smith Time Delay Matrix for Electromagnetics: Computational Aspects for Radiation and Scattering Analysis

The WS time delay matrix relates a lossless and reciprocal system's scattering matrix to its frequency derivative, and enables the synthesis of modes that experience well-defined group delays when interacting with the system. The elements of the WS time delay matrix for surface scatterers and antennas comprise renormalized energy-like volume integrals involving electric and magnetic fields that arise when exciting the system via its ports. Here, direct and indirect methods for computing the WS time delay matrix are presented. The direct method evaluates the energy-like volume integrals using surface integral operators that act on the incident electric fields and current densities for all excitations characterizing the scattering matrix. The indirect method accomplishes the same task by computing scattering parameters and their frequency derivatives. Both methods are computationally efficient and readily integrated into existing surface integral equation codes. The proposed techniques facilitate the evaluation of frequency derivatives of antenna impedances, antenna patterns, and scatterer radar cross sections in terms of renormalized field energies derived from a single frequency characterization of the system.

Performance assessment of multivariate process using time delay matrix

Researchers keep trying to find a way to reduce the requirement knowledge of Multivariate Process for performance assessments. Till now, the knowledge of interactor matrix or first several Markov parameter matrices are at least required to obtain the minimum variance benchmark in the performance assessment of Multivariate Process. In this paper, a novel minimum variance performance assessment technique is proposed for multivariate processes. Under the condition that the time delay indicator matrix can be written as row echelon form by row or/and column shift operations, the delay order is determined and the minimum variance (MV) benchmark can be straightforward obtained when the knowledge of time delay matrix is available. Comparing with the traditional approaches, the first several Markov parameter matrices and the knowledge of the plant are both not necessary required based on the proposed technique. It has been proved that the proposed technique can directly solve the problem of the performance assessment of Multivariate Process. The validity of the proposed algorithm will be verified through numerical examples, which include the practical industrial model - ‘Shell’ oil fractionator process.

Wigner–Smith matrix, exponential functional of the matrix Brownian motion and matrix Dufresne identity

We consider a multichannel wire with a disordered region of length L and a reflecting boundary. The reflection of a wave of frequency ω is described by the scattering matrix , encoding the probability amplitudes to be scattered from one channel to another. The Wigner–Smith time delay matrix is another important matrix, which encodes temporal aspects of the scattering process. In order to study its statistical properties, we split the scattering matrix in terms of two unitary matrices, (with in the presence of time reversal symmetry), and introduce a novel symmetrisation procedure for the Wigner–Smith matrix: , where k is the wave vector and v the group velocity. We demonstrate that can be expressed under the form of an exponential functional of a matrix Brownian motion. For semi-infinite wires, L → ∞, using a matricial extension of the Dufresne identity, we recover straightforwardly the joint distribution for ’s eigenvalues of Brouwer and Beenakker (2001 Physica E 9 463). For finite length L, the exponential functional representation is used to calculate the first moments , and . Finally we derive a partial differential equation for the resolvent in the large N limit.