Block Hankel Matrix Research Articles

Hankel Matrix Correlation Function-Based Subspace Identification Method for UAV Servo System

For the identification problem of closed-loop subspace model, we propose a zero space projection method based on the estimation of correlation function to fill the block Hankel matrix of identification model by combining the linear algebra with geometry. By using the same projection of related data in time offset set and LQ decomposition, the multiplication operation of projection is achieved and dynamics estimation of the unknown equipment system model is obtained. Consequently, we have solved the problem of biased estimation caused when the open-loop subspace identification algorithm is applied to the closed-loop identification. A simulation example is given to show the effectiveness of the proposed approach. In final, the practicability of the identification algorithm is verified by hardware test of UAV servo system in real environment.

Maximum Entropy vector kernels for MIMO system identification

Recent contributions have framed linear system identification as a nonparametric regularized inverse problem. Relying on ℓ2-type regularization which accounts for the stability and smoothness of the impulse response to be estimated, these approaches have been shown to be competitive w.r.t. classical parametric methods. In this paper, adopting Maximum Entropy arguments, we derive a new ℓ2 penalty; to do so we exploit the structure of the Hankel matrix, thus controlling at the same time complexity, measured by the McMillan degree, stability and smoothness of the identified models. As a special case, we recover the nuclear norm penalty on the squared block Hankel matrix. In contrast with the previous literature on reweighted nuclear norm penalties, our kernel is described by a small number of hyper-parameters, which are iteratively updated through marginal likelihood maximization; constraining the structure of the kernel acts as a (hyper)regularizer which helps controlling the effective degrees of freedom of our estimator. To optimize the marginal likelihood, we adapt a Scaled Gradient Projection (SGP) algorithm which is proved to be significantly computationally cheaper than other first and second order off-the-shelf optimization methods. The paper also contains an extensive comparison with many state-of-the-art methods on several Monte-Carlo studies, which confirms the effectiveness of our procedure.

A Fast, Memory-Efficient Discrete-Time Realization Algorithm for Reduced-Order Li-Ion Battery Models

Research into reduced-order models (ROM) for Lithium-ion batteries is motivated by the need for a real-time embedded model possessing the accuracy of physics-based models, while retaining computational simplicity comparable to equivalent-circuit models. The discrete-time realization algorithm (DRA) proposed by Lee et al. (2012, “One-Dimensional Physics-Based Reduced-Order Model of Lithium-Ion Dynamics,” J. Power Sources, 220, pp. 430–448) can be used to obtain a physics-based ROM in standard state-space form, the time-domain simulation of which yields the evolution of all the electrochemical variables of the standard pseudo-2D porous-electrode battery model. An unresolved issue with this approach is the high computation requirement associated with the DRA, which needs to be repeated across multiple SoC and temperatures. In this paper, we analyze the computational bottleneck in the existing DRA and propose an improved scheme. Our analysis of the existing DRA reveals that singular value decomposition (SVD) of the large Block–Hankel matrix formed by the system's Markov parameters is a key inefficient step. A streamlined DRA approach that bypasses the redundant Block–Hankel matrix formation is presented as a drop-in replacement. Comparisons with existing DRA scheme highlight the significant reduction in computation time and memory usage brought about by the new method. Improved modeling accuracy afforded by our proposed scheme when deployed in a resource-constrained computing environment is also demonstrated.

Retracted: Multi-step damped multichannel singular spectrum analysis for simultaneous reconstruction and denoising of 3D seismic data

Multichannel singular spectrum analysis (MSSA) is an effective approach for simultaneous seismic data reconstruction and denoising. MSSA utilizes truncated singular value decomposition (TSVD) to decompose the noisy signal into a signal subspace and a noise subspace and weighted projection onto convex sets (POCS)-like method to reconstruct the missing data in the appropriately constructed block Hankel matrix at each frequency slice. However, there still exists some residual noise in signal space due to two major factors: the deficiency of traditional TSVD and the iteratively inserted observed noisy data during the process of weighted POCS like iterations. In this paper, we first further extend the recently proposed damped MSSA (DMSSA) for random noise attenuation, which is more powerful in distinguishing between signal and noise, to simultaneous reconstruction and denoising. Then combined with DMSSA, we propose a multi-step strategy, named multi-step damped MSSA (MS-DMSSA), to efficiently reduce the inserted noise during the POCS like iterations, thus can improve the final performance of simultaneous reconstruction and denoising. Application of the MS-DMSSA approach on 3D synthetic and field seismic data demonstrates a better performance compared with the conventional MSSA approach.

A fast rank-reduction algorithm for three-dimensional seismic data interpolation

Rank-reduction methods have been successfully used for seismic data interpolation and noise attenuation. However, highly intense computation is required for singular value decomposition (SVD) in most rank-reduction methods. In this paper, we propose a simple yet efficient interpolation algorithm, which is based on the Hankel matrix, for randomly missing traces. Following the multichannel singular spectrum analysis (MSSA) technique, we first transform the seismic data into a low-rank block Hankel matrix for each frequency slice. Then, a fast orthogonal rank-one matrix pursuit (OR1MP) algorithm is employed to minimize the low-rank constraint of the block Hankel matrix. In the new algorithm, only the left and right top singular vectors are needed to be computed, thereby, avoiding the complexity of computation required for SVD. Thus, we improve the calculation efficiency significantly. Finally, we anti-average the rank-reduction block Hankel matrix and obtain the reconstructed data in the frequency domain. Numerical experiments on 3D seismic data show that the proposed interpolation algorithm provides much better performance than the traditional MSSA algorithm in computational speed, especially for large-scale data processing.

A Fast, Efficient Discrete-Time Realization Algorithm for Reduced-Order Battery Models

Research into reduced-order models (ROM) for lithium-ion batteries is motivated by need for a real-time embedded model with the accuracy of physics-based models but having computational simplicity comparable to that of equivalent-circuit models. An attractive approach to this problem was recently proposed by Lee et al [1] with a model order reduction method known as Discrete-time-Realization Algorithm (DRA). The ROM obtained in standard state-space representation could then be used to obtain the time-evolution of all the internal electrochemical quantities of the standard porous-electrode model. An unresolved issue of this reduced-order modelling approach is the high computation requirement associated with the DRA, which has to be performed multiple times to identify cell parameters at various SOC and temperatures. In this presentation, we analyse the computational bottleneck in the existing DRA and propose a significant improvement to the overall modelling procedure. Our analysis indicates the Singular Value Decomposition of a large Block-Hankel matrix of the system’s Markov parameters is a key inefficient step for the overall high computational requirement of DRA. A fast computational approach is presented that significantly reduces both the memory and the floating-point operation count by bypassing the redundant step of forming the large Block-Hankel matrix. Comparisons with the existing DRA method highlight the significant reduction in computation time and memory usage (Figure 1), as well as the accuracy improvement in key electrochemical quantities, afforded by our new method. With the aid of the improved modelling procedure, the transfer functions forming the porous electrode model is reformulated and suitably combined to arrive at a physics-informed equivalent circuit model of the Panasonic NCA 18650 BD cells used in automotive applications. Figure 1

Accelerated high-bandwidth MR spectroscopic imaging using compressed sensing.

To develop a compressed sensing (CS) acceleration method with a high spectral bandwidth exploiting the spatial-spectral sparsity of MR spectroscopic imaging (MRSI). Accelerations were achieved using blip gradients during the readout to perform nonoverlapped and stochastically delayed random walks in kx -ky -t space, combined with block-Hankel matrix completion for efficient reconstruction. Both retrospective and prospective CS accelerations were applied to (13) C MRSI experiments, including in vivo rodent brain and liver studies with administrations of hyperpolarized [1-(13) C] pyruvate at 7.0 Tesla (T) and [2-(13) C] dihydroxyacetone at 3.0 T, respectively. In retrospective undersampling experiments using in vivo 7.0 T data, the proposed method preserved spectral, spatial, and dynamic fidelities with R(2) ≥ 0.96 and ≥ 0.87 for pyruvate and lactate signals, respectively, 750-Hz spectral separation, and up to 6.6-fold accelerations. In prospective in vivo experiments, with 3.8-fold acceleration, the proposed method exhibited excellent spatial localization of metabolites and peak recovery for pyruvate and lactate at 7.0 T as well as for dihydroxyacetone and its metabolic products with a 4.5-kHz spectral span (140 ppm at 3.0 T). This study demonstrated the feasibility of a new CS approach to accelerate high spectral bandwidth MRSI experiments. Magn Reson Med 76:369-379, 2016. © 2016 Wiley Periodicals, Inc.

Covariance differencing‐based matrix decomposition for coherent sources localisation in bi‐static multiple‐input–multiple‐output radar

In this study, a covariance differencing-based matrix decomposition algorithm is proposed for locating coherent sources under spatially coloured noise in bi-static multiple-input–multiple-output (MIMO) radar. The method contains three steps. First, the covariance differencing technique is employed to eliminate sensor noise, especially the spatially coloured noise. Second, a block Toeplitz or block Hankel matrix is constructed for decorrelation with the covariance differenced matrix. The forward-only, backward-only and combined forward-backward block Toeplitz/Hankel matrix constructions are defined, respectively. Third, unitary estimation of signal parameters by rotational invariance techniques (ESPRIT) algorithm is applied to estimate directions-of-departure (DODs) and directions-of-arrival (DOAs) of sources. The proposed algorithm offers several advantages. First, it is more robust and provides better estimation performance than other methods. Then, the coloured noise problem is overcome in a simple and effective way. Further, the computational load is comparatively low. Simulation results demonstrate the validity of the proposed algorithm.

Modal parameter identification from output data only

To identify the modal parameters of a vibrating system from output data only we use a state space model and usually two approaches are considered: the block Hankel matrix and its shifted version and the block observability matrix and its shifted version. It is shown in the communication that these two approaches give the same results even in the noisy data case. We present numerical and experimental results, who prove the effectiveness of the procedure.

Modal Parameter Identification from Output Data Only: Equivalent Approaches

The problem of modal parameter identification from output data only is presented. To identify the modal parameters different algorithms are presented: the block Hankel matrix and its shifted version and the block observability and block controllability matrices and their shifted version. These algorithms are derived from properties of the subspace approach. It is shown in the paper that these algorithms give the same results even in the noisy data case. Numerical and experimental results are presented showing the effectiveness of the procedure. In particular a microsystem constituted of a perforated microplate is analysed.

A fast SVD for multilevel block Hankel matrices with minimal memory storage

Motivated by the Cadzow filtering in seismic data processing, this paper presents a fast SVD method for multilevel block Hankel matrices. A seismic data presented as a multidimensional array is first transformed into a two dimensional multilevel block Hankel (MBH) matrix. Then the Lanczos process is applied to reduce the MBH matrix into a bidiagonal or tridiagonal matrix. Finally, the SVD of the reduced matrix is computed using the twisted factorization method. To achieve high efficiency, we propose a novel fast MBH matrix-vector multiplication method for the Lanczos process. In comparison with existing fast Hankel matrix-vector multiplication methods, our method applies 1-D, instead of multidimensional, FFT and requires minimum storage. Moreover, a partial SVD is performed on the reduced matrix, since complete SVD is not required by the Caszow filtering. Our numerical experiments show that our fast MBH matrix-vector multiplication method significantly improves both the computational cost and storage requirement. Our fast MBH SVD algorithm is particularly efficient for large size multilevel block Hankel matrices.

Realization and identification of autonomous linear periodically time-varying systems

The subsampling of a linear periodically time-varying system results in a collection of linear time-invariant systems with common poles. This key fact, known as “lifting”, is used in a two-step realization method. The first step is the realization of the time-invariant dynamics (the lifted system). Computationally, this step is a rank-revealing factorization of a block-Hankel matrix. The second step derives a state space representation of the periodic time-varying system. It is shown that no extra computations are required in the second step. The computational complexity of the overall method is therefore equal to the complexity for the realization of the lifted system. A modification of the realization method is proposed, which makes the complexity independent of the parameter variation period. Replacing the rank-revealing factorization in the realization algorithm by structured low-rank approximation yields a maximum likelihood identification method. Existing methods for structured low-rank approximation are used to identify efficiently a linear periodically time-varying system. These methods can deal with missing data.

A novel recursive subspace identification approach of closed-loop systems

In this paper, a subspace model identification method under closed-loop experimental condition is presented which can be implemented to recursively identify and update the system model. The projected matrices play an important role in this identification scheme which can be obtained by the projection of the input and output data onto the space of exogenous inputs and recursively updated through sliding window technique. The propagator type method in array signal processing is then applied to calculate the subspace spanned by the column vectors of the extended observability matrix without singular value decomposition. The speed of convergence of the proposed method is mainly dependent on the number of block Hankel matrix rows and the initialization accuracy of the projected data matrices. The proposed method is feasible for the closed-loop system contaminated with coloured noises. Two numerical examples show the effectiveness of the proposed algorithm.

Parametric Solutions to the Generalized Discrete Yakubovich‐Transpose Matrix Equation

AbstractThis paper is concerned with the complete parametric solutions to the generalized discrete Yakubovich‐transpose matrix equation X − AXTB = CY. which is related with several types of matrix equations in control theory. One of the parametric solutions has a neat and elegant form in terms of the Krylov matrix, a block Hankel matrix and an observability matrix. In addition, the special case of the generalized discrete Yakubovich‐transpose matrix equation, which is called the Karm‐Yakubovich‐transpose matrix equation, is considered. The explicit solutions to the Karm‐Yakubovich‐transpose matrix equation are also presented by the so‐called generalized Leverrier algorithm. At the end of the paper, two examples are given to show the efficiency of the proposed algorithm.

Predictor-based multivariable closed-loop system identification of the EXTRAP T2R reversed field pinch external plasma response

The usage of computationally feasible overparametrized and nonregularized system identification signal processing methods is assessed for automated determination of the full reversed-field pinch external plasma response spectrum for the experiment EXTRAP T2R. No assumptions on the geometry of eigenmodes are imposed. The attempted approach consists of high-order autoregressive exogenous estimation followed by Markov block coefficient construction and Hankel matrix singular value decomposition. It is seen that the obtained ‘black-box’ state-space models indeed can be compared with the commonplace ideal magnetohydrodynamics (MHD) resistive thin-shell model in cylindrical geometry. It is possible to directly map the most unstable autodetected empirical system pole to the corresponding theoretical resistive shell MHD eigenmode.

Simultaneous seismic data denoising and reconstruction via multichannel singular spectrum analysis

We present a rank reduction algorithm that permits simultaneous reconstruction and random noise attenuation of seismic records. We based our technique on multichannel singular spectrum analysis (MSSA). The technique entails organizing spatial data at a given temporal frequency into a block Hankel matrix that in ideal conditions is a matrix of rank [Formula: see text], where [Formula: see text] is the number of plane waves in the window of analysis. Additive noise and missing samples will increase the rank of the block Hankel matrix of the data. Consequently, rank reduction is proposed as a means to attenuate noise and recover missing traces. We present an iterative algorithm that resembles seismic data reconstruction with the method of projection onto convex sets. In addition, we propose to adopt a randomized singular value decomposition to accelerate the rank reduction stage of the algorithm. We apply MSSA reconstruction to synthetic examples and a field data set. Synthetic examples were used to assess the performance of the method in two reconstruction scenarios: a noise-free case and data contaminated with noise. In both cases, we found extremely low reconstructions errors that are indicative of an optimal recovery. The field data example consists of a 2D prestack volume that depends on common midpoint and offset. We use the MSSA reconstruction method to complete missing offsets and, at the same time, increase the signal-to-noise ratio of the seismic volume.

Noise Issues of Modal Identification using Eigensystem Realization Algorithm

The eigensystem realization algorithm (ERA) is one of the most popular methods in civil engineering applications for estimating the modal parameters, including complex-valued modal frequencies and modal vectors, of dynamic systems. In dealing with noisy measurement data, the ERA partitions the realized model into principal (signal) and perturbational (noise) portions so that the noise portion can be disregarded. During the separation of signal and noise, a critical issue is the determination for the dimensions of the block Hankel matrix which is built from noisy measurement data. We show that the signal and noise matrices can be better separated when the number of blockrows and number of block-columns of the corresponding block Hankel matrix are chosen to be close to each other. We introduce the concept of using the Frobenius norm (L2-norm) of the signal and noise matrices to quantify the signal to noise ratio in the global sense (involving multiple signals). We also propose a verification procedure to justify that the estimated modal parameters are noise insensitive and thus indeed associated with the true system. The procedure involves artificially injecting random noise into the measured signals (which are noisy signals) to create noisy-noisy signals, then comparing the identification results obtained respectively from the measured and noisynoisy signals. Using experimental data collected from a test plate, we demonstrate that if signal and noise portions have been properly separated while using the measured data, then the artificial noise would almost completely accumulate to the noise portion. Therefore, the modal estimation based on the signal portion only would remain the same by using either the measured or the noisy-noisy signals.

Limit Distributions of Eigenvalues for Random Block Toeplitz and Hankel Matrices

Block Toeplitz and Hankel matrices arise in many aspects of applications. In this paper, we will research the distributions of eigenvalues for some models and get the semicircle law. Firstly we will give trace formulas of block Toeplitz and Hankel matrix. Then we will prove that the almost sure limit $\gamma_{T}^{(m)}$ $(\gamma_{H}^{(m)})$ of eigenvalue distributions of random block Toeplitz (Hankel) matrices exist and give the moments of the limit distributions where m is the order of the blocks. Then we will prove the existence of almost sure limit of eigenvalue distributions of random block Toeplitz and Hankel band matrices and give the moments of the limit distributions. Finally we will prove that $\gamma_{T}^{(m)}$ $(\gamma_{H}^{(m)})$ converges weakly to the semicircle law as m→∞.

ESPRIT-Like Two-Dimensional DOA Estimation for Coherent Signals

A second-order statistics-based algorithm is proposed for two-dimensional (2D) direction-of-arrival (DOA) estimation of coherent signals. The problem is solved by arranging the elements of the correlation matrix of the signal received from a uniform rectangular array to a block Hankel matrix. In noiseless cases, it is shown that the rank of the block Hankel matrix equals the number of the DOAs and is independent of the coherency of the incoming waves. Therefore, the signal subspace of the block Hankel matrix can be estimated properly and spans the same column space as the array response matrix. Two matrix pencil pairs containing the DOA parameters are extracted from the signal subspace. This matrix pencil-based estimation problem is then resolved using our previously proposed pairing-free 2D parameter estimation algorithm. Simulation results show that the proposed algorithm outperforms the spatial smoothing method in terms of mean square error (MSE).

Modal parameter identification based on ARMAV and state–space approaches

An accurate prediction for the response of civil and mechanical engineering structures subject to ambient excitation requires the information of dynamic properties of these structures including natural frequencies, damping ratios and mode shapes. Since the excitation force is not available as a measured signal, we need to develop techniques which are capable of accurately extracting the modal parameters from output-only data. This article presents the results of modal parameter identification using two time-domain methods as follows: the autoregressive moving average vector (ARMAV) method and the state–space method. These methods directly work with the recorded time signals and allow the analysis of structures where only the output is measured, while the input is unmeasured and unknown. The equivalence between ARMAV and state–space approaches for the problem of modal parameter identification of vibrating systems is shown in the article. Using only the singular value decomposition of a block Hankel matrix of sample covariances, it is shown that these two approaches give identical modal parameters in the case where the block Hankel matrix has full row rank. The time-domain modal identification algorithms have a serious problem of model order determination: when extracting structural modes these algorithms always generate spurious modes. A modal indicator to differentiate spurious and structural modes is presented. Numerical and experimental examples are given to show the effectiveness of the ARMAV or state–space approaches in modal parameter identification using response data only.