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The paper is devoted to the study and numerical solution of linear systems of integral equations with an identically singular matrix in the principal part (called integral-algebraic equations). The systems studied in the paper are fundamentally different from those previously studied in that the upper and lower limits of integration are variable. These systems of equations will be referred to as integral-algebraic equations with variable integration limits. The paper develops and justifies an algorithm for a numerical solution of the first order of accuracy and establishes conditions for the existence of a unique continuous solution to the given problem. There are numerical computations provided for model examples. [ABSTRACT FROM AUTHOR]

We generalize the theory of homotopic deviation of square complex matrices to regular matrix pencils. To this end, we study the existence and the analyticity of the resolvent of the matrix pencils whose matrices are under homotopic deviation with the deviation parameter $ t \in \mathbb {C} $ t ∈ C . Moreover, we investigate and identify the limits of both the resolvent and the spectrum of the deviated matrix pencils, as $ | t | \to \infty $ | t | → ∞. We also study the special cases where t tends to the eigenvalues of the related matrix pairs. We use the notions and the results of the generalized homotopic deviation theory to analyze the Weierstrass structure of the deviated matrix pencils under two different assumptions, in particular, either the eigenvalues of the deviated matrix pencils are independent parameters, or the deviation parameter t is an independent parameter. Numerical examples illustrate and support the theoretical results. [ABSTRACT FROM AUTHOR]

We study when functions of the eigenvalues of the pencil (1) Re (e − i t A) = cos ⁡ (t) Re A + sin ⁡ (t) Im A are constant functions of t. The results are then applied to questions regarding the numerical range, the higher rank numerical range and the c -numerical range, and we derive trace type conditions for when these numerical ranges are disks centered at 0. The theory of symmetric polynomials plays an important part in the proofs. [ABSTRACT FROM AUTHOR]

Using the spectral projection onto the deflating subspace of a regular matrix pencil corresponding to the eigenvalues inside a specified region of the complex plane, we proposed a new method to compute a corresponding canonical form. The study included a perturbation analysis of the method as well as examples to illustrate its numerical and theoretical merits. [ABSTRACT FROM AUTHOR]

In this research article, we propose a new matrix iterative method with a convergence order of five for computing the sign of a complex matrix by examining the different patterns and symmetry of existing methods. Analysis of the convergence of the method was explored on a global scale, and attraction basins were demonstrated. In addition to this, the asymptotic stability of the scheme was explored.Then, an algorithm for determing thegeneralized eigenvalues for the case of regular matrix pencils was investigated using the matrix sign computation. We performed a series of numerical experiments using numerous matrices to confirm the usefulness and superiority of the proposed method. [ABSTRACT FROM AUTHOR]

Using the spectral projection onto the deflating subspace of a regular matrix pencil corresponding to the eigenvalues inside a specified region of the complex plane, we proposed a new method to compute a corresponding canonical form. The study included a perturbation analysis of the method as well as examples to illustrate its numerical and theoretical merits.

Acoustic signal analysis is an important component of transformer online monitoring. Currently, traditional methods have problems such as low spectral resolution, imbalanced sample distribution, and unsatisfactory classification performance. This article first introduces the matrix pencil algorithm for time-frequency spectrum analysis of acoustic signals, and then uses the SMOTE algorithm to expand the imbalanced samples. Then, an ACmix hybrid deep neural network model is constructed to classify 11 types of transformer operation and environmental acoustic signals. Finally, detailed experiments were conducted on the method proposed in this paper, and the experimental results showed that the matrix pencil algorithm has high time-frequency resolution and good noise resistance performance. The SMOTE sample expansion method can significantly improve the recognition accuracy by more than 2%. Overall accuracy of the proposed method in acoustic signal classification tasks reaches 91.81%. [ABSTRACT FROM AUTHOR]

We obtain the generic complete eigenstructures of complex Hermitian n times n matrix pencils with rank at most r (with r leq n). To do this, we prove that the set of such pencils is the union of a finite number of bundle closures, where each bundle is the set of complex Hermitian n times n pencils with the same complete eigenstructure (up to the specific values of the distinct finite eigenvalues). We also obtain the explicit number of such bundles and their codimension. The cases r =n, corresponding to general Hermitian pencils, and r < n exhibit surprising differences, since for r < n the generic complete eigenstructures can contain only real eigenvalues, while for r = n they can contain real and nonreal eigenvalues. Moreover, we will see that the sign characteristic of the real eigenvalues plays a relevant role for determining the generic eigenstructures. [ABSTRACT FROM AUTHOR]

We are concerned with finding the set I ⪰ (A , B) of real values μ such that the matrix pencil A + μ B is positive semidefinite. If A , B are not simultaneously diagonalizable via congruence (SDC), I ⪰ (A , B) either is empty or has only one value μ. When A , B are SDC, I ⪰ (A , B) , if not empty, can be a singleton or an interval. Especially, if I ⪰ (A , B) is an interval and at least one of the matrices is nonsingular then its interior is the positive definite interval I ≻ (A , B). If A , B are both singular, then even I ⪰ (A , B) is an interval, its interior may not be I ≻ (A , B) , but A , B are then decomposed as block diagonals of submatrices A 1 , B 1 with B 1 nonsingular such that I ⪰ (A , B) = I ⪰ (A 1 , B 1). Applying I ⪰ (A , B) , the hard-case of the generalized trust-region subproblem (GTRS) can be dealt with by only solving a system of linear equations or reduced to the easy-case of a GTRS of smaller size. [ABSTRACT FROM AUTHOR]

Acoustic signal analysis is an important component of transformer online monitoring. Currently, traditional methods have problems such as low spectral resolution, imbalanced sample distribution, and unsatisfactory classification performance. This article first introduces the matrix pencil algorithm for time-frequency spectrum analysis of acoustic signals, and then uses the SMOTE algorithm to expand the imbalanced samples. Then, an ACmix hybrid deep neural network model is constructed to classify 11 types of transformer operation and environmental acoustic signals. Finally, detailed experiments were conducted on the method proposed in this paper, and the experimental results showed that the matrix pencil algorithm has high time-frequency resolution and good noise resistance performance. The SMOTE sample expansion method can significantly improve the recognition accuracy by more than 2%. Overall accuracy of the proposed method in acoustic signal classification tasks reaches 91.81%.

Given a matrix polynomial A (λ) of degree d and the associated vector space of pencils DL (A) described in Mackey et al. [12] , we construct a parametrization for the set of left and right transformations that preserve the block structure of such pencils. They form a special class of structure-preserving transformations (SPTs). An SPT in that class maps DL (A) to DL (A ˜) , where A ˜ (λ) is a new matrix polynomial that is still of degree d and whose finite and infinite eigenvalues and their partial multiplicities are the same as those of A (λ). Unlike previous work on SPTs, we do not require the leading matrix coefficient of A (λ) to be nonsingular. We show that additional constraints on the parametrization lead to SPTs that also preserve extra structures in A (λ) such as symmetric, alternating, and T -palindromic structures. Our parametrization allows easy construction of SPTs that are low-rank modifications of the identity matrix. The latter transform A (λ) into a matrix polynomial A ˜ (λ) whose j th matrix coefficient A ˜ j is a low-rank modification of A j. We expect such SPTs to be one of the key tools for developing algorithms that reduce a matrix polynomial to Hessenberg form or tridiagonal form in a finite number of steps and without the use of a linearization. [ABSTRACT FROM AUTHOR]

An accurate fault location estimation (FLE) helps in quick power system restoration in the event of transmission lines exposed to faults. However, the accuracy of FLE gets affected in shunt FACTS-compensated system. Therefore, it is indispensable to develop an efficient FLE algorithm which serves the purpose of accurate estimation of fault location in shunt-compensated system. Consequently, this paper proposes a traveling wave-based FLE algorithm utilizing short-time matrix pencil method (STMPM) to estimate the fault location in the aforementioned system. In the proposed algorithm, STMPM is used to decompose windowed signal into time-indexed complex frequencies. From the fact that damping factors extracted from complex frequencies are near or equal to zero when the arriving traveling waves after inception of fault are at the middle of sliding window, the arrival time of waves is estimated and fault location is obtained. The efficacy of the proposed algorithm is tested on 100 MVAR STATCOM-compensated 500 kV transmission system under several fault scenarios and diverse operating conditions. The performance of STMPM-based FLE algorithm is compared with that of the existing methods and found to be more robust and reliable. Further, real-time validation of the proposed algorithm is carried out in OPAL-RT environment. [ABSTRACT FROM AUTHOR]

In this research article, we propose a new matrix iterative method with a convergence order of five for computing the sign of a complex matrix by examining the different patterns and symmetry of existing methods. Analysis of the convergence of the method was explored on a global scale, and attraction basins were demonstrated. In addition to this, the asymptotic stability of the scheme was explored.Then, an algorithm for determing thegeneralized eigenvalues for the case of regular matrix pencils was investigated using the matrix sign computation. We performed a series of numerical experiments using numerous matrices to confirm the usefulness and superiority of the proposed method.

In this paper, we consider the numerical solution of large-scale discrete-time projected Lyapunov equations. We provide some reasonable extensions of the most frequently used low-rank iterative methods for linear matrix equations, such as the low-rank Smith method and the low-rank alternating-direction implicit (ADI) method. We also consider how to reduce complex arithmetic operations and storage when shift parameters are complex and propose a partially real version of the low-rank ADI method. Through two standard numerical examples from discrete-time descriptor systems, we will show that the proposed low-rank alternating-direction implicit method is efficient.

Different representations of linear dissipative Hamiltonian and port-Hamiltonian differential–algebraic equations (DAE) systems are presented and compared. Using global geometric and algebraic points of view, translations between different representations are presented. Characterizations are also derived when a general DAE system can be transformed into one of these structured representations. Approaches for computing the structural information and the described transformations are derived that can be directly implemented as numerical methods. The results are demonstrated with a large number of examples. [ABSTRACT FROM AUTHOR]

Harmonic and interharmonic content in power system signals is increasing with the development of renewable energy generation and power electronic devices. These multiple signal components can seriously degrade power quality, trip thermal generators, cause oscillations, and threaten system stability, especially the interharmonic tones with positive damping factors. The first step to mitigate these adverse effects is to accurately and quickly monitor signal features, including frequency, damping factor, amplitude, and phase. This paper proposes a concise and robust index to identify the number of modes present in the signal using the singular values of the Hankel matrix and discusses the scope of its application by testing the influence of various factors. Next, the simplified matrix pencil theory is employed to estimate the signal component frequency and damping factor. Then their estimates are considered in the modified least-squares algorithm to extract the wideband multi-component phasors accurately. Finally, this paper designs a series of scenarios considering varying signal frequency, damping factor, amplitude, and phase to test the proposed algorithm thoroughly. The results verify that the proposed method can achieve a maximum total vector error of less than 1.5%, which is more accurate than existing phasor estimators in various signal environments. The high accuracy of the proposed method is because it considers both the estimation of the frequency number and the effect of signal damping. [ABSTRACT FROM AUTHOR]

Purpose: To introduce dynamic mode decomposition (DMD) as a robust alternative for the assessment of pulmonary functional information from dynamic non‐contrast‐enhanced acquisitions. Methods: Pulmonary fractional ventilation and normalized perfusion maps were obtained using DMD from simulated phantoms as well as in vivo dynamic acquisitions of healthy volunteers at 1.5T. The performance of DMD was compared with conventional Fourier decomposition (FD) and matrix pencil (MP) methods in estimating functional map values. The proposed method was evaluated based on estimated signal amplitude in functional maps across varying number of measurements. Results: Quantitative assessments performed on phantoms and in vivo measurements indicate that DMD is capable of successfully obtaining pulmonary functional maps. Specifically, compared to FD and MP methods, DMD is able to reduce variations in estimated amplitudes across different number of measurements. This improvement is evident in the fractional ventilation and normalized perfusion maps obtain from phantom simulations with frequency variations and noise, as well as in the maps obtained from in vivo measurements. Conclusions: A robust method for accurately estimating pulmonary ventilation and perfusion related signal changes in dynamic acquisitions is presented. The proposed method uses DMD to obtain functional maps reliably, while reducing amplitude variations caused by differences in number of measurements. [ABSTRACT FROM AUTHOR]

Matrix pencil and Hankel total least squares (HTLS) are two popular ring-down electro- mechanical mode identification algorithms. The appeal of these algorithms can be attributed to faster execution due to the non-iterative procedure of model order determination based on singular value decomposition of the data matrix. In this paper, these two algorithms are shown to be equivalent – the data matrix in one being the transpose of that in the other. Although this equivalence is proved in the context of power systems, it is valid for other areas of system identification as well. Further, the performance of these algorithms is examined as noise level in the signal increases, and it is shown that these work right down to an SNR of 1 dB provided the signal has only poorly damped modes.

Appropriate signal representation is the fundamental issue of concern in all signal processing-based applications. For instance, in the context of signal compression one would require the signal representation to be such that most of the information is confined in the smallest subspace with least number of coefficients. In classification scenario, the representation has to be such that it accentuates differential information amongst classes. In the context of applications dealing with signal decomposition or denoising, one would require the representation such that it separates the input into its independent components so that individual components, i.e. the signal and noise lie in separate spaces. In this paper, we propose signal representation scheme that can be regarded as generalised KLT for multi-class scenario. We introduce an approach that would find the differential information between two classes rather than modelling individual classes separately. These classes are viewed on a common frame of reference in which one class would have a constant variance, unlike the other class which would have unequal variance along its basis vectors which would capture the differential information of one class over the other. This, when mathematically formulated, leads to the solution of the Matrix Pencil equation. This is borne out by illustrative examples on the classification of the MNIST (Deng in IEEE Signal Process Mag 29(6):141–142, 2012) and Google Speech Command Dataset (Pete in Software Engineer, G.B.T. Google Speech Command Dataset. https://ai.googleblog.com/2017/08/launching-speech-commands-dataset.html, 2017). Its applicability for biomedical data like brain state transition detection has also been explored and recorded. [ABSTRACT FROM AUTHOR]

Bundles of matrix pencils (under strict equivalence) are sets of pencils having the same Kronecker canonical form, up to the eigenvalues (namely, they are an infinite union of orbits under strict equivalence). The notion of bundle for matrix pencils was introduced in the 1990's, following the same notion for matrices under similarity, introduced by Arnold in 1971, and it has been extensively used since then. Despite the amount of literature devoted to describing the topology of bundles of matrix pencils, some relevant questions remain still open in this context. For example, the following two: (a) provide a characterization for the inclusion relation between the closures (in the standard topology) of bundles; and (b) are the bundles open in their closure? The main goal of this paper is providing an explicit answer to these two questions. In order to get this answer, we also review and/or formalize some notions and results already existing in the literature. We also prove that bundles of matrices under similarity, as well as bundles of matrix polynomials (defined as the set of m × n matrix polynomials of the same grade having the same spectral information, up to the eigenvalues) are open in their closure. [ABSTRACT FROM AUTHOR]

We revisit the notion of root polynomials, thoroughly studied in (Dopico and Noferini, 2020 [9]) for general polynomial matrices, and show how they can efficiently be computed in the case of a matrix pencil λ E + A. The method we propose makes extensive use of the staircase algorithm, which is known to compute the left and right minimal indices of the Kronecker structure of the pencil. In addition, we show here that the staircase algorithm, applied to the expansion (λ − λ 0) E + (A − λ 0 E) , constructs a block triangular pencil from which a minimal basis and a maximal set of root polynomials at the eigenvalue λ 0 , can be computed in an efficient manner. [ABSTRACT FROM AUTHOR]

[Display omitted] • The MPM can be used as a filter to analytically improve NMR sensitivity. • MPF is not susceptible to aliasing and resolution reduction like Fourier filters. • Useful for pure phase correction and filtration in a single step. Despite inherent sensitivity constraints, nuclear magnetic resonance (NMR) plays an indispensable role in probing molecular structures and dynamics across scientific disciplines. Remarkably, while extensive efforts have targeted instrumental and experimental sensitivity improvements, comparatively little focus has been dedicated to sensitivity enhancement through signal analysis. Amidst this present gap, the matrix pencil method (MPM) has emerged as a versatile algorithm that offers tunable filtering and phasing capabilities. Extensive prior research has established the MPM as an adept fitting tool in signal analysis. Here, the efficacy of the MPM is investigated by precisely modeling noisy data to separate information-bearing signals from noise, thereby expanding its utility in various magnetic resonance applications. Simulated data is used to confirm the ability of the MPM to discern and separate signals from noise. Comparative analyses against standard Fourier-based filtering methods highlight the superior performance of the matrix pencil filter (MPF) in preserving signal fidelity without introducing aliasing artifacts. A variety of experimental data is then explored to demonstrate the proficiency of the MPF in characterizing signal components and correcting phase distortions. Collectively, these case studies underscore the filtering capacity of the MPM, portending its use for analytical sensitivity improvements in a wide range of NMR applications. [ABSTRACT FROM AUTHOR]

Massive multiple-input multiple-output (MIMO) is believed to deliver unrepresented spectral efficiency gains for 5G and beyond. However, a practical challenge arises during its commercial deployment, which is known as the “curse of mobility”. The performance of massive MIMO drops alarmingly when the velocity level of user increases. In this paper, we tackle the problem in frequency division duplex (FDD) massive MIMO with a novel Channel State Information (CSI) acquisition framework. A joint angle-delay-Doppler (JADD) wideband precoder is proposed for channel training. Our idea consists in the exploitation of the partial channel reciprocity of FDD and the angle-delay-Doppler channel structure. More precisely, the base station (BS) estimates the angle-delay-Doppler information of the UL channel based on UL pilots using Matrix Pencil (MP) method. It then computes the wideband JADD precoders according to the extracted parameters. Afterwards, the user estimates and feeds back some scalar coefficients for the BS to reconstruct the predicted DL channel. Asymptotic analysis shows that the CSI prediction error converges to zero when the number of BS antennas and the bandwidth increases. Numerical results with industrial channel model demonstrate that our framework can well adapt to high speed (350 km/h), large CSI delay (10 ms) and channel sample noise. [ABSTRACT FROM AUTHOR]

Harmonic and interharmonic content in power system signals is increasing with the development of renewable energy generation and power electronic devices. These multiple signal components can seriously degrade power quality, trip thermal generators, cause oscillations, and threaten system stability, especially the interharmonic tones with positive damping factors. The first step to mitigate these adverse effects is to accurately and quickly monitor signal features, including frequency, damping factor, amplitude, and phase. This paper proposes a concise and robust index to identify the number of modes present in the signal using the singular values of the Hankel matrix and discusses the scope of its application by testing the influence of various factors. Next, the simplified matrix pencil theory is employed to estimate the signal component frequency and damping factor. Then their estimates are considered in the modified least-squares algorithm to extract the wideband multi-component phasors accurately. Finally, this paper designs a series of scenarios considering varying signal frequency, damping factor, amplitude, and phase to test the proposed algorithm thoroughly. The results verify that the proposed method can achieve a maximum total vector error of less than 1.5%, which is more accurate than existing phasor estimators in various signal environments. The high accuracy of the proposed method is because it considers both the estimation of the frequency number and the effect of signal damping.

Consider a higher order state space system and associated system matrix S(λ). The aim of this paper is to linearize the system preserving system characteristics. That is, linearization preserving system characteristics (e.g, controllability, observability, various zeros and transfer function) for analysis of higher order state-space systems. In particular, we introduce Fiedler-like linearizations (Fiedler linearizations, proper generalized Fiedler (PGF) linearizations) of the system matrix S(λ) to study zeros of higher order system. Further, we show that the linearized systems are strict system equivalent to the higher order systems and hence preserve system characteristics of the original systems. We show that the PGF pencils of S(λ) provide a class of structure-preserving linearizations of S(λ). We study recovery of zero directions of higher order state space system from those of the linearizations. That is, the zero directions of the transfer functions associated to higher order state space system are recovered from the eigenvectors of the Fiedler pencils without performing any arithmetic operations. [ABSTRACT FROM AUTHOR]

We study the prime decomposition of a quadratic monic matrix polynomial. From the prime decomposition of a quadratic matrix polynomial, we obtain a formula of the general solution to the corresponding second-order differential equation. For a quadratic matrix polynomial with pairwise commuting coefficients, we get a sufficient condition for the existence of a prime decomposition.

Despite the recent development of prescribed-time control theory, the highly desirable separation principle remains unavailable for nonlinear systems with only the output being measurable. In this paper, for the first time we establish such separation principle for a class of nonlinear systems, such that the prescribed-time observer and prescribed-time controller can be designed independently, and the parameter designs do not affect each other. Our method makes use of two parametric Lyapunov equations (PLEs) to generate two symmetric positive-definite matrices, aiming to avoid conservative treatments of nonlinear functions commonly associated with high-gain methods during the design process. Our work provides a stronger version of the matrix pencil formulation that is applicable when nonlinearities satisfy the so-called linear growth condition, even if the growth rate is unknown. In our method the selection of design parameters is straightforward as it involves only three parameters: one for the prescribed convergence time t f , and the other two are for the controller and the observer respectively, and the choice of the latter two parameters does not affect each other. Once the system order is determined, one can directly obtain reasonable ranges for these two parameters. Numerical simulations verify the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Subject and Purpose. The Matrix Pencil method is investigated as a means of increasing the accuracy of measuring the angles of low-altitude targets above sea level. The aim of the study was to evaluate its effectiveness in the conditions of interference created by multi-beam propagation, represented by mirror and diffuse reflection of radio waves from the sea surface, and thermal noise of radar receiving channels, and to justify measures to achieve high efficiency in different combinations. and obstacle levels. Methods and Methodology. The study was conducted by computer simulation. It covered the sea waves, which met the conditions of both specular and strong diffuse reflection, as well as different levels of thermal noise in the receiving channels of the radar. Typical conditions were used in the simulation: the length of the route – 2000 m, the height of the center of the receiving antenna at sea level – 12 m, the width of the antenna beam – 1°, the angular height of the target – 0.3°. The effectiveness of the method was determined by comparing it with the previously studied method of root-MUSIC under identical conditions. Results. The data on the errors of measuring the place angle when using different a priori parameters of the Matrix Pencil method are obtained. Recommendations for the choice of these parameters have been developed to minimize errors in measuring the angles of the place. The advantages and disadvantages of the Matrix Pencil method in comparison with the root-MUSIC method are noted. Conclusion. The obtained results can be used in the development of radars designed to support low-altitude objects above sea level.

• Porosity indications through frequency/wavenumber domain filtering. • Determination of mode- and wavenumber-specific damping using the matrix pencil method. • Principal component analysis of guided wavefield data. • Pore content distinguishability based on scattering observations of uniform accumulations. • Significance of pore morphology/orientation on scattering results. The present numerical study aims to analyze the effect of pores on propagating guided waves while outlining the relevance of pore formation and its significance to structural properties. The primary focus of this study was on the pores present in planar carbon fiber-reinforced polymer (CFRP) parts. The wavefield data were generated by applying the elastodynamic finite integration technique (EFIT) solver AE3D, allowing for a systematic analysis based on fully controllable configurations and noise-free data. An isotropic material model was used to simplify the modeling of the anisotropic CFRP by assuming the averaged quasi-isotropic properties of the CFRP in all directions. More than 220 specimens containing single pores or four basic configurations of pore accumulations were modeled and analyzed in terms of their effect on the out-of-plane component of the propagating guided waves. Analysis of the guided wavefield data focused on the direction-dependent maximum amplitude differences and deviations in the wavefield energy. Mode- and wavenumber-specific amplitude loss along the centerline was analyzed using a matrix pencil algorithm. Finally, principal component analysis (PCA) was conducted to examine deviations caused by the presence of pores. Analysis of the wavefield effects of the pores for each specific configuration showed good differentiation between the considered pore volume groups. At the same time, the effects of the pores were shown to be significantly dependent on the pore configuration in terms of orientation and morphological properties. Associated with this, the intersecting outcome ranges for different pore volume groups were evident when all the pore configurations were considered simultaneously. [ABSTRACT FROM AUTHOR]

Indoor localization using WiFi signal parameters is challenging, with encouraging decimeter localization results available with enough line-of-sight coverage and hardware infrastructure. This paper proposes a new 2-dimensional multiple packets based matrix pencil (2D M-MP) method to estimate the Angle of Arrival (AoA) and Time of Flight (ToF) based on WiFi channel state information (CSI). Compared with the conventional parameter estimation algorithms, this method has two advantages. First, 2D M-MP method uses the discrete Fourier transform (DFT) to convert the complex computation into real computation to reduce the computational complexity significantly without losing accuracy. Second, it accumulates multiple CSI packets to improve the parameter estimation accuracy effectively, especially at low values of signal-to-noise-ratio (SNR) environment. To verify the practicability of our proposed 2D M-MP method, we set up a localization system in an actual scenario using commodity WiFi cards which demonstrates that the performance of 2D M-MP method is better than conventional parameter estimation algorithms and can achieve a localization accuracy of 42 cm in indoor hall deployment. [ABSTRACT FROM AUTHOR]

A number of theoretical and computational problems for matrix polynomials are solved by passing to linearizations. Therefore a perturbation theory, that relates perturbations in the linearization to equivalent perturbations in the corresponding matrix polynomial, is needed. In this paper we develop an algorithm that finds which perturbation of matrix coefficients of a matrix polynomial corresponds to a given perturbation of the entire linearization pencil. Moreover we find transformation matrices that, via strict equivalence, transform a perturbation of the linearization to the linearization of a perturbed polynomial. For simplicity, we present the results for the first companion linearization but they can be generalized to a broader class of linearizations. [ABSTRACT FROM AUTHOR]

Introduction. In practical signal processing and its many applications, researchers and engineers try to find a number of harmonics and their frequencies in a time signal contaminated by noise. In this manuscript we propose a new approach to this problem. Aim. The main goal of this work is to embed the original time series into a set of multi-dimensional information vectors and then use shift-invariance properties of the exponentials. The information vectors are cast into a new basis where the exponentials could be separated from each other. Materials and methods. We derive a stable technique based on the singular value decomposition (SVD) of lagcovariance and cross-covariance matrices consisting of covariance coefficients computed for index translated copies of an original time series. For these matrices a generalized eigenvalue problem is solved. Results. The original time series is mapped into the basis of the generalized eigenvectors and then separated into components. The phase portrait of each component is analyzed by a pattern recognition technique to distinguish between the phase portraits related to exponentials constituting the signal and the noise. A component related to the exponential has a regular structure, its phase portrait resembles a unitary circle/arc. Any commonly used method could be then used to evaluate the frequency associated with the exponential. Conclusion. Efficiency of the proposed and existing methods is compared on the set of examples, including the white Gaussian and auto-regressive model noise. One of the significant benefits of the proposed approach is a way to distinguish false and true frequency estimates by the pattern recognition. Some automatization of the pattern recognition is completed by discarding noise-related components, associated with the eigenvectors that have a modulus less than a certain threshold.

A novel matrix-pencil (MP)-based interference mitigation approach for frequency-modulated continuous-wave (FMCW) radars is proposed in this article. The interference-contaminated segment of the beat signal is first cut out, and then, the signal samples in the cutout region are reconstructed by modeling the beat signal as a sum of complex exponentials and using the MP method to estimate their parameters. The efficiency of the proposed approach for the interference with different parameters (i.e., interference duration, signal-to-noise ratio (SNR), and different target scenarios) is investigated by means of numerical simulations. The proposed interference mitigation approach is intensively verified on experimental data. Comparisons of the proposed approach with the zeroing and other beat-frequency interpolation techniques are presented. The results indicate the broad applicability and superiority of the proposed approach, especially in low SNR and long interference duration situations. [ABSTRACT FROM AUTHOR]

In this work we study the change of the structure of a regular pencil when we perform small perturbations over some of its rows and the other rows remain unaltered. We provide necessary conditions when several rows are perturbed, and prove them to be sufficient to prescribe the homogenous invariant factors or the Weyr characteristic of the resulting pencil when one row is perturbed. [ABSTRACT FROM AUTHOR]

We study the characterization of several distance problems for linear differential-algebraic systems with dissipative Hamiltonian structure. Since all models are only approximations of reality and data are always inaccurate, it is an important question whether a given model is close to a 'bad' model that could be considered as ill-posed or singular. This is usually done by computing a distance to the nearest model with such properties. We will discuss the distance to singularity, the distance to the nearest high index problem, and the distance to instability for dissipative Hamiltonian systems. While for general unstructured differential-algebraic systems the characterization of these distances are partially open problems, we will show that for dissipative Hamiltonian systems and related matrix polynomials there exist explicit characterizations that can be implemented numerically. [ABSTRACT FROM AUTHOR]

We propose a connection between finite zero-sum stochastic games (henceforth stochastic games) and multiparameter eigenvalue problems. This connection, which relies on the theory developed by Shapley and Snow for matrix games, opens new possibilities in the study of stochastic games. In particular, we derive from this connection a handful of new results for stochastic games. [ABSTRACT FROM AUTHOR]

Out-of-band interference (OOBI) test is a bench-mark test for M-class phasor measurement units (PMUs) in IEEE/IEC 60255–118-1-2018 and C37.118.1a-2014. Because of the spectral interference produced by interharmonic, it is difficult for dynamic phasor estimators to satisfy the OOBI test requirements with a short observation time. To this end, this article proposed a Taylor weighted least squares (TWLS) method based on the matrix pencil for the dynamic phasor estimation of the OOBI contaminated signal. The proposed method first applies the matrix pencil algorithm to obtain rough estimations for the frequencies of the fundamental component and other components (i.e., harmonics, subharmonics, and interharmonics) within a few cycles. Then, the TWLS method was adopted to acquire more accurate estimations of the dynamic phasor and frequencies. The effectiveness of the proposed method is validated by simulations under steady and dynamic conditions that follow the requirements of the IEEE standard. [ABSTRACT FROM AUTHOR]

The objective of this study is to extract two pairs of left and right deflating subspaces of a regular matrix pencil A − λ B corresponding to the eigenvalues lying inside and outside the unit disk via computation of left and right disk functions d i s k L (A − λ B) and d i s k R (A − λ B) which are obtained from matrix sign evaluations of two suitable matrices. We propose a novel fifth order iterative method for evaluating matrix sign function which can be implemented to compute disk functions. Convergence analysis of the proposed method has been investigated globally along with the illustration of attraction basins. Asymptotic stability of the method is also presented. Then an application of deflating subspaces in solving generalized eigenvalue problem has been explored. Numerical test problems have been performed by considering matrices of various sizes to justify the efficiency and superiority of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we propose decentralized prescribed-time control schemes for a class of interconnected nonlinear systems by using time-varying high-gain domination approach. It is shown that prescribed-time exact regulation can be achieved by a linear observer based controller in the presence of completely unknown interactions. Besides, without significantly modifying the original controller structure, we develop a singularity-free controller with bounded time-varying feedback to drive the system state to converge to a compact set within a preset time, and ultimately converge to zero. In both of these approaches, the controller and observer designs have a dual architecture, thereby obviating the need for the separation principle (which might not be valid for nonlinear systems with mismatched uncertainties). In addition, the relationship between the upper bound of the time-varying gain and the stability of the closed-loop system is thoroughly analyzed. [ABSTRACT FROM AUTHOR]

We study almost inner derivations of 2-step nilpotent Lie algebras of genus 2, i.e., having a 2-dimensional commutator ideal, using matrix pencils. In particular we determine all almost inner derivations of such algebras in terms of minimal indices and elementary divisors over an arbitrary algebraically closed field of characteristic not 2 and over the real numbers. [ABSTRACT FROM AUTHOR]

Abstract Background The response of many biomedical systems can be modelled using a linear combination of damped exponential functions. The approximation parameters, based on equally spaced samples, can be obtained using Prony’s method and its variants (e.g. the matrix pencil method). This paper provides a tutorial on the main polynomial Prony and matrix pencil methods and their implementation in MATLAB and analyses how they perform with synthetic and multifocal visual-evoked potential (mfVEP) signals. This paper briefly describes the theoretical basis of four polynomial Prony approximation methods: classic, least squares (LS), total least squares (TLS) and matrix pencil method (MPM). In each of these cases, implementation uses general MATLAB functions. The features of the various options are tested by approximating a set of synthetic mathematical functions and evaluating filtering performance in the Prony domain when applied to mfVEP signals to improve diagnosis of patients with multiple sclerosis (MS). Results The code implemented does not achieve 100%-correct signal approximation and, of the methods tested, LS and MPM perform best. When filtering mfVEP records in the Prony domain, the value of the area under the receiver-operating-characteristic (ROC) curve is 0.7055 compared with 0.6538 obtained with the usual filtering method used for this type of signal (discrete Fourier transform low-pass filter with a cut-off frequency of 35 Hz). Conclusions This paper reviews Prony’s method in relation to signal filtering and approximation, provides the MATLAB code needed to implement the classic, LS, TLS and MPM methods, and tests their performance in biomedical signal filtering and function approximation. It emphasizes the importance of improving the computational methods used to implement the various methods described above.

The paper focuses on the problem of increasing the accuracy of radar measurements of the elevation angle of the low-altitude targets over the sea. The Matrix Pencil method, as one of the means to resolve this problem, is studied in the paper. This method refers to the methods of high spectral resolution, but in a number of publications, the possibility of its use for measuring the angles of arrival of radio waves is also considered. However, the extent of its suitability for solving the problem needs to be studied taking into account the specific conditions under which the measurement of the elevation angles of low-altitude targets is performed. The research was carried out by computer simulation. It covered the sea states with the conditions of both specular and strong diffuse reflection satisfied. One of the main goals of this work was to determine ultimate measurement accuracy achieved by the method when only the diffuse reflection from the sea is considered as a disturbance (a specular signal along with a direct signal from the target are considered as a measured signal). It was established that by this indicator the Matrix Pencil method does not concede to the well-known MUSIC method, but even exceeds it, especially for strong diffuse reflection (improvement reaches 20%). Another impor-tant task was to investigate the influence of the a priori parameters of the method on the measurement errors: the number of measured signals, the values of the so-called pencil-parameter and the number of spatial samples of the signal at the given antenna aperture. As a result, recommen-dations on the selection of these parameters have been developed, which make it possible to minimize the errors in the measurement of the elevation angles. This research expands the understanding of ways to increase the accuracy of mea-suring the angles of low-altitude targets over an agitated sea.

In acoustic logging, most published studies of extracting array data characteristics are difficult to avoid the interference from the false modes. The false modes contribute little to the acoustic waveforms but makes the true modes hard to identify. We introduce a threshold of the energy spectrum into the pole-calculation method to remove the false modes, thereby eliminating their interferences to the true modes. To avoid the separation of multiple crossings of pole distribution, we directly acquire the slowness dispersion of multiple modes by superposing the single-mode results calculated from the weighted spectral semblance method. We processed the slowness-frequency snapshot and design an extreme-point extraction formula to generate the multi-mode scatterplot, which provide a precise slowness dispersion result. In terms of four synthetic cases of acoustic logging, it has demonstrated that the proposed algorithm can provide a high-resolution slowness dispersion profile for multiple modes. Even for acoustic logging while drilling with strong tool wave components, the aliases and interferences of false modes do not exist. Finally, a test proves that the proposed algorithm also has a good anti-noise ability. [ABSTRACT FROM AUTHOR]