Kronecker Product Research Articles

Dominant subspaces of high-fidelity polynomial structured parametric dynamical systems and model reduction

In this work, we investigate a model order reduction scheme for high-fidelity nonlinear structured parametric dynamical systems. More specifically, we consider a class of nonlinear dynamical systems whose nonlinear terms are polynomial functions, and the linear part corresponds to a linear structured model, such as second-order, time-delay, or fractional-order systems. Our approach relies on the Volterra series representation of these dynamical systems. Using this representation, we identify the kernels and, thus, the generalized multivariate transfer functions associated with these systems. Consequently, we present results allowing the construction of reduced-order models whose generalized transfer functions interpolate these of the original system at pre-defined frequency points. For efficient calculations, we also need the concept of a symmetric Kronecker product representation of a tensor and derive particular properties of them. Moreover, we propose an algorithm that extracts dominant subspaces from the prescribed interpolation conditions. This allows the construction of reduced-order models that preserve the structure. We also extend these results to parametric systems and a special case (delay in input/output). We demonstrate the efficiency of the proposed method by means of various numerical benchmarks.

Upper bounds for the norm of the sum and Kronecker product of Hilbert space operators

Upper bounds for the norm of the sum and Kronecker product of Hilbert space operators

Constrained Normalized Subband Adaptive Filter Algorithm and Its Performance Analysis

AbstractConstrained least mean square (CLMS) algorithm is the most popular constrained adaptive filtering algorithm due to its simple structure and easy implementation. However, its convergence slows down when the input signal is colored. To address this issue, this paper firstly introduces the normalized subband adaptive filter (NSAF) into the constrained filtering problem and derives a constrained NSAF (CNSAF) algorithm using the Lagrange multiplier method. Benefiting from the good decorrelation capability of the NSAF, the proposed CNSAF algorithm significantly improves the convergence performance of the CLMS algorithm under colored inputs. Then, the mean and mean-square stability of the CNSAF algorithm is analyzed, and the theoretical models to characterize the transient and steady-state mean square deviation (MSD) behaviors of the CNSAF algorithm are derived utilizing the Kronecker product property and vectorization method. Further to extend the CNSAF algorithm to the problem of sparse system identification, a sparse version of the CNSAF algorithm (S-CNSAF) is derived. Finally, the validity of the derived theoretical MSD prediction models and the superiority of the proposed algorithms are confirmed by extensive computer simulations on system identification with colored inputs.

Time Aware Long Short-Term Memory and Kronecker Gated Intelligent Transportation for Smart Car Parking

Technology desires to improve quality of life and impart citizen’s health as well as happiness. The concept of Internet of Things (IoT) refers to smart world where prevailing objects are said to be embedded and hence interact with each other (i.e., between objects and human beings) to achieve an objective. In the period of IoT as well as smart city, there is requirement for Intelligent Transport System-based (ITS) ingenious smart parking or car parking space prediction (CPSP) for more feasible cities. With the increase in population and mushroom growth in vehicles are bringing about several distinct economic as well as environmental issues. One of pivotal ones is optimal parking space identification. To address on this problem, in this work, Time-aware Long Short-Term Memory and Kronecker product Gated Recurrent Unit (TLSTM-KGRU) for smart parking in internet of transportation things is proposed. The TLSTM-KGRU method is split into two sections. In the first section, smart parking occupancy is derived using Time-aware Long Short-Term Memory (Time-aware LSTM) for Kuala Lumpur Convention Centre car parking sensor dataset. Following which the resultant smart car occupancy results are subjected to Linear Interpolations and Kronecker product Gated Recurrent Unit for predicting smart parking. When compared against other predictive methods such as SGRU-LSTM and CPSP using DELM, our experimental outcomes denote that TLSTM-KGRU method has improved performance for smart parking occupancy forecast as it not only enhances sensitivity and specificity but also reduces prediction time with minimum delay.

Robust asymptotic stability analysis for fractional-order systems with commensurate time delays: The 1 

The robust asymptotic stability problem for fractional-order linear systems with commensurate time delays (FOSCDs) and structured perturbations is investigated in this manuscript. Based on the properties of Kronecker products and μ-analysis, the necessary and sufficient conditions are given to guarantee the asymptotic stability of FOSCDs. Then, the necessary and sufficient conditions for the robust asymptotic stability of FOSCDs with structured perturbations are investigated. To reduce the computing complexity and deal with the problems of systems with larger dimensions, a sufficient condition on the robust asymptotic stability of FOSCDs with structured perturbations is given based on the properties of Kronecker products and 2-norm transformations. Finally, several examples are illustrated to test the effectiveness of the proposed results.

GbyE: an integrated tool for genome widely association study and genome selection based on genetic by environmental interaction.

The growth and development of organism were dependent on the effect of genetic, environment, and their interaction. In recent decades, lots of candidate additive genetic markers and genes had been detected by using genome-widely association study (GWAS). However, restricted to computing power and practical tool, the interactive effect of markers and genes were not revealed clearly. And utilization of these interactive markers is difficult in the breeding and prediction, such as genome selection (GS). Through the Power-FDR curve, the GbyE algorithm can detect more significant genetic loci at different levels of genetic correlation and heritability, especially at low heritability levels. The additive effect of GbyE exhibits high significance on certain chromosomes, while the interactive effect detects more significant sites on other chromosomes, which were not detected in the first two parts. In prediction accuracy testing, in most cases of heritability and genetic correlation, the majority of prediction accuracy of GbyE is significantly higher than that of the mean method, regardless of whether the rrBLUP model or BGLR model is used for statistics. The GbyE algorithm improves the prediction accuracy of the three Bayesian models BRR, BayesA, and BayesLASSO using information from genetic by environmental interaction (G × E) and increases the prediction accuracy by 9.4%, 9.1%, and 11%, respectively, relative to the Mean value method. The GbyE algorithm is significantly superior to the mean method in the absence of a single environment, regardless of the combination of heritability and genetic correlation, especially in the case of high genetic correlation and heritability. Therefore, this study constructed a new genotype design model program (GbyE) for GWAS and GS using Kronecker product. which was able to clearly estimate the additive and interactive effects separately. The results showed that GbyE can provide higher statistical power for the GWAS and more prediction accuracy of the GS models. In addition, GbyE gives varying degrees of improvement of prediction accuracy in three Bayesian models (BRR, BayesA, and BayesCpi). Whatever the phenotype were missed in the single environment or multiple environments, the GbyE also makes better prediction for inference population set. This study helps us understand the interactive relationship between genomic and environment in the complex traits. The GbyE source code is available at the GitHub website ( https://github.com/liu-xinrui/GbyE ).

Exploiting high-quality reconstruction image encryption strategy by optimized orthogonal compressive sensing

Compressive sensing is favored because it breaks through the constraints of Nyquist sampling law in signal reconstruction. However, the security defects of joint compression encryption and the problem of low quality of reconstructed image restoration need to be solved urgently. In view of this, this paper proposes a compressive sensing image encryption scheme based on optimized orthogonal measurement matrix. Utilizing a combination of DWT and OMP, along with chaos, the proposed scheme achieves high-security image encryption and superior quality in decryption reconstruction. Firstly, the orthogonal optimization method is used to improve the chaotic measurement matrix. Combined with Part Hadamard matrix, the measurement matrix with strong orthogonal characteristics is constructed by Kronecker product. Secondly, the original image is sparsely represented by DWT. Meanwhile, Arnold scrambling is used to disturb the correlation between its adjacent pixels. Following this, the image is compressed and measured in accordance with the principles of compressive sensing and obtain the intermediate image to be encrypted. Finally, the chaotic sequence generated based on 2D-LSCM is used to perform on odd-even interleaved diffusion and row-column permutation at bit-level to obtain the final ciphertext. The experimental results show that this scheme meets the cryptographic requirements of obfuscation, diffusion and avalanche effects, and also has a large key space, which is sufficient to resist brute-force cracking attacks. Based on the sparse and reconstruction algorithm of compressive sensing proposed in this paper, it has better image restoration quality than similar algorithms. Consequently, the compressive sensing image encryption scheme enhances both security and reconstruction quality, presenting promising applications in the evolving landscape of privacy protection for network big data.

Handwritten online signature verification and Forgery Detection for Online Handwritten Signature using Hybrid Wavelet Transform-1 with HMM classifier

Online signature verification employs a distinctive biometric trait by utilizing both static and dynamic features extracted from 2D signature images. A hybrid wavelet transform, denoted as HWT-1 with a size of 256, is formed through the Kronecker product of two orthogonal transforms, such as DCT, DHT, Haar, Hadamard, and Kekre, each with sizes 4 and 64. This HWT facilitates signal analysis at both global and local levels, akin to traditional wavelet transforms. Specifically, HWT-1 processes the 256 samples of online handwritten signatures, yielding 128 samples that constitute the feature vectors for signature verification and forgery detection. These feature vectors are then inputted into the Left-Right and Ergodic Hidden Markov Model (HMM) classifiers for further analysis. The HMMs are trained using 10 randomly selected genuine signature samples and subsequently tested with the remaining 10 genuine signatures and 20 forged signatures from the SVC 2004 signature database, repeating this process 20 times to compute average values. Among all possible combinations of HWT-1 utilizing DCT, DHT, Haar, Hadamard, and Kekre transforms for the Left-Right HMM model, the combination of DCT 4 and Haar 64 demonstrates the best performance with a False Rejection Rate (FRR) and False Acceptance Rate (FAR) of 1.05% and 0.99%, respectively, at state 3. Similarly, considering all feasible combinations of HWT-1 for the Ergodic HMM model, the combination of DCT 4 and Kekre 64 yields the optimal performance with an FRR and FAR of 1.24% and 1.33%, respectively, at state 3.   

Handwritten Online Signature Verification and Forgery Detection Using Hybrid Wavelet Transform-2 with HMM Classifier

Online signature verification stands out as a distinctive biometric feature, offering both static and dynamic attributes within 2D signature images. A Hybrid Wavelet Transform-2 (HWT-2) with a size of 256 is constructed by employing the Kronecker product of two orthogonal transforms: DCT, DHT, Haar, Hadamard, and Kekre, each with sizes of 4 and 64. The HWT enables the analysis of signals at both global and local levels, akin to wavelet transforms. HWT-2 is applied to 256 samples of online handwritten signatures, and the first 128 samples of the output are utilized as feature vectors for the verification and forgery detection of online handwritten signatures. These feature vectors are inputted into Left-Right and Ergodic Hidden Markov Model (HMM) classifiers for analysis. The HMMs are trained using 10 randomly selected genuine signature samples, and subsequently tested on the remaining 10 genuine signatures and 20 forged signatures from 40 users of the SVC 2004 signature database. This process is repeated 20 times, and the average values are computed. Among all possible combinations of HWT-2 using DCT, DHT, Haar, Hadamard, and Kekre transforms for the Left-Right HMM model, the combination of DCT 4 and DHT 64 demonstrates the best performance, with False Rejection Rate (FRR) and False Acceptance Rate (FAR) values of 3.96% and 1.48%, respectively, for state 5. Similarly, for the Ergodic HMM model, the combination of DCT 4 and DHT 64 exhibits the best performance, with FRR and FAR values of 1.10% and 2.88%, respectively, for state 5. These results indicate that combinations of HWT-2 outperform individual orthogonal transforms, and further, that HWT-2 combinations within the Ergodic HMM model offer superior performance compared to the Left-Right HMM model.   

Comparative Analysis of Handwritten Online Signature Verification and Forgery Detection Using Hybrid Wavelet Transform-1 and 2 with HMM Classifier

Online signature verification is a unique biometric feature. Provides static and dynamic features for 2D signature images. Hybrid wavelet transform -1 and 2 (HWT-1 and HWT-2) of size 256 is created using the Kronecker product of two orthogonal transforms such as DCT, DHT, Haar, Hadamard and Kekre with size 4 and 64. HWT has the ability to analyze signals such as wavelet transform at global and local levels. HWT-1 and HWT-2 are used for the 256 samples of the online Handwritten signature and the first 128 samples of the output are used as feature vectors for handwritten online signature verification and forgery detection. This feature vector is given to the Left-Right and Ergodic modelled HMM classifier for analysis. HMM is trained using 10 randomly chosen genuine signature samples and is used to test remaining 10 genuine signatures and 20 forged signatures of 40 users of SVC 2004 signature database. This process is iterated 20 times and then average values are calculated. Considering all the possible combination of HWT-1 and HWT-2 for DCT, DHT, Haar, Hadamard and Kekre transform for Left Right HMM model, DCT 4 Haar 64 HWT-1 offers best performance of FRR, FAR of 1.05%, 0.99% respectively for state 3. Considering all the possible combination of HWT-1 and HWT-2 for DCT, DHT, Haar, Hadamard and Kekre transform for Ergodic HMM model, DCT 4 DHT 64 offers best performance of FRR, FAR of 1.10%, 2.88% respectively for state 5. We conclude that comparing HWT-1 and HWT-2 combinations for Left Right and Ergodic HMM model, HWT-1 for Left Right model offer better performance.   

The Conservative and Efficient Numerical Method of 2-D and 3-D Fractional Nonlinear Schrödinger Equation Using Fast Cosine Transform

This paper introduces a novel approach employing the fast cosine transform to tackle the 2-D and 3-D fractional nonlinear Schrödinger equation (fNLSE). The fractional Laplace operator under homogeneous Neumann boundary conditions is first defined through spectral decomposition. The difference matrix Laplace operator is developed by the second-order central finite difference method. Then, we diagonalize the difference matrix based on the properties of Kronecker products. The time discretization employs the Crank–Nicolson method. The conservation of mass and energy is proved for the fully discrete scheme. The advantage of this method is the implementation of the Fast Discrete Cosine Transform (FDCT), which significantly improves computational efficiency. Finally, the accuracy and effectiveness of the method are verified through two-dimensional and three-dimensional numerical experiments, solitons in different dimensions are simulated, and the influence of fractional order on soliton evolution is obtained; that is, the smaller the alpha, the lower the soliton evolution.

A Robot Positional Error Compensation Method Based on Improved Kriging Interpolation and Kronecker Products

This paper proposes a stable and high-accuracy model-free calibration method for unopened robotic systems, which can significantly improve the robot positional accuracy. Two improvements are made to the existing kriging-based error compensation method to achieve robustness/practicality enhancement goals: (1) The semi-variogram model is solved using the sequential quadratic programming (SQP) algorithm, and the distance weight coefficients are added to the objective function. The accuracy of semi-variogram modeling is guaranteed, thereby improving the performance and stability of error compensation. (2) The hand-eye (the pose matrix between the robot base and the laser tracker) and the tool center point (TCP) position are calibrated based on Kronecker products without relying on the spatial analyzer (SA) software to construct the robot base coordinate system separately. Thus, the calibration accuracy and efficiency are significantly improved. Experiments have been conducted, and the results reveal that the proposed method can significantly reduce the robot positional error. Furthermore, the proposed approach performs better than the existing kriging-based error compensation method without Kronecker products. The maximum, mean, and root mean square (RMS) values of the absolute positional errors are reduced by 34.48%, 8.09%, and 10.05%, respectively.

IHUVS: Infinite Homography-Based Uncalibrated Methodology for Robotic Visual Servoing

The construction of task functions in robotic manipulation is of paramount importance for uncalibrated visual servoing. The existing methods generally use image information as control variables and estimate the image Jacobian matrix online, thus possessing issues relating to convergence, and image Jacobian matrix singularities. Therefore, this work proposes a novel methodology dubbed infinite homography-based uncalibrated visual servoing (IHUVS), in which the visual control of the robot end-effector pose is decomposed into its rotational and translational components. The corresponding rotational controller designs the visual servoing task function using the relationship between the infinite homography matrix and rotation matrix, and employs the Kronecker product to derive linear equations for rotational control, as well as to conduct the associated task error analysis. Meanwhile, the translational controller utilizes Kalman filtering for online estimation of the Jacobian matrix that is required by the proportional control scheme. The robot end-effector motion in Cartesian space is generated via the IHUVS method, without knowing the camera's intrinsic parameters and the robot hand-eye relationship. A simulation analysis is carried out to assess the algorithm's numerical performance, while robotic visual servoing experiments are also conducted to verify the accuracy and efficacy of the proposed IHUVS method.

How do wavelength correlations affect transmission spectra? Application of a new fast and flexible 2D Gaussian process framework to transiting exoplanet spectroscopy

The use of Gaussian processes (GPs) is a common approach to account for correlated noise in exoplanet time series, particularly for transmission and emission spectroscopy. This analysis has typically been performed for each wavelength channel separately, with the retrieved uncertainties in the transmission spectrum assumed to be independent. However, the presence of noise correlated in wavelength could cause these uncertainties to be correlated, which could significantly affect the results of atmospheric retrievals. We present a method that uses a GP to model noise correlated in both wavelength and time simultaneously for the full spectroscopic dataset. To make this analysis computationally tractable, we introduce a new fast and flexible GP method that can analyse 2D datasets when the input points lie on a (potentially non-uniform) 2D grid - in our case a time by wavelength grid - and the kernel function has a Kronecker product structure. This simultaneously fits all light curves and enables the retrieval of the full covariance matrix of the transmission spectrum. Our new method can avoid the use of a `common-mode' correction, which is known to produce an offset to the transmission spectrum. Through testing on synthetic datasets, we demonstrate that our new approach can reliably recover atmospheric features contaminated by noise correlated in time and wavelength. In contrast, fitting each spectroscopic light curve separately performed poorly when wavelength-correlated noise was present. It frequently underestimated the uncertainty of the scattering slope and overestimated the uncertainty in the strength of sharp absorption peaks in transmission spectra. Two archival VLT/FORS2 transit observations of WASP-31b were used to compare these approaches on real observations. Our method strongly constrained the presence of wavelength-correlated noise in both datasets, and significantly different constraints on atmospheric features such as the scattering slope and strength of sodium and potassium features were recovered.

H∞ Controller Design for Networked Systems With Two-Channel Packet Dropouts and FDI Attacks.

In this article, the stochastic analysis and H∞ controller design problems of networked systems with packet dropouts and false data injection attacks are investigated. Different from the existing literature, we focus on the linear networked systems with external disturbances and both sensor-controller channel and controller-actuator channel are studied. First, we present a discrete-time modeling framework that leads to a stochastic closed-loop system with randomly varying parameters. To facilitate the analysis and H∞ control of resulting discrete-time stochastic closed-loop system, an equivalent yet analyzable stochastic augmented model is further constructed by matrix exponential computation. Based on this model, a stability condition is derived in the form of linear matrix inequality (LMI) with the aid of a reduced-order confluent Vandermonde matrix, Kronecker product operation, and law of total expectation. Specifically, the dimension of the LMI obtained in this article does not increase as the upper bound of consecutive packet dropouts does, which is also different from the existing literature. Subsequently, a desired H∞ controller is obtained such that the original discrete-time stochastic closed-loop system is exponentially mean-square stable with a prescribed H∞ performance. Finally, a numerical example and a direct current motor system are exploited to substantiate the effectiveness and practicability of the designed strategy.

One-Step Solving the Hand–Eye Calibration by Dual Kronecker Product

Abstract Hand–eye calibration is a typical research direction in robotics applications. The current methods can be divided into two categories according to whether the rotational and translational equations are decoupled for computation: two-step methods and one-step methods. Both one-step and two-step methods generally convert such problems to linear null space computations, which are implemented by the corresponding computational operators. Owing to the booming development of the rotation operators, the two-step methods have been more fully researched. However, due to the limitations of the research on computational operators integrating rotation and translation, the one-step methods still have much scope for research. Dual algebra, as effective mathematical entities for screws and wrenches, provides the theoretical basis for the development of the one-step methods for hand–eye calibration. In this paper, a computational operator for the dual matrices computation was first proposed, i.e., dual Kronecker product. Subsequently, a hand–eye calibration framework was proposed based on the dual Kronecker product, which allowed the screw motion to be represented as multiple dual vectors. Furthermore, the equivalence of this framework with the orthogonal-dual-tensor-based approach was derived, providing a more intuitive computational representation. The feasibility and superiority of the proposed computational framework were experimentally verified.

Transition Analysis of Stochastic Logical Control Networks

Transition analysis is essential for analyzing and regulating logical control networks (LCNs). The algebraic state-space representation method, which relies on the semi- tensor product of matrices, has been shown to allow deterministic LCNs to be represented as linear- like systems. However, due to the inherent uncertainty, it is hard to obtain the algebraic expression of a stochastic LCN. A unified paradigm for the transition analysis of LCNs with stochastic and deterministic dynamics is provided in this research. First, the algebraic expression of LCN with deterministic dynamics is reviewed. Second, the algebraic expression of LCN with stochastic dynamics is considered, where the non-equivalence between the dispersed form and the integrated form is proposed. Then the reason for the non-equivalence is provided. After that, a consistency condition is presented to bridge the gap between the independent model and the conditionally independent model. Lastly, we specifically point out that probabilistic LCN satisfies the consistency criteria, allowing one to calculate the probabilistic LCN transition matrix by using a power-reducing operator.

Stability Analysis and Synthesis for Networked Systems With Noisy Sampling Intervals, Random Transmission Delays, and Packet Dropouts

In this article, we study the stability analysis and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> controller synthesis problems for networked systems with random transmission delays and packet dropouts that are subject to noisy sampling intervals. The sampling errors and transmission delays are described by an independent and identically distributed sequence of continuous random variables and the transmission delays are assumed to be smaller than the noisy sampling intervals. First, a closed-loop discrete-time stochastic framework for networked system that allows investigating noisy sampling intervals, random transmission delays and packet dropouts simultaneously is established. To render the discrete-time stochastic system suitable for stability analysis and synthesis, an equivalent yet tractable stochastic augmented model is then reformulated with the help of matrix exponential computation. Based on this augmented model, a stability condition involving the expectation of a nonlinear and random coupling matrix is derived. By recurring to the law of total expectation and Kronecker product operation, the expectation operation of the coupling matrix subject to high nonlinearity and multiple randomness is decoupled. Subsequently, a controller algorithm is designed such that the exponential mean-square stability of the original discrete-time stochastic system with a prescribed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> performance is guaranteed. Finally, an example is provided to illustrate the effectiveness of the controller design algorithm.

SOLVING BIPOLAR FULLY FUZZY SYLVESTER MATRIX EQUATIONS WITH NEGATIVE FUZZY NUMBERS

Sylvester matrix equations play a crucial role in control theory for controller design. Bipolar Fully Fuzzy Sylvester Matrix Equations (FFSME), incorporating both positive and negative components, are employed in controller design to address uncertainties that may affect a system’s performance and stability. However, there is not much existing research on combining bipolar fuzzy numbers and FFSME,and most of them mainly deal with positive coefficients. Thus, this paper presents a method that enables solving the negative coefficient of bipolar FFSME in the form of Left-Right (LR) triangular fuzzynumbers using an Associated Bipolar Linear System (ABLS). To obtain the ABLS, bipolar FFSME is transformed into a bipolar Fully Fuzzy Linear System (FFLS) using the Kronecker product and Vecoperator. Subsequently, the solution is derived through the inverse method, and the equation of the ABLS is rearranged as a bipolar fuzzy matrix. Additionally, this paper provides two numerical examples to illustrate the applicability of the constructed method.

Stochastic stability robustness and H 2 performance robustness of continuous-time Markovian jump linear systems with uncertain transition rates via μ analysis

This paper considers the stability robustness and H 2 performance robustness of continuous-time Markovian jump linear systems with uncertain transition rates. Based on the Kronecker product, the robustness problems are transformed into problems of judging whether a class of uncertain matrices can maintain nonsingularity. For the robust stability problem, the method for computing the robust upper and lower bounds of transition rates is given first and then the necessary and sufficient condition for the investigated system to maintain robust stochastic stability is given when considering interval uncertainty. For the robust H 2 performance problem, the robustness bounds and sufficient and necessary conditions guaranteeing the robust H 2 performance are similarly given. The robustness bounds for analysing the tolerance of continuous-time Markovian jump linear systems to parameter uncertainties can be conveniently computed by utilising the proposed results. Numerical examples illustrate that the results are less conservative than existing results.