Realization Of Linear Systems Research Articles

Dual rectangular fuzzy complex matrix equations: extended solution, algebraic solution, solution and their calculating1

In this paper, we focus on generalized fuzzy complex numbers and propose a straightforward matrix method to solve the dual rectangular fuzzy complex matrix equations C · Z ˜ + L ˜ = R · Z ˜ + W ˜ , in which C and R are crisp complex matrices and Z ˜ , L ˜ and M ˜ are fuzzy complex number matrices. The existing methods for solving fuzzy complex matrix equations involve separately calculating the extended solution and the corresponding parameters of the real and imaginary parts, whereby we obtain the algebraic solution of the equations. By means of the interval arithmetic and embedding approach, the n × n dual rectangular fuzzy complex linear systems could be converted into 2n × 2n fuzzy linear systems, which are also equivalent to the 4n × 4n real linear systems. By directly solving the 4n × 4n real linear systems, the algebraic solutions can be obtained. The general dual rectangular fuzzy complex matrix equations and dual rectangular fuzzy complex linear systems are investigated by the generalized inverses of matrices. Finally, some examples are given to illustrate the effectiveness of method.

A Block Conjugate Gradient Method for Quaternion Linear

This study aims at the simultaneous solution of several quaternion linear systems with the same Hermitian and positive definite coefficient matrix by employing the conjugate gradient method. We consider the setting when the quaternion Hermitian positive definite coefficient matrix at hand is very large so that direct methods are not applicable. In the study, we first transform linear quaternion systems into real linear systems. The transformed real linear systems have special structure due to the fact that they are real representations of quaternion systems. Benefitting from the special structure, we further reduce the size of these linear systems. Then a block conjugate gradient method is applied to the resulting reduced real linear systems. The solution obtained after applying the conjugate gradient method is a real representation of the solution of the original quaternion problem. Thus, a conversion of this real solution to the quaternion setting is performed in the end.

A new method for computation of positive realizations of fractional linear continuous-time systems

A new method for computation of positive realizations of fractional linear continuous-time systems

3-D anisotropic modelling of geomagnetic depth sounding based on unstructured edge-based finite-element method

SUMMARY Geomagnetic depth sounding (GDS) is a geophysical electromagnetic (EM) method that studies the deep structure and composition of the Earth by using long-period EM signals from geomagnetic observatories and satellites. In this paper, a 3-D anisotropic GDS modelling algorithm is developed. The curl–curl equation is discretized using the edge-based finite-element method on unstructured tetrahedral grids. In order to solve the computationally demanding problem of EM modelling on a global scale, the complex linear system is first separated into the equivalent real linear systems and then the real system is iteratively solved by the flexible generalized minimum residual method with a block diagonal pre-conditioner. This will greatly reduce the condition number of the linear system and thus speed up the solution process. We verify the accuracy of the proposed algorithm by comparing our results with the existing methods. After that, we design a subduction zone model to simulate the EM responses under isotropic and anisotropic environments, respectively. The numerical results show the high efficiency of the proposed algorithm and the response differences between isotropic and anisotropic models. This research can provide theoretical and technical support for the high-accuracy and efficient inversion of GDS data for the geo-dynamic study.

A low‐dimensional realization algorithm for periodic input/output behavioral systems

The state‐space realization of linear systems is of utmost importance in linear systems theory. After the realization problem for the time‐invariant case was solved, particular attention was paid to the case of linear periodic systems. Recently, such systems have regained importance, for instance, in the context of coding theory where periodic convolutional encoders play an important role. The majority of the contributions within this area only concern the realization of transfer functions (or impulse responses), thus excluding the case of input/output linear systems without coprime representations. By the end of the eighties of the last century, Jan C. Willems suggested an approach (nowadays known as the behavioral approach) that considers a wider class of systems and allows to overcome this drawback. According to this approach, the central object in a system is its behavior which consists of all the signals that satisfy the system laws (also called system trajectories). Consequently, the behavior of a system with an input/output representation that is not coprime contains more trajectories than the set of input/output signals defined by the system transfer function. Our work takes this fact into account. In previous research conducted by the authors, some results on the realization of periodic behavioral systems were obtained, which, despite their importance, are conservative with respect to the dimension of the obtained realizations. In this paper, we revisit the behavioral periodic realization problem and give further insight into this question. This allows to derive new results and set up an algorithm to compute low‐dimensional state‐space realizations.

Algorithms for solving a class of real quasi-symmetric Toeplitz linear systems and its applications

<abstract><p>In this paper, fast numerical methods for solving the real quasi-symmetric Toeplitz linear system are studied in two stages. First, based on an order-reduction algorithm and the factorization of Toeplitz matrix inversion, a sequence of linear systems with a constant symmetric Toeplitz matrix are solved. Second, two new fast algorithms are employed to solve the real quasi-symmetric Toeplitz linear system. Furthermore, we show a fast algorithm for quasi-symmetric Toeplitz matrix-vector multiplication. In addition, the stability analysis of the splitting symmetric Toeplitz inversion is discussed. In mathematical or engineering problems, the proposed algorithms are extraordinarily effective for solving a sequence of linear systems with a constant symmetric Toeplitz matrix. Fast matrix-vector multiplication and a quasi-symmetric Toeplitz linear solver are proven to be suitable for image encryption and decryption.</p></abstract>

The Fuzzy Complex Linear Systems Based on a New Representation of Fuzzy Complex Numbers

Since the product of complex numbers and rectangular fuzzy complex numbers (RFCN) is not necessarily a RFCN in the former fuzzy complex linear system (FCLS), the scalar multiplication and addition operations of complex numbers and fuzzy complex numbers (FCN) based on a new representation of FCN are proposed. We also introduce a new method for solving FCLS, which can convert FCLS into two distinct linear systems. One is an n×n complex linear system, and the other is an (mn)×(mn) real linear system, where n is the number of unknown variables, and m is the number of substitutional cyclic sets composed of coefficients of FCLS. In particular, using this method to solve one-dimensional fuzzy linear systems, a (2n)×(2n) RLS is obtained, which is consistent with Friedman’s method. Finally, FCLS based on the RFCN as a special case are also investigated.

Minimal positive realizations for linear systems with two-different-pole transfer functions

The minimal positive realization problem for linear systems is to find positive systems with a state space of minimal dimensions while keeping their transfer functions unchanged. Here the necessary and sufficient condition is established for the linear systems with third-order two-different-pole transfer functions to have a three-dimensional minimal positive realization by developing a new approach. The approach is to identify a canonical form of positive realizations with the help of polyhedral cone theory, and the identification consists of a sequential edge transformations with some invariant feature. The analyses are then successfully extended to the linear systems with general nth-order two-different-pole transfer functions. At the same time, the approach can also be used to generate infinitely many distinct minimal positive realizations for some linear systems.

ParGeMSLR: A parallel multilevel Schur complement low-rank preconditioning and solution package for general sparse matrices

This paper discusses parGeMSLR, a C++/MPI software library for the solution of sparse systems of linear algebraic equations via preconditioned Krylov subspace methods in distributed-memory computing environments. The preconditioner implemented in parGeMSLR is based on algebraic domain decomposition and partitions the symmetrized adjacency graph recursively into several non-overlapping partitions via a p-way vertex separator, where p is an integer multiple of the total number of MPI processes. From a numerical perspective, parGeMSLR builds a Schur complement approximate inverse preconditioner as the sum between the matrix inverse of the interface coupling matrix and a low-rank correction term. To reduce the cost associated with the computation of the approximate inverse matrices, parGeMSLR exploits a multilevel partitioning of the algebraic domain. The parGeMSLR library is implemented on top of the Message Passing Interface and can solve both real and complex linear systems. Furthermore, parGeMSLR can take advantage of hybrid computing environments with in-node access to one or more Graphics Processing Units. Finally, the parallel efficiency (weak and strong scaling) of parGeMSLR is demonstrated on a few model problems arising from discretizations of 3D Partial Differential Equations.

Uniform Global Asymptotic Stabilization of Semilinear Periodic Discrete-Time Systems

Semilinear discrete-time control systems with periodic coefficients are considered. The problem of uniform global asymptotic stabilization of the zero equilibrium of the closed-loop system by state feedback is studied. It is assumed that the free dynamic system has a Lyapunov stable zero equilibrium. The method for constructing a damping control is extended from time-invariant systems to time varying periodic semilinear discrete-time systems. By using this approach, sufficient conditions for uniform global asymptotic stabilization for those systems are obtained. Moreover, the converse Lyapunov theorem on Lyapunov (nonasymptotic) stability is proved for complex and real linear periodic discrete-time systems. Finally, examples of using the obtained results are presented.

Pipelined Preconditioned Conjugate Gradient Methods for real and complex linear systems for distributed memory architectures

Preconditioned Conjugate Gradient (PCG) is a popular method for solving large and sparse linear systems of equations. The performance of PCG at scale is affected due to the costly global synchronization steps that arise in dot-products on distributed memory systems. Pipelined PCG (PIPECG) removes the costly global synchronization steps from PCG by only executing a single non-blocking allreduce per iteration and overlapping it with independent computations.In our previous work, we have developed a novel pipelined PCG algorithm called PIPECG-OATI (One Allreduce per Two Iterations) for real linear systems which executes a single non-blocking allreduce per two iterations and provides a large overlap of global communication with independent computations at higher number of cores. Our method achieves this overlap by using iteration combination and by introducing new recurrence and non-recurrence computations. We implement optimizations in the PIPECG-OATI method to use cache memory efficiently.In this work, we present PIPECG-OATI-c method for linear systems with complex Hermitian positive definite and complex symmetric matrices. We compare our method with various pipelined CG methods on a variety of problems and demonstrate that our method always gives the least run times. We performed experiments with our method using 20M and 30M unknowns on up to 16K cores and obtained up to 2.48X performance improvement over PCG and 2.14X performance improvement over PIPECG methods. We also experimented with up to 1-billion unknowns on 16K cores, the largest problem size explored for the CG problem, to our knowledge, and obtained about 25% improvement over PCG.

Distributionally Robust State Estimation for Linear Systems Subject to Uncertainty and Outlier

Parameter uncertainties and measurement outliers unavoidably exist in a real linear system. Such uncertainties and outliers make the true joint state-measurement distributions (induced by the true system model) deviate from the nominal ones (induced by the nominal system model) so that the performance of the optimal state estimator designed for the nominal model becomes unsatisfactory or even unacceptable in practice. The challenges are to quantitatively describe the uncertainties in the model and the outliers in the measurements, and then robustify the estimator in a right way. This article studies a distributionally robust state estimation framework for linear systems subject to parameter uncertainties and measurement outliers. It utilizes a family of distributions near the nominal one to implicitly describe the uncertainties and outliers, and the robust state estimation in the worst case is made over the least-favorable distribution. The advantages of the presented framework include: 1) it only uses a few scalars to parameterize the method and does not require the structural information of uncertainties; 2) it generalizes several classical filters (e.g., the fading Kalman filter, risk-sensitive Kalman filter, relative-entropy Kalman filter, outlier-insensitive Kalman filters) into a unified framework. We show that the distributionally robust state estimation problem can be reformulated into a linear semi-definite program and in some special cases it can be analytically solved. Comprehensive comparisons with existing state estimation frameworks that are insensitive to parameter uncertainties and measurement outliers are also conducted.

A new method for computation of positive realizations of linear discrete-time systems

A new method for computation of positive realizations of given transfer matrices of linear discrete-time linear systems is proposed. Sufficient conditions for the existence of positive realizations of transfer matrices are given. A procedure for computation of the positive realizations is proposed and illustrated by an example.

Asynchronous iterations of HSS method for non-Hermitian linear systems

A general asynchronous alternating iterative model is designed, for which convergence is theoretically ensured both under classical spectral radius bound and, then, for a classical class of matrix splittings for -matrices. The computational model can be thought of as a two-stage alternating iterative method, which well suits to the well-known Hermitian and skew-Hermitian splitting (HSS) approach, with the particularity here of considering only one inner iteration. Experimental parallel performance comparison is conducted between the generalized minimal residual (GMRES) algorithm, the standard HSS and our asynchronous variant, on both real and complex non-Hermitian linear systems, respectively, arising from convection–diffusion and structural dynamics problems. A significant gain on execution time is observed in both cases.

Solution of large-scale 3D controlled-source electromagnetic modeling problem using efficient iterative solvers

With geophysical surveys evolving from traditional 2D to 3D models, the large volume of data adds challenges to inversion, especially when aiming to resolve complex 3D structures. An iterative forward solver for the controlled-source electromagnetic (CSEM) method requires less memory than that for a direct solver; however, it is not easy to iteratively solve an ill-conditioned linear system of equations arising from finite-element discretization of Maxwell’s equations. To solve this problem, we have developed efficient and robust iterative solvers for frequency- and time-domain CSEM modeling problems. For the frequency-domain problem, we first transform the linear system into its equivalent real-number format, and then introduce an optimal block-diagonal preconditioner. Because the condition number of the preconditioned linear equation system has an upper bound of [Formula: see text], we can achieve fast solution convergence when applying a flexible generalized minimum residual solver. Applying the block preconditioner further results in solving two smaller linear systems with the same coefficient matrix. For the time-domain problem, we first discretize the partial differential equation for the electric field in time using an unconditionally stable backward Euler scheme. We then solve the resulting linear equation system iteratively at each time step. After the spatial discretization in the frequency domain, or space-time discretization in the time domain, we exploit the conjugate-gradient solver with auxiliary-space preconditioners derived from the Hiptmair-Xu decomposition to solve these real linear systems. Finally, we check the efficiency and effectiveness of our iterative methods by simulating complex CSEM models. The most significant advantage of our approach is that the iterative solvers we adopt have almost the same accuracy and robustness as direct solvers but require much less memory, rendering them more suitable for large-scale 3D CSEM forward modeling and inversion.

Newton’s method for the parameterized generalized eigenvalue problem with nonsquare matrix pencils

The l parameterized generalized eigenvalue problems for the nonsquare matrix pencils, proposed by Chu et al.in 2006, can be formulated as an optimization problem on a corresponding complex product Stiefel manifold. In this paper, an effective and efficient algorithm based on the Riemannian Newton’s method is established to solve the underlying problem. Under our proposed framework, to solve the corresponding Newton’s equation, it can be converted to solve a standard real symmetric linear system with a dimension reduction. By combining the Riemannian curvilinear search method with Barzilai–Borwein steps, a hybrid algorithm with both global and quadratic convergence is obtained. Numerical experiments are provided to illustrate the efficiency of the proposed method. Detailed comparisons with some latest methods are also provided to show the merits of the proposed approach.

On convergence of two-stage iterative scheme

Climent and Perea [Journal of Computational and Applied Mathematics 58:43–48, 2003; MR2013603] proposed first the convergence theory of two-stage iterative scheme for solving real rectangular linear systems. In this article, we revisit the same theory. The first main result provides some sufficient conditions which guarantee that the induced splitting from a two-stage iterative scheme is a proper weak regular splitting. We then establish a few comparison results. Out of these, many are even new in nonsingular matrix setting. Further, we study the monotone convergence theory of the two-stage iterative method. Besides these, we also prove the uniqueness of a proper splitting of a rectangular matrix under certain assumptions.

Application Of A Real-time Linear Array Ultrasound System To The Evaluation Of Live Cattle

A real time linear array ultrasound machine was assessed for bias accuracy and precision. A total of 51 heifers, and steers were scanned for fat thickness over the longissimus muscle, the shoulder and rump; for M. longissimus area between the 12th and 13th ribs; and marbling was subjectively evaluated from M. longissimus scans. Ultrasound data were compared with a live judging committee estimates and with carcass values obtained during USDA grading. Data adjusted for live weight were analysed by correlation and regression techniques. Ultrasound estimates of fat thickness were lower (P<.01) than carcass measurements. Shoulder fat thickness estimation was the lowest in precision and accuracy. The residual standard deviations of ultrasound estimates of fat thickness were not (P>.05) influenced by fat thickness and sex. Ultrasound estimates of M. longissimus area were not different (P>.05) from carcass values and were high in precision (RSD: 2.6 to 6.5cm2). The precision and accuracy of data collected by real time linear array ultrasound system indicate a potential for the application of the ultrasound technique to grading of livestock and carcasses.

Minimal realization and approximation of commensurate linear fractional-order systems via Loewner matrix method.

In this paper we propose a data driven realization and model order reduction (MOR) for linear fractional-order system (FoS) by applying the Loewner-matrix method. Given the interpolation data which obtained by sampling the transfer function of a FoS, the minimal fractional-order state space descriptor model that matching the interpolation data is constructed with low computational cost. Based on the framework, the commensurate order α of the fractional-order system is estimated by solving a least squares optimization in terms of sample data in case of unknown order-α. In addition, we present an integer-order approximation model using the interpolation method in the Loewner framework for FoS with delay. Finally, several numerical examples demonstrate the validity of our approach.

Robust State Estimation for Linear Systems Under Distributional Uncertainty

Modeling uncertainties for real linear systems are unavoidable. These uncertainties can significantly degrade the performance of optimal state estimators designed for nominal system models. The challenge is quantifying such uncertainties and devising robust estimators that are insensitive to them. This paper is therefore concerned with distributionally robust state estimation for linear Markov systems. We propose a new modeling framework that describes uncertainties using a family of distributions so that the worst-case robust estimate in the state space is made over the least-favorable distribution. This framework uses only one or two scalars to express the uncertainty set and does not require the structural information of model uncertainties. Furthermore, the moment-based ambiguity set is suggested to embody the distributional uncertainty family. As a result, the estimation problem transforms into a nonlinear semidefinite program with linear constraints, which can be analytically and efficiently solved. Intensive experiments illustrate the advantages of the proposed framework over existing methods.