Matrix Equation Approach Research Articles

Fast Direct Solution of 2D Scalar Volume Integral Equation via Tensor Train Decomposition for Scatterers of Arbitrary Shape

The method of moments (MoM) discretization of the volume integral equation (VIE) is an effective tool for calculating field responses in the presence of arbitrary scatterers. This method scales poorly with problem size, however, due to the inherent difficulty of inverting the relevant matrix equations. The application of tensor train (TT) decomposition techniques has recently shown promise in alleviating the complexity of such calculations, with dramatic efficiency boosts in both computational time and memory having been proven possible for very simple scatterers. For arbitrary scatterers, previous work has shown that a conjugate gradient- TT (CG- TT) procedure was still capable of producing favorable runtime and memory complexities, particularly in quasi-constant rank regimes. In this work, we consider a direct solution of TT decomposed MoM matrix equation approach that is largely insensitive to MoM matrix condition number, while maintaining the generality of CG- TT. Preliminary results indicate that this approach is capable of matching the efficiency of CG- TT in the quasi-constant rank regime for scatterers of arbitrary geometry.

Modeling a New AQM Model for Internet Chaotic Behavior Using Petri Nets

Formal modeling is considered one of the fundamental phases in the design of network algorithms, including Active Queue Management (AQM) schemes. This article focuses on modeling with Petri nets (PNs) a new scheme of AQM. This innovative AQM is based on a discrete dynamical model of random early detection (RED) for controlling bifurcations and chaos in Internet congestion control. It incorporates new parameters (α,β) that make possible better stability control over oscillations of an average queue length (AQL) at the router. The PN is validated through the matrix equation approach, reachability tree, and invariant analysis. The correctness is validated through the key properties of reachability, boundedness, reversibility, deadlock, and liveness.

Bandwidth modulation and centre wavelength shift of a single FBG for simultaneous water level and temperature sensing

Abstract A diaphragm-based hydrostatic level transducer with the use of a single Fibre Bragg grating (FBG) is presented for simultaneous measurement of water level and temperature. In order to distinguish the mismatch between water level and temperature sensitivity, a bandwidth modulation and centre wavelength shift measurement technique was introduced. The change in bandwidth is sensitive towards water level variations with a sensitivity of 0.0015 nm/cm. Meanwhile, the change of centre wavelength is sensitive towards water level and temperature variations, with sensitivities of 0.0008 nm/cm and 0.0365 nm/°C, respectively. By expressing the obtained sensitivities in a matrix equation approach, simultaneous water level and temperature measurement can be predicted accurately.

Perturbation analysis for the QX factorization for centrosymmetric matrices

The QX factorization of the centrosymmetric matrix is a structure-preserving QR factorization proposed by Konrad Burnik. In this paper, we first propose a sufficient condition for the uniqueness of this decomposition. Then, using the refined matrix equation approach, we derive the first-order normwise perturbation bounds convenient for computation for the QX factorization. Using the matrix-vector equation approach and the modified matrix-vector equation approach, we obtain the optimal first-order normwise perturbation bounds. Moreover, the normwise condition numbers for the QX factorization are also presented. Some numerical examples are given to illustrate our theoretical results.

Multiplicative perturbation bounds for the block Cholesky downdating problem

ABSTRACT In this paper, we give the explicit expressions for the strong and the weak rigorous multiplicative perturbation bounds for the block Cholesky downdating problem. First, we derived the strong rigorous multiplicative perturbation bounds by combining the modified matrix-vector equation approach with the strategy for Lyapunov majorant function and the Banach fixed point theorem. Then, the weak rigorous multiplicative bounds are obtained with the help of the matrix-equation approach. The obtained results are illustrated by numerical tests.

A natural rubber diaphragm based transducer for simultaneous pressure and temperature measurement by using a single FBG

Abstract A natural rubber diaphragm bonded with a single FBG is demonstrated to measure the pressure and temperature simultaneously. Half of the grating is bonded to the rubber diaphragm while the other half is bonded to the aluminium sheet. Under varying pressure at constant temperature, the sensitivity is recorded at 0.1007 nm/kPa with coefficient of 6.49 × 10−5 kPa−1. The temperature sensitivities at constant pressure is recorded at 0.0797 nm/°C and 0.0326 nm/°C, respectively. The experimentally obtained different sensitivities of the sensor in matrix equation approach capable to interpret the Bragg wavelength shifts into pressure and temperature information simultaneously.

An MSSS-preconditioned matrix equation approach for the time-harmonic elastic wave equation at multiple frequencies

In this work, we present a new numerical framework for the efficient solution of the time-harmonic elastic wave equation at multiple frequencies. We show that multiple frequencies (and multiple right-hand sides) can be incorporated when the discretized problem is written as a matrix equation. This matrix equation can be solved efficiently using the preconditioned IDR(s) method. We present an efficient and robust way to apply a single preconditioner using MSSS matrix computations. For 3D problems, we present a memory-efficient implementation that exploits the solution of a sequence of 2D problems. Realistic examples in two and three spatial dimensions demonstrate the performance of the new algorithm.

Linear quadratic regulation for discrete-time antilinear systems: An anti-Riccati matrix equation approach

In this paper, the linear quadratic regulation problem is investigated for discrete-time antilinear systems. Two cases are considered: finite time state regulation and infinite time state regulation. First, the discrete minimum principle is generalized to the complex domain. By using the discrete minimum principle and dynamic programming, necessary and sufficient conditions for the existence of the unique optimal control are obtained for the finite time regulation problem in terms of the so-called anti-Riccati matrix equation. Besides, the optimal value of the performance index under the optimal control is provided. Furthermore, the optimal regulation problem on an infinite interval is investigated under the assumption that the considered time-invariant antilinear system is controllable. The resulted closed-loop system under the optimal control turns out to be asymptotically stable.

Stability analysis of nonlinear observer for neutral uncertain time-delay systems

This note is concerned with the observer design problem for a class of nonlinear neutral systems with time delay. The problem addressed is to design a full-order observer that guarantees the asymptotic stability of an error dynamic system. Firstly, some sufficient conditions for the existence of observers of a class of nonlinear neutral systems with time-varying delay are presented. An effective algebraic matrix equation approach is developed. Then we give a design method of the observer that is dependent on the solution of a linear matrix inequality. Furthermore, robust observer designs for a class of nonlinear neutral systems with time delay and uncertainties are obtained. Finally, an example is given to show the effectiveness of our proposed approaches.

A technique to package Fiber Bragg Grating Sensors for Strain and Temperature Measurements

This paper reports an effective method of packaging a fiber Bragg grating (FBG) for the simultaneous measurement of temperature and strain. The technique consists of embedding two fiber Bragg grating sensors inside a polymeric material with different geometrical characteristics. The mechanical and thermal characterizations of the optical transducer were performed. Then a matrix equation used measurements of the wavelength shifts from the two sensors and yielded information about the temperature and strain coefficients. The preliminary results demonstrate the feasibility of the encapsulation technique allowing measurements of temperature and strain in smart structures and harsh environments. The experimental procedure provides robustness to the sensor and the matrix equation approach has the potential to determine simultaneously the strain and temperature coefficients.

Parametric Jordan Form Assignment Revisited

AbstractThe Jordan form assignment problem (complete and partial) for linear multivariable control system is formulated and completely solved by the unified Sylvester matrix equation approach. A new explicit parametrization of all state feedbacks assigning an admissible Jordan form through the minimum number of parameters is provided. The main advantage of this parametrization, unlike previous ones, is that it is explicit; thus, it can be used, not only for numeric, but also symbolic computations. This fact is demonstrated on the problem of parametrization of all output feedbacks assigning an admissible Jordan form, where Gröbner basis method is applied. Developed algorithms for Jordan form assignment by state and output feedback are implemented in the freely available Maple package.

Two-sided generalized hyperbolic QR factorization and its perturbation analysis

We first propose the two-sided generalized hyperbolic QR factorization of a matrix pair, which is a generalization of the generalized QR factorization. More explicitly, two orthogonal matrices in the generalized QR factorization are replaced with two corresponding J-orthogonal matrices, where J is a signature matrix. Then, we consider the perturbation analysis of the new matrix factorization. Some first order normwise perturbation bounds are derived using the refined matrix equation approach and the updated matrix–vector equation approach. The corresponding perturbation bounds for the generalized QR factorization are also obtained as the special case. In comparison, these bounds can be much tighter than the previous ones derived from Barrlund (1994) [A. Barrlund, Perturbation bounds for the generalized QR factorization, Linear Algebra Appl. 207 (1994) 251–271].

Perturbation analysis for the hyperbolic QR factorization

The hyperbolic QR factorization is a generalization of the classical QR factorization and can be regarded as the triangular case of the indefinite QR factorization proposed by Sanja Singer and Saša Singer. In this paper, the perturbation analysis for this factorization is considered using the classical matrix equation approach, the refined matrix equation approach, and the matrix–vector equation approach. The first order and rigorous normwise perturbation bounds with normwise or componentwise perturbations in the given matrix are derived. The obtained first order bounds can be much tighter than the corresponding existing ones. Each of the obtained rigorous bounds is composed of a small constant multiple of the corresponding first order bound and an additional term with simple form. In particular, for square matrix, the rigorous bounds for the factor R are just the 6+3 multiple of the corresponding first order bounds. These rigorous bounds can be used safely for all cases in comparison to the first order bounds.

Perturbation analysis for block downdating of the generalized Cholesky factorization

The generalized Cholesky factorization is a generalization of the classical Cholesky factorization and its block downdating problem means finding the downdated generalized Cholesky factorization when a matrix XXT is subtracted from the original matrix, where X is full column rank. In this paper, we consider the perturbation analysis of this problem. Some first order perturbation bounds are first obtained using the refined matrix equation approach and the matrix–vector equation approach. These results generalize the corresponding ones for the block downdating problem of the classical Cholesky factorization. Then, the rigorous perturbation bounds are also obtained using the combination of the classical and refined matrix equation approaches. Each of these bounds is composed of a small constant multiple of the first order term of the corresponding first order perturbation bound and an additional second order term.

Determination of pressure balance distortion coefficient and zero-pressure effective area uncertainties

The behaviour of piston-cylinder operated pressure balances is characterized by the distortion coefficientλand zero-pressure effective areaA0which model the variation of a pressure balance’s area in terms of the applied pressure. This paper determines the uncertainties inλandA0when utilizing the method of cross-floating with another pressure balance standard whose parameters and associated uncertainties are known. A limitation that is frequently encountered in many attempts of the uncertainty analysis for a pressure balance is that no readily accessible uncertainty quantification framework for the distortion coefficient is present. As a result the uncertainty in a pressure balance’s area at elevated applied pressures is typically underestimated in the absence of this uncertainty information. We firstly review the uncertainty formulation for a pressure balance generated pressure involving correlation effects in terms of an implicit multivariate matrix equation approach and then utilizing the resulting solution present the methodology to consistently perform the uncertainty analysis forλandA0.

Parametric Jordan Form Assignment Revisited

AbstractThe Jordan form assignment problem (complete and partial) for linear multivariable control system is formulated and completely solved by the unified Sylvester matrix equation approach. A new explicit parametrization of all state feedbacks assigning an admissible Jordan form trough the minimum number of parameters is provided. The main advantage of this parametrization, unlike previous ones, is that it is really explicit and therefore can be simply used not only for numeric but also symbolic computations. This fact is demonstrated on the problem of parametrization of all output feedbacks assigning an admissible Jordan form, where Gröbner basis method is applied. Developed algorithms for Jordan form assignment by state and output feedback are implemented in the free available Maple 13 Package.

Rigorous Perturbation Bounds of Some Matrix Factorizations

This article presents rigorous normwise perturbation bounds for the Cholesky, LU, and QR factorizations with normwise or componentwise perturbations in the given matrix. The considered componentwise perturbations have the form of backward rounding errors for the standard factorization algorithms. The used approach is a combination of the classic and refined matrix equation approaches. Each of the new rigorous perturbation bounds is a small constant multiple of the corresponding first-order perturbation bound obtained by the refined matrix equation approach in the literature and can be estimated efficiently. These new bounds can be much tighter than the existing rigorous bounds obtained by the classic matrix equation approach, while the conditions for the former to hold are almost as moderate as the conditions for the latter to hold.

A Stein matrix equation approach for computing coprime matrix fraction description

This paper is concerned with the coprime matrix fraction description of linear systems. Analytical solutions are obtained by solving a Stein matrix equation with nilpotent coefficient matrix. An explicit solution consisting of finite summations to the Stein matrix equation is also provided. The proposed approach can be easily transformed into a numerically reliable algorithm if the Stein matrix equation is solved in a numerically stable way. The parametric eigenstructure assignment problem is reconsidered to illustrate the application and efficiency of the proposed approach.

Legendre polynomial solutions of high-order linear Fredholm integro-differential equations

In this study, a Legendre collocation matrix method is presented to solve high-order Linear Fredholm integro-differential equations under the mixed conditions in terms of Legendre polynomials. The proposed method converts the equation and conditions to matrix equations, by means of collocation points on the interval [−1, 1], which corresponding to systems of linear algebraic equations with Legendre coefficients. Thus, by solving the matrix equation, Legendre coefficients and polynomial approach are obtained. Also examples that illustrate the pertinent features of the method are presented and by using the error analysis, the results are discussed.

시변 시간 지연을 포함하는 선형 뉴트럴 시스템의 관측기 설계

본 논문은 시변 시간 지연을 가지는 선형 뉴트럴 시스템에 관한 관측기 설계 및 안정도 해석에 관해서 논의한다. 시변 시간 지연을 가지는 선형 뉴트럴 시스템의 안정도를 판별하기 위하여 Lyapunov-Krasovskii의 이론을 도입한다. 오차 상태 방정식의 안정도 조건으로 시간 변동 시간 지연에 종속적인 충분조건을 제시한다. 선형 행렬 부등식의 해를 이용하여 관측기의 이득 값을 설계하며, 설계된 관측기를 이용하여 오차 상태 방정식의 안정도를 판별한다. 본 논문의 결과는 Luenberger가 제안했던 관측기의 일반적인 결과를 나타냄을 확인한다. 모의실험을 통해 제안된 이론을 입증한다. This paper is concerned with the observer design problem for linear neutral systems with time-varying delays. The problem addressed is that of designing a full-order observer that guarantees the exponential stability of the error system. An effective algebraic matrix equation approach is developed to solve this problem. In particular, both observer analysis and design problems are investigated. Sufficient conditions for a linear neutral system to be stable are first established. Furthermore, an illustrative example is used to demonstrate the validity of the proposed design procedure.