Coupled Matrix Equations Research Articles

Global quaternion generalized minimal residual method for generalized coupled Sylvester quaternion matrix equations with application to colour image encryption and decryption

ABSTRACT Sylvester matrix equations play important roles in control and system theory. We consider a class of the generalized coupled Sylvester quaternion matrix equations, which includes a lot of special matrix equations such as Stein matrix equation, Lyapunov matrix equation and Sylvester matrix equation on the quaternion skew field. First, the quaternion Krylov subspace and its F-orthogonal basis are constructed via a new quaternion matrix product and the global quaternion Arnoldi process. Then a global quaternion generalized minimal residual (Gl-QGMRES) method is proposed for solving the generalized coupled Sylvester quaternion matrix equations. The convergence of the proposed method is theoretically analysed. Finally, the effectiveness and feasibility of the proposal are verified by some numerical experiments. As an application of Gl-QGMRES method, we establish a kind of colour image encryption and decryption by embedding watermark images.

An Inversion-Free Iterative Algorithm With a Scalar Tuning Parameter for Coupled Riccati Matrix Equations Arising in LQ Optimal Control of Markov Jump Systems

An Inversion-Free Iterative Algorithm With a Scalar Tuning Parameter for Coupled Riccati Matrix Equations Arising in LQ Optimal Control of Markov Jump Systems

Adaptive Parameter Alternating Direction Algorithm for Centrosymmetric Solutions of a Class of Generalized Coupled Sylvester-transpose Matrix Equations

Adaptive Parameter Alternating Direction Algorithm for Centrosymmetric Solutions of a Class of Generalized Coupled Sylvester-transpose Matrix Equations

Active control of noise in plate-cavity coupled systems using the pole placement method

Structure-acoustic coupled systems have received great attention in engineering, and a typical representative is a plate coupled with a cavity. The sound in the cavity is excited by the vibration of the plate and when there is strong coupling between the acoustic modes of the cavity and the vibrational modes of the plate, the sound pressure can be enhanced dramatically. In this paper, active control based on the pole placement method is used for the first time to reduce the noise in a plate-cavity coupled system. The principle of the pole assignment is that the poles of a system represent the system’s dynamical behaviour, and their assignment allows one to modify the response of the system using sensors and actuators. In this work, the usage of pole assignment to modify the dynamics of the plate is theoretically derived, and the resulting sound in the cavity is analytically calculated by solving a coupled matrix equation. To access the performance of the pole placement method in reducing the noise in the coupled system, three example cases are considered. The first is a local control case which uses a single actuator to control the first two modes of the plate in order to attenuate the resulting sound in the cavity. The second case is imposing global control by using two actuators to modify the first four modes of the plate, weakening the strong interaction between the pate and the cavity by shifting their natural frequencies away from each other. The third is applying uncontrollability conditions to retain particular modes when the control method is applied. The analytical results show that the pole placement method can be effective to control both the vibration and the vibro-acoustics in the plate-cavity coupled system. The finite element (FE) method is used to validate the analytical models. The FE predictions show good agreement with the analytical results. The pole placement method is shown to be effective in controlling the noise in this plate-cavity coupled system, and there is the potential to extend the method to more complex structure-acoustic coupled systems.

A Hybridizable Discontinuous Galerkin Time-Domain Method With Robin Transmission Condition for Transient Thermal Analysis of 3-D Integrated Circuits

In this article, a discontinuous Galerkin time-domain (DGTD) method hybridized with Robin transmission condition (RTC-DGTD) is proposed for the transient thermal analysis of 3-D integrated circuits (ICs). As a kind of domain decomposition method (DDM), the DGTD method employs a term named numerical flux for the information exchange of solutions between neighboring subdomains. The numerical flux is a function of both temperature <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula> and heat flux q, and thus for the heat equation, apart from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula> , the variable q has to be solved as well. To reach this, the traditional DGTD method decomposes the second-order partial-differential thermal equation into two first-order ones, which results that both <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula> and q need to be solved in the whole computational domain, thereby the computational cost is extremely expensive. In this work, to get this trouble-treated, the proposed DGTD method is directly applied to discretize the second-order heat equation. An RTC is established at the interfaces of subdomains, which serves as the second governing equation and meanwhile bridges another connection of solutions in adjacent subdomains. In this way, the new variable q is now only present at the interfaces of subdomains. Consequently, the total number of unknowns is significantly reduced compared with that in the traditional DGTD method. To further speed up the time-marching scheme, an adaptive time-stepping method in terms of the proportional–integral–derivative (PID) control algorithm is employed. Besides, by introducing a restriction operator, the globally coupled matrix equation is torn into a number of small matrix equations pertinent to each subdomain, which is later solved via a finite-element tearing and interconnecting (FETI)-like method. To validate and verify the accuracy and efficiency of the proposed algorithm, several representative examples are studied and compared with COMSOL simulations.

Joint state and actuator fault estimation for networked systems under improved accumulation-based event-triggered mechanism

The joint state and actuator fault estimation problem is investigated in this paper for a type of networked systems subject to loss of the actuator effectiveness (LAE). A so-called improved accumulation-based event-triggered mechanism (ETM) is used to regulate the transmission of signals between the sensors and the estimator for the purpose of communication resource saving. Compared with the traditional ETM schemes, such accumulation-based ETM is robust against the “undesired” abrupt changes of signals (which would occur due to certain big noises). Different from the integral-based ETM for continuous-time systems, the improved accumulation-based ETM proposed in this paper is a “weighted” ETM, where a given weight coefficient is employed to “balance” the weights of output measurements in different time instants. The multiplicative LAE is described by an unknown diagonal matrix. The object of this paper is to design a remote estimator such that both the fault signals and system states can be simultaneously estimated in the sense of minimizing an upper bound of the corresponding estimation error covariance at each sampling instant. First, the upper bound of the estimation error covariance is given by means of the induction method. Then, the desired estimator gain is calculated recursively by solving two sets of coupled matrix equations. Finally, two simulation examples are given to verify the usefulness of the strategy we proposed subject to the LAE under the improved accumulation-based ETM.

Three Symmetrical Systems of Coupled Sylvester-like Quaternion Matrix Equations

The current study investigates the solvability conditions and the general solution of three symmetrical systems of coupled Sylvester-like quaternion matrix equations. Accordingly, the necessary and sufficient conditions for the consistency of these systems are determined, and the general solutions of the systems are thereby deduced. An algorithm and a numerical example are constructed over the quaternions to validate the results of this paper.

A systematic approach towards realization of transmission zeros in TE 01δ mode dielectric resonator filters with all iris couplings

This paper presents a novel method using a coupling matrix to realize dielectric resonator filters with Transmission Zeros (TZs). This method eliminates the necessity of capacitive coupling in conventional DR filter designs with TZs. Using the proposed technique, the cross-coupling sign can be changed systematically while the amplitude frequency response is kept unchanged. The method is independent of the filter physical structure and all design steps are based on coupling matrix equation and models. Therefore, unlike the extracted pole technique, this method not only does not require time consuming optimization in full wave simulations but also the extra resonator in this method helps in improving the group delay performance of the filter. After presenting the method, the calculated coupling matrix for a 7 × 4 DR filter will be realized using TE01δ mode of cylindrical dielectric resonators and simulation and measurement results will be compared.

Efficient Biexciton Preparation in a Quantum Dot-Metal Nanoparticle System Using On-Off Pulses.

We consider a hybrid nanostructure composed by semiconductor quantum dot coupled to a metallic nanoparticle and investigate the efficient creation of biexciton state in the quantum dot, when starting from the ground state and using linearly polarized laser pulses with on-off modulation. With numerical simulations of the coupled system density matrix equations, we show that a simple on-off-on pulse-sequence, previously derived for the case of an isolated quantum dot, can efficiently prepare the biexciton state even in the presence of the nanoparticle, for various interparticle distances and biexciton energy shifts. The pulse durations in the sequence are obtained from the solution of a transcendental equation.

Integrated modelling and optimal operation analysis of multienergy systems based on Stackelberg game theory

An integrated modelling and operation method for a multienergy system (MES) is proposed in this paper. First, a coupling matrix equation containing the energy flow in the process of energy production, transmission, conversion, storage and consumption is established. Second, the whole MES is selected as the game subject, and each energy subsystem is selected as the game follower. Then, an integrated model of an MES is modelled to quantify the complementary cold-heat-power-gas multi-energy and source-network-load-storage coordinated interactions based on Stackelberg game theory. Third, the day-ahead MES operation scheme is optimized based on the established model and the equilibrium solution is used to realize a reasonable balance of benefits between the whole MES and its energy subsystems. Numerical studies demonstrate that the proposed method increases the operation cost of the whole MES by ¥68.23 (0.917% increase) but reduces the operation cost of the heat and gas subsystems by ¥254.82 (3.29% decrease) and ¥289.4 (3.72%), respectively, with the objective of minimizing operation cost, and improves the whole system exergy efficiency of the power, heat and gas subsystems by 0.572%, 0.548% and 2.076%, respectively, with the objective of maximizing the exergy efficiency. Thus, one can take into account the different benefits among the whole MES and its energy subsystems and provide a multidimensional dispatch scheme for dispatchers, highlighting the potential for MES development.

On σ-entropy Analysis of Linear Stochastic Systems in State Space

In this paper the problem of random disturbance attenuation capabilities in linear continuous systems is studied. It is supposed that the system operates under random disturbances with bounded σ-entropy level. σ-entropy norm indicates a performance index of the continuous system on the set of the random signals with bounded σ-entropy. This paper presents a time-domain solution to the calculation of σ-entropy norm of the continuous linear time-invariant system. σ-entropy norm is defined after solving coupled matrix equations: one algebraic Riccati equation, one nonlinear equation over log determinant function, and two Lyapunov equations.

A Flexible Global GCRO-DR Method for Shifted Linear Systems and General Coupled Matrix Equations

In this paper, we mainly focus on the development and study of a new global GCRO-DR method that allows both the flexible preconditioning and the subspace recycling for sequences of shifted linear systems. The novel method presented here has two main advantages: firstly, it does not require the right-hand sides to be related, and, secondly, it can also be compatible with the general preconditioning. Meanwhile, we apply the new algorithm to solve the general coupled matrix equations. Moreover, by performing an error analysis, we deduce that a much looser tolerance can be applied to save computation by limiting the flexible preconditioned work without sacrificing the closeness of the computed and the true residuals. Finally, numerical experiments demonstrate that the proposed method illustrated can be more competitive than some other global GMRES-type methods.

FINITE ITERATIVE (R, S)-CONJUGATE SOLUTIONS OF THE GENERALIZED COMPLEX COUPLED SYLVESTER-TRANSPOSE EQUATIONS

The iterative method of generalized complex coupled Sylvester-transpose equations $ AXB+CY^TD = E,\; MX^TN+GYH = F $ over (R, S)-conjugate matrix solution $ (X,Y) $ is proposed. Usually, the type of matrix arises from some physical problems with some form of generalized reflexive symmetry. On the condition that the coupled matrix equations are consistent, we show the solution pair $ (X^*, Y^*) $ can be obtained by generalization of CG iterative method within finite iterative steps in the absence of roundoff-error for any initial guess chosen by the (R, S)-conjugate matrix. Moreover, the optimal approximation (R, S)-conjugate matrix solutions can be derived by searching the least Frobenius norm solution of the novel generalized complex coupled Sylvester-transpose matrix equations. Finally, some numerical examples are given to illustrate the presented iterative algorithm is efficient.

A numerical solution of a class of periodic coupled matrix equations

This paper studies the numerical solutions of a class of periodic coupled matrix equations. Based on the least square method, a finite iterative algorithm for a class of periodic coupled matrix equations is proposed, and the convergence of the algorithm is proved by theoretical derivation. For any initial value, the algorithm can converge to the solution in finite iterations. Since the equations considered in paper contain many variants, the proposed algorithm has a wide range of applications. Finally some numerical examples in practical systems are given to prove the effectiveness and efficiency of the algorithm.

Numerical simulation of shock wave problems with the two-phase two-fluid model

The accurate evaluation of the rapid pressure wave propagations in reactor flow systems is an important and challenge issue for reactor safety. In order to simulate pressure wave propagations in the two-phase flow system, this paper presents a new two-fluid numerical solution method for solving the two-phase flow process. A numerical convenient set of equations and the associated one-step coupled solution method are proposed for the model. The field equations are devised from the RELAP5 code, but they have great effectiveness in forming a coupled matrix equation. All the unknown variables of the flow system can be obtained through directly solving the coupled matrix equation. The proposed model is applied to simulate the shock wave problems, i.e. the classical shock tube problem and the water faucet problem. The stability and convergence performance of the simulation results have been verified with analytical data and RELAP5 results. The numerical results demonstrate that the new model is capable and accurate in capturing the transient shock wave propagation of single- and two-phase flows.

Fields and characteristic impedances of dipole and quadrupole cylindrical stripline kickers

We present semi-analytical methods for calculating the electromagnetic field in dipole and quadrupole stripline kickers with curved plates of infinitesimal thickness. Two different methods are used to solve Laplace's equation by reducing it either to a single or to two coupled matrix equations; they are shown to yield equivalent results. Approximate analytic solutions for the lowest order fields (dipole or quadrupole) are presented and their useful range of validity are shown. The kickers plates define a set of coupled transmission lines and the characteristic impedances of modes relevant to each configuration are calculated. The solutions are compared with those obtained from a finite element solver and found to be in good agreement. Mode matching to an external impedance determines the kicker geometry and this is discussed for both kicker types. We show that a heuristic scaling law can be used to determine the dependence of the characteristic impedance on plate thickness. The solutions found by semi-analytical methods can be used as a starting point for a more detailed kicker design.

Explicit Iterative Algorithms for Continuous Coupled Lyapunov Matrix Equations

In this article, two iterative algorithms are proposed to solve the coupled Lyapunov matrix equations associated with continuous-time Markovian jump linear systems. First, an explicit iterative algorithm is presented to solve the considered matrix equations via introducing a tuning parameter. A necessary and sufficient condition is established for the proposed algorithm to be convergent. Furthermore, the optimal value of the tuning parameter achieving the fastest convergence rate is given explicitly. Second, a modified version of the previous explicit algorithm is established by using a new information update scheme. Similarly, a necessary and sufficient condition is provided for the modified explicit iterative algorithm. Based on this convergence condition, some easily verifiable convergence results are further derived, and the optimal tuning parameter is given explicitly. Finally, two examples are presented to validate the obtained theoretical results.

A Novel Subdomain 2D/Q-2D Finite Element Method for Power/Ground Plate-Pair Analysis

Upon excitation by a surface magnetic current, a power/ground plate-pair supports only $\mathrm{TM}^{z}$ modes. This means that the magnetic field has only azimuthal components permitting a simple but effective domain decomposition method (DDM) to be used. In the proximity of an antipad, field interactions are rigorously modeled by a quasi-two-dimensional (Q-2D) finite element method (FEM) making use of three-dimensional (3D) triangular prism mesh elements. Since high-order $\mathrm{TM}^{z}$ modes are confined in the close proximity of the antipad, field interactions in the region away from the antipad only involve the fundamental mode and are rigorously modeled by a 2D FEM. This approach reduces 3D computation domain into a hybrid 2D/Q-2D domain. The discretization of this hybrid domain results in a global matrix system consisting of two globally coupled matrix equations pertinent to 2D and Q-2D domains. In this article, these two matrix equations are “decoupled” using a Riemann solver and the information exchange between the two domains is facilitated using numerical flux. The resulting decoupled two matrix equations are iteratively solved using the Gauss–Seidel algorithm. The accuracy, efficiency, and robustness of the proposed DDM are verified by four representative examples.

Implementation and validation of a one-step coupled solution method for the two-fluid model

Most of the system codes for reactor Thermal hydraulics simulation are based on the two-fluid model. The accurate and efficient simulations of the two-fluid model are therefore important. This paper introduces a numerically convenient set of equations and the associated one-step coupled solution method for the two-phase two-fluid model. These equations are devised from the RELAP5 difference equations, but they are cast into a new form having great numerical effectiveness in forming a coupled matrix equation. A one-step solution of the coupled matrix equation is performed to obtain the whole unknown variables of the flow system. Proper numerical treatments, closure relations and coupling of wall conduction are accounted for by modeling boiling phase change convective heat transfer behavior. Finally, the numerical solutions of the new two-fluid model equations were successfully implemented and validated using the water faucet problem and a subcooled flow boiling experiment, and good agreements were obtained.

Operational Matrix Approach for Second-Order Matrix Differential Models

The current paper contributes a new numerical algorithm for solving a class of second-order matrix differential equations. To do so, the operational matrix of integration based on the shifted Legendre polynomials together with the collocation method is used to reduce the main problem to coupled matrix equations. An error estimation is provided which verifies the exponential rate of convergence. Numerical experiments are reported to demonstrate the applicability and efficiency of the suggested scheme.