Reflexive Solutions Research Articles

Constrained solutions of generalized coupled discrete-time periodic matrix equations with application in state observer design for linear periodic systems

PurposeIt is desired to provide a diversified iterative scheme for solving the constrained solutions of the generalized coupled discrete-time periodic (GCDTP) matrix equations from the perspective of optimization.Design/methodology/approachThe paper considers generalized reflexive solutions of the GCDTP matrix equations by applying the Jacobi gradient-based iterative (JGI) algorithm, which is an extended variant of the gradient-based iterative (GI) algorithm.FindingsThrough numerical simulation, it is verified that the efficiency and accuracy of the JGI algorithm are better than some existing algorithms, such as the GI algorithm in Hajarian, the RGI algorithm in Sheng and the AGI algorithm in Xie and Ma.Originality/valueIt is the first instance in which the GCDTP matrix equations are solved applying the JGI algorithm.

Iterative algorithm for a generalized matrix equation with momentum acceleration approach and its convergence analysis

This article discusses a numerical gradient-based method for solving a generalized matrix equation. The iterative method mentioned includes two positive parameters, for which a range is determined to ensure the convergence of the introduced method. It has been shown that the optimal parameters of this method satisfy a constrained optimization problem. Then, specific solutions of this equation, such as the reflexive solution, are examined. Furthermore, to increase the rate of convergence of the proposed method, the idea of momentum is utilized, and a range for the momentum parameter is obtained to ensure the convergence. Finally, the efficiency of this method is investigated through numerical simulations. Additionally, in numerical results section, applications of mentioned matrix equation in image restoration and anti-linear systems are addressed.

RANKS OF SUBMATRICES IN THE REFLEXIVE SOLUTIONS OF SOME MATRIX EQUATIONS

Maximal and minimal ranks of the two submatrices X₁ and X₂ in the (skew-) Hermitian reflexive solution X=U[X₁ 00 X₂]U^{∗} of the matrix equation AXA^{∗}=C, in the reflexive solution of the matrix equation AXB=C are derived. Then necessary and sufficient conditions for these reflexive solutions to have special forms, and the general expressions of these reflexive solutions are achieved.

Matrix Equation’s Reflexive and Anti-Reflexive Solutions over Quaternions

We consider when the quaternion matrix equation AXB+CXD=E has a reflexive (or anti-reflexive) solution with respect to a given generalized reflection matrix. We adopt a real representation method to derive the solutions when it is solvable. Moreover, we obtain the explicit expressions of the least-squares reflexive (or anti-reflexive) solutions.

An Iterative Algorithm for the Generalized Reflexive Solution Group of a System of Quaternion Matrix Equations

In the present paper, an iterative algorithm is proposed for solving the generalized (P,Q)-reflexive solution group of a system of quaternion matrix equations ∑l=1M(AlsXlBls+ClsXl˜Dls)=Fs,s=1,2,…,N. A generalized (P,Q)-reflexive solution group, as well as the least Frobenius norm generalized (P,Q)-reflexive solution group, can be derived by choosing appropriate initial matrices, respectively. Moreover, the optimal approximate generalized (P,Q)-reflexive solution group to a given matrix group can be derived by computing the least Frobenius norm generalized (P,Q)-reflexive solution group of a reestablished system of matrix equations. Finally, some numerical examples are given to illustrate the effectiveness of the algorithm.

The BCR algorithms for solving the reflexive or anti-reflexive solutions of generalized coupled Sylvester matrix equations

This paper proposes the Lanczos version of the biconjugate residual algorithm to solve the reflexive or anti-reflexive solutions of a class of generalized coupled Sylvester matrix equations. We give a convergence analysis, and by constructing a special form of the initial matrices, the minimum-norm reflexive solutions can be obtained through finite-step iterations without considering rounding errors. At the end of the paper, we provide numerical examples to illustrate the feasibility and effectiveness of this algorithm.

Iterative method to the coupled operator matrix equations with sub-matrix constraint and its application in control

A finite iterative algorithm is presented for solving the numerical solutions to the coupled operator matrix equations in Zhang (2017b). In this paper, a new finite iterative algorithm is presented for solving the constraint solutions to the coupled operator matrix equations [Formula: see text], where the constraint solutions include symmetric solutions, bisymmetric solutions and reflexive solutions as special cases. If this system is consistent, for any initial constraint matrices, the exact constraint solutions can be obtained by the introduced algorithm within finite iterative steps in the absence of the roundoff errors. Also, if this system is not consistent, the least-norm constraint solutions can be obtained within the finite iteration steps in the absence of the roundoff errors. Furthermore, if a group of suitable matrices are given, the optimal approximation solutions can be derived. Finally, several numerical examples are given to show the effectiveness of the presented iterative algorithm.

Bioinspired Approaches for Autonomous Small-Object Detection and Avoidance

Small-object detection and avoidance in unknown environments is a significant challenge to overcome for small autonomous vehicles that are generally highly agile and restricted in payload and computational processing power. Typical machine-vision and range measurement-based solutions suffer either from restricted fields-of-view or significant computational complexity and are, hence, not easily portable to small platforms. In order to overcome these drawbacks, in this paper, two novel bioinspired approaches are proposed to extract information about small-field objects contained in planar optic flow. The first approach, which is analogous to the small-field extraction process hypothesized to occur in the lobula plate of the fly visual system, is based on the Fourier residual analysis of instantaneous optic flow. Alternatively, the flow-of-flow method is the engineering analogue of the small-field extraction process thought to occur in the fruit-fly's medulla, and extracts high-frequency content of optic flow by means of an elementary motion detector array. Both approaches extract instantaneous relative range and bearing of small-field obstacles from planar optic flow in a local environment characterized by small and wide-field obstacles, which is then combined with an artificial potential function-based low-order steering control law. The proposed sensing and control scheme is experimentally validated with a quadrotor vehicle that is able to effectively navigate an unknown environment laden with small-field clutter. This bioinspired approach is computationally efficient, which renders extraction of vehicle velocity and local environment structure superfluous, and thus, serves as a robust, reflexive solution to the problem of small-object detection, and avoidance for small autonomous robots.

Iterative algorithm for the reflexive solutions of the generalized Sylvester matrix equation

In this paper, the generalized Sylvester matrix equation AV + BW = EVF + C over reflexive matrices is considered. An iterative algorithm for obtaining reflexive solutions of this matrix equation is introduced. When this matrix equation is consistent over reflexive solutions then for any initial reflexive matrix, the solution can be obtained within finite iteration steps. Furthermore, the complexity and the convergence analysis for the proposed algorithm are given. The least Frobenius norm reflexive solutions can also be obtained when special initial reflexive matrices are chosen. Finally, numerical examples are given to illustrate the effectiveness of the proposed algorithm.

Convergence properties of BCR method for generalized Sylvester matrix equation over generalized reflexive and anti-reflexive matrices

ABSTRACTSolving the well-known Lyapunov and Sylvester matrix equations appears in a wide range of applications such as in control theory and signal processing. This article establishes the matrix form of the biconjugate residual (BCR) algorithm for computing the generalized reflexive solution X and the generalized anti-reflexive solution Y of the generalized Sylvester matrix equation It is proven that the proposed BCR algorithm converges within a finite number of iterations in the absence of round-off errors. At the end, various numerical implementations illustrating the effectiveness and accuracy of the proposed BCR algorithm are presented and discussed.

Hermitian and non-negative definite reflexive and anti-reflexive solutions to AX=B

ABSTRACTIn this paper, we establish some conditions for the existence and the representations for the Hermitian reflexive and Hermitian anti-reflexive, and non-negative definite reflexive solutions to the matrix equation AX=B with respect to a generalized reflection matrix P by using the Moore–Penrose inverse. Moreover, in corresponding solution sets of the equation, the explicit expressions of the nearest matrix to a given matrix in the Frobenius norm have been provided.

Convergence of HS version of BCR algorithm to solve the generalized Sylvester matrix equation over generalized reflexive matrices

In recent years, several linear matrix equations such as Lyapunov and Sylvester matrix equations have received considerable attention due to their important applications in engineering and applied mathematics. In this work, an iterative technique based on the Hestenes–Stiefel (HS) version of biconjugate residual (BCR) algorithm is introduced to solve the generalized Sylvester matrix equation∑i=1f(AiXBi)+∑j=1g(CjYDj)=E,over the generalized reflexive matrices X and Y. We show that the proposed iterative technique converges to the generalized reflexive solutions within a finite number of iterations in the absence of round-off errors. Numerical examples confirm the efficiency and accuracy of the proposed iterative technique and also comparisons with other algorithms are made.

The Least Squares Hermitian (Anti)reflexive Solution with the Least Norm to Matrix Equation AXB=C

For a given generalized reflection matrix J, that is, JH=J, J2=I, where JH is the conjugate transpose matrix of J, a matrix A∈Cn×n is called a Hermitian (anti)reflexive matrix with respect to J if AH=A and A=±JAJ. By using the Kronecker product, we derive the explicit expression of least squares Hermitian (anti)reflexive solution with the least norm to matrix equation AXB=C over complex field.

Finite iterative algorithms for the generalized reflexive and anti-reflexive solutions of the linear matrix equation AXB = C

In this paper, an iterative method is presented to solve the linear matrix equation AXB = C over the generalized reflexive (or anti-reflexive) matrix X (A ? Rpxn, B ? Rmxq, C ? Rpxq, X ? Rnxm). By the iterative method, the solvability of the equation AXB = C over the generalized reflexive (or anti-reflexive) matrix can be determined automatically. When the equation AXB = C is consistent over the generalized reflexive (or anti-reflexive) matrix X, for any generalized reflexive (or anti-reflexive) initial iterative matrix X1, the generalized reflexive (anti-reflexive) solution can be obtained within finite iterative steps in the absence of roundoff errors. The unique least-norm generalized reflexive (or anti-reflexive) iterative solution of AXB = C can be derived when an appropriate initial iterative matrix is chosen. A sufficient and necessary condition for whether the equation AXB = C is inconsistent is given. Furthermore, the optimal approximate solution of AXB = C for a given matrix X0 can be derived by finding the least-norm generalized reflexive (or anti-reflexive) solution of a new corresponding matrix equation AXB = C. Finally, several numerical examples are given to support the theoretical results of this paper.

GRADIENT BASED ITERATIVE ALGORITHM TO SOLVE GENERAL COUPLED DISCRETE-TIME PERIODIC MATRIX EQUATIONS OVER GENERALIZED REFLEXIVE MATRICES

The discrete-time periodic matrix equations are encountered in periodic state feedback problems and model reduction of periodic descriptor systems. The aim of this paper is to compute the generalized reflexive solutions of the general coupled discrete-time periodic matrix equations. We introduce a gradient-based iterative (GI) algorithm for finding the generalized reflexive solutions of the general coupled discretetime periodic matrix equations. It is shown that the introduced GI algorithm always converges to the generalized reflexive solutions for any initial generalized reflexive matrices. Finally, two numerical examples are investigated to confirm the efficiency of GI algorithm.

The Hermitian [formula omitted]-(anti-)reflexive solutions of a linear matrix equation

In this paper, we first establish the necessary and sufficient conditions for the existence and the explicit expressions of the Hermitian {P,k+1}-(anti-)reflexive solutions of the matrix equation AX=B, and meanwhile the best approximation solution is considered. Then, if the solvability conditions are not satisfied, the least squares Hermitian {P,k+1}-(anti-)reflexive solutions and the least squares Hermitian {P,k+1}-(anti-)reflexive solutions with the minimum norm of the above matrix equation are respectively derived. In addition, two algorithms are shown to compute the least squares Hermitian {P,k+1}-(anti-)reflexive solutions, and the corresponding numerical examples are also given to illustrate the feasibility of the algorithms.

Generalized reflexive and anti-reflexive solutions of the coupled Sylvester matrix equations via CD algorithm

The generalized reflexive and anti-reflexive matrices have several applications in engineering and scientific computations. In the present paper, first we propose a conjugate direction (CD) algorithm for finding the generalized reflexive solution X and the generalized anti-reflexive solution Y of the coupled Sylvester matrix equations { A 1 XB 1 + C 1 YD 1 = E 1 A 2 XB 2 + C 2 YD 2 = E 2 Then, we analyze the convergence properties of the CD algorithm under some plausible assumptions. Finally by applying two numerical examples, we compare the proposed algorithm with other popular iterative solvers in use today.

Comments on “The reflexive least squares solutions of the general coupled matrix equations with a submatrix constraint”

In this note, a technical error is pointed out in the equality (2) of Lemma 2.2 in the above mentioned paper, and the corresponding correction is presented.

A finite iterative algorithm for Hermitian reflexive and skew-Hermitian solution groups of the general coupled linear matrix equations

In this paper, we focus on the following coupled linear matrix equations $$\begin{aligned} \mathcal {M}_i(X,Y)={\mathcal {M}_{i1}(X)+\mathcal {M}_{i2}(Y)}=L_i, \end{aligned}$$ with $$\begin{aligned} {\mathcal {M}_{i \ell }(W)}&= \sum \limits _{j = 1}^{q } \left( {{\sum \limits _{\lambda = 1}^{t_1^{(\ell )} } {A_{ij\lambda }^{(\ell )} } } W_j B^{(\ell )}_{ij\lambda } + {\sum \limits _{\mu = 1}^{t_2^{(\ell )} } {C_{ij\mu }^{(\ell )} \overline{W} _j D^{(\ell )}_{ij\mu } } } + {\sum \limits _{\nu = 1}^{t_3^{(\ell )} } {E^{(\ell )}_{ij\nu } W_{j}^T F^{(\ell )}_{ij\nu } } }}\right) , \\&\ell =1,2. \end{aligned}$$ where $$A^{(\ell )}_{ij\lambda },B^{(\ell )}_{ij\lambda }$$ , $$C^{(\ell )}_{ij\mu }, D^{(\ell )}_{ij\mu }$$ , $$E^{(\ell )}_{ij\nu },F^{(\ell )}_{ij\nu }$$ and $$L_i$$ (for $$i \in I[1,p]$$ ) are given matrices with appropriate dimensions defined over complex number field. Our object is to obtain the solution groups $$X=(X_1,X_2,\ldots ,X_q)$$ and $$Y=(Y_1,Y_2,\ldots ,Y_q)$$ of the considered coupled linear matrix equations such that $$X$$ and $$Y$$ are the groups of the Hermitian reflexive and skew-Hermitian matrices, respectively. To do so, an iterative algorithm is proposed which stops within finite number of steps in the exact arithmetic. Moreover, the algorithm determines the solvability of the mentioned coupled linear matrix equations over the Hermitian reflexive and skew-Hermitian matrices, automatically. In the case that the coupled linear matrix equations are consistent, the least-norm Hermitian reflexive and skew-Hermitian solution groups can be computed by choosing suitable initial iterative matrix groups. In addition, the unique optimal approximate Hermitian reflexive and skew-Hermitian solution groups to given arbitrary matrix groups are derived. Finally, some numerical experiments are reported to illustrate the validity of our established theoretical results and feasibly of the presented algorithm.

Convergence of ADGI methods for solving systems of linear matrix equations

Purpose – The linear matrix equations have wide applications in engineering, physics, economics and statistics. The purpose of this paper is to introduce iterative methods for solving the systems of linear matrix equations. Design/methodology/approach – According to the hierarchical identification principle, the authors construct alternating direction gradient-based iterative (ADGI) methods to solve systems of linear matrix equations. Findings – The authors propose efficient ADGI methods to solve the systems of linear matrix equations. It is proven that the ADGI methods consistently converge to the solution for any initial matrix. Moreover, the constructed methods are extended for finding the reflexive solution to the systems of linear matrix equations. Originality/value – This paper proposes efficient iterative methods without computing any matrix inverses, vec operator and Kronecker product for finding the solution of the systems of linear matrix equations.