The Experts below are selected from a list of 13257 Experts worldwide ranked by ideXlab platform
Masoud Hajarian - One of the best experts on this subject based on the ideXlab platform.
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matrix iterative methods for solving the Sylvester transpose and periodic Sylvester matrix equations
2013Co-Authors: Masoud HajarianAbstract:Abstract The problem of solving matrix equations has many applications in control and system theory. This paper is concerned with the iterative solutions of the Sylvester-transpose matrix equation ∑ i = 1 k ( A i XB i + C i X T D i ) = E , and the periodic Sylvester matrix equation A ^ j X ^ j B ^ j + C ^ j X ^ j + 1 D ^ j = E ^ j for j = 1 , 2 , … , λ . The basic idea is to develop the conjugate gradients squared (CGS) and bi-conjugate gradient stabilized (Bi-CGSTAB) methods for obtaining matrix iterative methods for solving the Sylvester-transpose and periodic Sylvester matrix equations. Numerical test results are given to compare matrix iterative methods with other well-known methods.
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the generalised Sylvester matrix equations over the generalised bisymmetric and skew symmetric matrices
2012Co-Authors: Mehdi Dehghan, Masoud HajarianAbstract:A matrix P is called a symmetric orthogonal if P = P T = P −1. A matrix X is said to be a generalised bisymmetric with respect to P if X = X T = PXP . It is obvious that any symmetric matrix is also a generalised bisymmetric matrix with respect to I identity matrix. By extending the idea of the Jacobi and the Gauss–Seidel iterations, this article proposes two new iterative methods, respectively, for computing the generalised bisymmetric containing symmetric solution as a special case and skew-symmetric solutions of the generalised Sylvester matrix equation including Sylvester and Lyapunov matrix equations as special cases which is encountered in many systems and control applications. When the generalised Sylvester matrix equation has a unique generalised bisymmetric skew-symmetric solution, the first second iterative method converges to the generalised bisymmetric skew-symmetric solution of this matrix equation for any initial generalised bisymmetric skew-symmetric matrix. Finally, some numerical results are given to illustrate the effect of the theoretical results.
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an iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices
2010Co-Authors: Mehdi Dehghan, Masoud HajarianAbstract:Abstract The generalized coupled Sylvester matrix equations AXB + CYD = M , EXF + GYH = N , (including Sylvester and Lyapunov matrix equations as special cases) have numerous applications in control and system theory. An n × n matrix P is called a symmetric orthogonal matrix if P = P T = P - 1 . A matrix X is said to be a generalized bisymmetric with respect to P , if X = X T = PXP . This paper presents an iterative algorithm to solve the generalized coupled Sylvester matrix equations over generalized bisymmetric matrix pair [ X , Y ] . The proposed iterative algorithm, automatically determines the solvability of the generalized coupled Sylvester matrix equations over generalized bisymmetric matrix pair. Due to that I (identity matrix) is a symmetric orthogonal matrix, using the proposed iterative algorithm, we can obtain a symmetric solution pair of the generalized coupled Sylvester matrix equations. When the generalized coupled Sylvester matrix equations are consistent over generalized bisymmetric matrix pair [ X , Y ] , for any (spacial) initial generalized bisymmetric matrix pair, by proposed iterative algorithm, a generalized bisymmetric solution pair (the least Frobenius norm generalized bisymmetric solution pair) can be obtained within finite iteration steps in the absence of roundoff errors. Moreover, the optimal approximation generalized bisymmetric solution pair to a given generalized bisymmetric matrix pair can be derived by finding the least Frobenius norm generalized bisymmetric solution pair of new generalized coupled Sylvester matrix equations. Finally, a numerical example is given which demonstrates that the introduced iterative algorithm is quite efficient.
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efficient iterative method for solving the second order Sylvester matrix equation evf 2 avf cv bw
2009Co-Authors: Mehdi Dehghan, Masoud HajarianAbstract:The second-order Sylvester matrix equation EVF2−AVF−CV=BW (including the generalised Sylvester matrix equation, normal Sylvester matrix equation and Lyapunov matrix equation as special cases) over unknown matrix pair [V, W], has wide applications in many fields. In the present study, the authors propose an iterative method to solve the second-order Sylvester matrix equation. The proposed iterative method does not depend on the Jordan form of the matrix F. By this iterative method, the solvability of the matrix equation can be determined automatically over unknown matrix pair [V, W]≠0. When the matrix equation is solvable, its solution pair can be obtained within finite iterative steps, and its least Frobenius norm solution pair can be obtained by choosing suitable initial matrix pair. Furthermore, its optimal approximation solution pair to a given matrix pair can be derived by finding the least norm solution pair of a new matrix equation. A numerical example is given to show the efficiency of the proposed method.
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an iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation
2008Co-Authors: Mehdi Dehghan, Masoud HajarianAbstract:Abstract The generalized coupled Sylvester matrix equations ( AY - ZB , CY - ZD ) = ( E , F ) with unknown matrices Y , Z are encountered in many systems and control applications. Also these matrix equations have several applications relating to the problem of computing stable eigendecompositions of matrix pencils. In this work, we construct an iterative algorithm to solve the generalized coupled Sylvester matrix equations over reflexive matrices Y , Z . And when the matrix equations are consistent, for any initial matrix pair [ Y 0 , Z 0 ] , a reflexive solution pair can be obtained within finite iteration steps in the absence of roundoff errors, and the least Frobenius norm reflexive solution pair can be obtained by choosing a special kind of initial matrix pair. Also we obtain the optimal approximation reflexive solution pair to a given matrix pair [ Y ¯ , Z ¯ ] in the reflexive solution pair set of the generalized coupled Sylvester matrix equations ( AY - ZB , CY - ZD ) = ( E , F ) . Moreover, several numerical examples are given to show the efficiency of the presented iterative algorithm.
Guangren Duan - One of the best experts on this subject based on the ideXlab platform.
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the complete solution to the Sylvester polynomial conjugate matrix equations
2011Co-Authors: Gang Feng, Wanquan Liu, Guangren DuanAbstract:In this paper we propose two new operators for complex polynomial matrices. One is the conjugate product and the other is the Sylvester-conjugate sum. Then some important properties for these operators are proved. Based on these derived results, we propose a unified approach to solving a general class of Sylvester-polynomial-conjugate matrix equations, which include the Yakubovich-conjugate matrix equation as a special case. The complete solution of the Sylvester-polynomial-conjugate matrix equation is obtained in terms of the Sylvester-conjugate sum, and such a proposed solution can provide all the degrees of freedom with an arbitrarily chosen parameter matrix.
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Finite iterative solutions to coupled Sylvester-conjugate matrix equations
2011Co-Authors: Ying Zhang, Guangren DuanAbstract:Abstract This paper is concerned with solutions to the so-called coupled Sylveter-conjugate matrix equations, which include the generalized Sylvester matrix equation and coupled Lyapunov matrix equation as special cases. An iterative algorithm is constructed to solve this kind of matrix equations. By using the proposed algorithm, the existence of a solution to a coupled Sylvester-conjugate matrix equation can be determined automatically. When the considered matrix equation is consistent, it is proven by using a real inner product in complex matrix spaces as a tool that a solution can be obtained within finite iteration steps for any initial values in the absence of round-off errors. Another feature of the proposed algorithm is that it can be implemented by using original coefficient matrices, and does not require to transform the coefficient matrices into any canonical forms. The algorithm is also generalized to solve a more general case. Two numerical examples are given to illustrate the effectiveness of the proposed methods.
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unified parametrization for the solutions to the polynomial diophantine matrix equation and the generalized Sylvester matrix equation
2010Co-Authors: Bin Zhou, Zhibin Yan, Guangren DuanAbstract:The polynomial Diophantine matrix equation and the generalized Sylvester matrix equation are important for controller design in frequency domain linear system theory and time domain linear system theory, respectively. By using the so-called generalized Sylvester mapping, right coprime factorization and Bezout identity associated with certain polynomial matrices, we present in this note a unified parametrization for the solutions to both of these two classes of matrix equations. Moreover, it is shown that solutions to the generalized Sylvester matrix equation can be obtained if solutions to the Diophantine matrix equation are available. The results disclose a relationship between the polynomial Diophantine matrix equation and generalized Sylvester matrix equation that are respectively studied and used in frequency domain linear system theory and time domain linear system theory.
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finite iterative algorithms for the generalized Sylvester conjugate matrix equation formula
2010Co-Authors: Guangren DuanAbstract:This paper investigates the generalized Sylvester-conjugate matrix equation, which includes the normal Sylvester-conjugate, Kalman-Yakubovich-conjugate and generalized Sylvester matrix equations as its special cases. An iterative algorithm is presented for solving such a kind of matrix equations. This iterative method can give an exact solution within finite iteration steps for any initial values in the absence of round-off errors. Another feature of the proposed algorithm is that it is implemented by original coefficient matrices. By specifying the proposed algorithm, iterative algorithms for some special matrix equations are also developed. Two numerical examples are given to illustrate the effectiveness of the proposed methods.
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weighted least squares solutions to general coupled Sylvester matrix equations
2009Co-Authors: Bin Zhou, Guangren Duan, Yong WangAbstract:This paper is concerned with weighted least squares solutions to general coupled Sylvester matrix equations. Gradient based iterative algorithms are proposed to solve this problem. This type of iterative algorithm includes a wide class of iterative algorithms, and two special cases of them are studied in detail in this paper. Necessary and sufficient conditions guaranteeing the convergence of the proposed algorithms are presented. Sufficient conditions that are easy to compute are also given. The optimal step sizes such that the convergence rates of the algorithms, which are properly defined in this paper, are maximized and established. Several special cases of the weighted least squares problem, such as a least squares solution to the coupled Sylvester matrix equations problem, solutions to the general coupled Sylvester matrix equations problem, and a weighted least squares solution to the linear matrix equation problem are simultaneously solved. Several numerical examples are given to illustrate the effectiveness of the proposed algorithms.
Mehdi Dehghan - One of the best experts on this subject based on the ideXlab platform.
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the generalised Sylvester matrix equations over the generalised bisymmetric and skew symmetric matrices
2012Co-Authors: Mehdi Dehghan, Masoud HajarianAbstract:A matrix P is called a symmetric orthogonal if P = P T = P −1. A matrix X is said to be a generalised bisymmetric with respect to P if X = X T = PXP . It is obvious that any symmetric matrix is also a generalised bisymmetric matrix with respect to I identity matrix. By extending the idea of the Jacobi and the Gauss–Seidel iterations, this article proposes two new iterative methods, respectively, for computing the generalised bisymmetric containing symmetric solution as a special case and skew-symmetric solutions of the generalised Sylvester matrix equation including Sylvester and Lyapunov matrix equations as special cases which is encountered in many systems and control applications. When the generalised Sylvester matrix equation has a unique generalised bisymmetric skew-symmetric solution, the first second iterative method converges to the generalised bisymmetric skew-symmetric solution of this matrix equation for any initial generalised bisymmetric skew-symmetric matrix. Finally, some numerical results are given to illustrate the effect of the theoretical results.
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an iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices
2010Co-Authors: Mehdi Dehghan, Masoud HajarianAbstract:Abstract The generalized coupled Sylvester matrix equations AXB + CYD = M , EXF + GYH = N , (including Sylvester and Lyapunov matrix equations as special cases) have numerous applications in control and system theory. An n × n matrix P is called a symmetric orthogonal matrix if P = P T = P - 1 . A matrix X is said to be a generalized bisymmetric with respect to P , if X = X T = PXP . This paper presents an iterative algorithm to solve the generalized coupled Sylvester matrix equations over generalized bisymmetric matrix pair [ X , Y ] . The proposed iterative algorithm, automatically determines the solvability of the generalized coupled Sylvester matrix equations over generalized bisymmetric matrix pair. Due to that I (identity matrix) is a symmetric orthogonal matrix, using the proposed iterative algorithm, we can obtain a symmetric solution pair of the generalized coupled Sylvester matrix equations. When the generalized coupled Sylvester matrix equations are consistent over generalized bisymmetric matrix pair [ X , Y ] , for any (spacial) initial generalized bisymmetric matrix pair, by proposed iterative algorithm, a generalized bisymmetric solution pair (the least Frobenius norm generalized bisymmetric solution pair) can be obtained within finite iteration steps in the absence of roundoff errors. Moreover, the optimal approximation generalized bisymmetric solution pair to a given generalized bisymmetric matrix pair can be derived by finding the least Frobenius norm generalized bisymmetric solution pair of new generalized coupled Sylvester matrix equations. Finally, a numerical example is given which demonstrates that the introduced iterative algorithm is quite efficient.
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efficient iterative method for solving the second order Sylvester matrix equation evf 2 avf cv bw
2009Co-Authors: Mehdi Dehghan, Masoud HajarianAbstract:The second-order Sylvester matrix equation EVF2−AVF−CV=BW (including the generalised Sylvester matrix equation, normal Sylvester matrix equation and Lyapunov matrix equation as special cases) over unknown matrix pair [V, W], has wide applications in many fields. In the present study, the authors propose an iterative method to solve the second-order Sylvester matrix equation. The proposed iterative method does not depend on the Jordan form of the matrix F. By this iterative method, the solvability of the matrix equation can be determined automatically over unknown matrix pair [V, W]≠0. When the matrix equation is solvable, its solution pair can be obtained within finite iterative steps, and its least Frobenius norm solution pair can be obtained by choosing suitable initial matrix pair. Furthermore, its optimal approximation solution pair to a given matrix pair can be derived by finding the least norm solution pair of a new matrix equation. A numerical example is given to show the efficiency of the proposed method.
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an iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation
2008Co-Authors: Mehdi Dehghan, Masoud HajarianAbstract:Abstract The generalized coupled Sylvester matrix equations ( AY - ZB , CY - ZD ) = ( E , F ) with unknown matrices Y , Z are encountered in many systems and control applications. Also these matrix equations have several applications relating to the problem of computing stable eigendecompositions of matrix pencils. In this work, we construct an iterative algorithm to solve the generalized coupled Sylvester matrix equations over reflexive matrices Y , Z . And when the matrix equations are consistent, for any initial matrix pair [ Y 0 , Z 0 ] , a reflexive solution pair can be obtained within finite iteration steps in the absence of roundoff errors, and the least Frobenius norm reflexive solution pair can be obtained by choosing a special kind of initial matrix pair. Also we obtain the optimal approximation reflexive solution pair to a given matrix pair [ Y ¯ , Z ¯ ] in the reflexive solution pair set of the generalized coupled Sylvester matrix equations ( AY - ZB , CY - ZD ) = ( E , F ) . Moreover, several numerical examples are given to show the efficiency of the presented iterative algorithm.
Long Jin - One of the best experts on this subject based on the ideXlab platform.
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noise tolerant gradient oriented neurodynamic model for solving the Sylvester equation
2021Co-Authors: Bei Liu, Long Jin, Haoen HuangAbstract:Abstract Recursive neural networks are generally divided into dynamic neural networks and static neural networks to refer to the neural networks with one or more feedback links in the network structure. Inevitably, there exist some problems such as poor approximation performance and poor stable convergence performance due to complex network structure. The noise-tolerant gradient-oriented neurodynamic (NTGON) model proposed in this study is an improved model based on the traditional idea of a gradient neural network (GNN) model. The proposed NTGON model can obtain accurate and efficient results under the condition of various noises when computing the Sylvester equation, which is effectively used to solve various problems with noise pollution that are frequently encountered in practical engineering. Compared with the original GNN model for the Sylvester equation, the NTGON model exponentially converges to the theoretical solution starting from any initial state. It is demonstrated that the noise-polluted NTGON model converges to the theoretical solution globally no matter how large the unknown matrix-form noise is. Furthermore, simulation results show that the proposed NTGON model achieves a performance that is superior to that of the original GNN model for solving the Sylvester equation in the presence of noise.
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discrete computational neural dynamics models for solving time dependent Sylvester equation with applications to robotics and mimo systems
2020Co-Authors: Long Jin, Mei LiuAbstract:In this article, a neural dynamics model is constructed and investigated for solving time-dependent Sylvester equation with matrix inversion involved in the solving process. Besides, to eliminate the matrix inversion in the model, the quasi-Newton Broyden–Fletcher–Goldfarb–Shanno method is leveraged to construct a new model. Moreover, the global convergence performance and the effectiveness of the two discrete computational models are testified by providing theoretical analyses and numerical experiments with comparisons to the existing solutions, respectively. Two applications to robotics and the multiple-input multiple-output system are given to elucidate the feasibility of the proposed models for solving time-dependent Sylvester equation.
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rnn for solving time variant generalized Sylvester equation with applications to robots and acoustic source localization
2020Co-Authors: Long Jin, Jingkun Yan, Xiuchun XiaoAbstract:A generalized Sylvester equation is a special formulation containing the Sylvester equation, the Lyapunov equation and the Stein equation, which is often encountered in various fields. However, the time-variant generalized Sylvester equation (TVGSE) is rarely investigated in the existing literature. In this article, we propose a noise-suppressing recurrent neural network (NSRNN) model activated by saturation-allowed functions to solve the TVGSE. For comparison, the existing zeroing neural network (ZNN) models and some improved ZNN models are introduced. Additionally, theoretical analysis on the convergence and robustness of the NSRNN model is given. Furthermore, computer simulations on illustrative examples and applications to robots and acoustic source localization are carried out. Validation results synthesized by the NSRNN model and other ZNN models are provided to illustrate the ability in solving the TVGSE and dealing with noises of the NSRNN model, and the inaction of other ZNN models to noises.
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Novel Discrete-Time Recurrent Neural Networks Handling Discrete-Form Time-Variant Multi-Augmented Sylvester Matrix Problems and Manipulator Application.
2020Co-Authors: Yang Shi, Long Jin, Jipeng Qiang, Dimitrios K. GerontitisAbstract:In this article, the discrete-form time-variant multi-augmented Sylvester matrix problems, including discrete-form time-variant multi-augmented Sylvester matrix equation (MASME) and discrete-form time-variant multi-augmented Sylvester matrix inequality (MASMI), are formulated first. In order to solve the above-mentioned problems, in continuous time-variant environment, aided with the Kronecker product and vectorization techniques, the multi-augmented Sylvester matrix problems are transformed into simple linear matrix problems, which can be solved by using the proposed discrete-time recurrent neural network (RNN) models. Second, the theoretical analyses and comparisons on the computational performance of the recently developed discretization formulas are presented. Based on these theoretical results, a five-instant discretization formula with superior property is leveraged to establish the corresponding discrete-time RNN (DTRNN) models for solving the discrete-form time-variant MASME and discrete-form time-variant MASMI, respectively. Note that these DTRNN models are zero stable, consistent, and convergent with satisfied precision. Furthermore, illustrative numerical experiments are given to substantiate the excellent performance of the proposed DTRNN models for solving discrete-form time-variant multi-augmented Sylvester matrix problems. In addition, an application of robot manipulator further extends the theoretical research and physical realizability of RNN methods.
Lin Xiao - One of the best experts on this subject based on the ideXlab platform.
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design verification and robotic application of a novel recurrent neural network for computing dynamic Sylvester equation
2018Co-Authors: Lin Xiao, Zhijun Zhang, Zili ZhangAbstract:To solve dynamic Sylvester equation in the presence of additive noises, a novel recurrent neural network (NRNN) with finite-time convergence and excellent robustness is proposed and analyzed in this paper. As compared with the design process of Zhang neural network (ZNN), the proposed NRNN is based on an ingenious integral design formula activated by nonlinear functions, which are able to expedite the convergence speed and suppress unknown additive noises during the solving process of dynamic Sylvester equation. In addition, the global stability, finite-time convergence and denoising property of the NRNN model are theoretically proved. The upper bound of the finite convergence time for the NRNN model is also estimated in theory. Simulative results further verify the efficiency of the NRNN model, as well as its superior robust and finite-time performance to the conventional ZNN model for dynamic Sylvester equation in front of additive noises. At last, the proposed design method for establishing the NRNN model is successfully applied to kinematical control of robotic manipulator in front of additive noises.
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a finite time recurrent neural network for solving online time varying Sylvester matrix equation based on a new evolution formula
2017Co-Authors: Lin XiaoAbstract:Sylvester equation is widely used to study the stability of a nonlinear system in the control field. In this paper, a finite-time Zhang neural network (FTZNN) is proposed and applied to online solution of time-varying Sylvester equation. Differing from the conventional accelerating method, the design of the proposed FTZNN model is based on a new evolution formula, which is presented and studied to accelerate the convergence speed of a recurrent neural network. Compared with the original Zhang neural network (ZNN) for time-varying Sylvester equation, the FTZNN model can converge to the theoretical time-varying solution within finite time, instead of converging exponentially with time. Besides, we can obtain the upper bound of the finite convergence time for the FTZNN model in theory. Simulation results show that the proposed FTZNN model achieves the better performance as compared with the original ZNN model for solving online time-varying Sylvester equation.
