MATLAB Simulation and Comparison of Zhang Neural Network and Gradient Neural Network for Online Solution of Linear Time-Varying Matrix Equation AXB − C = 0

Chen, Ke; Yue, Shuai; Zhang, Yunong

doi:10.1007/978-3-540-85984-0_9

Cited by 19 publications

(5 citation statements)

References 14 publications

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“…In conclusion, the Simulink implementation of Algorithm 3 computes the outer inverse = ( ) † which satisfies condition (29) from the definition of the ( , )-inverse, but not condition (28) from the same definition. In other words, satisfies neither R( ) = R( ) nor N( ) = N( ).…”

Section: Matlab Fcnmentioning

confidence: 99%

“…We refer to [28,29] for further details. In the case of constant coefficient matrices , , , it is necessary to use the linear GNN of the forṁ= The generalized nonlinearly activated GNN model (GGNN model) is applicable in both time-varying and time-invariant case and possesses the forṁ…”

Section: Algorithms and Implementation Detailsmentioning

confidence: 99%

“…Solve the matrix equation ( ) ( ) ( ) ( ) ( ) = ( ) with respect to ( ) using an appropriate adaptation of the GGNN approach developed in [28,29] and restated in (43). In the particular case, the model becomeṡ…”

Section: Numerical Examplesmentioning

confidence: 99%

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Conditions for Existence, Representations, and Computation of Matrix Generalized Inverses

et al. 2017

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Conditions for the existence and representations of {2}-, {1}-, and {1, 2}-inverses which satisfy certain conditions on ranges and/or null spaces are introduced. These representations are applicable to complex matrices and involve solutions of certain matrix equations. Algorithms arising from the introduced representations are developed. Particularly, these algorithms can be used to compute the Moore-Penrose inverse, the Drazin inverse, and the usual matrix inverse. The implementation of introduced algorithms is defined on the set of real matrices and it is based on the Simulink implementation of GNN models for solving the involved matrix equations. In this way, we develop computational procedures which generate various classes of inner and outer generalized inverses on the basis of resolving certain matrix equations. As a consequence, some new relationships between the problem of solving matrix equations and the problem of numerical computation of generalized inverses are established. Theoretical results are applicable to complex matrices and the developed algorithms are applicable to both the time-varying and time-invariant real matrices.

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Section: Matlab Fcnmentioning

confidence: 99%

Section: Algorithms and Implementation Detailsmentioning

confidence: 99%

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Conditions for Existence, Representations, and Computation of Matrix Generalized Inverses

et al. 2017

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“…For comparative purpose, we could also develop and exploit the following GNN model for solving online equation (1) (see Appendix C and/or [12][17] [18][20] for more details):…”

Section: A Neural-solvers Descriptionmentioning

confidence: 99%

Modeling, verification and comparison of Zhang Neural Net and gradient neural net for online solution of time-varying linear matrix equation

Tan

Chen

Shi

et al. 2009

2009 4th IEEE Conference on Industrial Electronics and Applications

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A new recurrent neural network (or say, net), i.e., Zhang Neural Network (ZNN), is recently proposed by Zhang et al for online time-varying matrix equations solving. Theoretical analysis, blocks modeling and verification results of Zhang neural network are investigated in this paper, in addition to the neuralsolver design method and its comparable gradient neural network (GNN). Towards the final purpose of hardware realization, this paper highlights the model building and convergence illustration of ZNN model in comparison with GNN. The verification results substantiate the feasibility and efficacy of ZNN model for online time-varying linear matrix equations solving. Index Terms-time-varying linear matrix equations, recurrent neural networks (RNN), gradient-based neural networks, blocks modeling, verification and comparison.

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“…When coefficient A is non-singular, x(t) has an unique theoretical solution x * = A −1 b; when coefficient A is singular, linear equations (1) can have no solution or multiple solutions. A number of related problems such as matrix inversion [1], [2], [3], [4], [5], quadratic programming/minimisation [6], [7], [8], Sylvester equations [9], Lyapunov matrix equation [10], [11] and linear matrix equations AX(t)B = C [12], [13] can be transformed into linear equations (1) via vectorisation and Kronecker product [4]. Such a problem is widely encountered in other fields of science and engineering such as ridge regression in machine learning [14], [15], [16], [17], signal processing [18], optical flow in computer vision [19], [20], and robotic inverse kinematics [21], [22].…”

Section: Introductionmentioning

confidence: 99%

Implicit Gradient Neural Networks with a Positive-Definite Mass Matrix for Online Linear Equations Solving

Chen

2017

Preprint

Self Cite

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Motivated by the advantages achieved by implicit analogue net for solving online linear equations, a novel implicit neural model is designed based on conventional explicit gradient neural networks in this letter by introducing a positive-definite mass matrix. In addition to taking the advantages of the implicit neural dynamics, the proposed implicit gradient neural networks can still achieve globally exponential convergence to the unique theoretical solution of linear equations and also global stability even under no-solution and multi-solution situations. Simulative results verify theoretical convergence analysis on the proposed neural dynamics.

show abstract

MATLAB Simulation and Comparison of Zhang Neural Network and Gradient Neural Network for Online Solution of Linear Time-Varying Matrix Equation AXB − C = 0

Cited by 19 publications

References 14 publications

Conditions for Existence, Representations, and Computation of Matrix Generalized Inverses

Conditions for Existence, Representations, and Computation of Matrix Generalized Inverses

Modeling, verification and comparison of Zhang Neural Net and gradient neural net for online solution of time-varying linear matrix equation

Implicit Gradient Neural Networks with a Positive-Definite Mass Matrix for Online Linear Equations Solving

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