Abstract
The time-variant matrix inversion (TVMI) problem solving is the hotspot of current research because of its frequent appearance and application in scientific research and industrial production. The generalized inverse problem of singular square matrix and nonsquare matrix can be related to Penrose equations (PEs). The PEs implicitly define the generalized inverse of a known matrix, which is of fundamental theoretical significance. Therefore, the in-depth study of PEs might enlighten problem solving of TVMI in a foreseeable way. For the first time, we construct three different matrix error-monitoring functions based on PEs and propose the corresponding models for TVMI problem solving by using the substitution technique and ZNN design formula. In order to facilitate computer simulation, the obtained continuous-time models are discretized by using ZTD (Zhang time discretization) formulas. Furthermore, the feasibility and effectiveness of the novel Zhang neural network (ZNN) multiple-multiplication model for matrix inverse (ZMMMI) and the PEs-based Getz–Marsden dynamic system (PGMDS) model in solving the problem of TVMI are investigated and shown via theoretical derivation and computer simulation. Computer experiment results also illustrate that the direct derivative dynamics model for TVMI is less effective and feasible.
Highlights
During the past decades, scientists and engineers have encountered linear matrix equation problems, i.e., Lyapunov equation, Sylvester equation, and the variational problem, again and again in various scenarios. e matrix inversion problem is one of the most prominent subproblems in the linear matrix equation problems
From the discrete simulation results of two examples of each five discrete algorithms, it can be seen that the convergence of ZMMMI model (10) and PEs-based Getz–Marsden dynamic system (PGMDS) model (38) is good, which is completely consistent with the conclusion of Proposition 1 and eorems 1 and 2
We have shed some light on the matrix inversion solution models derivation, i.e., ZMMMI model (10), PGMDS model (38), and DDD model (49), from Penrose equations (PEs)
Summary
Scientists and engineers have encountered linear matrix equation problems, i.e., Lyapunov equation, Sylvester equation, and the variational problem, again and again in various scenarios. e matrix inversion problem is one of the most prominent subproblems in the linear matrix equation problems. Scientists and engineers have encountered linear matrix equation problems, i.e., Lyapunov equation, Sylvester equation, and the variational problem, again and again in various scenarios. Among the problems encountered in a variety of optimization problems, the fundamental one is the solution of matrix inversion, such as signal processing [1], biomedical prediction [2], image reconstruction [3], nonlinear optimization [4], and robot inverse kinematics [5,6,7,8]. Efforts were directed towards computational issues of time-variant matrix inversion (TVMI) and a wealth of algorithms were proposed and applied to solving this problem [9,10,11]. Zhang et al [12] discussed the solution of TVMI problem based on direct
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