General and Improved Five-Step Discrete-Time Zeroing Neural Dynamics Solving Linear Time-Varying Matrix Equation with Unknown Transpose

Abstract

In this paper, a general five-step discrete-time zeroing neural dynamics (DTZND) model is proposed to solve linear time-varying matrix equation with unknown transpose. Specifically, the explicit continuous-time zeroing neural dynamics (CTZND) model is derived from the time-varying matrix equation with unknown transpose via Kronecker product and vectorization technique. Furthermore, a general five-step discretization formula is designed to approximate the first-order derivative of the target point, and the convergence condition is given. Thus, the general five-step DTZND model is obtained by using the general five-step discretization formula to discretize the CTZND model. Theoretical analyses present the stability and convergence of the proposed general five-step DTZND model. Numerical experiment results substantiate that the proposed DTZND model for solving linear time-varying matrix equation is stable and convergent with the theoretically analyzed errors. In addition, the improved DTZND models are provided in terms of accuracy and computational complexity, and verified by numerical experiments.

Appendix

The lemma of transforming $\gamma$ into $\omega$ is given as follows.

Lemma 1

For transformation $\gamma =(\omega +1)/(\omega -1)$ with $\gamma$ and $\omega$ being complex numbers, $|\gamma |<1$ is satisfied if and only if $\text{ Re }(\omega )<0$ where $|\gamma |$ and $\text{ Re }(\omega )$ indicate the absolute value of $\gamma$ and the real part of $\omega$, respectively.

Proof

Let $\omega =\eta +\xi j$, where $\eta$ and $\xi$ are real numbers, and j is the imaginary unit. Thus, $\gamma$ is given by

$$\begin{aligned} \gamma =\frac{\eta +1+\xi j}{\eta -1+\xi j}. \end{aligned}$$

(28)

Firstly, if $\eta =\text{ Re }(\omega )<0$, then

$$\begin{aligned} (\eta +1)^2+\xi ^2<(\eta -1)^2+\xi ^2. \end{aligned}$$

(29)

According to (29), we can obtain

$$\begin{aligned} |\eta +1+\xi j|<|\eta -1+\xi j|. \end{aligned}$$

(30)

Therefore, $|\gamma |<1$.

Secondly, if $|\gamma |<1$, then (30) can be obtained, of which the solution is $\eta <0$.

Finally, it is concluded that $|\gamma |<1$ is satisfied for transformation $\gamma =(\omega +1)/(\omega -1)$ if and only if $\text{ Re }(\omega )<0$. Thus, the proof is completed. $\square$