A robust zeroing neural network and its applications to dynamic complex matrix equation solving and robotic manipulator trajectory tracking

Introduction

Complex matrix problems frequently arise in mathematics and engineering, since complex matrices are widely applied in signal processing (Liu et al., 2014), image quality assessment (Wang, 2012), joint diagonalization (Maurandi et al., 2013), and robot path tracking (Guo et al., 2019, 2020; Jin et al., 2020, 2022a,c,e; Shi et al., 2021, 2022a; Liu et al., 2022). Various numerical algorithms have been presented to solve the complex matrix problems, such as the Newton iterative method (Rajbenbach et al., 1987) and the Greville recursive method (Gan and Ling, 2008). However, the complexity of these iterative algorithms is proportional to the dimension of the matrix to be calculated, and these iterative algorithms are very effective in the calculation of low dimensional matrix. As the dimension of the matrix increases, the computational workload also increases dramatically. Moreover, with the development of big data science, the demand for large-scale computation is also inevitable. Owing to their serial-processing characteristic, the powerlessness of iterative algorithms in large-scale computation are gradually revealed.

To solve the above mentioned issue, the neural network method is proposed and deeply investigated due to its potential advantages of distributed-storage and parallel-computation in large-scale computation (Lin et al., 2022a,b; Zhou et al., 2022a,b). As a typical recurrent neural network (RNN), the gradient-based neural network (GNN) is widely used to solve matrix problems in recent years (Liu et al., 2021; Jin et al., 2022b). For example, an odd activation function (AF) activated GNN model is presented in Zhang (2005), and it solves matrix inversion problem effectively. Besides, an improved GNN model for solving linear inequalities is presented in Xiao and Zhang (2011). The GNN model can only approach the theoretical solutions of time-varying problems with fluctuation, rather than precisely converging to their theoretical solutions, and they are commonly used to solve static problems. However, time-varying problems are often encountered with the increasingly development engineering techniques, and it is urgent to develop a neural network model for solving time-varying problems.

It is worth to mention that the zeroing neural network (ZNN) model for solving dynamic problems has been proposed by Zhang and Ge (2005). As the time derivative of coefficient matrices is fully considered, the ZNN model achieves accurate solution to dynamic problems, which makes the ZNN model a powerful tool for solving dynamic problems. In Li et al. (2013), a sign-bi-power activation function (SBPAF) activated ZNN model achieves finite-time convergent to the theoretical solution of dynamic linear equation. In Jin (2021b), a finite time convergence recurrent neural network (FTCRNN) model is realized for solving time-varying complex matrix equation, and it has faster convergent speed than the conventional ZNN model. The above mentioned improved ZNN models guarantee accurate and fast solution to dynamic problems in ideal no noise environment. However, noises are unavoidable in reality, anti-noise ability must be considered for all the neural models. Hence, many anti-noise neural models have been reported to address this issue in recent years. In Jin et al. (2016), an anti-noise IEZNN model is reported for dynamic matrix inversion in noise polluted environment. Besides, in Jin et al. (2017), a NTZNN model is presented for solving dynamic problems in noisy environment. The existing anti-noise models work properly in noisy environment, but their finite-time convergent performance can be further improved. Thus, the improvement of the convergence and robustness of the existing neural models is still open. Moreover, the previous neural models focused on solving real domain dynamic problems (Li et al., 2020, 2021, 2022; Gong and Jin, 2021; Jin, 2021a; Jin and Qiu, 2022; Jin et al., 2022d,f; Shi et al., 2022b; Zhu et al., 2022), and the neural network research for solving complex domain dynamic problems is also indispensable. With the expansion of neural models to complex domain, various complex domain scientific and engineering problems can be solved easily.

Inspired by the above mentioned issues, a robust zeroing neural network (RZNN) model with fast convergence and robustness to noises for solving dynamic complex matrix equation (DCME) problems is proposed in this work. Its fast convergence irrelevant to system initial state and robustness to various noises are verified by rigorous mathematical analysis. Besides, the ZNN model activated by SBPAF are also applied to solve the DCME in same condition for the purpose of comparison, and the corresponding simulation results further demonstrate the superior convergence and robustness of the proposed RZNN model for solving dynamic complex domain problems.

The dynamic complex matrix equation and its transformation

Generally, DCME problem can be described by the following equation.

A⁢(t)⁢D⁢(t)=B⁢(t)∈ℂn×n(1)

where A(t) ∈Cn×n and B(t) ∈Cn×n are the known dynamic complex matrices, and D(t)∈Cn×n represents the unknown dynamic complex matrix to be solved.

As we know, it is very difficult to find the matrix D(t) directly from the above complex domain equation. However, any complex number contains real and imaginary parts, and we can solve the complex matrix D(t) through the transformation below.

[Are⁢(t)+j⁢Ai⁢m⁢(t)]⁢[Dre⁢(t)+j⁢Di⁢m⁢(t)]=Br⁢e⁢(t)+j⁢Bi⁢m⁢(t)(2)

Then, calculating Eq. 2 yields

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