Nonlinearly Activated Neural Network for Solving Time-Varying Complex Sylvester Equation

The Sylvester equation is often encountered in mathematics and control theory. For the general time-invariant Sylvester equation problem, which is defined in the domain of complex numbers, the Bartels–Stewart algorithm and its extensions are effective and widely used with an $O(n^{3})$ time complexity. When applied to solving the time-varying Sylvester equation, the computation burden increases intensively with the decrease of sampling period and cannot satisfy continuous realtime calculation requirements. For the special case of the general Sylvester equation problem defined in the domain of real numbers, gradient-based recurrent neural networks are able to solve the time-varying Sylvester equation in real time, but there always exists an estimation error while a recently proposed recurrent neural network by Zhang et al. [this type of neural network is called Zhang neural network (ZNN)] converges to the solution ideally. The advancements in complex-valued neural networks cast light to extend the existing real-valued ZNN for solving the time-varying real-valued Sylvester equation to its counterpart in the domain of complex numbers. In this paper, a complex-valued ZNN for solving the complex-valued Sylvester equation problem is investigated and the global convergence of the neural network is proven with the proposed nonlinear complex-valued activation functions. Moreover, a special type of activation function with a core function, called sign-bi-power function, is proven to enable the ZNN to converge in finite time, which further enhances its advantage in online processing. In this case, the upper bound of the convergence time is also derived analytically. Simulations are performed to evaluate and compare the performance of the neural network with different parameters and activation functions. Both theoretical analysis and numerical simulations validate the effectiveness of the proposed method.

I. Introduction

The general Sylvester equation problem, which is defined in the domain of complex numbers, is often encountered in mathematics and control theory and finds applications in linear least-squares regression [1] and eigen-structure assignment [2], etc. The Bartels–Stewart algorithm [3] and its extensions [4], [5] are widely used to numerically solve a static Sylvester equation without time-varying parameters. It has been proven that the Bartels–Stewart algorithm completes the calculation with a time complexity . When applied to the time-varying Sylvester equation, the Bartels–Stewart algorithm has to be executed in every sampling period, and the algorithm fails when the sampling rate is too high to allow the algorithm to complete the calculation in a single sampling period, not to mention the more challenging scenario to solve the equation continuously in real time.