Fixed-Time Convergent Gradient Neural Network for Solving Online Sylvester Equation

Abstract

This paper aims at finding a fixed-time solution to the Sylvester equation by using a gradient neural network (GNN). To reach this goal, a modified sign-bi-power (msbp) function is presented and applied on a linear GNN as an activation function. Accordingly, a fixed-time convergent GNN (FTC-GNN) model is developed for solving the Sylvester equation. The upper bound of the convergence time of such an FTC-GNN model can be predetermined if parameters are given regardless of the initial conditions. This point is corroborated by a detailed theoretical analysis. In addition, the convergence time is also estimated utilizing the Lyapunov stability theory. Two examples are then simulated to demonstrate the validation of the theoretical analysis, as well as the superior convergence performance of the presented FTC-GNN model as compared to the existing GNN models.

An accelerated zeroing neural network for solving continuous coupled Lyapunov matrix equations

“…So far, it can be clearly obtained that for the continuous CLME (5), the state matrix P i (t m ) will globally converge to the theoretical solution P * i . In summary, by selecting an appropriate activation function array Φ i (⋅), the improved ZNN model (16) can converge to the theoretical solution of the continuous CLME (5). Thus, the theorem is completed.…”

“…Consider the improved ZNN model (16), assume that the Markovian jump system (1-2) is stochastically stable. For any given initial condition P i (t 0 ) ∈ ℝ n×n , i ∈ 𝕀 [1, N ], select Φ i (⋅) as a monotonically increasing odd activation function array, then the state matrices P i (t m ) ∈ ℝ n×n , i ∈ 𝕀 [1, N ], globally converge to the unique positive definite solution of the continuous CLME (5).…”

“…In [15], by introducing additional nonlinearity, a GNN model was established for solving time-varying Lyapunov equation, and the proposed GNN model can achieve finite time convergence due to this improvement. Moreover, a fixed time convergent GNN model was developed to solve the Sylvester equation in [16], the advantage of this model is that an upper bound on the convergence time can be predetermined. However, there always exist lagging-behind errors between the GNN solutions and the theoretical solutions for matrix equations solving.…”

Application of Gradient Optimization Methods in Defining Neural Dynamics

“…The influence of AFs on the convergence performance of a GNN design for solving the matrix equation AXB + X = C was investigated in [31]. A fixed-time convergent GNN for solving the Sylvester equation was investigated in [32]. Moreover, noise-tolerant GNN models equipped with a suitable activation function (AF) able to solve convex optimization problems were developed in [33].…”

Zeroing Neural Networks Combined with Gradient for Solving Time-Varying Linear Matrix Equations in Finite Time with Noise Resistance

“…As a result, neural dynamic methods are now considered a powerful alternative to online computation of matrix problems due to their parallel distribution properties and the convenience of hardware implementation [15,16]. As a typical RNN, the gradient-based neural network (GNN) is designed to solve the minimization problems [8,[17][18][19]. However, when coping with time-dependent cases, GNN produces relatively large lag errors [20].…”