Zeroing Neural Network With Novel Arctan Activation Function and Parameter-Varying Dual Integral Structure for Solving Time-Varying Lyapunov Equations

The time-varying Lyapunov equation plays a central role in the stability analysis and control of dynamic systems across various fields. Traditional methods for dealing with time-varying Lyapunov equations often face challenges such as slow convergence rates and sensitivity to environmental noise. To address these challenges, this paper proposes a novel hybrid neural network model known as the dual integral structure ZNN model. The proposed model integrates an innovative arctangent activation function and designs corresponding arctangent exponential time-varying convergence parameters, known as the VPDIAZNN model, along with another model featuring linear exponential time-varying convergence parameters, referred to as the VPDIZNN model. Theoretical analysis proves the stability and robustness of the models under linear noise. Additionally, comparative experiments demonstrate that in bounded noise and linear noise environments, both the VPDIAZNN and VPDIZNN models can converge quickly, achieving accuracies of up to
${10}^{-6}$
, while the HTVPZNN and NTFZNN models fail to converge. When computing high-order matrices, our models achieve the highest precision and even reach up to
${10}^{-10}$
.