Application of Sliding Mode Control in Uncertain Fractional-Order Reaction-Diffusion Memristor Neural Networks

In recent years, as neural networks have been widely applied in various fields, the research on their control and stability has gained increasing attention. Fractional-order (FO) memristor neural networks (MNNs), due to their ability to simulate biological synapses, demonstrate unique advantages in information processing and learning. However, MNNs face numerous challenges in applications, such as system uncertainties, signal transmission delays, and complex spatiotemporal evolutionary characteristics. These factors may lead to network instability and performance degradation. Therefore, studying a robust control method to solve these problems has significant theoretical and practical implications.

In the background introduction section, it’s necessary to first introduce the basic concept of memristors and their application in neural networks. Memristors, as the fourth type of electronic component, alongside inductors, capacitors, and resistors, was proposed by Chua in 1971. The electrical behavior of memristors is very similar to the operational mechanism of biological synapses, thus providing an effective means to mimic biological synapses. Additionally, the advantages of fractional calculus theory over integer-order calculus, such as long memory, non-locality, and weak singularity, can be mentioned, contributing to good application prospects in viscoelastic theory, control theory, and neural networks.

Source of the Paper

This research was completed by Yue Cao, Yonggui Kao, Zhen Wang, Xinsong Yang, Ju H. Park, and Wei Xie, from the Department of Mathematics at Harbin Institute of Technology, the School of Science and Technology at Shandong University, the School of Electronic and Information Engineering at Sichuan University, and the Department of Electrical Engineering at Yeungnam University. The paper was submitted on January 27, 2024, revised multiple times, and finally accepted on May 20, 2024, and published in the journal Neural Networks.

Research Content

The main content of the paper includes designing a sliding mode control (SMC) method for a class of uncertain fractional-order reaction-diffusion memristor neural networks (FORDMNNs). Different from traditional fractional-order sliding mode control methods, this study constructs a linear sliding mode switching function for the first time and designs the corresponding sliding mode control law. The paper demonstrates the global asymptotic stability of the sliding mode dynamics in detail and proves that the sliding mode surface is reachable in finite time under the proposed control law. Additionally, numerical tests validate the effectiveness of the theoretical analysis.

Research Process

The research is broadly divided into the following steps:

1. Mathematical Model Construction: To describe MNNs considering uncertainties and time delays, a fractional-order mathematical model is first constructed. The model employs Riemann-Liouville (Fractional-Order, FO) and Caputo fractional-order derivatives.
2. Sliding Mode Control Method Design: The paper constructs a linear sliding mode switching function and designs the corresponding sliding mode control law, combining the system decomposition method.
3. Stability Analysis: Utilizing the Lyapunov function, the sliding mode dynamics are transformed into a form with known boundary and initial conditions to prove the global asymptotic stability of the sliding mode surface.
4. Numerical Simulation Verification: Through specific numerical tests, the effectiveness of the designed control law is verified, and the dynamic evolution of the closed-loop system state is observed.

Research Results

Firstly, a fractional-order memristor neural network model is constructed. Through the sliding mode switching function, the designed control law achieves global asymptotic stability for the system under the given control target. Moreover, numerical simulations verify the effectiveness of the sliding mode method in handling system uncertainties and time delay issues. The spatiotemporal evolution results of the open-loop system state show oscillations and instability, while in the closed-loop system, the sliding mode control law rapidly converges the system state to a stable state.

Research Conclusion

The conclusion of the paper is that the proposed sliding mode control method significantly enhances the stability and performance of uncertain fractional-order memristor neural networks. The designed linear sliding mode switching function reduces the complexity of the control law design, making the system more robust to external disturbances. The stability analysis provides a theoretical basis. Additionally, future research directions are suggested to improve the applicability of the system model by further investigating sliding mode methods with different types of time delays.

Research Highlights

1. Innovative Design: For the first time, a linear sliding mode switching function is constructed, simplifying the sliding mode control design process.
2. Theoretical Proof: The global asymptotic stability of the sliding mode dynamics is proven through the Lyapunov function.
3. Numerical Verification: Numerical experiments validate the effectiveness of the sliding mode control method in addressing system uncertainties and time delay issues.
4. Applicative Value: It provides new control methods and theoretical basis for similar fractional-order systems.