Scientific Report: On “Event-Triggered Fuzzy Adaptive Stabilization of Parabolic PDE–ODE Systems”

Research Background and Significance

In modern engineering systems, such as flexible manipulators, heat transfer devices, and reactor controllers, many complex systems must be modeled using partial differential equations (PDEs). Due to their unique reaction-diffusion characteristics, PDEs are often used to describe infinite-dimensional systems. However, when these systems cascade with ordinary differential equations (ODEs) to form reaction-diffusion control systems, the design of effective controllers becomes significantly more complicated, especially in the presence of coupling phenomena or nonlinear factors.

In particular, in industrial scenarios like metal rolling, flexible marine risers, and hypersonic aircraft thermal protection, these PDE–ODE system models play key roles. Although past studies have achieved certain results for standalone PDEs or linear ODE control systems, these traditional approaches often neglect the nonlinear characteristics in cascading systems and the impact of uncertainties on the controller design. Additionally, conventional periodic sampling control methods can result in resource wastage.

To overcome these challenges, this study introduces an intelligent control strategy leveraging the event-triggered control (ETC) mechanism along with fuzzy logic system (FLS) techniques from artificial intelligence (AI). This approach aims to optimize the use of network resources while achieving stable control of complex reaction-diffusion systems.

Authors and Publication Context

This paper was authored by Yuan-Xin Li, Bo Xu, and Xing-Yu Zhang, affiliated with Liaoning University of Technology, Qingdao University, and University of Science and Technology Beijing, respectively. It was published in the IEEE Transactions on Artificial Intelligence in December 2024. The research was supported by funding from China’s National Natural Science Foundation, the Liaoning Province Basic Research Program, and the Liaoning Provincial Education Department Key Project.

Research Content and Methodology

Research Objective

This paper proposes a novel fuzzy adaptive event-triggered control technique for a class of parabolic PDE–ODE cascade systems. The main objective is to design a controller that ensures the PDE subsystem stabilized by the ODE subsystem achieves asymptotic stability under nonlinear and uncertain control coefficients, while keeping all closed-loop signals bounded.

Research Process

1. Problem Modeling and Assumptions

The study begins by modeling a set of PDE–ODE dynamic systems. Here, the PDE subsystem demonstrates reaction-diffusion behavior, while the ODE subsystem provides boundary control signals to the PDE system. The goal is to ensure bounded closed-loop signals and asymptotic convergence of system states to zero via an event-triggered control mechanism.

2. Infinite-to-Finite Dimensional Transformation

The paper introduces an innovative infinite-dimensional transformation method that simplifies controller design. Through the Volterra integral transformation, the original PDE–ODE system is converted into a target form, reducing the complexity in handling coupling relationships between the PDE and ODE subsystems.

3. Adaptive Backstepping-Based Controller Design

To handle the nonlinearities and uncertainties in the ODE subsystem, the study utilizes an adaptive backstepping technique combined with Lyapunov stability theory for controller design. The steps include:

• Step 1: Introduce coordinate transformations to map system states to error spaces.
• Step 2: Approximate unknown nonlinear functions using fuzzy logic systems (FLSs).
• Step 3: Develop a novel Lyapunov function integrating the lower bound of ODE control coefficients, thereby reducing dependency on upper-bound system parameters.
• Step 4: Recursively design virtual controllers to compensate for the coupled states of the PDE–ODE system.

4. Event-Triggered Control Mechanism

To reduce the communication burden, an event-triggered strategy is proposed. The control signal of the ODE subsystem is updated only when a pre-set triggering condition is met. The triggering condition is based on a proportional gain and a fixed offset factor. This mechanism significantly reduces unnecessary updates while ensuring system stability.

5. System Stability Analysis

The paper provides rigorous theoretical proof that the proposed controller guarantees boundedness of all closed-loop signals. Furthermore, it demonstrates that system states and control inputs asymptotically converge to zero, avoiding infinite triggering events (known as “Zeno behavior”).

Simulation Experiments

To validate the effectiveness of the proposed control method, a practical Josephson connection circuit is selected as an experimental case. This system is reformulated into the PDE–ODE framework studied here. The experimental results include:

1. System Stability: The PDE system states (u(x,t)) and ODE subsystem states (x_1(t)) and (x_2(t)) achieve asymptotic convergence under the proposed control law.

2. Adaptive Parameter Adjustment: The adaptive parameters (\vartheta_i) and (\rho_i) in the fuzzy logic system are shown to remain bounded.

3. Event-Triggered Performance: The event-triggering mechanism effectively reduces update frequency and avoids Zeno behavior.

4. Comparison Experiments: Compared to a traditional boundary control method, the proposed approach outperforms by handling unknown nonlinear control coefficients with reduced control effort.

5. Parameter Optimization: Additional analyses explore how varying control parameters impact system performance. It is observed that increasing (C_i) or reducing trigger parameters (\gamma) and (\chi) enhances tracking performance but may lead to more frequent controller updates. Thus, a tradeoff between performance and resource efficiency is necessary.

Research Conclusions and Key Contributions

This study introduces a novel fuzzy adaptive control strategy leveraging an ETC mechanism to address the challenges of nonlinearities and uncertainties in reaction-diffusion PDE–ODE systems. Key contributions include:

1. Developing a new adaptive backstepping controller for PDE–ODE nonlinear cascade systems with unknown control coefficients, bridging gaps in existing research.
2. Proposing a relative-threshold-based ETC strategy to efficiently manage network resources.
3. Demonstrating high practical engineering potential through system analysis and numerical simulations.

Future research will focus on observer-based boundary feedback control and system robustness under spatiotemporal disturbances.