Adaptive Event-Triggered Control for PDE-ODE Cascade Systems via Hierarchical Sliding Mode

This article considers adaptive hierarchical sliding mode control (HSMC) for hyperbolic partial differential equation (PDE)-ordinary differential equation (ODE) cascade systems. Unlike existing PDE-ODE cascade systems, the ODE subsystem studied in this paper involves unknown nonlinear functions and unknown gain functions, and thus fuzzy logic systems (FLSs) are introduced to approximate them. Meanwhile, a projection algorithm is proposed to ensure that the designed controller does not have singularity issues. Subsequently, with the aid of integration by parts and Volterra integral (VI) transformation, the original system is equivalently transformed into a new target one. For this target system, an adaptive fuzzy control approach based on hierarchical sliding mode (HSM) is developed. Compared with the commonly used backstepping method, this method simplifies the controller design steps while also eliminating the algebraic loop and “complexity explosion” issues. In addition, a triggering mechanism with a switching threshold is introduced to alleviate communication burden. Ultimately, the proposed control algorithm ensures that all signals including system states and actuator states are bounded, and this fact is verified through a Josephson junction circuit simulation.