Adaptive output-feedback tracking control for heterodirectional hyperbolic PDE-ODE cascade systems: a swapping filter-based approach

This work addresses a class of cascade systems composed of partial differential equations (PDEs) and ordinary differential equations (ODEs). The parameters of the PDE system vary with space, and there are unknown functions in the domain. Through the boundary of the PDE, the ODE subsystem imposes a direct effect on the state of the whole cascade system, and the transmission direction of the 2×2 PDE system is spatially varying and opposite. The control and sensing modules of the cascade system are on the same side. The swapping filter is constructed for the observer error system by invertible transformation, and an adaptive law based on the modified least squares with a forgetting factor method design is proposed. An observer with integral compensation is designed under persistent excitation (PE) conditions, which ensures that both the state of the observation error system and the parameter estimation error converge exponentially to zero. In accordance with the separation theorem, the memoryless adaptive tracking controller designed on the basis of the observer not only ensures that all signals within the closed-loop system are bounded but also enables the cascaded side-boundary signals to track the reference signal exponentially. Finally, we demonstrate the theoretical results through an oil well model simulation.