Adaptive event-triggered tracking control for strict-feedback nonlinear ODE cascaded n+1 coupled hyperbolic PDE systems

This paper considers the adaptive event-triggered tracking control for n+1 coupled hyperbolic partial differential equation (PDE) cascaded with an uncertain nonlinear ordinary differential equation (ODE) in strict-feedback form. Such an ODE-PDE system arises in many applications such as crane systems with heavy rope and payload. Essentially different from the systems in most of the related literatures, we mainly consider that: (1) the event-triggered tracking control for the cascaded system rather than time-triggered stabilization control, (2) the control input only appears at one end of the ODE rather than directly at the boundary point of the PDE system, (3) the ODE possesses high-order nonlinear and uncertain dynamics rather than linear and deterministic ones. Due to the cascaded system structure and the presence of uncertainty and nonlinearity in the ODE, the input-to-state stable property which is important for the event-triggered control (ETC) is difficult to check. Additionally, how to solve the boundary event-triggered tracking problem for the system remains an open question so far. To this end, by combining the geometric design method, infinite- and finite-dimensional backstepping techniques, an adaptive tracking controller is first constructed. Further, a dynamic event-triggered mechanism is proposed to reduce the actuation frequency. Theoretical proof is rigorously given to show the asymptotic convergence of the tracking error of the PDE controlled output, and the existence of a minimal dwell-time. Finally, a numerical simulation consisting of a crane system in an experimental setting is presented to show the effectiveness of the proposed control scheme.