Delay-Adaptive Boundary Control of Coupled Hyperbolic PDE-ODE Cascade Systems

This article presents a delay-adaptive boundary control scheme for a 2 × 2 coupled linear hyperbolic partial differential equation (PDE)-ordinary differential equation (ODE) cascade system with an unknown and arbitrarily long input delay. To construct a nominal delay-compensated control law, assuming a known input delay, a three-step backstepping design is used. To build the delay-adaptive boundary control law, the nominal control action is fed with the estimate of the unknown delay, which is generated from a batch least-squares identifier that is updated by an event-triggering mechanism that evaluates the growth of the norm of the system states. As a result of the closed-loop system, the actuator and plant states can be regulated exponentially while avoiding Zeno occurrences. The prescribed-time identification of the unknown delay is also achieved. As far as we know, this is the first delay-adaptive control result for systems governed by heterodirectional hyperbolic PDEs. The effectiveness of the proposed design is demonstrated in the control application of a deep-sea construction vessel with cable-payload oscillations and subject to input delay.