Safe Adaptive Control of Hyperbolic PDE-ODE Cascades

Adaptive safe control employing conventional continuous infinite-time adaptation requires that the initial conditions be restricted to a subset of the safe set due to parametric uncertainty, where the safe set is shrunk in inverse proportion to the adaptation gain. The recent regulation-triggered adaptive control approach with batch least-squares identification (BaLSI, pronounced "ballsy") completes perfect parameter identification in finite time and offers a previously unforeseen advantage in adaptive safe control, which we elucidate in this paper. Since the true challenge of safe control is exhibited for CBF of a high relative degree, we undertake a safe BaLSI design in this paper for a class of systems that possess a particularly extreme relative degree: ODE-PDE-ODE sandwich systems. Such sandwich systems arise in various applications, including delivery UAVs (Unmanned Aerial Vehicles) with a cable-suspended load. Collision avoidance of the payload with the surrounding environment is required. The considered class of plants is $2\times2$ hyperbolic PDEs sandwiched by a strict-feedback nonlinear ODE and a linear ODE, where the unknown coefficients, whose bounds are known and arbitrary, are associated with the PDE in-domain coupling terms that can cause instability and with the input signal of the distal ODE. This is the first safe adaptive control design for PDEs, where we introduce the concept of PDE CBF whose non-negativity, as well as the ODE CBF's non-negativity, are ensured with a backstepping-based safety filter. Our safe adaptive controller is explicit and operates in the entire original safe set.