Safe Adaptive Control of Parabolic PDE-ODE Cascades

In this paper, we propose a safe adaptive boundary control strategy for a class of parabolic partial differential equation-ordinary differential equation (PDE-ODE) cascaded systems with parametric uncertainties in both the PDE and ODE subsystems. The proposed design is built upon an adaptive Control Barrier Function (aCBF) framework that incorporates high-relative-degree CBFs together with a batch least-squares identification (BaLSI)-based adaptive control that guarantees exact parameter identification in finite time. The proposed control law ensures that: (i) if the system output state initially lies within a prescribed safe set, safety is maintained for all time; otherwise, the output is driven back into the safe region within a preassigned finite time; and (ii) convergence to zero of all plant states is achieved. Numerical simulations are provided to demonstrate the effectiveness of the proposed approach.

💡 Research Summary

The paper addresses the challenging problem of safely controlling a class of cascaded systems composed of a parabolic partial differential equation (PDE) coupled with an ordinary differential equation (ODE), where both subsystems contain unknown constant parameters. Traditional boundary control methods for parabolic PDEs, largely based on backstepping, focus on stabilization but do not guarantee that the system output respects prescribed safety constraints during transients. To fill this gap, the authors develop a novel adaptive boundary‑control scheme that simultaneously ensures safety and asymptotic convergence to the origin.

The core of the approach is an adaptive Control Barrier Function (aCBF) framework that incorporates high‑relative‑degree CBFs. The safety constraint is expressed by a barrier function (h(y_{1},t)) that is assumed to be (n)‑times differentiable, where (y_{1}) is the first component of the ODE state. To handle the high relative degree, the authors recursively construct a set of barrier functions (h_{i}(z_{i},t)) (for (i=1,\dots,n)) using a series of transformations. The first transformation (Z = T_{z}Y) puts the ODE into a controllable canonical form, while the second transformation augments the barrier with a smooth auxiliary signal (\sigma(t)) that forces the system back into the safe set within a user‑prescribed finite time (t_{a}) if the initial condition lies outside.

Parameter uncertainties in the diffusion coefficient (\lambda) and the input gain (b) are identified in finite time by a batch least‑squares identification (BaLSI) algorithm. The BaLSI provides exact estimates after a finite identification interval, and these estimates are fed back into the adaptive control law.

To eliminate the destabilizing term in the PDE and to decouple the ODE dynamics, a backstepping transformation \