Boundary second-order sliding-mode control of an uncertain heat process with unbounded matched perturbation

The primary concern of the present paper is the regulation of an uncertain heat process with collocated boundary sensing and actuation. The underlying heat process is governed by an uncertain parabolic partial differential equation (PDE) with Neumann boundary conditions. It exhibits an unknown constant diffusivity parameter and it is affected by a smooth boundary disturbance, which is not available for measurements and which is possibly unbounded in magnitude. The proposed robust synthesis is formed by the linear feedback design and by the ”Twisting” second-order sliding-mode control algorithm, suitably combined and re-worked in the infinite-dimensional setting. A non-standard Lyapunov functional is invoked to establish the global asymptotic stability in a Sobolev space, involving spatial state derivatives of the same order as that of the plant equation. The stability proof is accompanied by a set of simple tuning rules for the controller parameters. The effectiveness of the developed control scheme is supported by simulation results.