Boundary second-order sliding-mode control of an uncertain heat process with spatially varying diffusivity

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 mixed boundary conditions. The process exhibits an unknown spatially varying diffusivity parameter, and is affected by a smooth uncertain boundary disturbance 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 prove the global asymptotic stability of the resulting closed-loop system in a suitable Sobolev space. The 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.