On the Second Order Sliding Mode Approach to Distributed and Boundary Control of Uncertain Parabolic PDEs

This chapter addresses the Lyapunov-based design of distributed and boundary second order slidingmode (2-SM) controllers in the domain of distributed parameters systems (DPSs). New distributed control results are given and, additionally, an already presented boundary control approach is briefly recalled to enhance the tutorial value of the chapter. Non-standard Lyapunov functionals are invoked to establish the relevant stability and convergence results in appropriate functional spaces. As the novelty, the state tracking problem for an uncertain reaction-diffusion process with spatially varying parameters and non-homogeneous mixed boundary conditions is first addressed. The reference profile is both time and space dependent, and the process is affected by a smooth distributed disturbance. The proposed robust synthesis of the distributed control input is formed by linear PI-type feedback design and the “Super-Twisting” second-order sliding-mode control algorithm, suitably combined and re-worked in the infinite-dimensional setting.

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