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.
In the second part of the chapter, recently achieved results  concerning the regulation of an uncertain heat process with collocated boundary sensing and actuation are recalled. The underlying heat process is governed by an uncertain parabolic partial differential equation (PDE) with controlled mixed boundary conditions, it exhibits an unknown spatially varying diffusivity parameter, and it is affected by a smooth boundary disturbance. The proposed robust synthesis is formed by combining linear PD-type feedback design and the “Twisting” second-order sliding-mode control algorithm. The stability of the twisting-based boundary controller can be investigated by means of a totally different family of Lyapunov functionals as compared to that used for the super-twisting-based distributed control scheme, and the approach is recalled in order to give a more complete overview of the available second-order sliding mode designs for PDEs. The presented control schemes are supported by simulation results.
KeywordsDistributed parameter systems Parabolic PDEs Heat equation Reaction-diffusion equation Boundary control Second-order sliding mode control Lyapunov analysis
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