Adaptive Sliding‐Mode Control of a Perturbed Diffusion Process With Pointwise In‐Domain Actuation

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Abstract
A sliding mode–based adaptive control law is proposed for a class of diffusion processes featuring a spatially‐varying uncertain diffusivity and equipped with several point‐wise actuators located at the two boundaries of the spatial domain as well as in its interior. The system is additionally perturbed by matched disturbances which are assumed to be uniformly bounded along with their time derivatives. The corresponding constant bounds are unknown , thus motivating the use of an adaptive control strategy. To achieve global asymptotic stability of the origin in the ‐sense, a control law consisting of a proportional and discontinuous term is proposed. The gain of the discontinuous term is continuously adjusted according to a gradient‐based adaptation algorithm. The stability and convergence analysis is Lyapunov‐based and it constructively yields simple tuning conditions for the controller gain parameters. Simulation results are finally discussed to support the theoretical findings.

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