Tracking Control of the Uncertain Heat and Wave Equation via Power-Fractional and Sliding-Mode Techniques

In the present paper, preliminary results towards the generalization to the infinitedimensional setting of some well-known robust finite-dimensional control algorithms are illustrated. More specifically, we deal with the tracking problem for some classes of linear uncertain infinitedimensional systems evolving in Hilbert spaces. We design distributed variable-structure stabilizers that are shown to be effective in the presence of external disturbances. The main focus of the present paper is on the rejection of nonvanishing external disturbances. The generalization to the infinitedimensional setting of the well-known finite-dimensional controllers, namely the “power-fractional” controller [S. P. Bhat and D. S. Bernstein, SIAM J. Control Optim., 38 (2000), pp. 751–766] and two “second-order sliding-mode” control algorithms (the “twisting” and “supertwisting” algorithms [L. Fridman and A. Levant, Higher order sliding modes as a natural phenomenon in control theory, in Robust Control via Variable Structure and Lyapunov Techniques, Springer-Verlag, Berlin, 1996, pp. 107–133; A. Levant, Internat. J. Control, 58 (1993), pp. 1247–1263]), is the main contribution of the present investigation. First, the “distributed twisting” control algorithm is developed to address the asymptotic state tracking of the perturbed wave equation. Next, the finite-time state tracking of the unperturbed heat equation is provided by means of a “distributed power-fractional” controller. Finally, the “distributed supertwisting” controller is suggested to address the asymptotic state tracking of the heat equation in spite of the presence of persistent disturbances. Constructive proofs of stability are developed via the Lyapunov functional technique, which leads to simple tuning rules for the controller parameters. Simulation results are discussed to verify the effectiveness of the proposed schemes.