An O(T2) boundary layer in sliding mode for sampled-data systems

By W. Su, Sergey V. Drakunov, Ü. Özgüner. IEEE Transactions on Automatic Control, 2000

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Chattering-free discrete-time sliding mode control (2016, methodology): “(5) Due to the unavailable information d(k), as that in Su et al. (2000), the disturbance can be estimated by using the so called delay estimate method, i.e., d̂(k) = x(k)− Φx(k − 1)− Γ u(k − 1) = d(k − 1), (6) which yields the final realizable controller u(k) = −(CΓ )−1[CΦx(k)− (1 − qh)s(k) + εh ·…”
Practical Nonsingular Terminal Sliding-Mode Control of Robot Manipulators for High-Accuracy Tracking Control (2009, background): “If boundedness and continuous condition of C ˙ q + G + Fv + τ d is satisfied, Δcon = O(L 2 ) [ 35 ], where big Oi s used to”
On the Discrete-Time Integral Sliding-Mode Control (2007, methodology): “Under practical considerations, the control cannot be implemented in the same form as in (6) because of the lack of prior knowledge regarding the discretized disturbance . However, with some continuity assumptions on the disturbance, can be estimated by its previous value , [ 1 ].”