New algorithms of adaptive switching gain for sliding mode control: Part I - Ideal case

Based on recent results on adaptive sliding mode control (ASMC) design for nonlinear systems with uncertainties, we propose some lemmas and theorems to discuss finite time convergence (FTC) and alternative approaches to smoothen the adaptation tuning algorithm of the ASMC design. These modifications are proposed to enhance accuracy without overestimation of the uncertainty magnitude and to suppress the chattering phenomenon. In fact, the new adaptation laws are designed in ways to assign a minimum admissible value to the switching gain. The robustness is proven using the Lyapunov stability criterion combined with an intuitive analysis of the control behavior. Simulation results are performed to demonstrate the effectiveness of the proposed algorithm. Part I introduces the new designs for ideal ASMC while the real case design will be shown in Part II.