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This paper pursues the construction of a valid strong Lyapunov function for second-order sliding mode algorithms, usually employed for control, observation, and differentiation purposes. The developed solution is based on combining convex representations of such algorithms with the direct Lyapunov method, which leads to stability conditions expressed in the form of linear matrix inequalities. The proposed methodology enlarges the set of gains for which the algorithms are guaranteed stable in prior literature.

In this paper, a control strategy is proposed for robust stabilization of nonlinear perturbed plants in the presence of actuator unmodeled dynamics. The block control technique is applied to design a nonlinear SM manifold for achieving the error tracking, and High Order Sliding Mode (HOSM) algorithm is implemented to ensure finite time convergence of the state vector to the designed SM manifold. A robust exact differentiator is designed to obtain the estimates of the sliding variable and its derivatives. The proposed method is applied to design robust controllers for a power electric system in the presence of the exciter system unmodeled dynamics.

The problem of angular velocity of rotor and voltage control in synchronous generator machines is considered. As a robust solution of this problem the block control technique with sliding mode is proposed. The physical implementations of the proposed control law for the real synchronous machines results in chattering due to the presence of exciter unmodelled dynamics. In order to reduce the chattering effect we design high order sliding mode controllers and exact sliding mode differentiators, ensuring the robustness of the closed-loop system.