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Lyapunov functions for Twisting and Terminal controllers

In this work, homogeneous Lyapunov functions for the Second Order Sliding Mode algorithms “Twisting” and “Terminal” are designed. Such functions are obtained using a systematic and constructive method. The designing method and the Lyapunov functions provide sets of gains that guarantee finite time stability of the system's origin in the disturbed case. It is also proved that the conditions for the controller's gains are even necessary and sufficient.