Sliding modes: from asymptoticity, to finite time and fixed time

This paper proposes a new fixed-time sliding mode (FSM) control, where the settling time for reaching the system origin is bounded to a constant independent of the initial condition; this is in contrast to the initial condition-dependent constants used in the traditional linear sliding mode (LSM) and terminal sliding mode (TSM) controls. First, a new sliding mode control with a single power term is discussed, where the power term can have any nonnegative value. Except for the traditional LSM and TSM controls, a new sliding mode control called power sliding mode (PSM) is proposed, whose power term is larger than 1. Then, a new FSM control with two power terms is investigated, whose design is based on the combination of TSM and PSM. In particular, the two power terms on the plane in the first quadrant are carefully discussed, and a detailed classification is provided. Here, the first quadrant can be classified into six categories, including LSM, generalized LSM, TSM, fast TSM (FTSM), PSM, and FSM. Furthermore, the analytical settling time is calculated, and three different estimation bounds of the settling time are given for reaching the origin under any initial condition. It is also interesting to derive the lowest bound for the settling time. Finally, FSM control design for general nonlinear dynamical systems with the relative degree from the control input to the output is also discussed.