SMC without the reaching phase ¿ the switching plane design for the third-order system

A new sliding mode control (SMC) algorithm for the third-order uncertain, nonlinear and time-varying dynamic system subject to acceleration constraint is proposed. The algorithm employs a time-varying switching plane. At the initial time t ¼ t0, the plane passes through the point determined by the system initial conditions in the error state space. Afterwards, the plane moves with a constant velocity to the origin of the state space. In order to select the switching plane parameters, the integral of the absolute error is minimised. Two types of the acceleration constraints are considered. First, a conventional constraint expressed by an inequality is analysed and then an elastic (or stretchable) constraint represented by a penalty function is taken into account. In the second case, we assume that the threshold value of the system acceleration is known and exceeding this value is undesirable, however possible if justified by essential improvement of the system performance. In both cases, the switching plane is chosen in such a way that the reaching phase is eliminated, insensitivity of the system to the external disturbance and the model uncertainty is guaranteed from the very beginning of the proposed control action and fast, monotonic error convergence to zero is achieved.