ITAE Optimal Sliding Modes for Third-Order Systems With Input Signal and State Constraints

In this note, the design of a time-varying switching plane for the sliding-mode control of the third-order system subject to velocity, acceleration and input signal constraints is considered. Initially, the switching plane passes through the system representative point (RP) in the error state space and then it moves with a constant velocity to the origin of the space. Having reached the origin the plane stops moving and remains fixed. The plane parameters are selected to minimize the integral of the time multiplied by the absolute error (ITAE) without violating velocity, acceleration and input signal constraints. Furthermore, the switching plane is chosen in such a way that the reaching phase is eliminated, insensitivity of the system with reference to the external disturbances and the model uncertainty is guaranteed from the very beginning of the control action and monotonic tracking error convergence to zero is ensured.