Design of nonlinear PID controllers based on hyper-stability criteria

In this paper, an approach to design a nonlinear Proportional-Integral-Derivative (PID) controller based on hyper-stability criteria is proposed. This approach improved the performances of conventional linear fixed-gain PID controller considerably, by incorporating a nonlinear gain in cascade with a conventional PID control structure. The controller parameters are designed under assumptions of a sector bounded nonlinear gain, to a proper choice that satisfies the inequality of Popov. Specifically, a control law is constructed by cascading an exponential base B function with a linear fixed-gain PID controller. This can be achieved by the proposed particular nonlinear gain k(e) as the hyperbolic cosine function obtained by a simple combination of particular solutions. Therefore, it's important to note that the method can guarantee an asymptotically stable error for a bounded non-linear gain. Finally, a numerical example is presented to show the applicability and performances of the proposed approach.