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) (e k 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.

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Abstract:In this paper, a graphical stabilization approach is proposed and analyzed for a class of unstable first order linear systems with time delay. We first show that the control designs based on time invariant models are unable to guarantee stability and asymptotic tracking for unstable first order linear systems in general case. So, the condition stability is analysed graphically by computing the first derivative and plotting the graph of a function with precision; the first derivative allows us to determine the critical points and several conditions of stability. Therefore, it's important to note that the method can guarantee the existence of a proportional gain to ensure the stability of the closed-loop system such that the time delay is small relatively to the time constant. Finally, a numerical example illustrates the efficiency and performances of the proposed approach.