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This paper presents a multi-objective optimal PID (Proportional-Integral-Derivative) controller with the derivative filter factor as the fourth design parameter. The complete design of the PID controller should involve tuning four parameters instead of three. However, most of the research papers consider only three parameters. The fourth parameter, the filter factor, is assigned to a default value or selected experimentally. In all cases, the choice of this factor filter will alter the closed-loop response’s characteristics that were assumed before inserting the filter in the control loop. Therefore in this study, we include the filter factor in the decision variable space from the early stage of the control system design. Also, we formulate the design problem as a multi-objective optimization problem in order to show all the trade-offs among the system speed of response, percentage overshoot, sensitivity to external load disturbances, and sensitivity to noises impacting the measurements as the four parameters of the PID control are tuned. The optimal trade-offs solutions are then introduced to the decision-maker who can choose any one of them.

We solve the problem of output feedback stabilization of a class of nonlinear systems, which may have unstable zero dynamics. We allow for any globally stabilizing full state feedback control scheme to be used as long as it satisfies a particular ISS condition. We show semi-global stability of the origin of the closed-loop system and also the recovery of the performance of an auxiliary system using a full-order observer. This observer is based on the use of an extended high-gain observer to provide estimates of the output and its derivatives plus a signal used by an extended Kalman filter to provide estimates of the remaining states. Finally, we provide a simulation example that illustrates the design procedure. stabilization was achieved under similar assumptions but with an extended-high-gain-observer-based output feedback controller.