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Guillermo Oaxaca

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In recent years the use of delayed controllers has increased considerably, since they can attenuate noise, replace derivative actions, avoid the construction of observers, and reduce the use of extra sensors. Also, delayed controllers have been shown to be more insensitive to high-frequency noise. However, currently, there are few criteria for tuning this type of controllers. This manuscript presents a rigorous study of performance, fragility, and robustness for a first-order system in closed-loop with a delayed controller, which leads to consider a quasi-polynomial q( a, k, s), where a ∈ R m denotes the system parameters, k ∈ R n are the controller gains, and s∈C. The best performance gains k ∗ , p are obtained for a fixed a. These gains provide the maximum exponential decay achievable in the system response to guarantee exponential convergence to a desired trajectory. Also, for a fixed a, criteria are given to obtain the least fragile gains k ∗ , f that ensure the desired trajectory tracking in the presence of controller’s gains variations. Meanwhile, for a fixed gains k, the greatest robustness parameters a ∗ , r are obtained. Thus, the desired trajectory tracking of the systems is ensured in the event of parametric variations. Finally, to illustrate and corroborate the proposed theoretical results, a real-time implementation is presented on a mobile prototype, known as omnidirectional mobile robot, studying a quasi-polynomial of degree 9 with three commensurable delays. These results offer convincing reasons to implement controllers with delayed-action and take into account an analysis of performance, fragility, and robustness to tune this type of controllers.