Performance, Fragility and Robustness for a Class of Quasi-Polynomials of Degree Two

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, while maintaining inherent insensitivity to high-frequency noise. Therefore, it is important to continue improving the tuning of these controllers, including properties such as performance, fragility and robustness that may be beneficial for this purpose. However, currently most studies prioritize tuning using only the performance property, some others only the fragility property, and some less only the robustness property. This work provides the first rigorous joint analysis of performance, fragility, and robustness for a class of systems whose characteristic equation is a quasi-polynomial of degree two, filling a gap in the current literature. Thus, necessary and sufficient conditions are proposed to improve the tuning of delayed-action controllers by ensuring a exponential decay rate on the convergence of the closed-loop system response (performance) and by ensuring stabilization and/or trajectory tracking in the face of changes in system parameters (robustness) and controllers gains (fragility). To illustrate and corroborate the effectiveness of the proposed theoretical results, a real-time implementation is presented on a mobile prototype consisting of an omnidirectional mobile robot, to streamline/guarantee trajectory tracking in response to variations in controller gains and robot parameters. This implementation and application of theoretical results are possible thanks to the proposal of a novel delayed nonlinear controller and some simple but strategic algebraic manipulations that reduce the original problem to the study of a quasi-polynomial of degree 9 with three commensurable delays. Finally, our results are compared with a classical proportional nonlinear controller showing that our proposal is relevant.