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.

The robustness of a linear system in the view of parametric variations requires a stability analysis of a family of polynomials. If the parameters vary in a compact set A, then obtaining necessary and sufficient conditions to determine stability on the family F A is one of the most important tasks in the field of robust control. Two interesting classes of families arise when A is a diamond or a box of dimension n+1. These families will be denoted by F D n and F B n , respectively. In this paper a study is presented to contribute to the understanding of Hurwitz stability of families of polynomials F A . As a result of this study and the use of classical results found in the literature, it is shown the existence of an extremal polynomial f ( α ∗ , x ) whose stability determines the stability of the entire family F A . In this case f ( α ∗ , x ) comes from minimizing determinants and sometimes f ( α ∗ , x ) coincides with a Kharitonov’s polynomial. Thus another extremal property of Kharitonov’s polynomials has been found. To illustrate the versatility/generality of our approach, this is addressed to families such as F D n and F B n , when n≤5. Furthermore, the study is also used to obtain the maximum robustness of the parameters of a polynomial. To exemplify the proposed results, first, a family F D n is taken from the literature to compare and corroborate the effectiveness and the advantage of our perspective. Followed by two examples where the maximum robustness of the parameters of polynomials of degree 3 and 4 are obtained. Lastly, a family F B 5 is proposed whose extreme polynomial is not necessarily a Kharitonov’s polynomial.