This paper presents a study on the fragility and performance of two linear controllers for second-order systems: proportional retarded control (PR) and proportional derivative control (PD). The study is carried out through an analysis of the corresponding characteristic equation of the closed-loop system. In the case of the PR control, Taylor’s theorem is used to show that if there exits a root of multiplicity three, it is necessarily dominant.
Let w be a monic polynomial of degree n+1 with roots xj in the interval [−1, 1]. We consider the problem of finding the roots xj for which the minimum of ¦w′(xj)¦, for 0≤j≤n, is as large as possible. We prove that the Clenshaw–Curtis points cos(jπ/n) are the only solution when n is even and that they get asymptotically close to the solution for odd values of n, as n goes to infinity. Our problem is related to the problem of minimizing the norm of inverse Vandermonde matrices.
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