Robust Stability of Hurwitz Polynomials Associated with Modified Classical Weights

Abstract

In this contribution, we consider sequences of orthogonal polynomials associated with a perturbation of some classical weights consisting of the introduction of a parameter t, and deduce some algebraic properties related to their zeros, such as their equations of motion with respect to t. These sequences are later used to explicitly construct families of polynomials that are stable for all values of t, i.e., robust stability on these families is guaranteed. Some illustrative examples are presented.

On Robust Stability for Hurwitz Polynomials via Recurrence Relations and Linear Combinations of Orthogonal Polynomials

“…As a consequence of the results presented here, together with those of the recent contribution [29], it is possible to use orthogonal polynomials to construct sequences of robustly stable Hurwitz polynomials in three different ways: by introducing an uncertain parameter on the orthogonality measure, by introducing uncertain parameters on the threeterm recurrence relation, and by using linear combinations of orthogonal polynomials. e results of this work can be extended in the following directions:…”

“…Observe that the previous theorem implies that we can construct a sequence of Hurwitz polynomials by using sequences P n 􏼈 􏼉 n≥0 and Q n 􏼈 􏼉 n≥0 , when the orthogonality measure is supported on R + ; i.e., the zeros of each P n are simple and positive. is property was used in [29] to construct robustly stable sequences of Hurwitz polynomials by using the Laguerre and Jacobi classical orthogonality measures supported on R + (see [25]). e main idea was to introduce a parameter t on the measure, in such a way that it remains positive.…”

An Application of Rouché’s Theorem to Delimit the Zeros of a Certain Class of Robustly Stable Polynomials

“…Indeed, the location of the roots is directly related to the performance of the system. It is important to notice that some families of robustly stable polynomials (defined in terms of orthogonal polynomials) have been proposed in the literature [17][18][19], and the behavior of the roots in terms of the uncertain parameter has been studied [19].…”

“…We point out that a similar procedure was used to construct robustly stable families of Hurwitz polynomials by using orthogonal polynomials on the real line [17,18], because there is a close connection between both theories [23,24]. General information about orthogonal polynomials on the real line can be found in [22,25].…”

New Stability Criteria for Discrete Linear Systems Based on Orthogonal Polynomials

“…In the latter, the authors construct sequences of Hurwitz polynomials from a sequence of orthogonal polynomials, and show several algebraic properties of the constructed family. Also, classical orthogonal polynomials are used in [35] to construct families of Hurwitz polynomials that are robustly stable.…”