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

Abstract

In this contribution, we use the connection between stable polynomials and orthogonal polynomials on the real line to construct sequences of Hurwitz polynomials that are robustly stable in terms of several uncertain parameters. These sequences are constructed by using properties of orthogonal polynomials, such as the well-known three-term recurrence relation, as well as by considering linear combinations of two orthogonal polynomials with consecutive degree. Some examples are presented.

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].…”

“…As in the Schur case, we need to use a known family of robustly stable Hurwitz polynomials. As a consequence, we use some robustly stable polynomials considered in [18], constructed by using recurrence relations satisfied by some sequences of orthogonal polynomials on the real line.…”

“…Example 4. Let us consider the Hurwitz robustly stable polynomial f (x, λ) = 170λ 6 + (119λ 5 + 80λ 4 )x + (10λ 2 + 56λ 3 + 17λ 4 )x 2 + (7λ + 8λ 2 )x 3 + x 4 , constructed as shown in [18]. Suppose that we want to know the values of λ for which the zeros of the polynomial f have real part in the interval (−3, −3/7).…”

“…and choosing R = min{R i , R s }, we can proceed to use ( 16) and obtain conditions in λ so that the real parts of the zeros of f (x, λ) satisfy (18). However, given that R must satisfy 0 < R < 1, it is clear that in order to apply this method we require a i ∈ (−∞, −1) and a s ∈ (−1, 0).…”