A Kharitonov-like theorem for robust stability independent of delay of interval quasipolynomials

Abstract

In this paper, a Kharitonov-like theorem is proved for testing robust stability independent of delay of interval quasipolynomials, p(s)+∑eqk(s), where p and qk's are interval polynomials with uncertain coefficients. It is shown that the robust stability test of the quasipolynomial basically reduces to the stability test of a set of Kharitonov-like vertex quasipolynomials, where stability is interpreted as stability independent of delay. As discovered in (IEEE Trans. Autom. Control 2008; 53:1219–1234), the well-known vertex-type robust stability result reported in (IMA J. Math. Contr. Info. 1988; 5:117–123) (See also (IEEE Trans. Circ. Syst. 1990; 37(7):969–972; Proc. 34th IEEE Conf. Decision Contr., New Orleans, LA, December 1995; 392–394) does contain a flaw. An alternative approach is proposed in (IEEE Trans. Autom. Control 2008; 53:1219–1234), and both frequency sweeping and vertex type robust stability tests are developed for quasipolynomials with polytopic coefficient uncertainties. Under a specific assumption, it is shown in (IEEE Trans. Autom. Control 2008; 53:1219–1234) that robust stability independent of delay of an interval quasipolynomial can be reduced to stability independent of delay of a set of Kharitonov-like vertex quasipolynomials. In this paper, we show that the assumption made in (IEEE Trans. Autom. Control 2008; 53:1219–1234) is redundant, and the Kharitonov-like result reported in (IEEE Trans. Autom. Control 2008; 53:1219–1234) is true without any additional assumption, and can be applied to all quasipolynomials. The key idea used in (IEEE Trans. Autom. Control 2008; 53:1219–1234) was the equivalence of Hurwitz stability and ℂ-o-stability for interval polynomials with constant term never equal to zero. This simple observation implies that the well-known Kharitonov theorem for Hurwitz stability can be applied for ℂ-o-stability, provided that the constant term of the interval polynomial never vanishes. However, this line of approach is based on a specific assumption, which we call the CNF-assumption. In this paper, we follow a different approach: First, robust ℂ-o-stability problem is studied in a more general framework, including the cases where degree drop is allowed, and the constant term as well as other higher-orders terms can vanish. Then, generalized Kharitonov-like theorems are proved for ℂ-o-stability, and inspired by the techniques used in (IEEE Trans. Autom. Control 2008; 53:1219–1234), it is shown that robust stability independent of delay of an interval quasipolynomial can be reduced to stability independent of delay of a set of Kharitonov-like vertex quasipolynomials, even if the assumption adopted in (IEEE Trans. Autom. Control 2008; 53:1219–1234) is not satisfied. Copyright © 2009 John Wiley & Sons, Ltd.