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

In this paper, a Kharitonov-like theorem is proved for testing. robust stability independent of delay of interval quasi polynomials. p(s) + Sigma(m)(k-=1) e(-hk) qk(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 Oyer Info 1988, 5 117-123) (See also (IEEE Dans Car 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. Anton:. 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 in quasipolynomial can be reduced to stability independent of delay of a set of Kharitonov-like vertex quasipolynomials In this impel. we show that the assumption made in (IEEE Trans Anion! 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 Tams Anton; Control 2008, 53 1219-1234) was the equivalence of Hurwitz stability and C(-o)-stability for in 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 dome 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 Thins Anton: 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 (C) 2009 John Wiley & Sons, Ltd