Edge Theorem for Multivariable Systems

Motivated by the seminal theorem of Kharitonov on robust stability of interval polynomials[1, 2], a number of papers on robustness analysis of uncertain systems have been published in the past few years[3, 4, 5, 6, 7, 8, 9, 10]. Kharitonov’s theorem states that the Hurwitz stability of the real (or complex) interval polynomial family can be guaranteed by the Hurwitz stability of four (or eight) prescribed critical vertex polynomials in this family. This result is significant since it reduces checking stability of infinitely many polynomials to checking stability of finitely many polynomials, and the number of critical vertex polynomials need to be checked is independent of the order of the polynomial family. An important extension of Kharitonov’s theorem is the edge theorem discovered by Bartlett, Hollot and Huang[4]. The edge theorem states that the stability of a polytope of polynomials can be guaranteed by the stability of its one-dimensional exposed edge polynomials. The significance of the edge theorem is that it allows some (affine) dependency among polynomial coefficients, and applies to more general stability regions, e.g., unit circle, left sector, shifted half plane, hyperbola region, etc. When the dependency among polynomial coefficients is nonlinear, however, Ackermann shows that checking a subset of a polynomial family generally can not guarantee the stability of the entire family[11, 12]. For Hurwitz stability of interval matrices, Bialas ’proved’ that in order to guarantee robust stability, it suffices to check all vertex matrices[13]. Later, it was shown by Barmish that Bialas’ result was incorrect[14]. Kokame and Mori eastblished a Kharitonov-like result on robust Hurwitz stability of interval polynomial matrices[15], and Kamal and Dahleh established some robust stability criteria for MIMO systems with fixed controllers and uncertain plants[16]. In this paper, we will study robustness of a class of MIMO systems with their transfer function matrices described by