Minkowski Geometric Algebra and the Stability of Characteristic Polynomials

A polynomial p is said to be Γ-stable if all its roots lie within a given domain Γ in the complex plane. The Γ-stability of an entire family of polynomials, defined by selecting the coefficients of p from specified complex sets, can be verified by (i) testing the Γ-stability of a single member, and (ii) checking that the “total value set” V * for p along the domain boundary ∂Γ does not contain 0 (V * is defined as the set of all values of p for each point on ∂Γ and every possible choice of the coefficients). The methods of Minkowski geometric algebra —the algebra of point sets in the complex plane — offer a natural language for the stability analysis of families of complex polynomials. These methods are introduced, and applied to analyzing the stability of disk polynomials with coefficients selected from circular disks in the complex plane. In this context, V * may be characterized as the union of a one-parameter family of disks, and we show that the Γ-stability of a disk polynomial can be verified by a finite algorithm (a counterpart to the Kharitonov conditions for rectangular coefficient sets) that entails checking that at most two real polynomials remain positive for all t, when the domain boundary ∂Γ is a given polynomial curve γ(t).Furthermore, the “robustness margin” can be determined by computing the real roots of a real polynomial.