Stability radius

The stability radius of a continuous function "f" (in a functional space "F") with respect to an open stability domain "D" is the distance between "f" and the set of unstable functions (with respect to "D"). We say that a function is "stable" with respect to "D" if its spectrum is in "D". Here, the notion of spectrum is defined on a case by case basis, as explained below.

Definition

Formally, if we denote the set of stable functions by "S(D)" and the stability radius by "r(f,D)", then::$r(f,D)=inf\_\{gin\; C\}\{|g|:f+g\; otin\; S(D)\},$where "C" is a subset of "F".

Note that if "f" is already unstable (with respect to "D"), then "r(f,D)=0" (as long as "C" contains zero).

Applications

The notion of stability radius is generally applied to special functions as polynomials (the spectrum is then the roots) and matrices (the spectrum is the eigenvalues). The case where "C" is a proper subset of "F" permits us to consider structured perturbations (e.g. for a matrix, we could only need perturbations on the last row). It is an interesting measure of robustness, for example in control theory.

Properties

Let "f" be a (complex) polynomial of degree "n", "C=F" be the set of polynomials of degree less than (or equal to) "n" (which we identify here with the set $mathbb\{C\}^\{n+1\}$ of coefficients). We take for "D" the open unit disk, which means we are looking for the distance between a polynomial and the set of Schur stable polynomials. Then::$r(f,D)=inf\_\{zin\; partial\; D\}frac\{|q(z),$where "q" contains each basis vector (e.g. $q(z)=(1,z,ldots,z^n)$ when "q" is the usual power basis). This result means that the stability radius is bound with the minimal value that "f" reaches on the unit circle.

Examples

* the polynomial $f(z)=z^8-9/10$ (whose zeros are the 8th-roots of "0.9") has a stability radius of 1/80 if "q" is the power basis and the norm is the infinity norm. So there must exist a polynomial "g" with (infinity) norm 1/90 such that "f+g" has (at least) a root on the unit circle. Such a "g" is for example $g(z)=-1/90sum\_\{i=0\}^8\; z^i$. Indeed "(f+g)(1)=0" and "1" is on the unit circle, which means that "f+g" is unstable.

ee also

* stable polynomial