Stability radius

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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.

1 Definition
2 Applications
3 Properties
4 Examples
5 See also

[edit] Definition

Formally, if we denote the set of stable functions by S(D) and the stability radius by r(f,D), then:

$r(f,D)=\inf_{g\in C}\{\|g\|:f+g\notin S(D)\},$

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).

[edit] 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.

[edit] 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:

$r(f,D)=\inf_{z\in \partial D}\frac{|f(z)|}{\|q(z)\|},$

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.

[edit] 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 $g(z)=-1/90\sum_{i=0}^8 z^i$ . Indeed (f+g)(1)=0 and 1 is on the unit circle, which means that f+g is unstable.