- Stability radius
The

**stability radius**of acontinuous function "f" (in afunctional space "F") with respect to an open stability domain "D" is thedistance 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 function s aspolynomial s (the spectrum is then the roots) and matrices (the spectrum is theeigenvalue s). 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 incontrol 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 Schurstable polynomial s. 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

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Stability**— can refer to: *Aircraft flight Stability (aircraft) *In atmospheric fluid dynamics, atmospheric stability, a measure of the turbulence in the ambient atmosphere *BIBO stability (Bounded Input, Bounded Output stability), in signal processing and… … Wikipedia**Numerical stability**— In the mathematical subfield of numerical analysis, numerical stability is a desirable property of numerical algorithms. The precise definition of stability depends on the context, but it is related to the accuracy of the algorithm. A related… … Wikipedia**Scrub radius**— Scrubradius 0 (top) positive (center) negative(bottom) The scrub radius is the distance in front view between the king pin axis and the center of the contact patch of the wheel, where both would theoretically touch the road. The kingpin axis is… … Wikipedia**Nyquist stability criterion**— The Nyquist plot for . When designing a feedback control system, it is generally necessary to determine whether the closed loop system will be stable. An example of a destabilizing feedback control system would be a car steering system that… … Wikipedia**Distal radius fracture**— Classification and external resources Colles fracture on X ray. ICD 10 S52.5 … Wikipedia**Cation-anion radius ratio**— Critical Radius Ratio In condensed matter physics the cation anion radius ratio is the ratio of the ionic radius of the cation to the ionic radius of the anion in a cation anion compound. This is simply given by rC / rA The allowed size of the… … Wikipedia**BIBO stability**— Bibo redirects here. For the Egyptian football player nicknamed Bibo, see Mahmoud El Khateeb. In electrical engineering, specifically signal processing and control theory, BIBO stability is a form of stability for signals and systems.BIBO stands… … Wikipedia**Marginal stability**— In the theory of dynamical systems, and control theory, a continuous linear time invariant system is marginally stable if and only if the real part of every eigenvalue (or pole) in the system s transfer function is non positive, and all… … Wikipedia**Info-gap decision theory**— is a non probabilistic decision theory that seeks to optimize robustness to failure – or opportuneness for windfall – under severe uncertainty,[1][2] in particular applying sensitivity analysis of the stability radius type[3] to perturbations in… … Wikipedia**List of mathematics articles (S)**— NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… … Wikipedia