Stability radius
Encyclopedia
The stability radius of an object (system, function, matrix, parameter) at a given nominal point is the radius of the largest ball
, centered at the nominal point, all whose elements satisfy pre-determined stability conditions. The picture of this intuitive notion is this:
where
denotes the nominal point, 
denotes the space of all possible values of the object 
, and the shaded area, 
, represents the set of points that satisfy the stability conditions.
where
denotes a closed ball
of radius
in 
centered at 
.
Relation to Wald's maximin model
It was shown that the stability radius model is an instance of Wald's maximin model
. That is,
where
The large penalty (
) is a device to force the 
player not to perturb the nominal value beyond the stability radius of the system. It is an indication that the stability model is a model of local stability/robustness, rather than a global one.
is a recent non-probabilistic decision theory. It is claimed to be radically different from all current theories of decision under uncertainty. But it has been shown that its robustness model, namely
is actually a stability radius model characterized by a simple stability requirement of the form
where 
denotes the decision under consideration, 
denotes the parameter of interest, 
denotes the estimate of the true value of 
and 
denotes a ball of radius 
centered at 
.
Since stability radius models are designed to deal with small perturbations in the nominal value of a parameter, info-gap's robustness model measures the local robustness of decisions in the neighborhood of the estimate
.
Sniedovich argues that for this reason the theory is unsuitable for the treatment of severe uncertainty characterized by a poor estimate and a vast uncertainty space.
More formally,
where
denotes the distance of 
from 
.
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.
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).
s (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
.
) 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
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 polynomial
s. Then:
where q contains each basis vector (e.g.
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.
Ball (mathematics)
In mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....
, centered at the nominal point, all whose elements satisfy pre-determined stability conditions. The picture of this intuitive notion is this:
where
denotes the nominal point, denotes the space of all possible values of the object , and the shaded area, , represents the set of points that satisfy the stability conditions.Abstract definition
The formal definition of this concept varies, depending on the application area. The following abstract definition is quite usefulwhere
denotes a closed ballBall (mathematics)
In mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....
of radius
in centered at .History
It looks like the concept was invented in the early 1960s. In the 1980s it became popular in control theory and optimization. It is widely used as a model of local robustness against small perturbations in a given nominal value of the object of interest. Relation to Wald's maximin modelWald's maximin modelIn decision theory and game theory, Wald's maximin model is a non-probabilistic decision-making model according to which decisions are ranked on the basis of their worst-case outcomes. That is, the best decision is one whose worst outcome is at least as good as the worst outcome of any other...
It was shown that the stability radius model is an instance of Wald's maximin modelWald's maximin model
In decision theory and game theory, Wald's maximin model is a non-probabilistic decision-making model according to which decisions are ranked on the basis of their worst-case outcomes. That is, the best decision is one whose worst outcome is at least as good as the worst outcome of any other...
. That is,
where
The large penalty (
) is a device to force the player not to perturb the nominal value beyond the stability radius of the system. It is an indication that the stability model is a model of local stability/robustness, rather than a global one.Info-gap decision theory
Info-gap decision theoryInfo-gap decision theory
Info-gap decision theory is a non-probabilistic decision theory that seeks to optimize robustness to failure – or opportuneness for windfall – under severe uncertainty, in particular applying sensitivity analysis of the stability radius type to perturbations in the value of a given estimate of the...
is a recent non-probabilistic decision theory. It is claimed to be radically different from all current theories of decision under uncertainty. But it has been shown that its robustness model, namely
is actually a stability radius model characterized by a simple stability requirement of the form
where denotes the decision under consideration, denotes the parameter of interest, denotes the estimate of the true value of and denotes a ball of radius centered at .Since stability radius models are designed to deal with small perturbations in the nominal value of a parameter, info-gap's robustness model measures the local robustness of decisions in the neighborhood of the estimate
.Sniedovich argues that for this reason the theory is unsuitable for the treatment of severe uncertainty characterized by a poor estimate and a vast uncertainty space.
Variations on a theme
There are cases where it is more convenient to define the stability radius slightly different. For example, in many applications in control theory the radius of stability is defined as the size of the smallest destabilizing perturbation in the nominal value of the parameter of interest. The picture is this:More formally,
where
denotes the distance of from .Stability radius of functions
The stability radius of a continuous functionContinuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
f (in a functional space F) with respect to an open
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
stability domain D is the distance
Distance
Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical 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: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 polynomialPolynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s (the spectrum is then the roots) and matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
(the spectrum is the eigenvalues). The case where C is a proper subset of F permits us to consider structured perturbations
Perturbation theory
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
(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
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
.
Properties
Let f be a (complexComplex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
) 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
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 polynomialStable polynomial
A polynomial is said to be stable if either:* all its roots lie in the open left half-plane, or* all its roots lie in the open unit disk.The first condition defines Hurwitz stability and the second one Schur stability. Stable polynomials arise in various mathematical fields, for example in...
s. Then:
where q contains each basis vector (e.g.
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
(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 
. Indeed (f+g)(1)=0 and 1 is on the unit circle, which means that f+g is unstable.