In mathematics, 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 of whose elements satisfy pre-determined stability conditions. The picture of this intuitive notion is this:

where ${\displaystyle {\hat {p}}}$ denotes the nominal point, ${\displaystyle P}$ denotes the space of all possible values of the object ${\displaystyle p}$, and the shaded area, ${\displaystyle P(s)}$, represents the set of points that satisfy the stability conditions. The radius of the blue circle, shown in red, is the stability radius.

Abstract definition

The formal definition of this concept varies, depending on the application area. The following abstract definition is quite useful^[1][2]

${\displaystyle {\hat {\rho }}({\hat {p}}):=\max \ \{\rho \geq 0:p\in P(s),\forall p\in B(\rho ,{\hat {p}})\}}$

where ${\displaystyle B(\rho ,{\hat {p}})}$ denotes a closed ball of radius ${\displaystyle \rho }$ in ${\displaystyle P}$ centered at ${\displaystyle {\hat {p}}}$.

History

It looks like the concept was invented in the early 1960s.^[3][4] In the 1980s it became popular in control theory^[5] and optimization.^[6] 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 model

It was shown^[2] that the stability radius model is an instance of Wald's maximin model. That is,

${\displaystyle \max \ \{\rho \geq 0:p\in P(s),\forall p\in B(\rho ,{\hat {p}})\}\equiv \max _{\rho \geq 0}\min _{p\in B(\rho ,{\hat {p}})}f(\rho ,p)}$

where

${\displaystyle f(\rho ,p)=\left\{{\begin{array}{cc}\rho &,\ p\in P(s)\\-\infty &,\ p\notin P(s)\end{array}}\right.}$

The large penalty (${\displaystyle -\infty }$) is a device to force the ${\displaystyle \max }$ 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 theory 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^[2] that its robustness model, namely

${\displaystyle {\hat {\alpha }}(q,{\tilde {u}}):=\max \ \{\alpha \geq 0:r_{c}\leq R(q,u),\forall u\in U(\alpha ,{\tilde {u}})\}}$

is actually a stability radius model characterized by a simple stability requirement of the form ${\displaystyle r_{c}\leq R(q,u)}$ where ${\displaystyle q}$ denotes the decision under consideration, ${\displaystyle u}$ denotes the parameter of interest, ${\displaystyle {\tilde {u}}}$ denotes the estimate of the true value of ${\displaystyle u}$ and ${\displaystyle U(\alpha ,{\tilde {u}})}$ denotes a ball of radius ${\displaystyle \alpha }$ centered at ${\displaystyle {\tilde {u}}}$.

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 ${\displaystyle {\tilde {u}}}$.

Sniedovich^[2] 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.

Alternate definition

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.^[7] The picture is this:

More formally,

${\displaystyle {\hat {\rho }}(q):=\min _{p\notin P(s)}dist(p,{\hat {p}})}$

where ${\displaystyle dist(p,{\hat {p}})}$ denotes the distance of ${\displaystyle p\in P}$ from ${\displaystyle {\hat {p}}}$.

Stability radius of functions

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.

See also

• stable polynomial
• Wald's maximin model

References