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 p ^ {\displaystyle {\hat {p}}} denotes the nominal point, P {\displaystyle P} denotes the space of all possible values of the object p {\displaystyle p} , and the shaded area, P ( s ) {\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
ρ ^ ( p ^ ) := max { ρ ≥ 0 : p ∈ P ( s ) , ∀ p ∈ B ( ρ , p ^ ) } {\displaystyle {\hat {\rho }}({\hat {p}}):=\max \ \{\rho \geq 0:p\in P(s),\forall p\in B(\rho ,{\hat {p}})\}}
where B ( ρ , p ^ ) {\displaystyle B(\rho ,{\hat {p}})} denotes a closed ball of radius ρ {\displaystyle \rho } in P {\displaystyle P} centered at p ^ {\displaystyle {\hat {p}}} .
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 model It was shown that the stability radius model is an instance of Wald's maximin model. That is,
max { ρ ≥ 0 : p ∈ P ( s ) , ∀ p ∈ B ( ρ , p ^ ) } ≡ max ρ ≥ 0 min p ∈ B ( ρ , p ^ ) f ( ρ , p ) {\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
f ( ρ , p ) = { ρ , p ∈ P ( s ) − ∞ , p ∉ P ( s ) {\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 max {\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 that its robustness model, namely
α ^ ( q , u ~ ) := max { α ≥ 0 : r c ≤ R ( q , u ) , ∀ u ∈ U ( α , u ~ ) } {\displaystyle {\hat {\alpha }}(q,{\tilde {u}}):=\max \ \{\alpha \geq 0:r_{c}\leq R(q,u),\forall u\in U(\alpha ,{\tilde {u}})\}}
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