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| The stability radius of an object (system, function, matrix, parameter) at a given nominal point is the radius of the largest [[ball (mathematics)|ball]], centered at the nominal point, all of whose elements satisfy pre-determined stability conditions. The picture of this intuitive notion is this:
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| [[Image:Radius of stability 1.png|500px]]
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| where <math>\hat{p}</math> denotes the nominal point, <math>P</math> denotes the space of all possible values of the object <math>p</math>, and the shaded area, <math>P(s)</math>, represents the set of points that satisfy the stability conditions.
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| == Abstract definition ==
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| The formal definition of this concept varies, depending on the application area. The following abstract definition is quite useful<ref name="zlobec09">Zlobec S. (2009). Nondifferentiable optimization: Parametric programming. Pp. 2607-2615, in ''Encyclopedia of Optimization,'' Floudas C.A and Pardalos, P.M. editors, Springer.</ref><ref name="MS10">Sniedovich, M. (2010). A bird's view of info-gap decision theory. ''Journal of Risk Finance,'' 11(3), 268-283.</ref>
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| :<math>\hat{\rho}(\hat{p}):= \max \ \{\rho\ge 0: p\in P(s), \forall p\in B(\rho,\hat{p})\}</math>
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| where <math>B(\rho,\hat{p})</math> denotes a closed [[ball (mathematics)|ball]] of radius <math>\rho</math> in <math>P</math> centered at <math>\hat{p}</math>. | |
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| == History ==
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| It looks like the concept was invented in the early 1960s.<ref name="wilf">Wilf, H.S. (1960). Maximally stable numerical integration. ''Journal of the Society for Industrial and Applied Mathematics,'' 8(3),537-540.</ref><ref name="milne">Milne, W.E., and Reynolds, R.R. (1962). Fifth-order methods for the numerical solution of ordinary differential equations. ''Journal of the ACM,'' 9(1), 64-70.</ref> In the 1980s it became popular in control theory<ref name="Hindrichsen86">Hindrichsen, D. and Pritchard, A.J. (1986). Stability radii of linear systems, ''Systems and Control Letters,'' 7, 1-10.</ref> and optimization.<ref name="zlobec88">Zlobec S. (1988). Characterizing Optimality in Mathematical Programming Models. ''Acta Applicandae Mathematicae,'' 12, 113-180.</ref> It is widely used as a model of local robustness against small perturbations in a given nominal value of the object of interest.
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| == Relation to Wald's maximin model ==
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| It was shown<ref name="MS10" /> that the stability radius model is an instance of [[Wald's maximin model]]. That is,
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| :<math>\max \ \{\rho\ge 0: p\in P(s), \forall p\in B(\rho,\hat{p})\} \equiv
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| \max_{\rho\ge 0}\min_{p\in B(\rho,\hat{p})} f(\rho,p)</math>
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| where
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| :<math>f(\rho,p) = \left\{\begin{array}{cc}\rho &, \ p\in P(s) \\ -\infty &,\ p\notin P(s)\end{array}\right.</math>
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| The large penalty (<math>-\infty</math>) is a device to force the <math>\max</math> 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. | |
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| == Info-gap decision theory ==
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| [[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<ref name="MS10" /> that its robustness model, namely
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| :<math>\hat{\alpha}(q,\tilde{u}):= \max\ \{\alpha\ge 0: r_{c} \le R(q,u),\forall u\in U(\alpha,\tilde{u})\}</math>
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| is actually a stability radius model characterized by a simple stability requirement of the form <math>r_{c}\le R(q,u)</math> where <math>q</math> denotes the decision under consideration, <math>u</math> denotes the parameter of interest, <math>\tilde{u}</math> denotes the estimate of the true value of <math>u</math> and <math>U(\alpha,\tilde{u})</math> denotes a ball of radius <math>\alpha</math> centered at <math>\tilde{u}</math>.
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| [[Image:Infogap robustness.png|500px]]
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| 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 <math>\tilde{u}</math>.
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| Sniedovich<ref name="MS10"/> 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.
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| == Variations on a theme ==
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| 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.<ref name="paice98">Paice A.D.B. and Wirth, F.R. (1998). Analysis of the Local Robustness of Stability for Flows. ''[[Mathematics of Control, Signals, and Systems]]'', 11, 289-302.</ref> The picture is this:
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| [[Image:Radius of stability 3.png|500px]]
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| More formally,
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| : <math>\hat{\rho}(q):= \min_{p\notin P(s)} dist(p,\hat{p})</math>
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| where <math>dist(p,\hat{p})</math> denotes the ''distance'' of <math>p\in P</math> from <math>\hat{p}</math>.
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| == Stability radius of functions ==
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| The '''stability radius''' of a [[continuous function]] ''f'' (in a [[functional space]] ''F'') with respect to an [[open set|open]] stability domain ''D'' is the [[distance]] between ''f'' and the [[Set (mathematics)|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.
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| === Definition ===
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| Formally, if we denote the set of stable functions by ''S(D)'' and the stability radius by ''r(f,D)'', then:
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| :<math>r(f,D)=\inf_{g\in C}\{\|g\|:f+g\notin S(D)\},</math>
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| where ''C'' is a subset of ''F''.
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| Note that if ''f'' is already unstable (with respect to ''D''), then ''r(f,D)=0'' (as long as ''C'' contains zero).
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| === Applications ===
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| The notion of stability radius is generally applied to [[special function]]s as [[polynomial]]s (the spectrum is then the roots) and [[matrix (mathematics)|matrices]] (the spectrum is the [[eigenvalue]]s). The case where ''C'' is a proper subset of ''F'' permits us to consider structured [[perturbation theory|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]].
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| === Properties===
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| Let ''f'' be a ([[complex number|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 <math>\mathbb{C}^{n+1}</math> 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:
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| :<math>r(f,D)=\inf_{z\in \partial D}\frac{|f(z)|}{\|q(z)\|},</math>
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| where ''q'' contains each basis vector (e.g. <math>q(z)=(1,z,\ldots,z^n)</math> 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.
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| === Examples ===
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| * the polynomial <math>f(z)=z^8-9/10</math> (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 <math>g(z)=-1/90\sum_{i=0}^8 z^i</math>. Indeed ''(f+g)(1)=0'' and ''1'' is on the unit circle, which means that ''f+g'' is unstable.
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| ==See also==
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| * [[stable polynomial]]
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| * [[Wald's maximin model]]
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| == References ==
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| <references />
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| {{DEFAULTSORT:Stability Radius}}
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| [[Category:Polynomials]]
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