Stability radius

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: 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. The formal definition of this concept varies, depending on the application area. The following abstract definition is quite useful where B ( ρ , p ^ ) {displaystyle B( ho ,{hat {p}})} denotes a closed ball of radius ρ {displaystyle ho } in P {displaystyle P} centered at p ^ {displaystyle {hat {p}}} . 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. It was shown that the stability radius model is an instance of Wald's maximin model. That is,