Kharitonov-like theorems for robust performance of interval systems (Q1874605)

When studying robustness of control systems, it is relevant to derive so-called vertex results: robust stability of an infinite number of systems (e.g. a polytope of systems) is ensured by checking the stability of a finite number of representative systems (e.g. vertices of the polytope). The Kharitonov theorem is the most well-known example of a vertex result: the stability of an interval system (modeled as a polynomial whose coefficients belong to independent intervals) is equivalent to the stability of four vertex polynomials, independently of the degree of the polynomial. The author is interested in deriving vertex results for the \(H_\infty\) norm analysis of interval systems. He shows that the maximum \(H_\infty\) norm of the sensitivity and complementary sensitivity functions (useful when analyzing robustness properties) is achieved at twelve Kharitonov vertices. Unfortunately, the result is very narrow in scope because only unit negative feedback is considered when closing the loop. In other words, if \(g(s)/f(s)\) denotes the open-loop system, then the closed-loop system will be \(f(s)/(f(s)+g(s))\). There are no indications in the paper on whether the result can be extended to more general (non-unit) feedback schemes.