Numerical techniques for the absolute stability problem of high-order systems: A conjecture

The absolute stability problem (ASP) is one of the oldest open problems in the theory of control. Even for the particular case of second-order systems a complete solution was presented only very recently. For third-order systems, the most general results so far were obtained by Barabanov, Pyatnitskiy and Rapoport. They derived an implicit characterization of the “most destabilizing” nonlinearity using the maximum principle. In a recent paper byMargaliot and Yfoulis [8] it has been shown that their approach leads to a simple and efficient numerical bisection scheme for solving the ASP with a single nonlinearity in the case of low-order systems Rⁿ, n ≤ 3, i.e. specifying the critical value where stability is lost in a tractable and accurate fashion.