Absolute stability of third-order systems: A numerical algorithm

The problem of absolute stability 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 theoretical results were obtained by Barabanov. He derived an implicit characterization of the “most destabilizing” nonlinearity using a variational approach. In this paper, we show that his approach yields a simple and efficient numerical scheme for solving the problem in the case of third-order systems. This allows the determination of the critical value where stability is lost in a tractable and accurate fashion. This value is important in many practical applications and we believe that it can also be used to develop a deeper theoretical understanding of this interesting problem.