Stability of Lur´e systems with interval plants and general sector-type nonlinearities

Robust absolute stability problems have been discussed for Lur´e systems which contain transfer functions with real parametric uncertainties. In dealing with this kind of systems, their linear parts with parameter perturbations are represented by interval plants. Popov´s theorem is known as a way of solving the classical absolute stability problem when nonlinear characteristics of Lur´e systems are time-invariant. It has been shown that this theorem can be extended to the case where the linear plant is replaced by interval plants. The absolute stability of such systems is checked by the geometrical relation between a Popov line and the Popov loci of several extreme plants in the complex plane. However, it is restricted to the case that the lower bound of the sector equals to zero. In this paper, an extension is made so that the nonlinear part has a general sector. This can be done by making the transformation for the original systems and applying Popov´s theorem to the transformed systems. The authors show that the absolute stability of original systems can be guaranteed if one has only to check Popov´s condition for the systems transformed from several extreme plants.