The Robustness of Strong Stability of Positive Homogeneous Difference Equations

Motivated by many applications in control engineering, problems of robust stability of dynamical systems have attracted a lot of attention of researchers during the last twenty years. In the study of these problems, the notion of stability radius was proved to be an effective tool, see 1–5 . In this paper, we study the robustness of strong stability of the homogeneous difference equation under parameter perturbations. The organization of this paper is as follows. In Section 2, we recall some results on nonnegative matrices and present preliminary results on homogeneous equations for later use. In Section 3, we study a complex strong stability radius under multiperturbations. Next, we present some results on strong stability radii of the positive class equations under parameter perturbations. It is shown that complex, real, and positive strong stability radii of positive systems coincide. More important, estimates and computable formulas of these stability radii are also derived. Finally, a simple example is given.