Characterization of Positive Invariance of Quadratic Convex Sets for Discrete-Time Systems Using Optimization Approaches

A positively invariant set is an important concept in dynamical systems. The study of positively invariant set conditions for discrete-time systems is one interesting topic in both theoretical studies and practical applications research. Different methods for characterizing the invariance of different types of sets have been established. For example, the ellipsoidal and the Lorenz cone, which are quadratic convex sets, have different properties from a polyhedral set. This paper presents an optimization method and a dual optimization method to characterize the positive invariance of the ellipsoidal and the Lorenz cone. The proposed methods are applicable to both linear and nonlinear discrete-time systems. Using nonlinear programming and an induced norm, the positive invariance condition problems are transformed into optimization problems, and the dual optimization method is also used to give equivalent dual forms. Fewer results on the positive invariance condition of Lorenz cones can be found than for the other type of set; this paper fulfills the results of this problem. In addition, the proposed methods in this paper provide more options for checking the positive invariance of quadratic convex sets from the perspective of optimization and dual optimization. The effectiveness of this method is demonstrated by numerical examples.