Closed-form Approximations for the Minimal Robust Positively Invariant Set using Constrained Convex Generators

Robust Positively Invariant (RPI) sets play a crucial role in constructing terminal constraints for Model Predictive Control (MPC) optimizations and the definition of the invariant sets to be used in Control Barrier Functions (CBFs). However, the solutions in the literature involve either an iterative method that is approaching the true set or an approximation using optimization programs. In this paper, by leveraging the fact that Constrained Convex Generators (CCGs) can represent both polytopes, ellipsoids and other sets, we propose closed-form expressions for outer and inner approximations. Moreover, the tightness can be defined by the system designer based on a straightforward analysis of the norm of the dynamics matrix. We then illustrate how our proposal fairs against the iterative approach highlighting how changing the horizon value influences the added conservatism.