On a Correspondence between Probabilistic and Robust Invariant Sets for Linear Systems

Dynamical systems with stochastic uncertainties are ubiquitous in the field of control, with linear systems under additive Gaussian disturbances a most prominent example. The concept of probabilistic invariance was introduced to extend the widely applied concept of invariance to this class of problems. Computational methods for their synthesis, however, are limited. In this paper we present a relationship between probabilistic and robust invariant sets for linear systems, which enables the use of well-studied robust design methods. Conditions are shown, under which a robust invariant set, designed with a confidence region of the disturbance, results in a probabilistic invariant set. We furthermore show that this condition holds for common box and ellipsoidal confidence regions, generalizing and improving existing results for probabilistic invariant set computation. We finally exemplify the synthesis for an ellipsoidal probabilistic invariant set. Two numerical examples demonstrate the approach and the advantages to be gained from exploiting robust computations for probabilistic invariant sets.