An enhanced hierarchy for (robust) controlled invariance

We revisit the problem of computing controlled invariant sets for controllable discrete-time linear systems. Inspired by previous work by the authors, our main idea works in two moves: the problem is lifted to a higher dimensional space, where we provide a closed-form expression for a set whose projection back onto the original space is proven to be controlled invariant. We propose two methods in which the key insight is computing controlled invariant sets by considering periodic control policies. The first method considers hyperboxes that are rendered recurrent and essentially improves computational performance of the authors' previous work, while computing the same sets. The second method relaxes the assumption of recurrent hyper-boxes and yields substantially larger controlled invariant sets as shown in case studies. These methods do not rely on iterative computations and their scalability is illustrated in several examples, which show that none of the methods is strictly better than the other.