An enhanced hierarchy for (robust) controlled invariance

On the Value of Preview Information For Safety Control

“…1 that the largest c stops increasing after p ≥ 6. This observation suggests that a disturbance set with c > 0.2222 may lead to an empty controlled invariant set for any preview time p. With some calculation, it can be verified that for c > 2/9, the necessary condition (18) does not hold for all p ≥ 0 and thus the maximal controlled invariant set is always empty no matter how large the p is.…”

“…Remark 1. Adopting the idea from [18], for a more general safe set in form of P × R, where P is a polytope, we can construct a controlled invariant set of Σ B,p within P × D p × R in 2 moves: First, we construct a polytope in a lifted space that encodes all hyperboxes B in P and all states (x, d 1:p ) within the maximal controlled invariant set within B × D p × R, based on the nonemptyness condition ( 17) and the closedform expression of C p . Then, we project this lifted set onto its first n(p + 1) coodinates, equal to the union of the maximal controlled invariant set within B × D p × R for all hyperboxes B contained by P. By construction, this set is a controlled invariant set in P × D p × R. Furthermore, as stated in Remark 1 of [19], any controllable system with a polytopic safe set (including input constraints) can be transformed into system in Brunovsky canonical form with a safe set in form of P × R. Thus, our results in this section can be used to compute controlled invariant sets for p-augmented systems of a controllable system.…”

“…] by a positive number c > 0. We are interested in the largest c for the augmented system Σ B,p to have nonempty controlled invariant sets within B × D p × R. According to Corollary 1, we can utilize the condition on nonempty controlled invariant set given by (18) to determine the largest possible c.…”

Invariant Sets for Assume-Guarantee Contracts

“…as described in [24], where a data-driven method for computing an approximation of a robust control invariant set from experimental data is proposed, or in [25], where the problem of invariant set computation for blackbox switched linear systems using merely a finite set of observations of system trajectories is investigated. Due to the necessity to deal with large-scale systems, scalable approaches for computing invariant sets are targeted too [26].…”

Polytope shaping while preserving invariance

“…In this case however, there is no clear mechanism that predicts or controls the complexity of involved intermediate operations. A number of works have been proposed towards controlling complexity, using a mixture of Lyapunov and set-based approaches [5], [6], [7], [8], [9], [10], [11], [12], [13], [14].…”