Safe approximations of chance constrained sets by probabilistic scaling

Chance-constrained sets approximation: A probabilistic scaling approach

“…Remark 2 motivates the approach presented in the next section, which builds upon the results presented in Alamo et al (2019). We show how the probabilistic scaling approach directly leads to approximations of user-chosen complexity, which can be directly used in applications instead of creating the need for a post-processing step to reduce the complexity of the sampled set.…”

“…The probabilistic scaling approach was presented in the conference papers (Alamo, Mirasierra, Dabbene, & Lorenzen, 2019;Mammarella, Alamo, Dabbene, & Lorenzen, 2020) and it is based on recent results on order statistics (Alamo, Manzano, & Camacho, 2018). The present work extends Alamo et al (2019), Mammarella et al (2020) in several directions. First, we perform here a thorough mathematical analysis of probabilistic scaling.…”

Probabilistic stabilizability certificates for a class of black-box linear systems

“…Proposition 1: Let n ≥ m, K ∈ N and δ K ∈ ∆ K be any given K-multisample with associated pairs of matrices {(A(δ (j) ), B(δ (j) )) j∈K }. Given any C-polytope S ⊆ X , the set L δ K in ( 10) is nonempty only if, for all (i, j) ∈ V × K, there exists an invertible submatrix G Q∪P ∈ R m×m of G(δ (j) ), with row indices as in Lemma 2, and subvector l Q∪P of l(δ (j) ), satisfying the conditions in (12).…”

“…Then, G I = col(H Q , F B P ) and l I = col(1 |Q| , 1 |P| −(F A(δ)v) P ), for any v ∈ vert(S). Finally, the conditions in (12) follow by splitting inequalities (G) k G −1 I l I ≤ l k , ∀k ∈ A \ I, between the two sets Q and P, and noting that (G) k = (H) k and l k = 1 for any k ∈ {1, . .…”

“…This work was partially supported through the Government's modern industrial strategy by Innovate UK, part of UK Research and Innovation, under Project LEO (Ref. 104781) probabilistic guarantees, frequently encountered in control, were discussed in [12], [13]. Finally, a scenario-based set invariance verification approach for black-box systems was proposed in [14], where the observation of system trajectory snapshots allowed to compute almost-invariant sets enjoying theoretical probabilistic invariance certificates.…”