On-the-fly Approximation of Multivariate Total Variation Minimization

In the context of change-point detection, addressed by Total Variation minimization strategies, an efficient on-the-fly algorithm has been designed leading to exact solutions for univariate data. In this contribution, an extension of such an on-the-fly strategy to multivariate data is investigated. The proposed algorithm relies on the local validation of the Karush-Kuhn-Tucker conditions on the dual problem. Showing that the non-local nature of the multivariate setting precludes to obtain an exact on-the-fly solution, we devise an on-the-fly algorithm delivering an approximate solution, whose quality is controlled by a practitioner-tunable parameter, acting as a trade-off between quality and computational cost. Performance assessment shows that high quality solutions are obtained on-the-fly while benefiting of computational costs several orders of magnitude lower than standard iterative procedures. The proposed algorithm thus provides practitioners with an efficient multivariate change-point detection on-the-fly procedure

Sampling-based Approximations with Quantitative Performance for the Probabilistic Reach-Avoid Problem over General Markov Processes

This article deals with stochastic processes endowed with the Markov (memoryless) property and evolving over general (uncountable) state spaces. The models further depend on a non-deterministic quantity in the form of a control input, which can be selected to affect the probabilistic dynamics. We address the computation of maximal reach-avoid specifications, together with the synthesis of the corresponding optimal controllers. The reach-avoid specification deals with assessing the likelihood that any finite-horizon trajectory of the model enters a given goal set, while avoiding a given set of undesired states. This article newly provides an approximate computational scheme for the reach-avoid specification based on the Fitted Value Iteration algorithm, which hinges on random sample extractions, and gives a-priori computable formal probabilistic bounds on the error made by the approximation algorithm: as such, the output of the numerical scheme is quantitatively assessed and thus meaningful for safety-critical applications. Furthermore, we provide tighter probabilistic error bounds that are sample-based. The overall computational scheme is put in relationship with alternative approximation algorithms in the literature, and finally its performance is practically assessed over a benchmark case study

Robust Region-of-Attraction Estimation

We propose a method to compute invariant subsets of the region-of-attraction for asymptotically stable equilibrium points of polynomial dynamical systems with bounded parametric uncertainty. Parameter-independent Lyapunov functions are used to characterize invariant subsets of the robust region-of-attraction. A branch-and-bound type refinement procedure reduces the conservatism. We demonstrate the method on an example from the literature and uncertain controlled short-period aircraft dynamics