Data-driven computation of invariant sets of discrete time-invariant black-box systems

We consider the problem of computing the maximal invariant set of discrete-time black-box nonlinear systems without analytic dynamical models. Under the assumption that the system is asymptotically stable, the maximal invariant set coincides with the domain of attraction. A data-driven framework relying on the observation of trajectories is proposed to compute almost-invariant sets, which are invariant almost everywhere except a small subset. Based on these observations, scenario optimization problems are formulated and solved. We show that probabilistic invariance guarantees on the almost-invariant sets can be established. To get explicit expressions of such sets, a set identification procedure is designed with a verification step that provides inner and outer approximations in a probabilistic sense. The proposed data-driven framework is illustrated by several numerical examples.Comment: A shorter version with the title "Scenario-based set invariance verification for black-box nonlinear systems" is published in the IEEE Control Systems Letters (L-CSS

Computation of the maximal invariant set of discrete-time linear systems subject to a class of non-convex constraints

We consider the problem of computing the maximal invariant set of discrete-time linear systems subject to a class of non-convex constraints that admit quadratic relaxations. These non-convex constraints include semialgebraic sets and other smooth constraints with Lipschitz gradient. With these quadratic relaxations, a sufficient condition for set invariance is derived and it can be formulated as a set of linear matrix inequalities. Based on the sufficient condition, a new algorithm is presented with finite-time convergence to the actual maximal invariant set under mild assumptions. This algorithm can be also extended to switched linear systems and some special nonlinear systems. The performance of this algorithm is demonstrated on several numerical examples.Comment: Accepted in Automatic

Invariant Sets Analysis for Constrained Switching Systems

We study discrete time linear constrained switching systems with additive disturbances, in the general setting where the switching acts on the system matrices, the disturbance sets, and the state constraint sets. Our primary goal is to extend the existing invariant set constructions when the switching signal is constrained by a given automation. We achieve it by working with a relaxation of invariance, namely the multi-set invariance. By exploiting recent results on computing the stability metrics for these systems, we establish explicit bounds on the number of iterations required for each construction. Last, as an application, we develop new maximal invariant set constructions for the case of linear systems in far fewer iterations compared to the state-of-the-art