Inner Approximations of the Maximal Positively Invariant Set for Polynomial Dynamical Systems

Abstract

The Lasserre or moment-sum-of-square hierarchy of linear matrix inequality relaxations is used to compute inner approximations of the maximal positively invariant set for continuous-time dynamical systems with polynomial vector fields. Convergence in volume of the hierarchy is proved under a technical growth condition on the average exit time of trajectories. Our contribution is to deal with inner approximations in infinite time, while former work with volume convergence guarantees proposed either outer approximations of the maximal positively invariant set or inner approximations of the region of attraction in finite time. * *The research was partly funded by Réseaux de Transport d'Électricité (RTE). †1 Antoine Oustry is withÉcole Polytechnique, Route de Saclay,

Computing Controlled Invariant Sets from Data Using Convex Optimization

“…Inner approximation A linear program whose feasible solutions provide inner approximations to the MPI set can be obtained by characterizing the complement of the MPI set. This idea was first used for the related problem of region of attraction estimation in [13] and later for the MPI set in [23].…”

“…Historically, the idea of using infinite-dimensional linear programming to address nonlinear optimal control problems originated, to the best of our knowledge, with the work of Rubio [23], closely followed by the works of Vinter and Lewis [24,16]. The work of Rubio [23] is in itself a follow-up on his earlier work [22] that use the infinite-dimensional linear-programming embedding to study calculus of variations problems within the framework of generalized curves introduced by Young in [25].…”

“…Historically, the idea of using infinite-dimensional linear programming to address nonlinear optimal control problems originated, to the best of our knowledge, with the work of Rubio [23], closely followed by the works of Vinter and Lewis [24,16]. The work of Rubio [23] is in itself a follow-up on his earlier work [22] that use the infinite-dimensional linear-programming embedding to study calculus of variations problems within the framework of generalized curves introduced by Young in [25]. Without control, the fact that certain objects evolving along the flow of a nonlinear dynamical system obey linear relationships is classical; for example, the evolution of a density (or more generally a Borel measure) defined on the state space is governed by the so-called continuity equation whereas values of a function defined on the state-space evolve according to the transport equation.…”

Controlled invariant sets: implicit closed-form representations and applications

“…One line of work [KHJ14,SG12,OTH19] focuses on outer and inner approximations of the Maximal CIS by solving either LPs or QPs. The resulting sets, however, are not always invariant.…”

A simple hierarchy for computing controlled invariant sets

“…Unavoidably, these techniques suffer from the efficiencyaccuracy tradeoff, and the curse of dimensionality. Some of the state-of-the-art methods propose inner and outer approximations of the MCIS, by solving Semi-Definite Programs (SDPs) [20,28], or Linear Programs (LPs) [33]. Naturally, outer approximations are not invariant, but even inner approximations are not guaranteed to be.…”