Computation of discriminating kernel and robust capture basin with regulation map by interval methods

Abstract

This paper proposes a novel algorithm that characterizes the robust capture basin and the discriminating kernel for constrained nonlinear systems with uncertainties based on viability theory. For nonlinear systems with constrained inputs and bounded uncertainties, the viability kernel is the largest set of states possessing a possibility to be viable in a set, and the capture basin is the largest set of states possessing a possibility to reach a target in a finite time, and keeping viable in a set before reaching the target. However, in the viability theory, both control and uncertainty in a parameterized system are considered as parameters:the discriminating kernel and the proposed robust capture basin link viability theory with robust control, which take both control and uncertainties into account. For the constrained uncertain nonlinear systems, the discriminating kernel is the largest set of states that is robust invariant in a set with proper control, and the robust capture basin is the largest set of states reaching their target in finite time with proper control despite of uncertainties and keeping viable in a set before reaching the target. Furthermore, we map all the states to optimal regulatory control such that the systems are regulated by a regulation map. To compute the robust capture basin and the discriminating kernel, we use interval methods to provide guaranteed solutions. The proposed algorithms in this paper approximate an outer approximation of the minimum reachable target and inner approximations of the robust capture basin and the discriminating kernel in a guaranteed way.

Differential Graphical Games for Constrained Autonomous Vehicles Based on Viability Theory

“…represents an inclusion function which is expressed in [29]; T D k (q k,i ) stands for Bouligand's tangent cone [28], and…”

“…Our algorithm is composed of off-line backward set propagation and on-line cooperative learning. We approximate the capture basin in a guaranteed way by using backward propagation on the target set, which improves the propagation approach proposed in [28], as the value function of the differential games can be approximated during propagation. Moreover, our propagation approach can be applied to the set with arbitrary shape.…”