On Ellipsoidal Techniques for Reachability Analysis. Part II: Internal Approximations Box-valued Constraints

Following Part I, this article continues to describe the calculation of the reach sets and tubes for linear control systems with time-varying coefficients and ellipsoidal hard bounds on the controls and initial states. It deals with parametrized families of internal ellipsoidal approximations constructed such that they touch the reach sets at every point of their boundary at any instant of time. The reach tubes are thus touched internally by ellipsoidal tubes along some curves. The ellipsoidal tubes are chosen here in such a way that the touching curves do not intersect and that the boundary of the reach tube would be entirely covered by such curves. This allows exact parametric representation of reach tubes through unions of tight internal ellipsoidal tubes as compared with earlier methods based on constructing one or several isolated approximating tubes. The method of external and internal ellipsoidal approximations is then propagated to systems with box-valued hard bounds on the controls and initial states. It appears that the proposed technique may well work for nonellipsoidal, box-valued constraints. This broadens the range of applications of the approach and opens new routes to the arrangement of efficient numerical algorithms.