On Ellipsoidal Techniques for Reachability Analysis. Part I: External Approximations

The paper describes 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. This is achieved by parametrized families of external and internal ellipsoidal approximations constructed such that they touch the reach sets at every point of their boundary at any instant of time (both from outside and inside, respectively). The surface of the reach tube would then be entirely covered by curves that belong to the approximating tubes. It is further shown how such approximations may be expressed through ordinary differential equations with coefficients given in explicit analytical form. This allows exact parametric representation of reach tubes through families of external and internal ellipsoidal tubes as compared with earlier methods based on constructing one or several isolated approximating tubes. The approach opens new routes to the arrangement of efficient numerical algorithms. The present Part I deals with external approximations.