Ellipsoidal Techniques for Reachability Analysis

This report describes the calculation of the reach sets and tubes for linear control systems with time-varying coe cients and hard bounds on the controls through tight external and internal ellipsoidal approximations. These approximating tubes touch the reach tubes or their neighborhoods from outside and inside respectively at every point of their boundary so that the surface of the reach tube would be totally covered by curves that belong to the approximating tubes. The described move allows to devise the approximation scheme so as to decrease the computational burden in calculating the reach sets as compared with other methods. It is particularly shown how such approximations may be expressed through ordinary di erential equations with coe cients given in explicit analytical form. This yields exact parametric representation of reach tubes through families of external and internal ellipsoidal tubes. The given techniques if further combined with calculation of external and internal approximates for intersections of ellipsoids, allow to approach reachability problems for hybrid dynamics that combine continuous-time performance with resets of the systems. The suggested techniques allow e ective graphic animation.