Smallest Ellipsoid Containing $p$-Sum of Ellipsoids With Application to Reachability Analysis

Abstract

In this article the problem of ellipsoidal bounding of convex set-valued data, where the convex set is obtained by the <inline-formula><tex-math notation= LaTeX >$p$</tex-math></inline-formula>-sum of finitely many ellipsoids, for any real <inline-formula><tex-math notation= LaTeX >$p\\geq 1$</tex-math></inline-formula> is studied. The notion of <inline-formula><tex-math notation= LaTeX >$p$</tex-math></inline-formula>-sum appears in the Brunn–Minkowski–Firey theory in convex analysis, and generalizes several well-known set-valued operations, such as the Minkowski sum of the summand convex sets (here, ellipsoids). We derive an outer ellipsoidal parameterization for the <inline-formula><tex-math notation= LaTeX >$p$</tex-math></inline-formula>-sum of a given set of ellipsoids, and compute the tightest such parameterization for two optimality criteria: minimum trace and minimum volume. For such optimal parameterizations, several known results in the system-control literature are recovered as special cases of our general formula. For the minimum volume criterion, our analysis leads to a fixed point recursion over a scalar that parameterizes the shape matrix of the outer ellipsoid. This recursion is proved to be contractive, and found to converge fast in practice. We apply these results to compute the forward reach sets for a linear control system subject to different convex set-valued uncertainty models for the initial condition and control, generated by varying <inline-formula><tex-math notation= LaTeX >$p\\in [1,\\infty ]$</tex-math></inline-formula>. Our numerical results show that the proposed fixed point algorithm offers more than two orders of magnitude speed-up in computational time for <inline-formula><tex-math notation= LaTeX >$p=1$</tex-math></inline-formula>, compared to the existing semidefinite programming approach without significant effect on the numerical accuracy. For <inline-formula><tex-math notation= LaTeX >$p>1$</tex-math></inline-formula>, the reach set computation results reported here are novel. Our results are expected to be useful in real-time safety critical applications, such as decision making, for collision avoidance of autonomous vehicles, where the computational time scale for reach set calculation needs to be much smaller than the vehicular dynamics time scale.