Efficient Numerical Approximation of Finite-Horizon Controllability Gramians and Their Cholesky Factors

This article proposes an efficient numerical method for computing finite-horizon controllability Gramians in Cholesky-factored form, without computing the full Gramian product. The method is a generalization of the scaling-and-squaring approach for approximating the matrix exponential, and provides a rigorous backward error analysis guaranteeing accuracy to the round-off error level in double precision.