Analytical Expression and Deconstruction of the Volume of the Controllability Ellipsoid

In this article, we present three theorems and develop an effective analytical method to compute analytically the volume of the controllability ellipsoid for the linear discrete-time (LDT) systems with $n$ different eigenvalues. Furthermore, by deconstructing the analytical expression of the volume, some factors on the shape of the ellipsoid, the side length of its circumscribed rhomboid, the evenness of the eigenvalue distribution of the LDT system are constructed. Based on the analytical expression of the volume and these factors, the control ability can be defined, computing, analyzed, and optimized for the LDT systems.