Inverse and Direct Solutions of the Discrete Time Lyapunov Equation With System Matrix in Companion Form

The discrete time Lyapunov equation is used in many applications and there is interest in its inverse and direct solutions. New methods are proposed to obtain solutions for cases where the system matrix is in controllable canonical form. The approach is based on the relationship between the discrete Lyapunov equation and the entries of one of the stability tables presented by Jury. It is shown that the inverse solution, which is based on this stability table, can be obtained using
$LDL^{t}$
decomposition. Also the direct solution of the discrete Lyapunov equation can be obtained directly from the entries of this stability table. The proposed algorithms are illustrated by numerical examples.