2-D Lyapunov equation and filter design based on the Fornasini-Marchesini second model

A two-dimensional (2-D) Lyapunov equation with constant coefficients is considered for the Fornasini-Marchesini second local state-space (LSS) model. First, a novel criterion relating to the Lyapunov equation is presented that sufficiently guarantees the asymptotic stability. A sufficiency condition that ensures the absence of limit cycles is also given. Next, the above stability condition is incorporated into the 2-D filter structure to design 2-D state-space digital filters with guaranteed asymptotic stability. An efficient method is then developed for computing the characteristic polynomial and the inverse of the system matrix. Finally, two numerical examples are given to design 2-D stable state-space digital filters