Numerical solution of the two-dimensional Lyapunov equations and application in order reduction of recursive digital filters

An iterative method for the computation of 2-D quasi-grammians is developed, where each iteration involves solving two 1-D Lyapunov equations. For a stable 2-D system, the iterative algorithm converges to the desired solutions very quickly. Further, an algorithm based on unconstrained optimization is developed for the computation of structured grammians. The algorithm involves two steps. The first step is to minimize the norm of the system matrix by 2-D similarity transformations; the second step is to use a scaling technique to accomplish the optimization. Two examples are given to evaluate the performance of the reduced-order systems that are obtained from the balanced approximations using different grammians