Joint optimization of high-order error feedback and realization for roundoff noise minimization in the Fornasini-Marchesini second model

For two-dimensional (2-D) state-space digital filters described by the Fornasini-Marchesini second local state-space model, the joint optimization of high-order error feedback and realization for minimizing roundoff noise at filter output subject to l₂-scaling constraints is investigated. We present linear-algebraic techniques that convert the problem at hand into an unconstrained optimization problem, and present an efficient quasi-Newton algorithm to solve the unconstrained optimization problem iteratively, in which closed-form formulas are derived for fast and accurate gradient evaluation. A numerical example is presented to illustrate the utility and effectiveness of the proposed algorithm.