Minimization of Frequency-Weighted $l_{2}$-Sensitivity Subject to $l_{2}$-Scaling Constraints for Two-Dimensional State-Space Digital Filters

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This paper investigates the problem of frequency-weighted l2-sensitivity minimization subject to l2-scaling constraints for two-dimensional (2-D) state-space digital filters described by the Roesser model. It is shown that the Fornasini-Marchesini second model can be imbedded in the Roesser model. Two iterative methods are developed to solve the constrained optimization problem encountered. The first iterative method introduces a Lagrange function and optimizes it using some matrix-theoretic techniques and an efficient bisection method. The second iterative method converts the problem into an unconstrained optimization formulation by using linear-algebraic techniques and solves it by applying an efficient quasi-Newton algorithm. The optimal filter structure with minimum frequency-weighted l2-sensitivity and no overflow is then synthesized by an appropriate coordinate transformation. Case studies are presented to demonstrate the validity and effectiveness of the proposed techniques.