Frequency-weighted L2-sensitivity minimization for 2-D state-space digital filters subject to L2-scaling constraints by a quasi-Newton method

This paper considers the problem of minimizing a frequency-weighted l
₂
-sensitivity measure subject to l
₂
-scaling constraints for 2-D state-space digital filters. First, the frequency-weighted l
₂
-sensitivity is analyzed for 2-D state-space digital filters described by the Roesser local state-space model. Next, the minimization problem of the frequency-weighted l
₂
-sensitivity subject to l
₂
-scaling constraints is formulated. The constrained optimization problem is then converted into an unconstrained optimization formulation by using linear-algebraic techniques. An efficient quasi-Newton algorithm with closed-form formula for gradient evaluation is applied to solve the unconstrained optimization problem. The optimal state-space filter structure with minimum frequency-weighted l
₂
-sensitivity and no overflow oscillations is constructed by applying the optimal coordinate transformation. Finally, a numerical example is presented to demonstrate the validity and effectiveness of the proposed technique.