Minimization of frequency-weighted l2-sensitivity for state-space digital filters subject to l2-scaling constraints

The minimization problem of a frequency-weighted l
2
-sensitivity measure subject to l
2
-scaling constraints for state-space digital filters is investigated. To this end, an expression for evaluating the frequency-weighted l
2
-sensitivity is introduced and a frequency-weighted l
2
-sensitivity minimization problem subject to the l
2
-scaling constraints is formulated. An efficient iterative algorithm is then explored to solve the constraint optimization problem directly. The proposed solution method utilizes a Lagrange function, some matrix-theoretic techniques, and an efficient bisection method. The optimal state-space model structure with minimum frequency-weighted l
2
-sensitivity and no overflow oscillations is constructed by applying an appropriate coordinate-transformation matrix. A numerical example is also presented to illustrate the utility of the proposed technique.