Minimization of L₂-sensitivity for state-space digital filters subject to L₂-dynamic-range scaling constraints

The problem of minimizing an L₂-sensitivity measure subject to L₂-norm dynamic-range scaling constraints for state-space digital filters is formulated. It is shown that the problem can be converted into an unconstrained optimization problem by using linear-algebraic techniques. Next, the unconstrained optimization problem is solved by applying an efficient quasi-Newton algorithm with closed-form formula for gradient evaluation. The coordinate transformation matrix obtained is then used to construct the optimal state-space filter structure that minimizes the L₂-sensitivity measure subject to the scaling constraints. A numerical example is presented to illustrate the utility of the proposed technique.