Closed Form Solutions to L2-Sensitivity Minimization Subject to L2-Scaling Constraints for Second-Order State-Space Digital Filters with Real Poles

This paper proposes closed form solutions to the L 2 -sensitivity minimization subject to L 2 -scaling constraints for second-order state-space digital filters with real poles. We consider two cases of second-order digital filters: distinct real poles and multiple real poles. The proposed approach reduces the constrained optimization problem to an unconstrained optimization problem by appropriate variable transformation. We can express the L 2 -sensitivity by a simple linear combination of exponential functions and formulate the L 2 -sensitivity minimization problem by a simple polynomial equation. As a result, L 2 -sensitivity is expressed in closed form, and its minimization subject to L 2 -scaling constraints is achieved without iterative calculations.

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