Closed Form Solutions to L2-Sensitivity Minimization of Second-Order State-Space Digital Filters with Real Poles

This letter proposes closed form solutions to the L2sensitivity minimization of 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. In case of second-order digital filters, we can express the L2-sensitivity of second-order digital filters by a simple linear combination of exponential functions and formulate the L2-sensitivity minimization problem by a simple polynomial equation. As a result, the minimum L2-sensitivity realizations can be synthesized by only solving a fourth-degree polynomial equation, which can be analytically solved. key words: state-space digital filters, L2-sensitivity, minimization, closed form solutions