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 L₂- sensitivity 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 L ₂-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.