On the limit cycles in the minimum L2-sensitivity realizations subject to L2-scaling constraints of second-order digital filters

This paper gives conjecture on the absence of limit cycles of the minimum L₂-sensitivity realizations subject to L₂-scaling constraints for second-order digital filters. We design second-order digital filters with various pole-zero configurations, synthesize the minimum L₂-sensitivity realizations subject to L₂-scaling constraints, and examine if their coefficient matrices satisfy a sufficient condition for the absence of limit cycles. As a result, in the range of practical pole radii, it is shown that the minimum L₂-sensitivity realizations subject to L₂-scaling constraints of second-order digital filters satisfy a sufficient condition for the absence of limit cycles. Furthermore, we demonstrate the absence of limit cycles of the minimum L₂-sensitivity realizations of a second-order digital filter by observing its zero-input response.