On the Existence of Stable Real nth-Order Transfer Functions With Desired Phase at n Frequencies

We consider the following question: Given n distinct non-negative frequencies {ω
₁
, ω
₂
, ... ω
_n
}, does there always exist a stable real nth-order transfer function (TF) with a desired phase at these frequencies? For n even, we prove that the answer is yes and propose a numerical algorithm for obtaining such TFs. For n > 1 odd, we conjecture that the answer is no. These results are useful for designing lower-order output feedback compensators that solve the multi-agent consensus problem for high-order linear agent dynamics.