Nyquist Plots for MIMO Systems Under Frequency Transformations

In this Letter, the Nyquist plot for linear, continuous-time MIMO systems under odd frequency transformations is dealt with. Given a system <inline-formula> <tex-math notation="LaTeX">$\mathrm {G}(s)$ </tex-math></inline-formula>, a frequency transformation of the type <inline-formula> <tex-math notation="LaTeX">$s\leftarrow F(s)$ </tex-math></inline-formula> allows one to obtain a transformed system <inline-formula> <tex-math notation="LaTeX">$\tilde {\mathrm {G}}(s)=\mathrm {G}(F(s))$ </tex-math></inline-formula> with useful properties. For instance, in analog filter design this operation allows to obtain multi-bandpass/bandstops filters from lossless frequency transformations applied to prototype lowpass filters. We prove that under these transformations the Nyquist plot is transformed into a locus having the same shape of that of the original system. As the number of encirclements of any point in the complex plane performed by the curves in the Nyquist plot can be also related to that of the original system, we conclude that closed-loop stability of the transformed system can be inferred from the original system.