Signal-to-noise ratio vs. unstable pole, non-minimum phase zero and relative degree in mean-square stabilizability

In this work, the mean-square stabilizability via output feedback of a networked feedback system over an unreliable communication channel is studied. The unreliability of the channel is modeled by multiplicative noises. The admissible signal-to-noise ratio (SNR) of the unreliable channel with which the mean-square stabilizability of the networked feedback system is preserved is studied in terms of characteristics of the plant in the system. It is found that the infimum of the admissible SNR is determined by the inner of the plant's non-minimum phase zeros, the balanced realization of the inner of plant's unstable poles and the relative degree of the plant. Moreover, it is shown that this infimum of the admissible SNR is an exponential function of the relative degree with the square product of the unstable poles as the base. Our result analytically describes the inherent constraint between the channel SNR and the unstable poles, non-minimum zeros, relative degree of the plant in the mean-square stabilization problem of the network feedback system.