Channel Random Transmission Delays and Colored Mean-Square Small Gain Theorem

This work mainly investigates the mean-square stability for a single-input single-output networked linear feedback system. The control signal in the networked system is transmitted over an unreliable channel with data transmission delays and packet dropouts. The transmission delays and packet dropouts are modeled by an independent and identically distributed random process with a known probability mass function. At the channel terminal, a linear receiving strategy is adopted. Based on the unit impulse response of the unreliable channel, we propose a new model of the channel uncertainty induced by the data transmission delays. It is found that the relation between the second-order statistics of the input and output of the channel uncertainty is time-invariant. To explore the effect of the channel uncertainty on the mean-square stability of this system, we introduce a concept called as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">frequency response of variation</i> of the unreliable channel. With this new concept, a necessary and sufficient condition of the mean-square stability is established for the networked feedback system. It reveals that the mean-square stability is determined by the interaction between the frequency response of variation and the feedback system with a nominal channel. The channel uncertainty induced by data transmission delays is a colored multiplicative uncertainty. When the plant in the networked feedback system is minimum phase, an analytic necessary and sufficient condition is presented for its mean-square stabilizability via output feedback. It turns out that the stabilizability is only determined by the interaction between the frequency response of variation of the channel and unstable poles of the plant. Finally, numerical examples are given to illustrate our results.