Minimum Data Rate for Mean Square Stabilizability of Linear Systems With Markovian Packet Losses

This paper investigates the minimum data rate for mean square stabilizability of linear systems over a lossy digital channel. The packet dropout process of the channel is modeled as a time-homogeneous Markov process. To overcome the difficulties induced by the temporal correlations of the packet dropout process and stochastically time-varying data rate due to packet dropouts, a randomly sampled system approach is developed to study the minimum data rate for mean square stabilizability. It is shown that the minimum data rate for scalar systems can be explicitly given in terms of the magnitude of the unstable mode and the transition probabilities of the Markov chain. The number of additional bits required to counter the effect of Markovian packet losses on stabilizability is exactly quantified. Our result contains existing results on data rate and packet dropout rate for stabilizability of linear scalar systems as special cases and provides a means for better bandwidth utilization by jointly considering bits per sample and an effective sampling. Necessary and sufficient conditions on the minimum data rate problem for mean square stabilizability of vector systems are provided respectively and shown to be optimal under some special cases.