Infimum Data Rates for Stabilising Markov Jump Linear Systems

In recent years, the problem of stabilising linear dynamical systems with low feedback data rates has been intensively investigated. A particular focus has been the characterisation of the infimum data rate for stabilisability, which specifies the smallest rate, in bits per unit time, at which information can circulate in a stable feedback loop. This paper extends this line of research to the case of fully-observed, finite-dimensional, linear systems without process noise but with control-independent, Markov parameters. Unlike previous formulations, the coding alphabet is permitted to be random and time-varying via a possible dependence on the observed Markov modes. Using quantisation techniques and real Jordan forms, it is shown that the smallest asymptotic mean data rate for stabilisability in r-th absolute output moment, over all coding and control schemes, is given by an exponent which measures the asymptotic mean growth rate of unstable eigenspace volumes. An explicit formula for it is obtained in the case of antistable dynamics. For scalar systems, this expression is quite different from an earlier one derived assuming a constant alphabet, in particular being independent of the moment order.