Internal Stability of Dynamically Quantised Control for Stochastic Scalar Plants

Abstract

In this paper we study conditions under which an unstable stochastic scalar linear plant with unbounded noise can be internally stabilised using ‘zooming'-like coding and control schemes having dynamic, finite-dimensional internal states. Such structures are known to be needed in communication-constrained control when no bound on the plant noise is available. However, previous schemes were based on coders and controllers starting with identical internal states. In this paper, we remove this assumption and explicitly construct a finite-dimensional coding and control policy that yields mean square stability of all state variables, for a random initial plant state and arbitrary initial encoder and controller states. This holds for any bit rate down to the universal minimum of the Data Rate Theorem. Furthermore, we show that despite the unbounded noise, the error and proportional errors between the scaling factors of the encoder and controller tend to zero in mean square and almost sure senses respectively. This suggests that the policy will still maintain mean square internal stability in the presence of channel bit errors, provided the bit error rate is sufficiently low. We support these conclusions with simulations.