Internal stability of dynamic quantised control for stochastic linear plants.

*(English)*Zbl 1166.93380Summary: Existing analyses of ‘zooming’ quantisation schemes for bit-rate-limited control systems rely on the encoder and controller being initialised with identical internal states. Due to the quantiser discontinuity and the plant instability, it was not clear if closed-loop stability was possible if the encoder and controller commenced from different initial conditions. In this article, we consider partially observed, unstable linear time-invariant plants, with unbounded and possibly non-Gaussian noise, and propose a modified zooming-like scheme with finite-dimensional internal encoder and controller states that may not initially be identical. Using a stochastic pseudo-norm, we prove that this scheme yields mean-square stability in all closed-loop state variables, not just the plant state, under a sufficient condition involving this initial error, the plant dynamics and the channel data rate. With diminishing initial error, this condition approaches a known universal lower bound on data rates and becomes tight. Furthermore, we show that the scheme automatically corrects itself, in the sense that the errors between the internal states of the encoder and controller tend to zero stochastically with time. This suggests that the policy will maintain stability in the presence of channel errors, for sufficiently low bit error rates. We support these conclusions with simulations.

##### MSC:

93E15 | Stochastic stability in control theory |

93C05 | Linear systems in control theory |

93E03 | Stochastic systems in control theory (general) |

94A40 | Channel models (including quantum) in information and communication theory |

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\textit{A. Gurt} and \textit{G. N. Nair}, Automatica 45, No. 6, 1387--1396 (2009; Zbl 1166.93380)

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