### Abstract

We study, for kth-order Markov sources, the structure of optimal causal encoders minimizing total rate subject to a mean-square distortion constraint over a finite horizon. The class of sources considered is general alphabet sources, and the encoder is allowed to be variable-rate. This leads to an optimization problem in an infinite-dimensional space, for which a solution exists. We show that the optimal causal encoder for a kth-order Markov source uses only the most recent k source symbols and the information available at the receiver. We also consider the infinite-horizon version, but for k=1, provide an existence result for an optimal stationary solution, discuss the effect of randomization on performance, and show that a recurrence-based time sharing of at most two deterministic quantizers is optimal. We further argue that for real-valued processes innovation encoding is not in general optimal, but for coding of a stable linear or non-linear source, the quantization of the innovation is an almost optimal scheme in the limit of low-distortion. This class of problems has natural applications in remote control of linear and non-linear systems with quantization.

Original language | English (US) |
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Title of host publication | Analysis and Design of Nonlinear Control Systems |

Subtitle of host publication | In Honor of Alberto Isidori |

Publisher | Springer Berlin Heidelberg |

Pages | 91-108 |

Number of pages | 18 |

ISBN (Print) | 9783540743576 |

DOIs | |

State | Published - Dec 1 2008 |

### ASJC Scopus subject areas

- Engineering(all)

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## Cite this

*Analysis and Design of Nonlinear Control Systems: In Honor of Alberto Isidori*(pp. 91-108). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-540-74358-3_7