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Directed Data-Processing Inequalities for Systems with Feedback.

Milan S Derpich1, Jan Østergaard2

  • 1Department of Electronic Engineering, Universidad Técnica Federico Santa María, Av. España 1680, Valparaíso 2390123, Chile.

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Summary

This study introduces new data-processing inequalities for feedback systems, establishing a law of information flow conservation. It defines an in-the-loop (ITL) transmission rate crucial for channel coding with internal messages.

Keywords:
data-processing inequalitydirected informationfeedback capacitymutual informationnetworked control

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Area of Science:

  • Information Theory
  • Control Systems Engineering
  • Networked Systems

Background:

  • Feedback systems involve complex information flow dynamics.
  • Traditional information measures like mutual information may be insufficient for systems with internal feedback loops.
  • Understanding information processing within closed-loop systems is critical for advancements in communication and control.

Purpose of the Study:

  • To develop novel data-processing inequalities for systems with feedback, relating mutual and directed information.
  • To introduce and characterize a new metric, the in-the-loop (ITL) transmission rate, for channel coding scenarios within feedback loops.
  • To demonstrate the inadequacy of conventional transmission rates and channel capacity for these specific scenarios.

Main Methods:

  • Derivation of novel data-processing inequalities for causal, non-linear, time-varying, and randomized mappings within feedback systems.
  • Application of these inequalities to establish relationships between mutual information and directed information.
  • Characterization of the ITL transmission rate and its properties in relation to directed information.

Main Results:

  • A law of conservation of information flow is established.
  • New data-processing inequalities are presented for nested random sequences within closed loops.
  • The ITL transmission rate is shown to be the unique relevant rate for zero error probability in channel coding with internal messages.
  • The supremum of achievable ITL transmission rates is upper bounded by the directed information rate across the communication channel.

Conclusions:

  • The developed data-processing inequalities provide fundamental insights into information flow in feedback systems.
  • The ITL transmission rate offers a more accurate measure for channel coding in systems with internal messages.
  • The findings generalize existing inequalities in networked control literature and offer new avenues for research.