# On Information Transfer in Control Dynamical Systems

## TL;DR

This paper introduces a new measure of information transfer in control dynamical systems that accurately captures causal interactions, overcoming limitations of existing measures like directed information and transfer entropy.

## Contribution

The paper proposes a novel definition of information transfer based on dynamical systems theory, applicable to various signals and extending to multiple time steps, with analytical expressions for linear systems.

## Key findings

- New information transfer measure captures true causal interactions.
- Analytical expressions derived for linear systems.
- Re-derives Bode fundamental limitations using the new measure.

## Abstract

In this paper, we show through examples, how the existing definitions of information transfer, namely directed information and transfer entropy fail to capture true causal interaction between states in control dynamical system. We propose a new definition of information transfer, based on the ideas from dynamical system theory, and show that this new definition can capture true causal interaction between states. The information transfer measure is generalized to define transfers between the various signals in a control dynamical system and analytical expression for information transfer between state-to-state, input-to-state, state-to-output, and input-to-output are provided for linear systems. There is a natural extension of our proposed definition to define information transfer over $n$ time steps and average information transfer over infinite time step. We show that the average information transfer in feedback control system between plant output and input is equal to the entropy of the open loop dynamics thereby re-deriving the Bode fundamental limitation results using the proposed definition of transfer.