Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Output Redundancy of LTI Systems: A Geometric Approach with Application to Privacy

Guitao Yang, Alexander J. Gallo, Angelo Barboni, Riccardo M. G. Ferrari, Andrea Serrani, Thomas Parisini

Abstract

This paper examines the properties of output-redundant systems, that is, systems possessing a larger number of outputs than inputs, through the lenses of the geometric approach of Wonham et al. We begin by formulating a simple output allocation synthesis problem, which involves ``concealing" input information from a malicious eavesdropper having access to the system output, while still allowing for a legitimate user to reconstruct it. It is shown that the solvability of this problem requires the availability of a redundant set of outputs. This very problem is instrumental to unveiling the fundamental geometric properties of output-redundant systems, which form the basis for our subsequent constructions and results. As a direct application, we demonstrate how output allocation can be employed to effectively protect the information of input information from certain output eavesdroppers with guaranteed results.

On the Output Redundancy of LTI Systems: A Geometric Approach with Application to Privacy

Abstract

This paper examines the properties of output-redundant systems, that is, systems possessing a larger number of outputs than inputs, through the lenses of the geometric approach of Wonham et al. We begin by formulating a simple output allocation synthesis problem, which involves ``concealing" input information from a malicious eavesdropper having access to the system output, while still allowing for a legitimate user to reconstruct it. It is shown that the solvability of this problem requires the availability of a redundant set of outputs. This very problem is instrumental to unveiling the fundamental geometric properties of output-redundant systems, which form the basis for our subsequent constructions and results. As a direct application, we demonstrate how output allocation can be employed to effectively protect the information of input information from certain output eavesdroppers with guaranteed results.
Paper Structure (22 sections, 11 theorems, 68 equations, 14 figures, 1 table)

This paper contains 22 sections, 11 theorems, 68 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. Brief Literature Review
  3. Contributions and outline of the paper
  4. Notation and Preliminaries
  5. Notation
  6. Preliminaries on Geometric Approach
  7. Problem formalization and the necessity of output redundancy
  8. Strong Output Redundancy
  9. Properties of SOR Systems
  10. Design of $\mathscr{D}$ for SOR systems
  11. Attacker's sensitivity to disturbance
  12. Illustrating example for SOR systems
  13. Weak Output Redundancy
  14. Properties of WOR systems
  15. Design of $\mathscr{D}$ for WOR systems
  16. ...and 7 more sections

Key Result

Proposition 1

Assume that $D$ is monic. Then, system eq:sys is left-invertible with respect to $u$ under the output disturbance $d$ if and only if it is left-invertible with respect to the aggregate input $u_{a}=u\oplus d$.

Figures (14)

  • Figure 1: Schematic diagram of our considered setup: the upper part presents the exogenous input, the system, and the obfuscation strategy; the middle section represents the network over which $y_d$ is broadcast, with the eavesdropper; the lower part of the diagram illustrates the actions taken by a legitimate agent to reconstruct an estimate of $u$, denoted by $\hat{u}$.
  • Figure 2: Relation between the dimension of input space, image of output map, and output space for: (a) strong output redundancy (SOR); (b) weak output redundancy (WOR).
  • Figure 3: Commutative diagram for strongly output-redundant systems, where $\mathop{\mathrm{Im}}\nolimits C|C : \mathscr{X} \to \mathop{\mathrm{Im}}\nolimits C$ is the codomain restriction of $C$ to $\mathop{\mathrm{Im}}\nolimits C$, $i : \mathop{\mathrm{Im}}\nolimits C \to \mathscr{Y}$ is the insertion map, and $\pi : \mathscr{Y} \to \mathscr{Y}/\mathop{\mathrm{Im}}\nolimits C$ is the canonical projection $\mathrm{mod} \mathop{\mathrm{Im}}\nolimits{C}$.
  • Figure 4: Example of a commutative diagram for system \ref{['eq:sys']} as discussed in the proof of Theorem \ref{['thm:SOR']}.
  • Figure 5: Strategy to reconstruct $u$ for SOR systems.
  • ...and 9 more figures

Theorems & Definitions (29)

  • Remark 1
  • Definition 1
  • Remark 2
  • Remark 3
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Definition 2
  • Definition 3
  • ...and 19 more