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Behaviors, trajectories and data: A novel perspective on the design of unknown-input observers

Giorgia Disarò, Maria Elena Valcher

TL;DR

This paper reframes unknown-input observer design for discrete-time LTI systems with disturbances within Willems' behavioral theory, using kernel and projection concepts to derive necessary and sufficient solvability conditions. It develops both model-based and data-driven formulations, linking acceptor behavior to asymptotic state tracking via a Schur-stable $A_{UIO}$ and providing constructive algorithms grounded in kernel/image representations. The data-driven section leverages historical trajectories and a persistence-of-excitation-like assumption to formulate a practical solvability criterion and observer construction directly from data. A numerical example demonstrates a valid, Schur-stable $UIO$ with explicit matrices, illustrating the approach's feasibility and potential for broader generalization.

Abstract

The purpose of this paper is to propose a novel perspective, based on Willems' "behavior theory", on the design of an unknown-input observer for a given linear time-invariant discrete-time state-space model, with unknown disturbances affecting both the state and the output equations. The problem is first addressed assuming that the original system model is known, and later assuming that the model is unknown but historical data satisfying a certain assumption are available. In both cases, fundamental concepts in behavior theory, as the projection of a behavior, the inclusion of a behavior in another one, and the use of kernel and image representations, provide quite powerful tools to determine necessary and sufficient conditions for the existence of an unknown-input observer (UIO), as well as algorithms to design one of them, if it exists.

Behaviors, trajectories and data: A novel perspective on the design of unknown-input observers

TL;DR

This paper reframes unknown-input observer design for discrete-time LTI systems with disturbances within Willems' behavioral theory, using kernel and projection concepts to derive necessary and sufficient solvability conditions. It develops both model-based and data-driven formulations, linking acceptor behavior to asymptotic state tracking via a Schur-stable and providing constructive algorithms grounded in kernel/image representations. The data-driven section leverages historical trajectories and a persistence-of-excitation-like assumption to formulate a practical solvability criterion and observer construction directly from data. A numerical example demonstrates a valid, Schur-stable with explicit matrices, illustrating the approach's feasibility and potential for broader generalization.

Abstract

The purpose of this paper is to propose a novel perspective, based on Willems' "behavior theory", on the design of an unknown-input observer for a given linear time-invariant discrete-time state-space model, with unknown disturbances affecting both the state and the output equations. The problem is first addressed assuming that the original system model is known, and later assuming that the model is unknown but historical data satisfying a certain assumption are available. In both cases, fundamental concepts in behavior theory, as the projection of a behavior, the inclusion of a behavior in another one, and the use of kernel and image representations, provide quite powerful tools to determine necessary and sufficient conditions for the existence of an unknown-input observer (UIO), as well as algorithms to design one of them, if it exists.
Paper Structure (7 sections, 6 theorems, 45 equations, 1 figure)

This paper contains 7 sections, 6 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Some basic notions about behaviors
  4. Behavioral approach to the design of a UIO
  5. Data-driven approach to the design of a UIO
  6. Example
  7. Conclusions

Key Result

Proposition 4

$\hat{\Sigma}$ is an acceptor for $\Sigma$ (i.e, condition P_incl holds) if and only if the matrices $A_{UIO}, B_{UIO}^u, B_{UIO}^y, D_{UIO}^u$ and $D_{UIO}^y$ satisfy the following conditions:

Figures (1)

  • Figure 1: Dynamics of the state estimation error

Theorems & Definitions (18)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Theorem 6
  • Theorem 7
  • proof
  • ...and 8 more