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Distributed Unknown Input Observers for Discrete-Time Linear Time-Invariant Systems

Franco Angelo Torchiaro, Gianfranco Gagliardi, Francesco Tedesco, Alessandro Casavola

TL;DR

This work addresses distributed state estimation for discrete-time LTI systems subjected to unknown inputs and measurement noise. It introduces a Distributed Unknown Input Observer (D-UIO) that leverages node-wise detectability decomposition to split the state into locally detectable and undetectable components, enabling separate SDP-based design of local output injections and diffusive consensus gains. The design yields two LMIs-based procedures to compute stabilizing gains, guaranteeing bounded estimation error in both subspaces without requiring a connected communication topology. Numerical experiments on a ring network and a multi-room heat-transfer model demonstrate that all nodes can recover the full state despite unknown disturbances and measurement noise, highlighting the method’s scalability and robustness.

Abstract

This paper introduces a Distributed Unknown Input Observer (D-UIO) design methodology that uses a technique called node-wise detectability decomposition to estimate the state of a discrete-time linear time-invariant (LTI) system in a distributed way, even when there are noisy measurements and unknown inputs. In the considered scenario, sensors are associated to nodes of an underlying communication graph. Each node has a limited scope as it can only access local measurements and share data with its neighbors. The problem of designing the observer gains is divided into two separate sub-problems: (i) design local output injection gains to mitigate the impact of measurement noise, and (ii) design diffusive gains to compensate for the lack of information through a consensus protocol. A direct and computationally efficient synthesis strategy is formulated by linear matrix inequalities (LMIs) and solved via semidefinite programming. Finally, two simulative scenarios are presented to illustrate the effectiveness of the distributed observer when two different node-wise decompositions are adopted.

Distributed Unknown Input Observers for Discrete-Time Linear Time-Invariant Systems

TL;DR

This work addresses distributed state estimation for discrete-time LTI systems subjected to unknown inputs and measurement noise. It introduces a Distributed Unknown Input Observer (D-UIO) that leverages node-wise detectability decomposition to split the state into locally detectable and undetectable components, enabling separate SDP-based design of local output injections and diffusive consensus gains. The design yields two LMIs-based procedures to compute stabilizing gains, guaranteeing bounded estimation error in both subspaces without requiring a connected communication topology. Numerical experiments on a ring network and a multi-room heat-transfer model demonstrate that all nodes can recover the full state despite unknown disturbances and measurement noise, highlighting the method’s scalability and robustness.

Abstract

This paper introduces a Distributed Unknown Input Observer (D-UIO) design methodology that uses a technique called node-wise detectability decomposition to estimate the state of a discrete-time linear time-invariant (LTI) system in a distributed way, even when there are noisy measurements and unknown inputs. In the considered scenario, sensors are associated to nodes of an underlying communication graph. Each node has a limited scope as it can only access local measurements and share data with its neighbors. The problem of designing the observer gains is divided into two separate sub-problems: (i) design local output injection gains to mitigate the impact of measurement noise, and (ii) design diffusive gains to compensate for the lack of information through a consensus protocol. A direct and computationally efficient synthesis strategy is formulated by linear matrix inequalities (LMIs) and solved via semidefinite programming. Finally, two simulative scenarios are presented to illustrate the effectiveness of the distributed observer when two different node-wise decompositions are adopted.

Paper Structure

This paper contains 16 sections, 2 theorems, 56 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem Formulation
  4. Structure of the Distributed Unknown Input Observer
  5. Rank condition for UIO existence
  6. Detectability decomposition
  7. Distributed Unknown Input Observer Algorithm
  8. Design of the D-UIO parameters
  9. Analyses of the error evolution
  10. Design of the stabilizing gain for the detectable subspace
  11. Design of the stabilizing gain for the undetectable subspace
  12. Numerical Example
  13. Example based on the detectability decomposition
  14. Example based on the observability decomposition
  15. Conclusions
  16. ...and 1 more sections

Key Result

Proposition 1

Let Assumption ass:bounded_meas_err hold true. Given the closed-loop error dynamics (eq:eid_step2) and the $H_{\infty}$ performance index (eq:h_inf_p), if matrices $P^* = (P^*)^T > 0$, $Y_d^*$ and a scalar $\beta_{id}^*$ exist that solve the following semidefinite programming (SDP) optimization prob then the gain $K_{id} = (P^*)^{-1}Y_d^*$ ensures that the closed-loop error dynamics (eq:eid_step2)

Figures (8)

  • Figure 1: Distributed Observer Scheme: Each node $N_i$ collects locally available data, including the sensor output $y_i$ from its corresponding sensor $S_i$ and the local input $u_i$.
  • Figure 2: Evolution of the system's state affected by the disturbance $w(k)$.
  • Figure 3: Error evolution for each node of the $H_{\infty}$ optimal distributed unknown input observer.
  • Figure 4: Error evolution for each node recorded when $K_{id}$ are determined with a non-optimal pole-placement strategy.
  • Figure 5: Observers configuration considered in the two scenarios. (a) a connected topology (b) a partitioned topology.
  • ...and 3 more figures

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Proposition 1
  • Proposition 2
  • Remark 3