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Distributed Estimation and Control for LTI Systems under Finite-Time Agreement

Camilla Fioravanti, Evagoras Makridis, Gabriele Oliva, Maria Vrakopoulou, Themistoklis Charalambous

TL;DR

This work tackles distributed stabilization of a discrete-time LTI system over a strongly connected network, assuming joint observability and controllability but only partial local information at each agent. It introduces a finite-time ratio consensus mechanism between estimation-control steps to compute the network-average state, enabling arbitrary placement of the estimation-error eigenvalues via distributed gains. A three-stage initialization and a token-passing protocol allow all agents to compute locally feasible controller gains $K_i$ (and observer gains $L_i$) in finite time, removing centralized design requirements. The resulting fully distributed scheme demonstrates stability guarantees and improved performance over asymptotic consensus methods in simulations. The framework offers resilience and scalability for multi-agent systems requiring coordinated estimation and control without centralized coordination.

Abstract

This paper considers a strongly connected network of agents, each capable of partially observing and controlling a discrete-time linear time-invariant (LTI) system that is jointly observable and controllable. Additionally, agents collaborate to achieve a shared estimated state, computed as the average of their local state estimates. Recent studies suggest that increasing the number of average consensus steps between state estimation updates allows agents to choose from a wider range of state feedback controllers, thereby potentially enhancing control performance. However, such approaches require that agents know the input matrices of all other nodes, and the selection of control gains is, in general, centralized. Motivated by the limitations of such approaches, we propose a new technique where: (i) estimation and control gain design is fully distributed and finite-time, and (ii) agent coordination involves a finite-time exact average consensus algorithm, allowing arbitrary selection of estimation convergence rate despite the estimator's distributed nature. We verify our methodology's effectiveness using illustrative numerical simulations.

Distributed Estimation and Control for LTI Systems under Finite-Time Agreement

TL;DR

This work tackles distributed stabilization of a discrete-time LTI system over a strongly connected network, assuming joint observability and controllability but only partial local information at each agent. It introduces a finite-time ratio consensus mechanism between estimation-control steps to compute the network-average state, enabling arbitrary placement of the estimation-error eigenvalues via distributed gains. A three-stage initialization and a token-passing protocol allow all agents to compute locally feasible controller gains (and observer gains ) in finite time, removing centralized design requirements. The resulting fully distributed scheme demonstrates stability guarantees and improved performance over asymptotic consensus methods in simulations. The framework offers resilience and scalability for multi-agent systems requiring coordinated estimation and control without centralized coordination.

Abstract

This paper considers a strongly connected network of agents, each capable of partially observing and controlling a discrete-time linear time-invariant (LTI) system that is jointly observable and controllable. Additionally, agents collaborate to achieve a shared estimated state, computed as the average of their local state estimates. Recent studies suggest that increasing the number of average consensus steps between state estimation updates allows agents to choose from a wider range of state feedback controllers, thereby potentially enhancing control performance. However, such approaches require that agents know the input matrices of all other nodes, and the selection of control gains is, in general, centralized. Motivated by the limitations of such approaches, we propose a new technique where: (i) estimation and control gain design is fully distributed and finite-time, and (ii) agent coordination involves a finite-time exact average consensus algorithm, allowing arbitrary selection of estimation convergence rate despite the estimator's distributed nature. We verify our methodology's effectiveness using illustrative numerical simulations.
Paper Structure (12 sections, 2 theorems, 19 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 12 sections, 2 theorems, 19 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and Graph Theory
  4. Average (Minimum-Time) Consensus in Directed Graphs
  5. Distributed Termination and Synchronization
  6. Problem Statement
  7. Finite-time Average Consensus as an Enabler for Distributed Estimation and Control
  8. Distributed selection of the Gains
  9. Single Eigenvalue Placement
  10. Proposed Protocol for Choosing the Gains
  11. Illustrative Examples
  12. Conclusions and Future Directions

Key Result

Theorem 1

Let Assumption ass:1 hold. Then, the local estimation errors, ${\bm e}_i[k]$, can be brought to zero asymptotically, by selecting suitable gain matrices $L_i$ that assign arbitrary eigenvalues to $A-\frac{1}{N}\sum_{l=1}^N L_lC_l$.

Figures (3)

  • Figure 1: Finite-time average consensus iterations $m$ (blue ticks) within distributed estimation and control iterations $k$ (red ticks).
  • Figure 2: Proposed distributed estimation and control scheme.
  • Figure 3: Comparison of the estimation error between the proposed approach (solid line, $\overline{m}=11$) and the one in savas2022separation when $m\in\{6,11,22\}$, for different communication duration $\tau \in \{0.1, 1, 10\}$.

Theorems & Definitions (8)

  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Theorem 2
  • proof
  • Remark 3
  • Remark 4