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Exact Leader Estimation: A New Approach for Distributed Differentiation

Rodrigo Aldana-Lopez, David Gomez-Gutierrez, Elio Usai, Hernan Haimovich

TL;DR

The paper solves the distributed leader-observer problem by introducing a homogeneous, Levant-inspired differentiator that enables each follower to estimate the leader's state and its derivatives up to order $m$ using only scalar neighbor outputs, even when the leader's identity is unknown and the leader is driven by a bounded input with unknown higher-order derivative. It proves finite-time, exact convergence in the noise-free continuous-time setting, and extends the framework to bounded measurement noise and sampled-data communication, providing explicit steady-state error bounds that scale homogeneously with the noise level. Gains are designed either explicitly for $m=1$ or recursively via a Redundant Exponential Differentiator (RED) style approach for $m>1$, ensuring contraction of the error dynamics. Numerical results for second- and high-order cases validate finite-time tracking of the leader signal and its derivatives, and demonstrate robustness to measurement noise and discretization. The approach reduces communication load by exchanging only scalar observer outputs while guaranteeing accurate leader differentiation, with practical implications for formation control and multi-agent coordination.

Abstract

A novel strategy aimed at cooperatively differentiating a signal among multiple interacting agents is introduced, where none of the agents needs to know which agent is the leader, i.e. the one producing the signal to be differentiated. Every agent communicates only a scalar variable to its neighbors; except for the leader, all agents execute the same algorithm. The proposed strategy can effectively obtain derivatives up to arbitrary $m$-th order in a finite time under the assumption that the $(m+1)$-th derivative is bounded. The strategy borrows some of its structure from the celebrated homogeneous robust exact differentiator by A. Levant, inheriting its exact differentiation capability and robustness to measurement noise. Hence, the proposed strategy can be said to perform robust exact distributed differentiation. In addition, and for the first time in the distributed leader-observer literature, sampled-data communication and bounded measurement noise are considered, and corresponding steady-state worst-case accuracy bounds are derived. The effectiveness of the proposed strategy is verified numerically for second- and fourth-order systems, i.e., for estimating derivatives of up to first and third order, respectively.

Exact Leader Estimation: A New Approach for Distributed Differentiation

TL;DR

The paper solves the distributed leader-observer problem by introducing a homogeneous, Levant-inspired differentiator that enables each follower to estimate the leader's state and its derivatives up to order using only scalar neighbor outputs, even when the leader's identity is unknown and the leader is driven by a bounded input with unknown higher-order derivative. It proves finite-time, exact convergence in the noise-free continuous-time setting, and extends the framework to bounded measurement noise and sampled-data communication, providing explicit steady-state error bounds that scale homogeneously with the noise level. Gains are designed either explicitly for or recursively via a Redundant Exponential Differentiator (RED) style approach for , ensuring contraction of the error dynamics. Numerical results for second- and high-order cases validate finite-time tracking of the leader signal and its derivatives, and demonstrate robustness to measurement noise and discretization. The approach reduces communication load by exchanging only scalar observer outputs while guaranteeing accurate leader differentiation, with practical implications for formation control and multi-agent coordination.

Abstract

A novel strategy aimed at cooperatively differentiating a signal among multiple interacting agents is introduced, where none of the agents needs to know which agent is the leader, i.e. the one producing the signal to be differentiated. Every agent communicates only a scalar variable to its neighbors; except for the leader, all agents execute the same algorithm. The proposed strategy can effectively obtain derivatives up to arbitrary -th order in a finite time under the assumption that the -th derivative is bounded. The strategy borrows some of its structure from the celebrated homogeneous robust exact differentiator by A. Levant, inheriting its exact differentiation capability and robustness to measurement noise. Hence, the proposed strategy can be said to perform robust exact distributed differentiation. In addition, and for the first time in the distributed leader-observer literature, sampled-data communication and bounded measurement noise are considered, and corresponding steady-state worst-case accuracy bounds are derived. The effectiveness of the proposed strategy is verified numerically for second- and fourth-order systems, i.e., for estimating derivatives of up to first and third order, respectively.

Paper Structure

This paper contains 17 sections, 15 theorems, 56 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main result
  3. Problem Statement: Distributed leader-observer proposal
  4. Main Theorem
  5. Proof of the main result
  6. Proof of Theorem \ref{['th:main']} for $m=1$
  7. Proof of Theorem \ref{['th:main']} for arbitrary $m>1$
  8. Performance Analysis
  9. Robustness analysis under bounded measurement noise
  10. Sampled data implementation
  11. Numerical example
  12. Second order system
  13. High order system
  14. Conclusions
  15. Auxiliary results
  16. ...and 2 more sections

Key Result

Proposition 1

Let Assumption as:graph hold. Assume that $\mathbf{e}_\mu(t)=\mathbf{0}$ for all $\mu\in\{0,\dots,m\}$ at a given $t\in\mathbb{R}$. Then, it follows that $x_{i,\mu}(t)=u^{(\mu)}(t)$.

Figures (2)

  • Figure 1: Trajectories for the leader position $u(t)$ and its derivative $\dot{u}(t)$ (red), as well as the trajectories for $x_{i,0}(t), x_{i,1}(t)$ for all agents $i\in\mathcal{I}$ (blue) with $m=1$. Two scenarios are considered, namely $\overline{\varepsilon}=0$ (noiseless) and $\overline{\varepsilon} = 0.1$. Due to the sampled-data implementation, only $t=0, \Delta, 2\Delta, \dots$ with $\Delta=10^{-3}s$ are depicted.
  • Figure 2: Trajectories for the leader position $u(t)$ and its derivatives $\dot{u}(t),\ddot{u}(t),{u}^{(3)}(t)$ (red), as well as the trajectories for $x_{i,0}(t), x_{i,1}(t), x_{i,2}(t), x_{i,3}(t)$ for all agents $i\in\mathcal{I}$ (blue) with $m=3$. Two scenarios are considered, namely $\overline{\varepsilon}=0$ (noiseless) and $\overline{\varepsilon} = 0.1$. Due to the sample-data implementation, only $t=0, \Delta, 2\Delta, \dots$ with $\Delta=10^{-3}$ are depicted.

Theorems & Definitions (28)

  • Proposition 1
  • proof
  • Definition 1
  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 18 more