# Precise Dynamic Consensus under Event-Triggered Communication

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Problem Statement and Protocol Proposal

#### 2.1. Notation

#### 2.2. Problem Statement

#### 2.3. Protocol Proposal

**Event-triggered communication**: To reduce the communication burden in the network, an event-triggered approach is used to decide when an agent communicates with its neighbours. Denote with ${\widehat{z}}_{i}\left[k\right]$ the estimation that agent $i\in \mathcal{V}$ has for the signal $\overline{z}\left(t\right)$ at $t=k\Delta $. Then, the communication link $(i,j)\in \mathcal{E}$ is updated at the sequence of instants ${\left\{{\ell}_{k}^{ij}\right\}}_{k=0}^{\infty}$ provided by the recursive triggering rule:

Algorithm 1: Exact dynamic consensus with event-triggered communication |

**Assumption**

**A1.**

**Theorem**

**1.**

**Corollary**

**1.**

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

## 3. Protocol Convergence

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

#### 3.1. Proof of Theorem 1

## 4. Numerical Experiments

## 5. Comparison to Related Work

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. The EDCHO Protocol

**Proposition**

**A1**

**Corollary**

**A1.**

## Appendix B. Taylor Theorem with Integral Remainder

**Proposition**

**A2**

**Corollary**

**A2.**

**Proof.**

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**Figure 1.**Block diagram for each agent $i\in \mathcal{V}$. The internal states that the agent computes are updated using the last transmitted estimates of the agent and its neighbours $j\in {\mathcal{N}}_{i}$, which are denoted by ${\widehat{z}}_{i}\left[{\ell}_{k}^{ij}\right]$ and ${\widehat{z}}_{j}\left[{\ell}_{k}^{ij}\right]$, respectively. The current estimate ${\widehat{z}}_{i}\left[k\right]$ is computed using the internal state ${\mathsf{\chi}}_{i,0}\left[k\right]$ and the local signal ${z}_{i}\left[k\right]$. This estimate is evaluated by the event trigger, which decides if the current value should be transmitted to the neighbours.

**Figure 3.**Consensus results for our protocol with $\Delta ={10}^{-4},\epsilon =0$ (full communication). By using a small time step and communication at every step, our protocol obtains highly precise results. (

**a**) Consensus estimates for the average signal using our proposal. The desired value $\overline{z}\left(t\right)$ is plotted as a dashed red line. The estimates quickly converge to the average signal. (

**b**) Consensus error of the estimates with respect to the average signal. The error converges to a neighbourhood of zero, which can be made arbitrarily small by tuning the value of the sampling period.

**Figure 4.**Consensus error for our protocol with $\Delta ={10}^{-4},\epsilon =0.01$. The addition of event-triggered communication increases the steady-state consensus error with respect to the full communication case (see Figure 3) and can be tuned with $\epsilon $.

**Figure 5.**Consensus results for our protocol with $\Delta ={10}^{-4},\epsilon =1$. Due to the increase in the event-triggering threshold, the error is higher than in the case with $\epsilon =0.01$, shown in Figure 4. (

**a**) Consensus estimates for the average signal using our proposal. The desired value $\overline{z}\left(t\right)$ is plotted as a dashed red line. The estimates quickly converge to the average signal, but with an increased steady-state error due to the event-triggered communication. (

**b**) Consensus error of the estimates with respect to the average signal. Increasing the event-triggering threshold $\epsilon $ causes a higher steady-state error.

**Figure 6.**Consensus error for our protocol with $\Delta ={10}^{-3},\epsilon =0.01$. Compared to the case shown in Figure 4, with $\Delta ={10}^{-4}$, the consensus error has increased.

**Figure 7.**Trade-off between the consensus error and the communication rate (with fixed $\Delta ={10}^{-4}$) and trade-off with the sampling period (with fixed $\epsilon =0.01)$. As the value of the triggering threshold $\epsilon $ increases, the communication through the network is reduced, at the cost of a higher consensus error. However, using the event-triggering mechanism, communication can be significantly reduced with respect to the full communication case (communication rate of 100%), with a relatively small increase in the steady-state error. The consensus error also increases with the sampling period $\Delta $. When $\Delta \to 0$, the parameter $\epsilon $ has a higher impact than $\Delta $ on the magnitude of the steady-state error.

**Figure 8.**Comparison of consensus error with linear protocol, FOSM, and our protocol, under full communication ($\Delta ={10}^{-4},\epsilon =0$). With the linear protocol, there exists a permanent steady-state error, which is higher than for the other protocols. Both the FOSM and our protocol can eliminate the error in a continuous-time implementation, but in the discretized setup our method improves the error caused by chattering with respect to the FOSM protocol.

**Figure 9.**Consensus results for the linear protocol with $\Delta ={10}^{-4},\epsilon =0$, showing the estimates of the derivatives of first (

**a**), second (

**b**), and third (

**c**) order. The derivatives ${\overline{z}}^{\left(1\right)}\left(t\right),{\overline{z}}^{\left(2\right)}\left(t\right),{\overline{z}}^{\left(3\right)}\left(t\right)$ are plotted as a dashed red line. High-order derivatives are not accurately computed using a linear protocol.

**Figure 10.**Consensus results for our protocol with $\Delta ={10}^{-4},\epsilon =0.001$, showing the estimates of the derivatives of first (

**a**), second (

**b**), and third (

**c**) order. The derivatives ${\overline{z}}^{\left(1\right)}\left(t\right),{\overline{z}}^{\left(2\right)}\left(t\right),{\overline{z}}^{\left(3\right)}\left(t\right)$ are plotted as a dashed red line. The error induced by the discretization and event-triggered communication is more apparent in higher-order derivatives, but the estimates are greatly improved with respect to the linear protocol (compare to Figure 9).

**Table 1.**Comparison of numerical results for different protocols. Our proposal reduces the estimation error with respect to linear and FOSM protocols in the full communication case. Moreover, adding event-triggered communication to our proposal produces a significant reduction in communication, while still achieving better error values than other protocols.

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**MDPI and ACS Style**

Perez-Salesa, I.; Aldana-Lopez, R.; Sagues, C.
Precise Dynamic Consensus under Event-Triggered Communication. *Machines* **2023**, *11*, 128.
https://doi.org/10.3390/machines11020128

**AMA Style**

Perez-Salesa I, Aldana-Lopez R, Sagues C.
Precise Dynamic Consensus under Event-Triggered Communication. *Machines*. 2023; 11(2):128.
https://doi.org/10.3390/machines11020128

**Chicago/Turabian Style**

Perez-Salesa, Irene, Rodrigo Aldana-Lopez, and Carlos Sagues.
2023. "Precise Dynamic Consensus under Event-Triggered Communication" *Machines* 11, no. 2: 128.
https://doi.org/10.3390/machines11020128