Pull-Based Distributed Event-Triggered Circle Formation Control for Multi-Agent Systems with Directed Topologies

Round 1

Reviewer 1 Report

This paper investigates a circle formation control problem for multi-agent systems with directed topologies by using pull-based distributed event-triggered control principles.

This paper contains some interesting results. The following comments are provided to help improve the quality of this paper.

1) The contribution of this paper is clear. It is suggested to further highlight the challenges of the investigated problem.

2) Compared with other existing event-triggered approaches, what is the key advantage of the pull based strategy?

3) In simulations, h is set to 0.2s, how to determine this value? If set h to a larger value, what will happen?

4) This paper is concerned with multi-agent control. The following related references may deserve the authors' attention. A complete literature review is suggested.
1: Prescribed finite-time consensus tracking for multi-agent systems with nonholonomic chained-form dynamics, IEEE Transactions on Automatic Control, vol. 64, no. 4, pp. 1686-1693, April 2019.
2: Collective behaviors of mobile robots beyond the nearest neighbor rules with switching topology, IEEE Transactions on Cybernetics, vol. 48, no. 5, pp. 1577-1590, May 2018.

5) The presentation of Lemma 2 is confusing. It requires the directed graph (strongly connected) contains one single spanning tree? If so, in Theorems 1 and 2,
the assumption of strongly connected directed graph should be changed to directed graph containing a spanning tree.

6) The language of this paper should be carefully polished. For example,
Page 8, "there are no trigger time for" ---> "there is no trigger time for"
Page 8, "Theorm 2" ---> "Theorem 2"
Page 9, "we further prove an estimate" ---> "we further provide an estimate"
Page 10, "then at any given time instant t, agent i can obtain..." ---> "at any given time instant t agent i can obtain..."

Author Response

We thank the editors and reviewers for their relevant and useful comments. Your careful reviews and insightful comments have greatly helped us in improving the technical quality and presentation quality of the paper. We have studied these comments carefully and have made the following revisions, which we hope will meet with approval.

In this document, we quote in purple words from the referees’ reports. Our replies follow in black words after “Response”. The modifications in the revised paper are highlighted in red.

Detailed answers to the reviewers’ comments and explanations of the corrections in the revised version of the manuscript are given as follows.

The contribution of this paper is clear. It is suggested to further highlight the challenges of the investigated problem.

Response: Thank you for providing this concern. The challenges of the investigated problem are described as follows: How to design the pull-based event-triggered condition for determining the pulling instants $t_{i}^{k}$? Under the designed control law (8) and event-triggered condition (36), can the circle formation problem of MASs over interactive topology be solved? As you suggested, we have highlighted the challenges of the investigated problem in the revised manuscript.

2) Compared with other existing event-triggered approaches, what is the key advantage of the pull based strategy?

Response: Thank you for your comment. We know that agent $i$ triggers at time $t_{i}^{k}$ means agent $i$ renews its control value at time $t_{i}^{k}$ and sends its state to all its out-neighbors immediately.  This does not mean that agent $i$ has to send a request to its in-neighbors at $t_{i}^{k}$ in order to get its in-neighbors’ states at $t_{i}^{k}$. Thus, the key advantage of the pull-based strategy is to avoid continuous transmission requests. In order to distinguish it from event-triggered, we name this sort of feedback as pull-based. We have highlighted this point in the revised manuscript.

3) In simulations, h is set to 0.2s, how to determine this value? If set h to a larger value, what will happen?

Response: Thank you for your comments. In order to easily display the main advantages of event-triggered control strategy, it is worth noting that the event detection of all those simulations is implemented in a sampled-data fashion. From Theorem 5 in [26], h should satisfy the condition (44) $ψ(k) = 1/(hλ_{N-k})$. If h set to larger than the permitted range, the circle formation problem is not solvable. We have made explanations in the revised manuscript.  

4) This paper is concerned with multi-agent control. The following related references may deserve the authors' attention. A complete literature review is suggested.
1: Prescribed finite-time consensus tracking for multi-agent systems with nonholonomic chained-form dynamics, IEEE Transactions on Automatic Control, vol. 64, no. 4, pp. 1686-1693, April 2019.
2: Collective behaviors of mobile robots beyond the nearest neighbor rules with switching topology, IEEE Transactions on Cybernetics, vol. 48, no. 5, pp. 1577-1590, May 2018.

Response: Thank you for providing these relevant papers for our reference. We have studied them carefully and cited them in the revised manuscript.

5) The presentation of Lemma 2 is confusing. It requires the directed graph (strongly connected) contains one single spanning tree? If so, in Theorems 1 and 2, 
the assumption of strongly connected directed graph should be changed to directed graph containing a spanning tree.

Response: Thank you for bringing this concern. In fact, for any two distinct nodes $i$ and $j$, we require the directed graph (strongly connected) contains one single spanning tree, and there exists a directed path from $j$ to $i$. We have modified the original expression in Theorems 1 and 2.

6) The language of this paper should be carefully polished. For example,
Page 8, "there are no trigger time for" ---> "there is no trigger time for"
Page 8, "Theorm 2" ---> "Theorem 2"
Page 9, "we further prove an estimate" ---> "we further provide an estimate"
Page 10, "then at any given time instant t, agent i can obtain..." ---> "at any given time instant t agent i can obtain..."

Response: Thank you so much for pointing this out. We have corrected all of them in the revised manuscript. In addition, we have gone through the whole paper carefully and tried our best to correct grammar mistakes and typos.

Author Response File: Author Response.pdf

Reviewer 2 Report

The authors have done a fine work in explaining the use of agents in this particular problem area. If there are areas to be considered for further improvement, I would suggest a short description for the readers on Zeno behaviour. Another area that could be re-worked are the the four separate figures identified as Figure 3 by making them more clear either by increasing the scale or the size so that it can be clearly evaluated.

Author Response

We thank the editors and reviewers for their relevant and useful comments. Your careful reviews and insightful comments have greatly helped us in improving the technical quality and presentation quality of the paper. We have studied these comments carefully and have made the following revisions, which we hope will meet with approval.

In this document, we quote in purple words from the referees’ reports. Our replies follow in black words after “Response”. The modifications in the revised paper are highlighted in red.

Detailed answers to the reviewers’ comments and explanations of the corrections in the revised version of the manuscript are given as follows.

The authors have done a fine work in explaining the use of agents in this particular problem area.

If there are areas to be considered for further improvement, I would suggest a short description for the readers on Zeno behaviour.

Another area that could be re-worked are the the four separate figures identified as Figure 3 by making them more clear either by increasing the scale or the size so that it can be clearly evaluated.

Response: Thank you so much for your interest in our work.

Zeno behavior is a phenomenon in hybrid systems that is of special interest, which can be described informally as the system making an infinite number of jumps in a finite amount of time; it exists when an infinite number of discrete transitions occur in a finite time interval.

As you suggested, we have incorporated the description of Zeno Behavior in the revised manuscript. In addition, we have increased the scale of the four separate figures to make it more clearly.

Author Response File: Author Response.pdf