Event-Triggered Control Protocols for Achieving Bipartite Consensus in Switched Multi-Agent Systems

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper proposes an event-triggered approach for achieving bipartite consensus in switching MASs. After reviewing this paper, I have the following comments:

1. The abstract states that the paper extends bipartite consensus to switched systems using event-triggered control. Could you more explicitly delineate the novel theoretical contributions compared to existing works on ETC for switched multi-agent systems?
2. How are the parameters in (14) and (22) chosen to balance between communication reduction and consensus performance? Is there a systematic tuning procedure?
3. Remark 4 mentions enforcing a minimum inter-event time to prevent Zeno behavior. Could you provide a proof or sufficient conditions that ensure the existence of such a minimum interval for both linear and nonlinear cases?
4. Could you discuss how your method compares to recent works such as “Prescribed-time fully distributed Nash equilibrium seeking of nonlinear multi-agent systems over unbalanced digraphs” and “Dynamic event-triggered reinforcement learning-based consensus tracking of nonlinear multi-agent systems”, particularly in handling nonlinear dynamics and unbalanced structures?
5. Based on your results, what are the most promising extensions of this work?

Author Response

Comment 1:

 

The comparison with existing work is insufficient, and the literature review is not thorough enough. It is necessary to clearly point out the differences and advantages of this paper compared to existing work.

 

Response:

 

We sincerely thank the reviewer for this insightful comment. We fully agree with the observation. To address this, we have revised the contribution statement in both the abstract and the introduction to clearly emphasize that the core novelty of this work lies in extending bipartite consensus to switched multi-agent systems—where the system dynamics switch among subsystems—within an event-triggered control (ETC) framework.

 

Specifically, we have added the following comparison in the Introduction:

 

“Notably, prior works on event-triggered control (ETC) for multi-agent systems have mainly considered switching communication topologies [35–38]. In contrast, this paper investigates ETC for achieving bipartite consensus in switched multi-agent systems where the switching arises from changes in the agents’ intrinsic dynamics. Research of this form remains relatively scarce.”

 

This clarification highlights the key distinction between topology switching (common in prior work) and dynamic switching (the focus of our paper), thereby strengthening the novelty claim.

 

Comment 2:

 

The paper assumes that the system topology always satisfies structural balance (Assumption 4) and that all subsystems meet the connectivity condition (Assumption 5). These assumptions are overly idealized in practical applications. The lack of consideration for imbalance greatly reduces the practical value of the results.

 

Response:

 

We acknowledge the reviewer’s concern. Structural balance is indeed a standard prerequisite for bipartite consensus, as established in foundational works such as Altafini (2013). Our primary focus in this paper is on the control design under switching dynamics and event-triggered communication, rather than relaxing topological assumptions.

 

That said, we fully agree that handling unbalanced or time-varying signed graphs is an important and practical direction. Therefore, we have explicitly added this topic to the “Future Work” section in the Conclusion:

 

“How to investigate bipartite consensus under unbalanced network structures will be the focus of our future research.”

 

Comment 3:

 

There is a lack of sufficient explanation of the experimental results, and the conclusion section lacks discussion of future work.

 

Response:

 

Thank you for your valuable comment. We have made two key improvements:

 

Added detailed parameters to each subsection of the Simulation Results section (e.g., controller gains, triggering thresholds, initial conditions, switching sequences).

Included a new paragraph titled “Future Work” at the end of the Conclusion, which outlines promising extensions such as unbalanced topologies and integration with learning-based methods.

These additions enhance the interpretability of the simulations and provide clear directions for future research.

 

Comment 4:

 

Figure 2, 3, 5, 6 is not clear enough; the agent states and trigger times should be separated. The individual figures are provided in the supplementary material.

 

Response:

 

Thank you for this suggestion. We have regenerated all figures with improved resolution and clearer legends. Specifically:

 

Agent state trajectories are now plotted with distinct line styles or colors.

Triggering instants are marked with vertical dashed lines or discrete markers (e.g., circles or stars) on a separate axis or layer.

Figure captions explicitly describe what each curve and marker represents.

These updated figures ensure that state evolution and communication events are easily distinguishable.

 

Comment 5:

 

Spelling issues: some agent systems are missing the plural 's', e.g., "multi-agent system" should be "multi-agent systems".

 

Response:

 

Thank you for catching this. The entire manuscript has been carefully proofread, and all instances of “multi-agent system(s)” have been corrected for grammatical consistency. For example:

 

“a multi-agent system” → used when referring to a single instance

“multi-agent systems” → used in general or plural contexts

This ensures linguistic accuracy throughout the paper.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

1. The abstract states the paper studies bipartite consensus "in multi-agent systems" but the title and introduction specify switched MAS. The introduction highlights a gap for "disturbed switched systems," yet disturbances (`C_i(t)`) appear in the problem statement for all system types. Is the core novelty the extension to switched systems, to disturbed systems, or the combination of both within an ETC framework? The contribution statement should be sharpened to clarify this.
2. The introduction claims that extending ETC to "achieving bipartite consensus in disturbed switched systems has not been examined." However, several cited references (e.g., [35], [36], [38]) deal with switching topologies or nonlinear switched MASs under ETC. What specifically distinguishes your "switched system" (where `A_σ, B_σ, f_σ` switch) from "switching topologies" in these references, and why is your combination with bipartite consensus and disturbances a novel contribution?
3. In Assumption 1, the LMI (5) is central to the linear controller design. What is the explicit method or procedure for finding the positive definite matrix `P_σ` and the parameter `ξ` that satisfies this LMI for each subsystem? The paper would benefit from mentioning a standard approach (e.g., using LMI solvers).
4. The proof of Theorem 1 (Section 3) concludes that `V̇₁(t) < 0` under the triggering condition (14). However, the derivation uses `ξ = λ_min(L)` from the transformed Laplacian `L`. Is it guaranteed that `L` (or `L_T`) is always positive definite for the considered structurally balanced, connected graphs, including a leader? A brief justification or citation is needed here, as the standard Laplacian of a connected graph is only positive semi-definite.
5. In the switched system analysis (Section 4), Assumption 6 and the proof require `P_a ≤ μ P_b` for all switching pairs. How are these matrices `P_σ` chosen to ensure that this uniform bound `μ` exists across all subsystems? This is a non-trivial condition for switched system stability, and the design process should be elaborated.
6. For the nonlinear systems (Sections 5 & 6), the controller gain is a scalar `k` (or `K_σ`). How is this gain parameter selected or designed to ensure consensus, given the Lipschitz condition (Assumption 7) and the matrix `Γ`? The relation between `k`, `ρ`, and the eigenvalues of `Γ ⊗ L` seems crucial but is not detailed.
7. In the nonlinear switched system section (6), the proof is omitted as it is "analogous" and "similar to established methods." However, Assumption 8 introduces a family of Lyapunov functions `V_i(x)` with specific decrease and bounded jump conditions. Does the chosen Lyapunov candidate `V = X_e^T X_e` actually satisfy Assumption 8 for the nonlinear switched system (42)-(43)? This should be verified, not assumed by analogy.
8. The event-triggering condition (7) states that the next event occurs when `Q ≥ 0` or at a topology switching instant `t_δ`. Does a forced communication at every switching instant (`t_δ`) not potentially increase communication burden significantly, especially under fast switching? What is the rationale for this mandatory update, and is there a trade-off analysis?
9. Remark 4 mentions enforcing a "minimum inter-event time `t` (with `t > 0`)" to prevent Zeno behavior. What is the explicit lower bound for this minimum time, and how is it derived from the system parameters (e.g., `A, B, K`, Lipschitz constant `ρ`)? A theoretical exclusion of Zeno behavior is essential for ETC schemes.
10. The simulation sections present results but lack key implementation details. What were the specific parameters `a₁, a₂` used in the triggering function `Q` (Eq. 14, 22) for the simulations? How were they selected to satisfy the inequality `a₁β₁ + a₂β₂ - ξ < 0`?
11. In the nonlinear examples (Figs. 5 & 6), the agents seem to use the Lorenz system dynamics. Are these chaotic dynamics (`α=10, β=8/3, γ=28`) acting as the nonlinear intrinsic dynamics `f(x)` or as an external disturbance? If it's the intrinsic dynamics, the initial states appear very far from the attractor; how does this affect the consensus manifold's interpretation?

1. The simulation results effectively validate the proposed method. However, providing more details on the simulation environment or MATLAB program (at least in review purpose) would enhance the credibility of the results.

13. There is a significant formatting issue: The manuscript header "Version December 3, 2025 submitted to Journal Not Specified" appears on multiple pages, and the citation placeholder ("Lastname, F.;...") is still in the text. This must be corrected before any review process.
14. In Section 3 (Linear Time-Invariant), the text abruptly states: "Suppose the MASs described by (1) satisfy Assumptions 1, 2, and 4. Then, under the event-triggered controller given by (11)... the consensus... is achieved. The triggering function Q is designed as follows:" This is followed by a sentence beginning "Suppose the multi-agent system described by (19)..." which belongs to Section 4. This appears to be a cut-and-paste error or misplaced text. Please clarify and correctly segment Theorem 1 and its proof.
15. Figure, Table, and Reference calls are inconsistent. The text refers to "Figure 1", "Figure 2", etc., but the PDF content shows figure captions without numbers (e.g., "topology of linear time-invariant system"). Are the figure placements and numbering in the submitted manuscript correct and sequential? Also, references [25] and [41] are duplicates.
16. The Data Availability Statement says "data are not available due to technical limitations." What "data" does this refer to? The simulations presumably generate state trajectories and event times from deterministic equations. Does this mean the simulation code is not shared? This statement should be more precise.

 

Comments on the Quality of English Language

N/A

Author Response

Comment 1:
The abstract states the paper studies bipartite consensus "in multi-agent systems" but the title and introduction specify switched MAS. The introduction highlights a gap for "disturbed switched systems," yet disturbances (C_i(t)) appear in the problem statement for all system types. Is the core novelty the extension to switched systems, to disturbed systems, or the combination of both within an ETC framework? The contribution statement should be sharpened to clarify this.

Response:
We sincerely appreciate this insightful comment. To address this concern, we have revised both the abstract and the introduction to explicitly emphasize that the core novelty of this work lies in integrating switched system dynamics and external disturbances within an event-triggered control (ETC) framework for achieving bipartite consensus. Specifically:

•  

The first sentence of the abstract now reads:
“This paper investigates the bipartite consensus problem for multi-agent systems subject to both switching dynamics and external disturbances within an event-triggered control (ETC) framework.”

•  
•  

In the introduction, we added the following contributions:
• Research on multi-agent consensus under dynamically varying system dynamics remains relatively scarce; this paper provides incremental contributions to this direction.
• By integrating switched systems and disturbed systems within the ETC framework, the proposed approach better reflects real-world application scenarios, thereby enhancing the generality and practical relevance of the theoretical results.
• This work extends the study of multi-agent systems to the domain of bipartite consensus. Specifically, by introducing a transformation matrix, the bipartite consensus problem is reformulated as a standard consensus problem.

•  

Comment 2:
The introduction claims that extending ETC to "achieving bipartite consensus in disturbed switched systems has not been examined." However, several cited references (e.g., [35], [36], [38]) deal with switching topologies or nonlinear switched MASs under ETC. What specifically distinguishes your "switched system" (where A_σ, B_σ, f_σ switch) from "switching topologies" in these references, and why is your combination with bipartite consensus and disturbances a novel contribution?

Response:
Thank you for this important clarification request. As noted in the revised manuscript (Lines 69–71), prior works such as [35]–[38] primarily focus on switching communication topologies under ETC. In contrast, our work considers switching intrinsic system dynamics, i.e., each subsystem possesses distinct state matrices A_σ, input matrices B_σ, and/or nonlinear functions f_σ(·).

This distinction introduces additional stability challenges beyond topology changes, as the agents’ internal dynamics themselves evolve over time. The integration of dynamic switching + external disturbances + bipartite consensus + ETC constitutes the key novelty of our approach. We have added the following clarification in the Introduction:
“Notably, prior works on event-triggered control (ETC) for multi-agent systems have mainly considered switching communication topologies [35–38]. In contrast, this paper investigates ETC for achieving bipartite consensus in switched multi-agent systems where the switching arises from changes in the agents’ intrinsic dynamics. Research of this form remains relatively scarce.”

Comment 3:
In Assumption 1, the LMI (5) is central to the linear controller design. What is the explicit method or procedure for finding the positive definite matrix P_σ and the parameter ζ that satisfies this LMI for each subsystem? The paper would benefit from mentioning a standard approach (e.g., using LMI solvers).

Response:
Thank you for this helpful suggestion. The LMI in (5) can be solved numerically using standard convex optimization toolboxes such as MATLAB’s LMI Control Toolbox or YALMIP with SDP solvers (e.g., SeDuMi or MOSEK). For completeness, we include a representative code snippet used in our simulations:

L_eig = eigs(L);min_eigenvalue = eigs(L, 1, 'smallestreal');epsilon_min = 0.2679;Q = epsilon_min * eye(3);R = (1 / (2 * epsilon_min)) * eye(1);X = icare(A, B, Q, R);K = B' * X;

This information has been added to the revised manuscript.

Comment 4:
The proof of Theorem 1 (Section 3) concludes that V₁(t) < 0 under the triggering condition (14). However, the derivation uses ζ = λ_min(L̃) from the transformed Laplacian L̃. Is it guaranteed that L̃ (or L̃_T) is always positive definite for the considered structurally balanced, connected graphs, including a leader? A brief justification or citation is needed here, as the standard Laplacian of a connected graph is only positive semi-definite.

Response:
We thank the reviewer for this careful observation. Indeed, the standard Laplacian is only positive semi-definite. However, in our setting, the matrix used is L̃ = L + G, where G = diag{g₁, …, gₙ} encodes leader-follower communication links, with at least one gᵢ > 0. Since the signed graph is connected and structurally balanced, and there exists at least one connection to the leader, L̃ is positive definite. This ensures ζ = λ_min(L̃) > 0, which justifies the use of ζ in the Lyapunov analysis.

(Note: In Word, you can type L̃ using Insert → Symbol or by typing "L" followed by Unicode U+0303 for the tilde. Subscripts like g₁ can be formatted using Word’s subscript feature.)

Comment 5:
In the switched system analysis (Section 4), Assumption 6 and the proof require that P_σa ≤ μ P_σb for all switching pairs. How are these matrices P_σ chosen to ensure that this uniform bound μ exists across all subsystems? This is a non-trivial condition for switched system stability, and the design process should be elaborated.

Response:
We appreciate this excellent point. After independently solving the LMI in Assumption 1 for each subsystem to obtain P_σ, we compute the pairwise comparison metric for all subsystem pairs (a, b). The constant μ is then selected as:
μ = max_{a,b ∈ P} { λ_max(P_a P_b⁻¹) }
This guarantees P_a ≤ μ P_b for all switching pairs. We have added this explanation after Remark 6, which now reads:

Remark 6: When switching occurs in the system, energy transitions arise between the pre-switching and post-switching states. From an energetic perspective, the parameter μ represents the multiplicative factor of energy growth. Across multiple switching transitions, the energy growth factor admits a finite upper bound. Specifically, after independently solving for each subsystem's Lyapunov matrix P_σ, the discrepancy between any two matrices P_a and P_b can be quantified by the maximum eigenvalue of their product P_a P_b⁻¹. The parameter μ is then defined as above.

Comment 6:
For the nonlinear systems (Sections 5 & 6), the controller gain is a scalar k (or k_σ). How is this gain parameter selected or designed to ensure consensus, given the Lipschitz condition (Assumption 7) and the matrix Γ? The relation between k, p, and the eigenvalues of Γ ⊗ L̃ seems crucial but is not detailed.

Response:
Thank you for highlighting this. In the revised manuscript, we clarify that Γ = diag{k₁, k₂, …, kₙ}, where kᵢ > 0 for all i. The gain parameters kᵢ are chosen such that the matrix Γ ⊗ L̃ yields sufficient coupling strength to dominate the nonlinear term governed by the Lipschitz constant p. By tuning kᵢ, we effectively adjust the eigenvalues of Γ ⊗ L̃, ensuring the overall system satisfies the convergence condition derived from the Lyapunov analysis. This clarification has been added to Section 5.

Comment 7:
In the nonlinear switched system section (6), the proof is omitted as it is "analogous" and "similar to established methods." However, Assumption 8 introduces a family of Lyapunov functions V_i(x) with specific decrease and bounded jump conditions. Does the chosen Lyapunov candidate V = X_eᵀ X_e actually satisfy Assumption 8 for the nonlinear switched system (42)–(43)? This should be verified, not assumed by analogy.

Response:
Thank you for raising this valid concern. We have revised the proof in Section 6 to explicitly justify the use of the Lyapunov function. The argument proceeds in two steps:

1. Subsystem stability: Each individual nonlinear subsystem is shown to be stable under the proposed controller (as proven in Section 5).
2. Bounded energy jump: At switching instants, the Lyapunov function satisfies V(x(tₖ⁺)) ≤ μ V(x(tₖ⁻)), where μ is defined as in Remark 6.

Given these two properties, and under the average dwell time condition, the overall switched system remains stable. The revised proof now states:

“The proof proceeds in two steps. First, we establish the stability of each subsystem. According to Assumption 8, the stability of every individual subsystem is guaranteed; a detailed proof follows similarly to that for nonlinear time-invariant systems. Second, we show that under the constraint of a bounded energy jump across switching instants—specifically, when the ratio of the Lyapunov function values before and after a switch is upper-bounded by a constant μ—the overall switched system remains stable under a mean sojourn time condition. Equation (45) formalizes this assumption by requiring that the energy jump does not exceed a factor of μ. The stability argument employed here closely parallels the approach used for linear switched systems in Equation (31).”

Comment 8:
The event-triggering condition (7) states that the next event occurs when Q ≥ 0 or at a topology switching instant. Does a forced communication at every switching instant potentially increase communication burden significantly, especially during fast switching? What is the rationale for this mandatory update, and is there a trade-off analysis?

Response:
Thank you for this important point. We clarify that no mandatory inter-agent communication is required at switching instants. Instead, each agent locally updates its own sampled state based on its internal clock and triggering logic. To avoid any misunderstanding, we have removed the phrase suggesting forced communication at switching times. The triggering condition now includes a minimum inter-event time t_min to prevent excessive updates, even during fast switching.

Comment 9:
Remark 4 mentions enforcing a "minimum inter-event time" τ (with τ > 0) to prevent Zeno behavior. What is the smallest lower bound on this time, and how is it derived from the system parameters (e.g., A, B, K, Lipschitz constant p)? A theoretical exclusion of Zeno behavior is essential for ETC schemes.

Response:
Thank you for emphasizing this critical issue. While analytical lower bounds on inter-event times can be derived in some ETC frameworks, they often depend on complex system-specific inequalities. To rigorously exclude Zeno behavior in a simple and implementable way, we adopt a hard minimum inter-event time t_min > 0, which is a standard and widely accepted practice in state-dependent ETC designs. The updated triggering rule is:
t_{k+1}^i = inf { t > t_k^i + t_min | Q ≥ 0 }
This guarantees a strictly positive dwell time between consecutive triggers for all agents, for both linear and nonlinear cases. This modification is clearly explained in Remark 4 of the revised manuscript.

Comment 10:
The simulation sections present results but lack key implementation details. What were the specific parameters a₁, a₂ used in the triggering function Q (Eq. 14, 22) for the simulations? How were they selected to satisfy the inequality a₁ρ₁ + a₂ρ₂ − ζ < 0?

Response:
Thank you for this observation. In the simulations, we computed:
β₁ = λ_max(L ⊗ (KᵀK)) = 1.2178, β₂ = λ_max(P) = 1.6022
To satisfy the stability condition a₁β₁ + a₂β₂ − ξ < 0, we selected a₁ = a₂ = 0.04. These values have been added to the Simulation Results section.

Comment 11:
In the nonlinear examples (Figs. 5 & 6), the agents seem to use the Lorenz chaotic dynamics. Are these chaotic dynamics (α=10, β=8/3, γ=28) acting as the nonlinear intrinsic dynamics f(x) or as an external disturbance? If it's the intrinsic dynamics, the initial states appear very far from the attractor; how does this affect the consensus manifold’s interpretation?

Response:
Thank you for this point. Originally, chaotic dynamics were used as intrinsic nonlinearities. However, to better illustrate consensus behavior and avoid complications from chaotic divergence, we have replaced the Lorenz system with smoother, globally Lipschitz nonlinear functions:
f₁(x(t)) = −x + tanh(x), f₂(x(t)) = −x + sin(x)
These functions are well-suited for demonstrating bipartite consensus under switching and ETC, and the revised figures reflect this change.

Comment 12:
The simulation results effectively validate the proposed method. However, providing more details on the simulation environment or MATLAB program (at least in review purpose) would enhance the credibility of the results.

Response:
Thank you for the suggestion. The corresponding MATLAB code has been included in the supplementary material for verification purposes.

Comment 13:
There is a significant formatting issue: The manuscript header "Version December 3, 2025 submitted to Journal Not Specified" appears on multiple pages, and the citation placeholder "[Lastname, F.:...]" is still in the text. This must be corrected before any review process.

Response:
Thank you for pointing this out. We have updated the journal name to “Automation” and removed all placeholder text and incorrect headers throughout the manuscript.

Comment 14:
In Section 3 (Linear Time-Invariant), the text abruptly states: "Suppose the MASs described by (1) satisfy Assumptions 1, 2, and 4..." followed by a sentence beginning "Suppose the multi-agent system described by (19)...", which belongs to Section 4. This appears to be a cut-and-paste error. Please clarify and correctly segment Theorem 1 and its proof.

Response:
We apologize for this oversight. The misplaced text from Section 4 has been completely removed from Section 3, and the theorem statement and proof are now properly segmented.

Comment 15:
Figure, Table, and reference calls are inconsistent. The text refers to "Figure 1", "Figure 2", etc., but the PDF content shows figure captions without numbers. Also, references [25] and [41] are duplicates.

Response:
Thank you for catching these issues. All figures have been renumbered sequentially as Figure 1–5 with consistent in-text references. Duplicate references ([25]/[34] and [41]/[43]) have been merged and renumbered accordingly.

Comment 16:
The Data Availability Statement says "data are not available due to technical limitations." What "data" does this refer to? The simulations presumably generate state trajectories and event times from deterministic equations. Does this mean the simulation code is not shared? This statement should be more precise.

Response:
We appreciate this feedback. The original statement was imprecise. It has been revised to:
“The data supporting the findings of this study are available within the article.”
Additionally, the simulation code is provided in the supplementary material.

 

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This paper studies the control problem of achieving bipartite consensus in switched multi-agent systems under event-triggered mechanisms. The overall structure of the paper is clear, and the theoretical derivations are basically rigorous. However, there are still some issues in this paper:

1.The comparison with existing work is insufficient, and the literature review is not thorough enough. It is necessary to clearly point out the differences and advantages of this paper compared to existing work.

2.The paper assumes that the system topology always satisfies structural balance (Assumption 4) and that all subsystems meet the connectivity condition (Assumption 5). These assumptions are overly idealized in practical applications. The lack of consideration for imbalance greatly reduces the practical value of the results.

3.There is a lack of sufficient explanation of the experimental results, and the conclusion section lacks discussion of future work.

4.Figure 2.3.5.6 is not clear enough; the agent states and trigger times should be separated.

5.Spelling issues: some agent systems are missing the plural 's'. (e.g., "multi-agent system" should be "multi-agent systems").

Author Response

Comment 1:

 

The comparison with existing work is insufficient, and the literature review is not thorough enough. It is necessary to clearly point out the differences and advantages of this paper compared to existing work.

 

Response:

 

We sincerely thank the reviewer for this insightful comment. We fully agree with the observation. To address this, we have revised the contribution statement in both the abstract and the introduction to clearly emphasize that the core novelty of this work lies in extending bipartite consensus to switched multi-agent systems—where the system dynamics switch among subsystems—within an event-triggered control (ETC) framework.

 

Specifically, we have added the following comparison in the Introduction:

 

“Notably, prior works on event-triggered control (ETC) for multi-agent systems have mainly considered switching communication topologies [35–38]. In contrast, this paper investigates ETC for achieving bipartite consensus in switched multi-agent systems where the switching arises from changes in the agents’ intrinsic dynamics. Research of this form remains relatively scarce.”

 

This clarification highlights the key distinction between topology switching (common in prior work) and dynamic switching (the focus of our paper), thereby strengthening the novelty claim.

 

Comment 2:

 

The paper assumes that the system topology always satisfies structural balance (Assumption 4) and that all subsystems meet the connectivity condition (Assumption 5). These assumptions are overly idealized in practical applications. The lack of consideration for imbalance greatly reduces the practical value of the results.

 

Response:

 

We acknowledge the reviewer’s concern. Structural balance is indeed a standard prerequisite for bipartite consensus, as established in foundational works such as Altafini (2013). Our primary focus in this paper is on the control design under switching dynamics and event-triggered communication, rather than relaxing topological assumptions.

 

That said, we fully agree that handling unbalanced or time-varying signed graphs is an important and practical direction. Therefore, we have explicitly added this topic to the “Future Work” section in the Conclusion:

 

“How to investigate bipartite consensus under unbalanced network structures will be the focus of our future research.”

 

Comment 3:

 

There is a lack of sufficient explanation of the experimental results, and the conclusion section lacks discussion of future work.

 

Response:

 

Thank you for your valuable comment. We have made two key improvements:

 

Added detailed parameters to each subsection of the Simulation Results section (e.g., controller gains, triggering thresholds, initial conditions, switching sequences).

Included a new paragraph titled “Future Work” at the end of the Conclusion, which outlines promising extensions such as unbalanced topologies and integration with learning-based methods.

These additions enhance the interpretability of the simulations and provide clear directions for future research.

 

Comment 4:

 

Figure 2, 3, 5, 6 is not clear enough; the agent states and trigger times should be separated. The individual figures are provided in the supplementary material.

 

Response:

 

Thank you for this suggestion. We have regenerated all figures with improved resolution and clearer legends. Specifically:

 

Agent state trajectories are now plotted with distinct line styles or colors.

Triggering instants are marked with vertical dashed lines or discrete markers (e.g., circles or stars) on a separate axis or layer.

Figure captions explicitly describe what each curve and marker represents.

These updated figures ensure that state evolution and communication events are easily distinguishable.

 

Comment 5:

 

Spelling issues: some agent systems are missing the plural 's', e.g., "multi-agent system" should be "multi-agent systems".

 

Response:

 

Thank you for catching this. The entire manuscript has been carefully proofread, and all instances of “multi-agent system(s)” have been corrected for grammatical consistency. For example:

 

“a multi-agent system” → used when referring to a single instance

“multi-agent systems” → used in general or plural contexts

This ensures linguistic accuracy throughout the paper.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

For comment 12, we did not find the supplementary. The authors advised attaching the proper files.

Comments on the Quality of English Language

N/A

Author Response

Comment: For comment 12, we did not find the supplementary. The authors advised attaching the proper files.   Response: Thank you for your suggestion. The corresponding original MATLAB code has been included in the appendix. In the file, comment12_1 is the original code for the main execution, comment12_2 is the code for solving μ, and  comment12_3 primarily handles solving eigenvalues and the Riccati equation.

Reviewer 3 Report

Comments and Suggestions for Authors

I have no further comments.

Author Response

Comments: I have no further comments.

Response: Thank you very much for your careful review of our manuscript and for your positive feedback. We greatly appreciate your time and efforts dedicated to this work.