Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper has potential relevance for gravitational wave detector control systems, but it requires significant revisions before publication. Specifically:

• Clarify motivation in the introduction - what gaps to be filled in current H₂/H_inf methods, why non smooth optimization is to be preferred, more recent references to current status of suspension control il ET/CE • Provide more details for reproducibility - numerical parameters are not reported, only referred to, no information about optimization settings, tolerances in the practical matlab implementation • Improve caption/text/figures consistency • Include at least one quantitative comparison or validation case   Comments on the Quality of English Language

Revise the English to improve readability - overly elaborate language sometimes obscures the actual concept.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The paper proposes a non-smooth controller optimization method and shows the results of ongoing research on the implementation of this method for gravitational wave applications. Typical performance requirements of these type of suspensions are defined in terms of both H2- and H∞-type constraints. A non-smooth optimization approach is investigated. Besides the controller, the distribution of the actuation is integrated with the optimization to investigate 
the feasibility of simultaneous controller and actuator optimization. The results shows that the proposed non-smooth optimization method is able to find suitable solution. The paper is sound but few improvements are recommended: The citations in the text are not in a crescent order, this should be corrected; The phrase in line 24: ...classical loop-shaping methods. requires a reference; the terms g in equation 3 require a explanation. With the corrections the paper can be published.

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

We are pleased to read this interesting article, and would support its publication. Nevertheless, there are some questions / comments that we hope the authors can follow up on. 

• Line 60. "The values for masses and lengths are taken from [19]." After reading the reference, we do not know what values are actually used  in the study here. Just to explain our confusion, the referenced paper has a figure showing "platform", "marionette", "cage" and "mirror". Our understanding is that "cage" is irrelevant for this study. while the other three are the masses m1, m2 and m3 in this paper. Is that correct? The suspension length for "platform" is unclear. The reference shows the "mirror" suspended by TWO fibers from the "marionette". It is presumably modeled as one here. Is that correct? It is perhaps simplest and clearest to just provide the actual numbers for masses and lengths explicitly here, and say they come from this reference. 
• Paragraph starting at line 164. "Figure 4 shows the cumulative x_RMS...  The value of x_RMS is reduced to a value of 9.88 10^-14 m." First of all, this is presumably in the band 0.02 to 0.2 Hz - we are referring back to lines 85 - 88 - and not the entire frequency range shown in Figure 4. If so, it may be worth repeating just to be clear. Secondly, Figure 4 goes up to ~4 10^-13 m just above 0.2 Hz. We do not believe that there is anything 'magic' happening at exactly 0.2 Hz. So while the statement is technically correct, we hope the language can be more nuanced and better reflect the situation. Also, are we sure that we know 9.88 well enough to report three decimal places? Something like "Figure 4 shows the cumulative x_RMS...  The value of x_RMS does not exceed 1 10^-13 m in the frequency range 0.02 to 0.2 Hz."
• Given that 1 10^-13 m is the target - line 85 - it would be interesting to know the design margin. We are thinking here not so much of the margins in the controls but the margins for masses and lengths. If we are the builder of the Einstein Telescope, we might want to know if the mirror can be 200 kg instead of 211 kg. We totally understand that this paper is about controls rather than telescope design, but we think it would be extremely useful information. 

One other point. We usually respect the citation format preferred by the cited article. The format for [19] is non-standard for Physical Review, and should turn into "Phys. Rev. D 108 123009 (2023)", where 108 is bold and 2023 is not bold. There are others. 

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The revised manuscript is overall in good shape. A few minor comments:

1. Figures 4 and 5 are now correct, but still a bit visually heavy. If possible, slightly bigger labels or fewer line styles would help readability.

2. In the Discussion, some sentences remain a bit long and formal. The meaning is clear, so this is only a stylistic point. Shorter sentences could help the flow.

3. Since the control requirements for ET/CE suspensions are still evolving – and this could be mentioned – it might be useful to add one sentence acknowledging that future refinements of the method may be needed once more detailed specifications become available.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf