Correct Stability Condition and Fundamental Performance Analysis of the α-β-γ-δ Filter
Round 1
Reviewer 1 Report
Allow me to be frank. I am unfamiliar with the topic of alpha-beta-gamma-delta filter and therefore I would not evaluate the correctness and the scientific contribution, although I was interested in the abstract and has agreed to review this work --- Yes, the manuscript indeed turns out to be very interesting to me. The work seems theoretically sound.
My suggestions/questions for improving the work is on the simulation/numerical study are as follows.
1. Please write the observation/data model straightforward by an equation, rather than simply by a single sentence in line 275.
2. Following the above suggestion, please make it clear what are the measurement noises for different scenarios, sec. 6.1.1, 6.1.2, and 6.2, respectively. (I can only find one statement in line 275 that, the standard derivation of the position measurement noise is 250m)
3. Can the proposed approach be compared with the observation-only inference as suggested by the following paper:
T. Li, et al. Effectiveness of Bayesian filters: An information fusion perspective, Information Sciences, vol. 329, 2016, 670-689
4. If the standard derivation of the position measurement noise is 250m, the RMS predictor errors of both the alpha-beta filter and the alpha-beta-gamma filter are much larger in Fig.4(a) (please also check Fig.6 (a) and (b)). In the view of the following paper, they are ineffective. "Effectiveness of Bayesian filters: An information fusion perspective, Inf. Sci., 2017" Anyway, please provide a comparison of the three mentioned filters with the measurement x_o that is made directly on the position of the target. This is very meaningful to simply tell: whether it is necessary to apply each of the three filters.
5. please use different colors for different curves in Fig.6(a). Right now, I can not tell the difference between alpha-beta-gamma-delta curve and alpha-beta curve.
6. Please provide a comparison of the computing time of different filters.
Author Response
We are grateful for your careful reading of the manuscript. Our responses to your comments are in an attached file.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The authors present a theoretical performance analysis of an α-β-γ-δ filter and compare it too standard α-β-γ and α-β filter . The authors also show the analogy of this filter to a Kalman filter. The research extends previously published studies concurring with them. The research extends previously published studies concurring with them. Furthermore, a theoretical performance analysis has not been done so far.
In general, the research poses a thorough analytical work although of little originality. As the authors said, l. 298. The work "clarifies" some aspects of the filter. I believe the entire analysis is also just of interest for a limited audience since - and that is a problem with every analysis - narrows down the scope to manageable parameters, e.g., linear move along one axis, no measurement errors, etc.
The quality of the presentation is good. Some sentences still sound 'interesting', but they are understandable. The paper also repeats several statements multiple times, and some typos slipped. I believe another thorough read will fix that.
The paper is in general scientific sound. However, there is room for improvement. First, the organization of the article requires one to jump between pages since the authors often refer to equations that they introduce pages before. Means some information gets lost along the path. Compressing the layout or labeling the equations (name them, not just numbers) would simplify reading; e.g., instead of (4) to (8), the authors could say 'boundary conditions 4-8)'. That would help this reviewer.
Secondly, I missed the point in Section 6. Although the authors provide a thorough analytical review, the impact of gain in controls is not new. Did the authors see something that they did not expect (besides the numbers which meaning is limited with a limited set of examples.)? Or why did the authors select a constant acceleration target, a constant gain target, and a high maneuvering target? What differences should the reader see? If the authors have a specific intention, they should point this out.
Also, the authors partially disagree with previously published results, e.g., l. 90 and Section 3, Example 1. Here, it would be scientific appropriate to add the calculation into an appendix instead of just saying the 'other paper made a mistake.'
Additionally, I would recommend softening the language here, or the authors should add a statement that confirms that the authors' calculation complies with all boundary conditions and limitations as stated in [13].
The same limitation applies for l. 300 "First, the incorrectness of the conventionally derived stability conditions was pointed out, and the correct stability conditions were derived using the Jury’s test," the authors should explicitly state that they comply with all previous conditions.
Author Response
We are grateful for your careful reading of the manuscript. Our responses to your comments are in an attached file.
Author Response File: Author Response.pdf
Reviewer 3 Report
This study present alpha-beta-gamma-delta filter as a natural continuum of alpha-beta and alpha-beta-gamma filters, and contracting by the seminal work on the filter, change the smoothness of filter to be consistent with lower order filters. The derivation of the formulation and the analogy to the Kalman filter was neat and desired for special readers, and the use of Jury's stability test is interesting to support the claim of the paper.
It would be great if the authors include the properties of Jury's test and an intuition of what physical aspect of stability does it actually proves, and why the work of Wu et al. that doesn't pass this test is incorrect. It would be interesting to contrast the results of scenarios 6.1.* with the Wu's alpha-beta-gamma-delta model to show its weakness.
Another small issue is in Figure 2, which the gain comparison is not fair. The x-axis must be beta for all trackers and alpha, gamma and delta should be calculated based on that to give a correct sense of the performance of the filter.
Author Response
We are grateful for your careful reading of the manuscript. Our responses to your comments are in an attached file.
Author Response File: Author Response.pdf
Reviewer 4 Report
The authors did a good job on the derivation of this filter. But there is no experiments or test that can demonstrate the performance of this filter. I recommend the authors to do some real ground test to verify the filter.
Author Response
We are grateful for your careful reading of the manuscript. Our responses to your comments are in an attached file.
Author Response File: Author Response.pdf
Round 2
Reviewer 4 Report
Thanks for your reply.
