Dynamic System Modeling of a Hybrid Neural Network with Phase Space Reconstruction and a Stability Identification Strategy

Round 1

Reviewer 1 Report

There are some parts that could be better explained.

I added some highlights

Comments for author File: Comments.pdf

Author Response

Paper: machines-1543864

Title: Dynamic system modeling of hybrid neural network with phase space reconstruction and stability identification strategy

Thank you very much for the suggestions for our draft. We have replied point by point to your comments, along with a clear indication of red color at the location of the revised paper.

The comments and replies can be summarized as follows:

• Q: There are some parts that could be better explained. “Calculate the proportion of false nearest neighbor points with gradually increasing the dimension m.”
• A: According to the False Nearest Neighbors function (FNN) method, by increasing the value of m from 2, the proportion of the false nearest point is calculated at each value of m. Repeat the computation procedure until the proportion of false nearest neighbors is less than 5% or no longer decreases with the increase of m.
• Q: Explain: “The quantitative identification on chaotic characteristics is depended on the calculation characteristic parameters of the singular attractor in the chaotic signal.”
• A: By calculating the characteristic parameter of the system, we can judge whether the system is a chaotic sequence or not. Here the maximum Lyapunov exponent is used as the characteristic parameter. This paper mainly judges that the system is a chaotic sequence by the positive maximum Lyapunov exponent.
• Q: Delete the “And”.
• A: This word “and” is deleted as your comment.
• Q: Explain: “The method is applied an optimization strategy with minimizing the objective function to realize the adjustment of hidden nodes、the expansion constant and output weight.”
• A: Here the method refers to RBF neural network modeling based on the gradient descent algorithm. The GD-RBF algorithm determines the optimal parameters iteratively in the training process by setting the objective function and objective error.
• Q: Explain: “…the space mode is taken…..”
• A: In order to solve the problem of inaccurate identification of pressure data at a single measuring point or multiple measuring points, we transformed the pressure data of each measuring point into spatial mode in Section 2.2. The process of rotating stall in system is presented by the variation of modal wave energy, which is a reflection of the essence and physical characteristic of the system. To avoid misunderstanding, it is replaced by “spatial mode”.
• Q: Explain: “It’s pointed out that in reference [20], the selection of delay time was not required a specific restriction for the noiseless chaotic sequence according to Takens theorem.”
• A: From the reference [20], It can be concluded that there is no specific limit on the value of delay time in the reconstruction of noise free chaotic sequence in phase space. Therefore, in order to reduce the amount of calculation, the prediction model algorithm in this paper selects the delay time .
• Q: Explain: “…With being the initialization value of GD algorithm, the output weight is initialized randomly.”
• A: In the chaotic phase space, the K-means algorithm is used to cluster the sample input to determine the data center and expansion constant of hidden nodes of RBF network. After the computation procedure, assign the obtained data center and extension constant to GD algorithm as its initial value. Then the output weight of is GD algorithm initialized. The explanation is added in the revision.

Thanks very much for taking your time to review this manuscript. I really appreciate all your comments and suggestions!

Author Response File: Author Response.pdf

Reviewer 2 Report

1. Is it accurate to use the first-order phase signal to represent the entire signal when judging stall?
2. When proposing the new method, the necessary formulas are lack which is inconvenient to understand.
3. When comparing the results of Chaos-GD-RBF neural network and Chaos-K-means-GD-RBF neural network, the residuals are similar. Is it necessary to increase the cost of calculation?
4. When calculating key values, such as delay time=8, embedding dims on=20, S=18, etc, it is lack of corresponding formula.
5. When comparing the one-step prediction method and the multi-step prediction method, only the stall time and computation cost are It is confused that, in the previous section, the residual is used as the judgment basis, regardless of the calculation time, and this section uses the opposite criteria.
6. What is the definition of the difference quotient used in the stall judgment, and what is the standard for judging the stall according to the difference quotient?
7. Compared with other stall prediction methods, what are the advantages of the method proposed in this paper in terms of stall prediction?

Author Response

Paper: machines-1543864

Title: Dynamic system modeling of hybrid neural network with phase space reconstruction and stability identification strategy

Thank you very much for the suggestions for our draft. We have replied point by point to your comments, along with a clear indication of red color at the location of the revised paper.

The comments and replies can be summarized as follows:

• Q: Is it accurate to use the first-order phase signal to represent the entire signal when judging stall?
• A: According to our previous research, it is inaccurate to use the pressure of a single measuring point to identify stall, so we use the first-order spatial modal amplitude to explore the recognition method. And it is shown from the results the first-order spatial modal amplitude can accurately reflect the essential characteristics of compressor rotating stall process. So it is a good way for the first-order phase signal to represent the entire signal when judging stall.
• Q: When proposing the new method, the necessary formulas are lack which is inconvenient to understand.
• A: Based on the reconstruction of phase space, a new Chaos-K-means-GD-RBF fusion neural network model is proposed to intelligently predict the rotating stall inception of compressor. The methodology of the modeling with neural network is explained in the section 3 in the manuscript. The descriptions on K-means and gradient descent algorithm are listed in 3.1 and 3.2 with equations. In the chaotic phase space, the K-means algorithm is adopted to cluster the sample input to determine the data center and expansion constant of the hidden nodes in the RBF network. After the computation procedure, assign the obtained data center and extension constant to GD algorithm as its initial value. Then the output weight of GD algorithm is initialized. Thus, the Chaos-K-means-GD-RBF hybrid neural network model is established with the above procedure. The explanation is added in the revision of page 11.
• Q: When comparing the results of Chaos-GD-RBF neural network and Chaos-K-means-GD-RBF neural network, the residuals are similar. Is it necessary to increase the cost of calculation?
• A: With comparing the two models, it can be seen from Table 1 that the residuals as mean absolute error (MAE) and root mean square error (RMSE) in Chaos-K-means-GD-RBF neural network is smaller than that of Chaos-GD-RBF neural network. Simultaneously, the cost of calculation is compared By extracting the average time of five steps in computation process, the time cost for determining three parameters in Chaos-GD-RBF model is 32.8s, and that of Chaos-K-means-GD-RBF model is 20.43s. In terms of calculation cost, it is shown an obvious advantage in the processing efficiency in Chaos-K-means-GD-RBF model. The explanation is added on page 13 of the revision.
• Q: When calculating key values, such as delay time=8, embedding dimson=20, S=18, etc, it is lack of corresponding formula.
• A: The corresponding formulas are supplemented in the revision page 6、 Within the limitation of phase difference threshold, the maximum number of prediction steps are searched through step-by-step calculation for the optimal choice with relative small errors. Finally, the best number of steps S=18 is obtained.
• Q: When comparing the one-step prediction method and the multi-step prediction method, only the stall time and computation cost are. It is confused that, in the previous section, the residual is used as the judgment basis, regardless of the calculation time, and this section uses the opposite criteria.
• A: The purpose of the previous comparison of residuals is to highlight the advantages of the proposed Chaos-K-means-GD-RBF model. And both the single step prediction and multi-step prediction are all based on Chaos-K-means-GD-RBF model. In fact, in the process of prediction, due to the existence of calculation time, one-step prediction is difficult to meet the demand of actual compressor rotating stall prediction, so it is necessary to seek multi-step prediction. Multi-step prediction is also obtained within a certain error threshold. The calculation time cost in the Chaos-K-means-GD-RBF model is only used to determine the prediction time for the inception.
• Q: What is the definition of the difference quotient used in the stall judgment, and what is the standard for judging the stall according to the difference quotient?
• A: The definition of difference quotient is added in the revision of page 14. When the rotating stall is about to occur, the difference quotient increases. And the rapid increase of the difference quotient can be regarded as a significant feature of the system approaching the rotating stall. Under setting the difference quotient threshold, the first peak exceeding the threshold is considered to be the pre-stall time.
• Q: Compared with other stall prediction methods, what are the advantages of the method proposed in this paper in terms of stall prediction?
• A: (1) In this paper, the analysis on the stall prediction is based on the first-order spatial modal amplitude representing rotating stall, which can effectively avoid the inaccurate identification of single measuring point or multiple measuring points. (2) An intelligent prediction model for rotating stall is proposed in this paper. Compared with traditional methods, not only the circumferential traveling wave is connected with the physical phenomena, but also a new Chaos-K-means-GD-RBF fusion modelling algorithm is established. The calculation accuracy and time of the fusion model are significantly improved. (3) Therefore, by taking the strategy of global sample entropy and difference quotient criterion identification, a warning of inception can be suggested accurately in advance with Chaos-K-means-GD-RBF hybrid neural network.

Thanks very much for taking your time to review this manuscript. I really appreciate all your comments and suggestions!

Author Response File: Author Response.pdf

Reviewer 3 Report

it is a good paper. In my opinion it would be better to include some recommendation for future improvement of the algorithms.

Author Response

Paper: machines-1543864

Title: Dynamic system modeling of hybrid neural network with phase space reconstruction and stability identification strategy

Thank you very much for the suggestions for our draft. We have replied point by point to your comments, along with a clear indication of red color at the location of the revised paper.

The comments and replies can be summarized as follows:

• Q: It is a good paper. In my opinion it would be better to include some recommendation for future improvement of the algorithms.
• A: In the future research, the algorithm with system modeling needs to be further improved in accuracy of multi-step prediction, so as to strive for a greater margin for rotating stall identification. The technology of deep neural network modeling is worthy of further exploration. Suggestion for improvement of the algorithm has been added in the discussion section of this revision.

Thanks very much for taking your time to review this manuscript. I really appreciate all your comments and suggestions!

Author Response File: Author Response.pdf