An Adaptive Estimation Model for the States and Loads in Electro-Hydraulic Actuation Systems

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

1. Although the ABSTRACT mentions "experimental verification has proven the effectiveness of the method", it lacks specific quantitative indicators (such as the reduction factor of RMSE and the error range mentioned in the text). It is recommended to supplement the ABSTRACT with key result data to enhance its persuasiveness.
2. The introduction section provides a general description of the limitations of existing methods (such as extended Kalman filters, particle filters, etc.), and it is necessary to more explicitly compare the core differences between the "structure-adaptive Kalman filter" proposed in this study and traditional methods, in order to highlight the innovative points.
3. The regression model of formula (12) relies on coefficients k_t, k_d, and k_p. The parameters are identified using the least squares method in the text, but the coverage range of experimental data (such as load range and speed range) and the adaptability of parameters under different operating conditions are not specified.
4. In the update of the adaptive covariance matrix, the update of the model error covariance matrix is based on the variance calculation of derivative differences (formulas 44-46), but the selection criteria for time window N and its impact on the results are not explained. It is recommended to supplement the explanation of the selection criteria.
5. In the experimental conditions, only two conditions, "linearly increasing load" and "step load", have been verified so far, and more complex dynamic scenarios (such as random fluctuating loads and high-frequency variable loads) have not been covered. It is recommended to supplement other experimental conditions to verify the universality of the model.
6. The relevant equipment models (hydraulic system components and control systems) were not specified in the experiment.
7. The authorsmentioned in the article that the sampling period of the algorithm is 50Hz (Δ t=0.02s). Please indicate if it can meet the real-time control requirements?

Author Response

Reviewer #1:

 

Dear reviewer,

We would like to appreciate your in-depth comments and useful guidance. In the revised version of the manuscript, we have considered all the remarks made, and below we present our answers and clarifications.

 

Reviewer comment:

Although the ABSTRACT mentions "experimental verification has proven the effectiveness of the method", it lacks specific quantitative indicators (such as the reduction factor of RMSE and the error range mentioned in the text). It is recommended to supplement the ABSTRACT with key result data to enhance its persuasiveness.

Authors' response

Thank you for your note. Quantitative results of the experimental verification have been added to the annotation, which makes it more convincing and clearly emphasizes the effectiveness of the proposed method. Reflected in L20 to L27.

Reviewer comment:

The introduction section provides a general description of the limitations of existing methods (such as extended Kalman filters, particle filters, etc.), and it is necessary to more explicitly compare the core differences between the "structure-adaptive Kalman filter" proposed in this study and traditional methods, in order to highlight the innovative points.

Authors' response

Thank you for your remark. In line with it, additional text has been added to the introduction, which outlines the main differences between the proposed structure-adaptive Kalman filter and traditional approaches (EKF, UKF, EnKF). Reflected in L78 to L82. This text is written more briefly, as we considered a recommendation of another reviewer to reduce the volume of the introduction. On the other hand, Section 6 includes additional experimental results that directly compare the work of:

• classic Kalman filter with fixed covariance matrices,
  • adaptive filter with empirical update based on observed errors,
  • proposed structure-adaptive Kalman filter, which combines both adaptive Ψ update and hybrid strategy for updating R.

The results are presented through various graphical and statistical forms and show that the proposed method demonstrates the highest accuracy, repeatability, and stability: the mean values of the errors are closest to zero, the standard deviations are minimal, and the distributions are symmetric and close to the normal law. At the same time, the classic fixed-matrix filter shows lagging and smoothing of transitions, while the adaptive filter with empirical dependencies is faster, but unstable and sensitive to variations. In this way, clearly defined differences are given both in quantitative and metrological terms. Reflected in L1057 to L1192.

Reviewer comment:

The regression model of formula (12) relies on coefficients k_t , k_d and k_p .The parameters are identified using the least squares method in the text, but the coverage range of experimental data (such as load range and speed range) and the adaptability of parameters under different operating conditions are not specified.

Authors' response

Thank you for your remark. It should be clarified that the determination of the coefficients k_t , k_d and k_p ​ is not the result of pre-fixed experiments, but is carried out through an automated identification procedure implemented in the algorithm itself (Section 3.2, Fig. 3). The algorithm simultaneously uses the measured quantities – the angular velocity of the hydraulic motor, the control angular velocity, the pressure difference, and the set load torque – and on this basis builds a regression matrix in real time and identifies the parameters.

On this remark, an additional part is included in the revised version of the article in section 3.2 detailing the experimental data used and the identification methodology. The studies cover different characteristic operating modes, of which three typical examples (linear increasing load, step modification and asymmetrical saw load) are presented in the article, providing a wide range of dynamic conditions. Reflected in L503 to L544.

To ensure sustainability and adaptability, a weighting procedure has been applied, whereby each trio of assessments is considered depending on the quality of the respective mode, measured by the criteria of standard error (MSE) and coefficient of determination R² . In this way, the modes with the best compliance have more weight, and the influence of the weaker ones is reduced. This eliminates the effect of random distortion and provides stable and reliable final parameter estimates.

We hope that the included addendum provided complete clarity on the coverage of experimental data and the adaptability of parameters in different modes of operation.

Reviewer comment:

In the update of the adaptive covariance matrix, the update of the model error covariance matrix is based on the variance calculation of derivative differences (formulas 44-46), but the selection criteria for time window N and its impact on the results are not explained. It is recommended to supplement the explanation of the selection criteria.

Authors' response

Thanks to the reviewer for the remark made. She correctly identified our omission, which we correct by further clarification in the text. Two parameters are used in the proposed structure – N and L.

• N represents the total time window (number of iterations) within which the differences between the real and estimated derivatives of the parameters are calculated. It sets the maximum range of data used to form a statistical base.

1. L is the effective number of values that are used to estimate variances at a given time. That is, L is a dynamic subset of N, selected adaptively depending on the ratio between the current standard errors (MSE) and the maximum error in the window under consideration.

In this way, final assessments are avoided: too low several points would lead to unstable assessments, and too high – to a loss of sensitivity to rapid dynamic changes. Adaptive L detection  provides a balance between robustness and sensitivity, which is key to reliable operation of the algorithm in a dynamic environment. Reflected in L718 to L725.

Reviewer comment:

In the experimental conditions, only two conditions, "linearly increasing load" and "step load", have been verified so far, and more complex dynamic scenarios (such as random fluctuating loads and high-frequency variable loads) have not been covered. It is recommended to supplement other experimental conditions to verify the universality of the model.

Authors' response

We thank the reviewer for his remark. The revised version of the article includes additional research – with an asymmetrical sawtooth load. Reflected in L1145 to L1192. In fact, more experiments have been conducted that cover a wider range of dynamic modes, but due to the limited volume of the article, only the most indicative ones are presented, which clearly demonstrate the influence of individual aspects of the filter structure.

It should be clarified that due to the non-reversible nature of the electric motor used, experiments with high-frequency sinusoidal loads cannot be realized. On the other hand, the asymmetrical saw-like load is fully applicable and was used as an additional experiment. It reproduces key features of more complex dynamic loads – such as asymmetry, uneven transitions, and dynamic fluctuations – and thus provides a reliable basis for assessing the robustness and universality of the proposed method.

Reviewer comment:

The relevant equipment models (hydraulic system components and control systems) were not specified in the experiment.

Authors' response

Thanks to the reviewer for the remark made. In accordance with it, the necessary data on the equipment used – hydraulic components and control system – have been added to section 6 of the revised version, which provides complete clarity about the experimental conditions.

Reviewer comment:

The authors mentioned in the article that the sampling period of the algorithm is 50Hz (Δ t=0.02s). Please indicate if it can meet the real-time control requirements?

Authors' response

Thank you for your comment. The selected sampling period of 50 Hz (Δt = 0.02 s) is consistent with the dynamic characteristics of the electro-hydraulic system used and provides reliable real-time process tracking. The experimental studies conducted have shown that this frequency is quite sufficient to reflect the characteristic dynamic changes, while allowing the algorithm to be executed without computational delays. The system's characteristic native frequencies are significantly lower than 50 Hz, which ensures that the selected sampling is adequate and provides the necessary balance between accuracy and computational efficiency, meeting the requirements of real-time control.

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Dear Authors,

I have several serious concerns regarding the publication of the current version of “An Adaptive Estimation Model for the States and Loads in Electro-Hydraulic Actuation Systems” in the journal actuators.

First, your submitted version is currently way too long with a total of 31 pages. For example, the introduction alone takes about two pages, which is very unusual. This total length is linked to an in my opinion lack of focus in terms of content. In general, you describe in great detail three key aspects that distinguish according to your extractions your Kalman filter approach from the conventional use of Kalman filters:

• algebraic calculation of a state variable (angular acceleration of the hydraulic motor)
• adaptive update of the covariance matrix of the model
• hybrid strategy for updating the covariance matrix of measurement errors

However, these three aspects are insufficiently experimentally validated. These aspects are not examined individually in section 6, which is why the scope of the description does not seem justified. It is with the current version not possible to differentiate to what extent, for example the algebraic solution of the acceleration actually adds value. At the moment there are only two measurements presented, which were carried out with the whole approach and no measurements were carried out with individual aspects of the solution approach, for example with and without the algebraic solution of the acceleration. In my opinion, it would be better if the very detailed descriptions were shortened and the individual aspects were also examined experimentally. The conclusions regarding the individual aspects in section 6 are not tenable as they stand.

Regarding section 3, it is not clear which tests were carried out for determining the coefficients kt, kd and kp. Please provide a description for the used experimental data.

Fundamentally, the current scientific value of the publication is not evident for the reader. Since the early 2000s, there have been numerous publications on the use of Kalman filters to determine parameters or state variables in hydraulic systems. Please provide in a concise manner how the individual aspects such as the adaptive update of the covariance matrix of the model and the hybrid strategy for updating the covariance matrix of measurement errors, differ from other publications and where the degree of novelty is. Furthermore, the necessity of the algebraic solution of the velocity is from an academic point of view fundamentally unclear. As previous described, such a statement has to be individually examined, regarding the solution accuracy and memory/computing power requirements.

Last but not least, the used state space is from a hydraulic perspective deficient. The included factor 𝜂_𝑝𝑢𝑚𝑝 combines numerous non-linearities, such as the speed- and pressure-dependent efficiencies of the hydraulic units, in one single factor. Modelling the pressure respectively the derivation of the pressure in the pump line and integrating it into the state space seems much more appropriate here.

Author Response

Reviewer #2:

 

Dear reviewer,

We want to express our sincere gratitude for your valuable comments and remarks. Your observations have helped us significantly improve our work. Our responses to your comments are included below.

Reviewer comment:

Regarding the note about the volume and focus of the introductory part:

Authors' response

We thank the reviewer for the comment. We agree that the presentation should be more dynamic and targeted. As a result of this recommendation, the introductory part was shortened and restructured, with a clearer emphasis on scientific novelty and the contribution of the research, without unnecessarily lengthening the theoretical explanations.

Reviewer comment:

Regarding the note of lack of experimental validity of the algebraic calculation of acceleration:

Authors' response

Thanks to the reviewer for the remark made. It gave us the opportunity to qualitatively improve the article, and in the revised version in section 6 a separate analysis is included, in which the algebraic calculation of ω_m(t)/dt is considered independently and compared with the classical method of numerical differentiation. The results presented in Fig. 11 and Fig. 12 show distinct differences in error distributions. Thanks to the algebraic approach to calculating this variable, a complete and internally consistent state vector is provided, in which the interdependence between the constituent variances can be realistically estimated. The absence of such a structural element in Fig. 12, where acceleration is not derived algebraically, leads to incomplete information on error distributions and limits the possibilities for full metrological traceability of variables. This has a direct impact on the adequacy of the covariance matrix and its ability to reflect the stochastic properties of the system.

An additional quantitative argument is presented in Fig. 13, where the combined metric introduced shows a simultaneous reduction of both systematic and random errors. The results are derived based on the histograms of the errors on which normal approximations are superimposed. The resulting sample estimates of the mean and standard deviation clearly show that the proposed structure leads to increased accuracy, repeatability, and metrological reliability of the estimates. This provides a significant methodological advantage over the classical approach. We hope that the additions made clearly show the scientific value and validity of the algebraic approach. Reflected in L982 to L1039.

Reviewer comment:

Regarding the remark about the lack of experimental validity of the adaptive update of the covariance matrix of the model (Ψ) and the hybrid strategy for updating the measurement error covariance matrix (R):

Authors' response

Thanks to the reviewer for the remark made. In the revised version, the studies have been tested in two different modes of operation – with stepwise change of the load moment and with an asymmetrical saw-like load, and the second mode has been introduced additionally to the initial version of the article in order to more fully validate the effectiveness of the proposed approach.

For each of the two modes, a comparison was made between three algorithms: (1) a classic Kalman filter with fixed Ψ and R, (2) an adaptive filter with an empirical update based on the observed errors and (3) a proposed method that combines both an adaptive update of the Ψ and a hybrid strategy for updating the R. The results obtained are presented through various graphical and statistical forms:

• dynamic graphs of the system's response ( 18, Fig. 21),
• temporal evolution of errors ( 19, Fig. 22),
• histograms and probability density functions ( 20, Fig. 23),
• quantitative characteristics of the errors (Table 1 and Table 2).

These results indicate that the proposed method demonstrates the highest accuracy, repeatability, and stability. In it, the mean values of the errors are closest to zero, the standard deviations are minimal, and the distributions are symmetric and close to the normal law. In contrast, the classic fixed-matrix filter shows lagging and smoothing of transitions, and the adaptive filter with empirical dependencies is faster, but unstable and sensitive to variations.

We hope that the additions made represent a clear and experimentally supported justification for the effectiveness and scientific novelty of the proposed adaptive scheme. We hope that this presentation corresponds to the remark made and convincingly demonstrates the advantages of the method. Reflected in L1057 to L1192.

Reviewer comment:

Regarding the observation that the initial version presented only two measurements with the whole approach and did not address the individual aspects separately:

Authors' response

In the revised version of the article, the experimental part was supplemented with a new study, so that the results now cover a wider range of dynamic modes and provide a more complete validation of the proposed method. Reflected in L1145 to L1192. Due to the limited volume of the article, it is not possible to include all the experiments conducted, but we have selected the most indicative ones, which clearly demonstrate the influence of the individual aspects of the filter structure. In addition, Section 6 has been shortened and restructured by removing secondary conclusions and text, making the presentation more focused and focused. We hope that the additions and edits made correspond to the remark and give a clearer idea of the scientific contribution and effectiveness of the proposed approach.

Reviewer comment:

Regarding Section 3, it is not clear which tests were conducted to determine the coefficients k_t​, k_d and k_p​. Please provide a description of the experimental data used.

Authors' response

Thank you for your note. It should be clarified that the determination of the coefficients k_t​, k_d and k_p​ is not the result of pre-fixed experiments, but is carried out through an automated identification procedure implemented in the algorithm described in section 3.2 and illustrated by the flowchart of Fig. 3.

The algorithm simultaneously uses the measured quantities – the angular velocity of the hydraulic motor, the control angular velocity and the pressure difference, as well as the set load torque. On this basis, a regression matrix is built in real time and adaptive identification of parameters is performed.

Coefficients k_t​, k_d and k_p  are selected in a way that ensures that the widest possible range of dynamic features of the system is covered, ensuring stability and consistency of assessments under variable modes.

An additional correction component k_p  is also included in the structure, based on the measured pressure difference. Its purpose is to compensate for inconsistencies arising from dynamic losses, nonlinearities or inertial effects that can lead to deviations between the reference load and the actual realized load.

This achieves:

• higher algorithm immunity to random interference and dynamic changes,
• more accurate monitoring of the reference signal in real time,

• minimizing systematic errors associated with unavoidable nonlinearities and inertial effects in the hydraulic system.

At the same time, we would like to thank the reviewer for his remark. Because based on it, an additional text has been introduced in section 3.2, which clarifies and supplements the methodology for determining the coefficients. The procedure is based on a weighting scheme, in which each trio of assessments is reported depending on the quality of the respective operating mode.

The quality of compliance is assessed by two criteria – standard error (MSE) and the coefficient of determination R². Thus, in the calculation of the final values, those modes in which the model provides a better correspondence with the experimental data, while the influence of weaker modes is reduced, have greater weight.

This approach eliminates the influence of random distortions characteristic of individual modes and provides more stable and reliable final odds estimates. Reflected in L503 to L544.

Reviewer comment:

Fundamentally, the current scientific value of the publication is not evident for the reader. Since the early 2000s, there have been numerous publications on the use of Kalman filters to determine parameters or state variables in hydraulic systems. Please provide in a concise manner how the individual aspects such as the adaptive update of the covariance matrix of the model and the hybrid strategy for updating the covariance matrix of measurement errors, differ from other publications and where the degree of novelty is. Furthermore, the necessity of the algebraic solution of the velocity is from an academic point of view fundamentally unclear. As previously described, such a statement must be individually examined, regarding the solution accuracy and memory/computing power requirements.

Authors' response

Thank you for your note and for the opportunity to clarify more clearly the scientific novelty of our research. We agree that since the early 2000s, numerous developments have been published dedicated to the application of the Kalman filter in hydraulic systems. However, our contribution consists in the integration of new concepts and specific algorithmic solutions that have not been considered in this combination and in this context so far. For clarity, we have qualified the new points in this project in two main aspects:

1. Structural novelty

Structurally, the novelty of the proposed method is expressed in the following complementary elements that make up a complete and coherent architecture of the algorithm:

1. The first essential element is the introduction of an algebraic calculation of the acceleration of a hydraulic motor instead of classical numerical differentiation. With this, our model differs from the standard Kalman filter, since one of the states is not modeled by a differential equation, but is determined directly in each iteration of the predictive step of the algorithm. This avoids the accumulation of errors and the high sensitivity to noise inherent in discrete derivatives. Additionally, the algebraic approach provides an internally consistent vector of states in which the interdependence between variables and their variances can be correctly estimated. This structural solution creates a reliable basis for metrological traceability of the evaluated quantities and simultaneously reduces the computational complexity of the algorithm.

The introduced computational structure has a direct impact on the accuracy of the model, and this effect is reflected in the covariance matrix of the model. To respond to the reviewer's recommendation, we also performed additional experimental validation presented in Section 6. From the results obtained in metrological aspect, the effect of this element leads to:

• reduction of the systematic component of error (the average bias of the error distribution approaches zero),
• reduction of the random component (the range/standard deviation of errors is reduced, which increases repeatability),
• a more adequate form of empirical distributions (lower asymmetry and absence of pronounced deformations), which improves the reliability of estimates and the validity of statistical assumptions.

This structural solution creates a reliable basis for metrological traceability of the evaluated quantities and at the same time reduces computational complexity, as there is no need for additional procedures for smoothing derivatives. It also lowers memory and computing power requirements, making it easier to implement the method in systems with limited hardware resources and increasing its real-world applicability.

2. Adaptive update of the covariance matrix of noise and errors in the model.

The novelty here lies in the way the covariance matrix is defined and updated in real time. The main aspects are:

1. Using parameter derivatives instead of their instantaneous values
   The variances in the matrix are determined based on the differences between the real and forecast changes in the parameters, and the real changes are calculated using their derivatives. In this way, the algorithm considers not only the instantaneous values, but also the rate of change, which is especially important for systems operating in dynamic modes.

In the metrological aspect, this solution has several advantages:

• resistance to noise and deviations – measurements of derivatives are less sensitive to random interference compared to direct measurements of the parameters themselves,

• reduced dependence on the absolute accuracy of the sensors – the filter works with changes in the values, which reduces the effect of systematic shifts in their values,
• better predicting future behavior – by focusing on trends of change rather than static values,
• connection with the physical nature of the processes – the inclusion of derivatives reflects the dynamic phenomena (e.g., changes in pressure or acceleration) that are decisive for the control and adaptation of the system.

1. Adaptive selection of the number of iterations for estimating variances – in order to avoid volatile or skewed estimates, the number of values used to calculate variances is dynamically determined, adapting to the current ratio between RMS errors and maximum error in the considered interval, thus balancing susceptibility to sudden changes and immunity to random disturbances.

We thank the reviewer for the valuable remark that pointed to us the need for a more complete experimental validation of the action of adaptively determined covariance matrices. As a result, further research is included in Section 6 that clearly shows the advantages of the proposed approach. They prove that the method provides both lower systematic error, reduced scattering, and higher repeatability of estimates in dynamic modes compared to classical structures. In this way, the article acquires a clearer and more complete appearance, and the reliability of the results is supported by additional experimental evidence.

3. Hybrid strategy for the covariance matrix of measurement errors

The third novelty is related to the way in which the covariance matrix of measurement errors is determined and updated. In classic implementations, it is assumed to be preset and fixed, which makes it insensitive to real changes in sensor characteristics. In the proposed method, a hybrid algorithm for adaptive dispersion estimation has been developed. The main aspects of this contribution are expressed in the following:

1. The first is the use of two parallel procedures to estimate the variances of measurement signals. The first procedure is based on exponential smoothing of the difference between measured and forecast values, which provides sensitivity to current changes. The second procedure is based on local regression on a sequence of previous values, which provides a reference estimate resistant to random fluctuations and noise. Depending on the ratio between the current and reference estimates, the algorithm goes through one of two adaptive logics: for limited deviations, an adaptive sensitivity coefficient is used, and for more significant differences, the variance is updated with the value from the regression analysis. Adaptive sensitivity factors provide differentiated adaptation for each measurement channel, considering individual changes in the characteristics of the respective sensors. This ensures both stability and sensitivity in a dynamically changing measuring environment.
2. The algorithm uses a polynomial model on a sequence of measurements of length G, and the degree of the polynomial is selected adaptively – a model of the first degree in case of smooth and monotonous change and a model of the second degree in case of more complex but regular dynamics. This provides a reference value that adequately describes the general trend of signal development and minimizes the impact of random fluctuations and local disturbances.

In the metrological aspect, this hybrid algorithm has significant advantages: it provides differentiated adaptation for each measurement channel, considering the individual characteristics of the respective sensors; increases sensitivity to real changes in the measuring environment; and at the same time guarantees resistance to accidental deviations. All this leads to more accurate and reliable estimates, which increases the efficiency of the Kalman filter under operating conditions.

For greater clarity and on the recommendation of the reviewer, an experimental analysis has been additionally introduced in Section 6, which validates the performance of the algorithm under dynamically changing conditions.

4. Inclusion of identification coefficients k_t​, k_d and k_p.

The fourth element of the contribution relates to the introduction of identification coefficients k_t​, k_d and k_p that summarize the influence of internal hydrodynamic processes and adapt the model to real working conditions. These parameters are automatically identified by an adaptive procedure based on regression analysis and weighted aggregation of the results of different regimes (described in Section 3.2). In this way, the model retains its computational efficiency and at the same time reflects nonlinear effects and dynamic losses, without the need to build a full-scale hydraulic model. Details of the methodology for determining them and the relationship with the coefficient η_pump​ are discussed in the replies to the previous remarks and in the supplemented text of the article.

1. Methodological contribution to the mathematical formulation of the method

From a methodological point of view, the contribution does not consist in the creation of new mathematical formulas, but in the way in which established mathematical statements are integrated and coherent within the structure of the Kalman filter. This includes the use of regression dependencies, parameter derivatives, weighting procedures, and local approximation, which in combination build a consistent algorithm for adaptive state estimation.

1. Integration of well-known productions in a new context. The contribution is in the methodological approach – classical tools such as regression, derivatives and smoothing are used not alone, but as complementary parts in the filter structure. Thus, both simplicity of calculations and the ability to adapt to different modes of operation are achieved.
2. Metrological justification. The staging is designed to ensure correct estimation of variances, traceability, and repeatability of the results. Each of the procedures – whether it is the identification of coefficients or the adaptive renewal of covariance matrices – is formulated in such a way as to preserve the validity of the statistical premises and to ensure metrological reliability of the estimates.
3. Balance between accuracy and computational efficiency. The proposed methodological framework provides increased accuracy of estimates while reducing computational complexity and hardware resource requirements. This clarifies the role of the algebraic calculation of acceleration, and thanks to the note made, we emphasize that it is part of a broader formulation that guarantees both higher accuracy and the possibility of real-time implementation.

Reviewer comment:

Finally, the used state space is from a hydraulic perspective deficient. The included factor  ?_???? combines numerous non-linearities, such as the speed- and pressure-dependent efficiencies of the hydraulic units, in one single factor. Modelling the pressure respectively the derivation of the pressure in the pump line and integrating it into the state space seems much more appropriate here.

Authors' response

Thank you for your remark. We fully agree that the introduction of internal hydrodynamic quantities such as pressure in the pump pipeline can lead to a more detailed physical model of the system. But one of the main goals of this project was to create a system in which the dynamics of the electro-hydraulic system could be described by input-output mechanical variables, without directly entering internal hydrodynamic parameters such as flow rate, pressure, or other quantities characteristic of internal flows. Such a solution is dictated by the need for the model to be sufficiently simplified and suitable for implementation by a linear Kalman filter, while preserving the main dependencies between the controlling and observed quantities. Instead of building a full-scale hydraulic model, the influence of internal processes is accounted for by empirically determined coefficients in the load variation equation. These coefficients – k_t​, k_d and k_p – are identified experimentally based on real measurements and reflect the dynamic characteristics of the system under variable loads and dynamic conditions. After the identification of the ?_???? ​ the coefficients k_t​, k_d and k_p are determined ​, which further specify and refine the dynamic description of the system, taking into account the internal mechanisms of dissipation and the influence of pressure. Thus, the three coefficients can be considered as building parameters that adapt the model to real conditions and guarantee its reliability under different load modes.

Secondly, it should be emphasized that the accuracy of the estimates obtained from the theoretical model in the state space depends on the covariance matrix of noise and errors in the model. In the structure under consideration, the variances are calculated in real time by analyzing the differences between the measured and predicted changes in the states. A key point is that the matrix is based on the derivatives of the parameters, and not on their absolute values in the individual iterations. This choice is dictated by the need to ensure greater accuracy and adaptability of the filter in a dynamic environment, since the derivatives reflect the speed of change of processes.

Thus, the model considers and adapts the behavior of the system in real time to changing conditions, while reducing dependence on the absolute accuracy of sensors. The use of derivatives also increases immunity to noise and deviations, as measurements of variations are often more dependable than direct measurements of static quantities. In addition, the calculation of derivatives makes it possible to better predict the future behavior of the system, with an emphasis on trends, not just momentary values.

Thus, the inclusion of derivatives in the model not only more clearly defines the physical nature of the processes (e.g., pressure changes or accelerations) but also plays a crucial role in increasing the accuracy and adaptability of the system under dynamic conditions.

We would also like to thank the reviewer for the guideline that the determination of the quantitative value of the coefficient ?_???? was not sufficiently clearly presented ​. As a result, in this version, we have supplemented the text with a detailed description of the procedure by which the ?_????​ is automatically identified by an algorithm similar to that used for coefficients k_t​, k_d and k_p including the application of weighted aggregation of the results of different regimes and quality assessment through criteria MSE and R². This clarification ensures that ?_???? It is not introduced as a fixed correction factor, but as an empirically determined parameter reflecting the operating conditions of the system. Reflected in L545 to L577.

 

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Dear Authors,

thank you for the extensive changes and your comments on them. I have no serious objections to this publication. Nevertheless, given the length of the paper, I would have considered it more appropriate to divide the content into several publications.