Novel Adaptive Extended State Observer for Dynamic Parameter Identification with Asymptotic Convergence

Round 1

Reviewer 1 Report

Despite time of identification of PIESO with state dependent regressor is significantly extended (ten times) in comparison with the case with the regressor independent on the system state and there are some discrepancy discrepancies between the simulation results in the experimental results (positional signal) I have no requests to the authors.

Author Response

We would like to thank the Reviewer for positive evaluation of the manuscript. 

While it is true that in the case of state dependent regressor the adaptation time may be increased, and we agree that it may be seen as a drawback of the proposed method, it can be reduced by the proper choice of adaptation gains. Simulations for both state independent and state dependent cases have been repeated with different adaptation gains and are now reported in the paper - see Fig. 6 and Fig. 8 in Section 4.1. It is shown that an increase of the adaptation gains leads to faster convergence at the cost of increased oscillations of the parameter estimates. Time of convergence of the state dependent case can thus be made comparable with the adaptation speed of the scenario with state independent regressor.

Reviewer 2 Report

This paper deals with parameter identification problems using ESO technique. The topic is interesting. The proposed method is validated by both simulation and experiments. Some issues are found and shown as follows. 

1. The definition on the third term on the right-hand side of equation 1 is confusing. What does b_n\psi(t,x)\theta stand for ? When I first saw equation 1, I could imagine it stands for uncertainties or input disturbances. But the authors claim that \theta is defined by “parameters of system”, what is the “system” in exact ? I would then think \theta is composed by the parameters in matrix A_n ! The authors should rewrite the paragraphs under equation 1 and make all definitions clear.

 

2. Before starting all derivations of the paper, the authors should give such an assumption that the uncertain/disturbing term in equation 1 is parameterizable.

 

3. The \gamma in equation 14 should be in written in majuscule \Gamma.

 

4. Lemma 1 is questionable since the authors seem to consider M\Gamma\psi^T\psi as symmetric in the derivation from equation 21 to 22. Note that even matrices A and B are symmetric, AB is not necessarily symmetric.

 

5. On the other hand, small updating gains of \hat{\theta} will lead to slow convergence, which decrease the control performance.

 

6. Some comparisons with different \gamma should be given in simulation.

Author Response

We would like to thank the Reviewer for their valuable comments. Multiple changes have been made to improve the paper according to the Reviewer's suggestions and are marked in orange in the revised manuscript.

1. The term b_n\psi(t,x)\theta may indeed incorporate linear parameters of the system (which could be otherwise also incorporated into A_n matrix). This term may also contain time dependent disturbances which may be independent of the state itself. In order to clarify this matter, Section 2 has been rewritten and an additional diagram of the system dynamics has been added, see Fig. 1.
2. The equation (1) has been rewritten into a more general form, and a specific parametrization of the term \delta is now given as the separate assumption.
3. We would like to thank for pointing out this typesetting error. It has been corrected and the paper has been thoroughly checked for any similar mistakes. 
4. We think that the issue pointed by the Reviewer was not clarified in the first version of the manuscript. Please note, that even while the term M\Gamma\psi^T\psi is not symmetric, it is guaranteed to be square and bounded. Thus, \theta^T*M\Gamma\psi^T\psi\theta is a scalar value which is no greater than its norm. The norm of this first non-symmetric term of (23) can then be upper-bounded by recalling the Cauchy-Schwartz inequality, while the second term is lower-bounded by the properties of positive definite matrices and Corollary 1. The bound on the first term can then be made arbitrarily small by choice of \Gamma. Short explanation of this derivation has been added to the manuscript to clarify the proof, please see the text below Eq. (23).
5. We fully agree with the Reviewer's comment. The constraints imposed on the adaptation gains can be seen as a drawback of the proposed method. On the other hand, as shown in Section 4.1, even if the adaptation time is increased by the choice of lower adaptation gains, the quality of state and disturbance estimation is not significantly affected and is comparable to the standard approach using the ESO observer. Namely, state estimation errors can be made arbitrarily small in a short time period if ESO gains are selected high enough. In this sense, the adaptation process can be seen as an action that aims to progressively improve the performance of the observer as the parameter estimates converge to the true values. This allows one to expect an improvement in quality in control systems using ADRC methodology.
6. Simulations for both state independent and state dependent cases have been repeated with different adaptation gains and are now reported in the paper, see Fig. 6 and Fig. 8 in Section 4.1. It is shown that an increase of the adaptation gains leads to faster convergence at the cost of increased oscillations of the parameter estimates. Moreover, incorrect values reported in Tab. 2 and 3 due to the typesetting errors have been amended.

Reviewer 3 Report

This paper studied a nonlinear algorithm for parameter identification.  The topic is scientifically studied and the work is worthy. The study has a good background in analysis for the parameter identification as well as the stability analysis.

However, the language and the poor, and the content are not scientifically designed. This decreased the readability of the paper.

The reviewer recommends an extensive language review as well as the design of the manuscript. It would be great if the extended observers and the whole identification algorithm were illustrated in a block diagram.

 

Author Response

We would like to thank the Reviewer for their comments. Multiple changes have been incorporated into the paper and are marked in orange in the revised manuscript.

Several paragraphs have been rewritten in order to increase the readability of the manuscript and the paper has been thoroughly checked for any language errors. In particular, the notation have been better clarified (cf. the last paragraph of Section 1, page 3) and the particular parametrization of the control system has been assumed (cf. Assumption 1, page 3). Moreover, three block diagrams have been introduced (cf. Fig. 1, Fig. 2 and Fig. 3) to clarify the structure of the system itself and of the adaptation algorithm in both state independent and state dependent cases.

In addition, new simulations have been conducted to compare the convergence of estimation errors for different values of adaptation gains, see Fig. 6 and Fig. 8 in Section 4.1 

Round 2

Reviewer 3 Report

The manuscript needs more effort to be sufficient for publishing. 

Author Response

We would like to thank the Reviewer for their comments. The paper has been revised and proofread thoroughly by considering the remarks in the review and language improvement suggested by Academic Editor. In addition, we have rearranged the structure of the document by adding the new subsections 2.1 and 2.2 where the notation and the system model, respectively, have been introduced.

The essential changes made in the text have been highlighted in orange.