L2-Gain-Based Practical Stabilization of an Underactuated Surface Vessel

Round 1

Reviewer 1 Report

The topic of the paper is within the scope of the journal and this paper is quite interesting. Overall, the paper is well written and organized with a proper length. My overall opinion of this paper is quite good, however, some contents presented are not clear enough. 
+ The contributions need to be more highlighted in the abstract. The novelty of the proposed control system should be emphasized. Also, the limitations of the existing method and new ideas for improvement should be explained more.
+ In the second simulation which applied to “Blue Lady”, some simulation results need to be added in the paper, such as the disturbance forces acting on the ship and control input or thruster forces and moments. Also, the explanations and analysis of simulation results should be explained more to show the validity of the data.
+ The units of control forces and moment, as well as the disturbance forces, should be checked. N or kN? Nm or kNm?
+ The manuscript writing can be further polished with professional English. The manuscript can be thoroughly revised for grammar check.

 

 

Author Response

First of all, the authors would like to express their sincere appreciation to the reviewer for his/her help in improving the paper. In the revised manuscript, authors revise the paper carefully according to the comments raised by the reviewer.

Comments:

The topic of the paper is within the scope of the journal and this paper is quite interesting. Overall, the paper is well written and organized with a proper length. My overall opinion of this paper is quite good, however, some contents presented are not clear enough. 

Concern 1: The contributions need to be more highlighted in the abstract. The novelty of the proposed control system should be emphasized. Also, the limitations of the existing method and new ideas for improvement should be explained more.

Response: Thanks for the comment. In the abstract of revised manuscript, the contributions have been explained in more details. The novelty of the proposed method is explained by comparing the commonly used disturbance observer based approach. The limitations of existing method and new ideas for improvement are explained by comparing a classic robust controller, sliding mode controller.

Concern 2: In the second simulation which applied to “Blue Lady”, some simulation results need to be added in the paper, such as the disturbance forces acting on the ship and control input or thruster forces and moments. Also, the explanations and analysis of simulation results should be explained more to show the validity of the data.

Response: Thanks for the comment. In the revised manuscript, for “Blue Lady”, the disturbance forces acting on the ship have been added in the simulation results. Also, the control inputs have been added. Moreover, explanation of the added plots and corresponding analysis have been added before Figure 10.

Concern 3: The units of control forces and moment, as well as the disturbance forces, should be checked. N or kN? Nm or kNm?

Response: Thanks for the comment. In simulation, the units of variables in the mathematical model of ship dynamics, the units of main particulars of ship, and the units of initial states adopt SI units. Therefore, the authors think that the units of control forces and moment, as well as the disturbance forces, are N and Nm.

 

Concern 4: The manuscript writing can be further polished with professional English. The manuscript can be thoroughly revised for grammar check.

Response: Thanks for the suggestion. The authors revise the manuscript carefully by checking English writing. Grammatical errors are corrected and some expressions are modified, as follows:

• In many places, the article “the” has been added to nouns in the revision.
• Page 1, “Marine vessels are usually underactuated systems which implies…” is corrected as “Marine vessels are usually underactuated systems, which implies….”
• Page 2, “A transverse function approach was proposed to practical stabilisation...” is modified as “A transverse function approach was proposed for practical stabilisation…”
• Page 2, “Moreover, the disturbances are often assumed as constant…” is corrected as “Moreover, the disturbances are often assumed as constants…”
• Page 2, “…well dealt with by disturbance observer.” is corrected as “…well dealt with by disturbance observers.”
• Page 3, “v the sway speed; r the yaw rate.” has been revised as “v is the sway speed; r is the yaw rate.”
• Page 4, “where constants are defined as” has been revised as “where the constants a,β,γ are defined as”
• Page 5, “The state equation in terms of. z_i has the form” has been revised as “The state equation w.r.t. z_i has the form”
• Page 5, “while z₁, z₅ ” has been revised as “while z₁, z₅ do not.”
• Page 7, “…this disturbance or decreasing the influence of disturbance when inequality (22) satisfies.” has been revised as “…this disturbance or to decrease the influence of disturbance till inequality (22) satisfies.”
• Page 8, “based on definition given by (11) and (12)” has been revised as “based on the definitions given by (11) and (12)”
• Page 9, “Moreover, the chattering phenomenon with sliding mode approach exist…” has been corrected as “Moreover, the chattering phenomenon with sliding mode approach exists…”
• Page 13, “Stabilizers based on L2-gain design are presented for underactuated surface ship…” has been corrected as “Stabilizers based on L2-gain design are presented for underactuated surface ships…”

Reviewer 2 Report

The authors are clear explaining the aim and purpose of the paper. It is proposed a stabilizer for an underactuated surface vessel affected by disturbances based on L2-gain design. Diffeomorphism transformation and Lyapunov function are employed in controller design to derive a robust adaptive controller. Simulation results demonstrate the validity of the proposed controllers.

The paper is well organized presenting the state-of-the-art of stabilization of underactuated surface vessels and the mathematical model of an underactuated surface vessel motion. In addition, the controller design is presented and finally the simulation results are reported and respective conclusions are presented in the last section.

The main conclusions based on the controller design and analysis process are:

• The proposed control scheme can be grouped into discontinuous feedback approach since the sign function is included in the controller;
• No asymptotic stability can be achieved but general stability instead. Errors can be made small as possible by decreasing the L2-gain index.
• The disturbances taken into account in the study are uncertain and nonlinear in a real sense, which is consistent with the essence of environmental disturbances induced by wind, waves and current.

In general the paper is very clear and the results are well stated and presented. The paper presents interesting results and comparisons that can be extended to experimental results and the method can be applied to other control issues.

As recommendation the text of the Introduction section should be reviewed because some sentences are too short and there are not sequence between them.

Author Response

First of all, the authors would like to express their sincere appreciation to the reviewer for his/her help in improving the paper. In the revised manuscript, authors revise the paper carefully according to the comments raised by the reviewer.

The authors are clear explaining the aim and purpose of the paper. It is proposed a stabilizer for an underactuated surface vessel affected by disturbances based on L2-gain design. Diffeomorphism transformation and Lyapunov function are employed in controller design to derive a robust adaptive controller. Simulation results demonstrate the validity of the proposed controllers.

The paper is well organized presenting the state-of-the-art of stabilization of underactuated surface vessels and the mathematical model of an underactuated surface vessel motion. In addition, the controller design is presented and finally the simulation results are reported and respective conclusions are presented in the last section.

The main conclusions based on the controller design and analysis process are:

The proposed control scheme can be grouped into discontinuous feedback approach since the sign function is included in the controller;

No asymptotic stability can be achieved but general stability instead. Errors can be made small as possible by decreasing the L2-gain index.

The disturbances taken into account in the study are uncertain and nonlinear in a real sense, which is consistent with the essence of environmental disturbances induced by wind, waves and current.

In general the paper is very clear and the results are well stated and presented. The paper presents interesting results and comparisons that can be extended to experimental results and the method can be applied to other control issues.

As recommendation the text of the Introduction section should be reviewed because some sentences are too short and there are not sequence between them.

Response: Thanks for the comments and suggestion. The introduction has been revised to improve the readability. Some words, phrases and sentences have been added, as follows

• Page 1, “two approaches are available.” has been extended as “two approaches are available for the stabilization of USVs”
• Page 2, “It is noted that in the aforementioned studies the plant…” has been extended as “It is noted that in the aforementioned studies the plant of USV…”
• Page 2, “..the practical feasibility of the controller.” has been revised as “…the practical feasibility of the controller for stabilization.”
• Page 2, statement “Different from the disturbance observer based approach, disturbance suppression based controller does not require the estimation of disturbance and corresponding compensation, but emphasize the achievement of required performance associated with the plant even affected by the worst disturbance. Among disturbance suppression based approaches,” has been added before the sentence “H-infinity control provides an effective way to suppress uncertain disturbances.”
• Page 2, “L2-gain design has been successfully…” has been extended as “L2-gain design applies to nonlinear systems with disturbance and has been successfully…”
• Page 2, statement “By this method, the influence of disturbance on required system performance can be suppressed.” has been added before the sentence “Moreover, by appropriately defining evaluation signals…”
• Page 2, phrase “In the study” has been added before the sentence “the variables to be stabilized are incorporated into…”; phrase “Besides L2-gain design” has been added before the sentence “other techniques adopted in the study include…”.
• Page 2, statement “To deal with the underactuation in the plant,” has been added before the sentence “both time invariant discontinuous and continuous feedback…”.

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

Overall, the paper is well-structured and quite interesting. In my opinion, it is a study to deal with a necessary problem, however, there are a lot of things that need to be addressed to meet the quality publication. Some of the concerns are as follows:

+ In the abstract, the original contributions have to be more highlighted. The limitations and the new ideas for improvement should be explained more. The novelty of the proposed system should be emphasized. The sentence “ Errors can be small as possible…” is not the contribution or new ideas of the paper.

+ In the introduction, it is suggested that the novel index of this paper should be explained in detail. And the introduction should be added to do a better job of explaining the existing methods and why they are or are not valuable. Moreover, in line 28, the more attention of USV should be added is docking or berthing, the authors can refer as: https://dx.doi.org/10.5574/JAROE.2016.2.4.192;

+ In section 2. Problem Formulations, a sub-section “Assumptions” should be added to make the problem clearer. The author can refer to the “Assumptions” section of the following paper:

“Study on Dynamic Behavior of Unmanned Surface Vehicle-Linked Unmanned Underwater Vehicle System for Underwater Exploration” Sensor, 2020. https://doi.org/10.3390/s20051329

+ Simulations analysis is insufficient, the superiority of the proposed control algorithm is unclear, the author is required to elaborate on the analysis of the simulation results. More simulation results need to be added in the paper, such as the disturbance forces and control input or thruster forces and moments. The explanations and analysis of simulation results should be enriched to show the validity of data.

+ The analysis in this paper should be supported by experimental results. The authors should use practical systems to validate the proposed methods with experiment results.

+ In this study, the motion control is applied the L2-gain control, to estimate the external disturbance acting on the system, the reviewer thinks that the disturbance observer can be designed to the system.

+The reviewers recommend that more future work should be added on Conclusion Section.

+ The English writing of this paper should be thoroughly polished, especially some grammatical errors and formula format errors are required to be revised carefully.

 

Author Response

First of all, authors sincerely appreciate the reviewer for his/her help with the paper. Authors revise the manuscript carefully according to the comments. As follows are the point-by-point responses.

comments: Overall, the paper is well-structured and quite interesting. In my opinion, it is a study to deal with a necessary problem, however, there are a lot of things that need to be addressed to meet the quality publication. Some of the concerns are as follows:

Concern 1: In the abstract, the original contributions have to be more highlighted. The limitations and the new ideas for improvement should be explained more. The novelty of the proposed system should be emphasized. The sentence “ Errors can be small as possible…” is not the contribution or new ideas of the paper.

Response：Thanks for the comment. In the abstract of revised manuscript, the contribution of the study has been explained in more details, stated as “To evaluate the controller performance of suppressing disturbances, two error signals are defined into which the variables to be stabilized are incorporated. Both time invariant discontinuous and continuous feedback laws are proposed in obtaining the control system.”

Authors agree with the reviewer. The sentence “ Errors can be small as possible…” is not the contribution or new ideas of the paper. This sentence has been removed in the revised manuscript.

Concern 2: In the introduction, it is suggested that the novel index of this paper should be explained in detail. And the introduction should be added to do a better job of explaining the existing methods and why they are or are not valuable. Moreover, in line 28, the more attention of USV should be added is docking or berthing, the authors can refer as: https://dx.doi.org/10.5574/JAROE.2016.2.4.192;

Response：Thanks for the suggestion. In the introduction of revised manuscript, more explanation has been added to make clearer the novelty of the paper, in the third paragraph of introduction section. Moreover, in line 30 (previous line 28), docking/berthing has been added for USV. The recommended reference has also been added in the revised manuscript.

Concern 3: In section 2. Problem Formulations, a sub-section “Assumptions” should be added to make the problem clearer. The author can refer to the “Assumptions” section of the following paper: “Study on Dynamic Behavior of Unmanned Surface Vehicle-Linked Unmanned Underwater Vehicle System for Underwater Exploration” Sensor, 2020. https://doi.org/10.3390/s20051329

Response：Thanks for the suggestion. In the revised manuscript, subsection “2.1 Assumption” and subsection “2.2 Mathematical Model” have been added. The recommended reference has also been added in the revised manuscript.

Concern 4: Simulations analysis is insufficient, the superiority of the proposed control algorithm is unclear, the author is required to elaborate on the analysis of the simulation results. More simulation results need to be added in the paper, such as the disturbance forces and control input or thruster forces and moments. The explanations and analysis of simulation results should be enriched to show the validity of data.

Response：Thanks for the comment. In the revised manuscript, more simulation results including disturbance forces and control inputs are added, as shown in Fig.4 and Fig.8 respectively. Moreover, the explanation and analysis of simulation results are enriched in the revised manuscript, in the paragraphs above the figures on simulation results.

Concern 5: The analysis in this paper should be supported by experimental results. The authors should use practical systems to validate the proposed methods with experiment results.

Response：Thanks for the comment. Unfortunately, at present authors cannot perform the experiments due to the lack of facility, ship model and ample financial support. Nevertheless, authors do hope in future they could use practical systems to validate the proposed methods with experiment results. In the revised manuscript, this has been mentioned in future work in the conclusion section.

Concern 6: In this study, the motion control is applied the L2-gain control, to estimate the external disturbance acting on the system, the reviewer thinks that the disturbance observer can be designed to the system.

Response：Thanks for the comment. Generally, the approaches to dealing with a system with external disturbances can be categorized into two types. One is to suppress disturbance while the other aims to compensate disturbance by designing a disturbance observer. In the study, the L2-gain control is designed to suppress external disturbances. In the revised manuscript, remarks on the methods of dealing with disturbances have been added in the introduction section, i.e. the third paragraph.

Concern 7: The reviewers recommend that more future work should be added on Conclusion Section.

Response：Thanks for the suggestion. In the revised manuscript, future work has been added in the conclusion section, mainly referring to experimental validation and the extension of this method to other control issues on the USVs such as path following, trajectory tracking, berthing/docking etc.

Concern 8: The English writing of this paper should be thoroughly polished, especially some grammatical errors and formula format errors are required to be revised carefully.

Response：Thanks for the comment. In revising the manuscript, authors tried their best to improve the readability and quality of the paper by checking the English writing and formula. Revised grammatical errors and English writing include:

• Line 25, “the sway motion cannot be controller” has been corrected as “the sway motion cannot be controlled”.
• Line 84 and line 99, “a surface vessel” has been replaced by “an underactuated surface vessel”.
• Line 112, “the second equation” has been revised as “the second equation of (3)”.
• Line 150, “Furthermore, It” has been corrected as “Furthermore, it”.
• Line 155, “state” has been replaced by “state variable”.
• Line 165 and line 200, “derivative” has been replaced by “time derivative”.
• Line 250, “plotted dashed lines” has been revised as “plotted by dashed lines”.
• Line 305, “backstepping techniques” has been replaced by “backstepping technique”.
• Line 309, “the sign function are” is replaced by “the sign function is”.

Reviewer 2 Report

See attachment

Comments for author File: Comments.pdf

Author Response

First of all, authors sincerely appreciate the reviewer for his/her help with the paper. Authors revise the manuscript carefully according to the comments. As follows are the point-by-point responses.

Controller Design is unconvincing.

Concern 1: The explanations here are insufficient end often erroneous.

For example the statement “obviously the smaller J is, the smaller ||z||₂ is” - is not correct because the whole matter refers to sup of the relation ||z||₂/||w||₂ ( see (22)). The issue is a little bit sophisticated and cannot be trivialized this way.

Response: Thanks for the comment. Authors agree with the reviewer. J = sup(||z||₂/||w||₂), (w≠0) means ||z||₂/||w||₂≤J satisfies according to the definition of sup. For given disturbance signal w, a smaller J means a smaller ||z||₂. That is what authors want to express in the original manuscript. However, it is noted that both w and z are variables while J is constant, which means that z changes with w under ||z||₂/||w||₂≤J. In other words, a large ||w||₂ possibly relates to a large ||z||₂, although a small ||z||₂ also satisfies ||z||₂/||w||₂≤J. In that sense, we could not simply state that “obviously the smaller J is, the smaller ||z||₂ is”. To avoid confusion, in the manuscript, the statement has been removed. The description of L2-gain performance by inequality are kept, which can be found in reference [25].

Concern 2: The statement “Combining the L2-gain with the derivative of Lyapunov function (21)” is not sufficiently explained to the reader making the whole procedure unclear.

Response: Thanks for the comment. As can be seen, inequality (22) involves integral representation. To guarantee the inequality (22) holds, the integrands in (22) are first considered. Moreover, the integrands can be combined with the time derivative of Lyapunov function (20) since the controller u₂ are involved in the time derivative (21). To make the controller design procedure more clear, in the revised manuscript, the above explanation has been added.

Concern 3: The backstepping method seems somehow poorly visible.

Response: Thanks for the comment. In the revised manuscript, to make the backstepping method more clear, explanation has been added, before section 3. In short, the controller u₂ are first designed, then u₁, then the controllers in system (8), finally the controllers in original system (1).

 

Minor mistakes:

• Page 6, line 149 : ||w||₂<∞? - signal norm must be finite by definition

Response: Thanks for the comment. In the revised manuscript, ||w||₂<∞ has been removed. As replied to the concern 1, the description of J = sup(||z||₂/||w||₂) is not enough. To avoid confusion, Eq.(22) in original manuscript and corresponding explanation have been removed in the revised manuscript.

• Page 6, line 150 „An equivalent inequality to can be defined as” should be ‘inequality can be’

Response: Thanks for pointing out the grammatical error. In the revised manuscript, such a statement has been removed because it relates to the description of J = sup(||z||₂/||w||₂)

• Page 6, line 154: „smaller ||z|| is” should be ‘smaller ||z||₂ is’

Response: Thanks for the comment. The L2-gain performance can be evaluated by the inequality (22) in revised manuscript. In the inequality, Euclidean norm is adopted. Therefore, ||z|| is used instead of ||z||₂. In the manuscript, it has been mentioned that ||•|| represents Euclidean norm.

• Page 6, line 163: „By properly choosing parameters r₁, λ₁ and α₁” - what means ‘properly choosing’ ?

Response: Thanks for the comment. As can be seen from (25), if the square of r₁<α₁/λ₁, the right hand side of inequality (25) can be guaranteed negative. This is can be achieved by choosing parameters r₁, λ₁ and α_1. In the revised manuscript, explanation of the selection of parameters has been added in the statement. Moreover, the word “properly ” has been removed since it is difficult to define how is “properly”.

Round 2

Reviewer 1 Report

In a general way, most of my comments were answered by the authors. The manuscript now is acceptable for publishing.

Author Response

Comment: In a general way, most of my comments were answered by the authors. The manuscript now is acceptable for publishing. Response: Thanks.

Reviewer 2 Report

comments are attached

Comments for author File: Comments.pdf

Author Response

Concern 1: Section 3. Controller Design is unconvincing.
The text (line 175 -177)
“As can be seen, inequality (22) involves integral representation. To guarantee the inequality (22) holds, the integrands in (22) are first considered. Moreover, the integrands can be combined with the time derivative of Lyapunov function since the controller u2 are involved in the time derivative (21).”

Response: Thanks for the comments. In the revised manuscript, authors revised the statement in a simpler way. "As can be seen, equality (21) contains controller u₂ and disturbance term  while inequality (22) describes the performance of suppressing the disturbance . One can combine (21) with (22) in designing the controller u₂ to guarantee that inequality (22) holds."

Concern 2: The backstepping method seems still poorly visible. The method consist in a several steps where, among other things, virtual control is designed etc. It is hardly to see these steps. The explanation than has been added, before section 3 is unsatisfactory.

Response: Thanks for the comments. Indeed, the backstepping method in the paper is not a standard backstepping method. The main difference is that the second Lyapunov function doesn't contain the first Lyapunov function. This is because we obtain a decoupled system by diffeomorphism transformation. However, standard backstepping method is applicable to a cascaded system. Although the controller design process starts from the decoupled system, ends at original system, it is not a standard backstepping method in a strictly sense. In the revised manuscript, the term "backstepping" has been removed throughout the paper.  

Round 3

Reviewer 2 Report

File in attachment

Comments for author File: Comments.pdf

Author Response

Authors appreciate the reviewer for his/her help with the paper. Please see the attachment for replies.

Author Response File: Author Response.pdf