Projection onto the Set of Rank-Constrained Structured Matrices for Reduced-Order Controller Design
Round 1
Reviewer 1 Report
The paper presents a synthesis procedure for the computation of reduced order output-feedback controllers for linear systems. The proposed is solved iteratively and is based on LMIs. Some numerical examples are presented to illustrate the approach in the context of stabilizing and H-infinity controllers.
The problem investigated is well known and the literature has a great number of methods, some based on non-smooth optimization and some based on convex optimization (LMIs). In the present reviewer's opinion, the paper cannot be accepted in the present form. See my comments in what follows.
-Section 3 is rather confuse. It is not clear what algorithms are new and, withing the proposed techniques, what is the best. In the numerical examples, it is mentioned that the proposed algorithm is in section 3.3.2. But what exactly is being solved in section 3.3.2? At least the main contribution of the paper should be established in terms of an algorithm, where it is clear what must programmed and solved iteratively.
-In the numerical examples, the present reviewer considers a good idea to evaluate the performance of the proposed method in the Complib database, considering many systems. However, the comparisons are not considering other important methods that address the problem of reduced order synthesis. For instance, hinfstruct:
https://www.mathworks.com/help/robust/ref/lti.hinfstruct.html
and hifoo
https://cs.nyu.edu/~overton/software/hifoo/
Moreover, other LMI-based methods, as
https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=618250
could also be used since the proposed method is based on LMIs.
-In the numerical examples, an important information is missing: computational times. Since all available methods for reduced order design are only sufficient (conservative), the relevant information is the trade-off between accuracy and computational complexity.
Minors:
-In the abstract, suggestion: change 'the problem is NP_hard' to 'the problem is nonconvex'.
-Keywords: there is an missing ')' at the end.
-Seubsection -> Subsection
As conclusion, in the present form the paper does not show clearly the advantage of the proposed method when compared with the existing techniques.
Author Response
The paper presents a synthesis procedure for the computation of reduced order output-feedback controllers for linear systems. The proposed is solved iteratively and is based on LMIs. Some numerical examples are presented to illustrate the approach in the context of stabilizing and H-infinity controllers.
The problem investigated is well known and the literature has a great number of methods, some based on non-smooth optimization and some based on convex optimization (LMIs). In the present reviewer's opinion, the paper cannot be accepted in the present form. See my comments in what follows.
Thank you very much for reviewing our paper and giving valuable comments. We have revised the paper according to your comments.
Section 3 is rather confuse. It is not clear what algorithms are new and, withing the proposed techniques, what is the best. In the numerical examples, it is mentioned that the proposed algorithm is in section 3.3.2. But what exactly is being solved in section 3.3.2? At least the main contribution of the paper should be established in terms of an algorithm, where it is clear what must programmed and solved iteratively.
Thank you for the valuable comments. To clarify and focus on the novelty of the proposed method, we omitted explanations on nuclear norm minimization and approximate alternating projection. In particular, to clarify the difference between our method and [13], we added Remark 2 in the last paragraph of Section 2 (pp. 5-6). We believe the proposed algorithm and its novelty are now clear by these revisions.
In the numerical examples, the present reviewer considers a good idea to evaluate the performance of the proposed method in the Complib database, considering many systems. However, the comparisons are not considering other important methods that address the problem of reduced order synthesis. For instance, hinfstruct:
https://www.mathworks.com/help/robust/ref/lti.hinfstruct.html
and hifoo
https://cs.nyu.edu/~overton/software/hifoo/
Moreover, other LMI-based methods, as
https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=618250
could also be used since the proposed method is based on LMIs.
Thank you for the valuable suggestion. We have conducted new numerical experiments of stabilizing static controller design for the 95 benchmark models with "hinfstuct" in MATLAB, which uses the nonsmooth H-infinity synthesis ([36] in the new manuscript), and the cone complementarity linearization algorithm [12]. We have completely rewritten Subsection 4.1 (p. 6) accordingly. We summarized the results in Table 1. As in this table, the "hinfstruct" is the best method among the five methods including the proposed method. However, our method is still effective since there are some plants that can be solved by the proposed method but not by other methods. We emphasize that our method is still an effective method that may provide a solution to a reduced-order controller design problem that cannot be solved by standard methods such as nonsmooth H-infinity, cone complementarity linearization, and nuclear norm minimization.
In the numerical examples, an important information is missing: computational times. Since all available methods for reduced order design are only sufficient (conservative), the relevant information is the trade-off between accuracy and computational complexity.
Thank you for the valuable comment. We have added the average CPU time in Table 1. The full list of the CPU time for each problem is available at [28].
-In the abstract, suggestion: change 'the problem is NP_hard' to 'the problem is nonconvex'.
-Keywords: there is an missing ')' at the end.
-Seubsection -> Subsection
Thank you very much. We have corrected these.
Reviewer 2 Report
I have the following observations:
1. In this work, it is proposed an efficient numerical computation method of reduced-order controller design for linear time-invariant systems.
2. I suggest inserting more references from last 5 years ISI journals type.
3. The paper it is well written, but in my opinion missing a section where it’s proved the stability of the new numerical computation method or it's made some references to others proof for this algorithm.
Comments for author File: 
Comments.docx
Author Response
Thank you very much for carefully reading our manuscript and giving valuable comments. We have revised our manuscript according to your comments as follows:
>2. I suggest inserting more references from last 5 years ISI journals type.
Thank you very much for this suggestion. We have added 5 recent papers [21]-[25], which are referred to in the third paragraph of Section 1.
>3. The paper it is well written, but in my opinion missing a section where it’s proved the stability of the new numerical computation method or it's made some references to others proof for this algorithm.
Thank you for this valuable comment. We have added a result of stability as a lemma in Subsection 3.1 (Lemma 1 on p.4 and its proof in Appendix A). Also, for the convergence of the ADMM algorithm proposed in Subsection 3.2 (p. 5), we refer to reference [33].
Reviewer 3 Report
The paper presents a numerical computation method of reduced-order controller design for linear time-invariant systems. Based on some numerical examples, the solution is compared with the nuclear norm minimization and the approximated alternating projection method to illustrate the effectiveness of the proposed algorithm. To highlight the difference between the proposed method and the other methods, 95 benchmark models whose order are less than 1000 from COMPLeib library.
Observations and recommendations for improving the manuscript:
- Typos in line 96 (Seubsection 3.3.2.) and in line 198 (Appendix F… Why F?). Please correct.
- Some supplementary details about the COMPLeib library would be useful. For example, about the referred models (AC4, NN1, NN11…).
- The conclusions should be expanded, specifying the advantages of the proposed method. Also, it should be detailed what better performance means.
- Maybe some more up-to-date (more recent) reference titles would be useful.
Author Response
Thank you very much for carefully reading our manuscript and giving valuable comments. We have revised our manuscript according to your comments as follows:
>Typos in line 96 (Seubsection 3.3.2.) and in line 198 (Appendix F… Why F?). Please correct.
Thank you very much. We have corrected these.
>Some supplementary details about the COMPLeib library would be useful. For example, about the referred models (AC4, NN1, NN11…).
We have added explanations for AC4, NN1, NN12, and HE6 in Subsection 4.2 with references [35]-[38].
>The conclusions should be expanded, specifying the advantages of the proposed method. Also, it should be detailed what better performance means.
Thank you very much for your advice. We have extended the conclusions according to your suggestion (Section 5, p. 7).
>Maybe some more up-to-date (more recent) reference titles would be useful.
Thank you very much for this suggestion. We have added 5 recent papers [21]-[25], which are referred to in the third paragraph of Section 1.
Round 2
Reviewer 1 Report
The paper has been revised, and more useful information has been included in the paper. For instance, numerical comparisons with other techniques.
The reviewer has one additional observation: the proposed method has not been evaluated in the context of reduced-order controllers with orders greater than zero (all examples are based on static feedback). The authors could present more experiments (for instance, in section 4.2) testing orders 0, 1, and 2, for example, and show that the method can improve the results as the order grows. It is known that some methods are suitable for static feedback but not for higher orders.
Finally, without the codes available in [28], the present reviewer could not figure out how to implement the proposed algorithm. Please, provide a detailed sequence of steps that must be performed in order to solve the proposed technique.
Minors:
-Please rephrase "... a few models with which these methods take very long time"
-Please rephrase "... and the proposed method has shown better performance in that the proposed algorithm successfully solved more benchmark problems than the other methods".
Author Response
The paper has been revised, and more useful information has been included in the paper. For instance, numerical comparisons with other techniques.
Thank you very much for reviewing our paper again and giving valuable comments. We have revised the paper according to your comments.
The reviewer has one additional observation: the proposed method has not been evaluated in the context of reduced-order controllers with orders greater than zero (all examples are based on static feedback). The authors could present more experiments (for instance, in section 4.2) testing orders 0, 1, and 2, for example, and show that the method can improve the results as the order grows. It is known that some methods are suitable for static feedback but not for higher orders.
Thank you for the valuable comments. We have examined 1st and 2nd order stabilizing controllers for the plant model TF1 (Section 4.2, pp. 7-8). We observed that hinfstruct (with default options) still fails to solve the problem while the other methods, including the proposed method, can solve it. We have added a new section (Section 4.2) for these results.
Finally, without the codes available in [28], the present reviewer could not figure out how to implement the proposed algorithm. Please, provide a detailed sequence of steps that must be performed in order to solve the proposed technique.
Thank you for this suggestion. We have summarized the proposed algorithm in Algorithm 1 (p. 6).
-Please rephrase "... a few models with which these methods take very long time"
-Please rephrase "... and the proposed method has shown better performance in that the proposed algorithm successfully solved more benchmark problems than the other methods".
Thank you for pointing out these. We have rephrased as
"a few models (e.g., EB5 and JE1) for which it takes a very long CPU time by these methods"
" the proposed algorithm successfully solved benchmark problems that other methods could not."
Reviewer 3 Report
The authors have taken into account the remarks and comments received and the overall quality of the manuscript has been improved. Only a small suggestion: please explicitly refer to Appendix B in the paper (line 86). I consider that the paper can be published.
Author Response
The authors have taken into account the remarks and comments received and the overall quality of the manuscript has been improved. Only a small suggestion: please explicitly refer to Appendix B in the paper (line 86). I consider that the paper can be published.
Thank you very much for reviewing our paper again and giving valuable comments. We have explicitly referred to Appendix B in the paper (p. 5).
