Approach for the Assessment of Stability and Performance in the s- and z-Complex Domains

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This manuscrit presents a systematic approach for rapid assessment of the performance and robustness of linear control systems through geometric analysis in the complex plane. The main issue is that the sicentific contribution is not clearly stated. What is the relevance of the proposed work given the well-stablished state of the art in linear control systems?. Can authors derive more general gemetric interpretations: see https://doi.org/10.1109/CDC.2005.1583206  In addition, some issues to be addresed are the following:

1. Introduction. A table sumary with similar approaches or literature results related to the presented work. Please make clear differences between past contributions and the ones authors propose.
2.  Gamma regions. It is quite hard to follow the manuscript since some equations are not in standard form (e.g. Eq. 1) and in some others, its parameters are not clearly defined (e.g.  in Eq. 4, ai,bj, a^ref_i, b^ref_j are not defined). Moreover, some acronyms are not defined in its first apearance in the manuscript (MISO, IPI, DIP etc.).
3. Robustness in the s-plane. Authors state that: "An original geometric interpretation of the conditions for Robust Stability 𝑅𝑆 and Robust Performance 𝑅𝑃, defined in the frequency domain by equations (31) and (32),..." Please include a close reference to this afirmation. Figures 4-7, are confusing, there are regions clossin the real part of the complex plane (instability), please make an effort to better explain them.
4.  Parametric Uncertainty Models. It is not clear the process of generating and how are related the 𝑸,𝑸𝒔,,𝑸𝒛,𝑸𝝎𝒔,𝑸 boxes, please make an effort to better describe this, flow diagram might be helpful. It also might be helpful to use an academic linear control system (e.g. dc motor) and to perform the numerical simulation to validate the authors proposal.
5. Conclusion. This reviewer believes by considering the above comments author sould render sharpen conclusions.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

NOTES:
• This paper examines the robustness of linear control systems through geometric analysis of poles in the complex plane. Robust control is an important issue in applications.
• Consider a class of linear or linearized systems (noise-free, delay-free) with multiple inputs and a single output (MISO systems). See (47) or (48) for the transfer function.
• Appropriate control systems are considered. For the characteristic equation of the closed-loop system, see (52). See (80) for discrete systems.
• For interesting, original examples, see Chapters: 4.2.1 Numerical Example, 4.3.1 Numerical Example (Discrete), 4.3.2 Investigating the Effect of Sampling Time Ts, 4.4.1. Numerical Example, 4.4.2. Frequency-Domain Robustness. 
• Discrete-time systems from continuous-time systems were obtained using different discretization methods (ods, 𝑍𝑂𝐻, 𝑇𝑢𝑠𝑡𝑖𝑛), and the influence of sampling time 𝑇𝑠 was investigated through an original modeling approach that treats it as an a priori uncertainty.
• The work is well-written. I SUGGEST PUBLISHING IT.

POSSIBLE ADDITIONS:
• In further considerations, I suggest drawing attention to the works by S. Białas and M. Busłowicz:
[1] Białas, S. (1985). A necessary and sufficient condition for the stability of convex combinations of stable polynomials or matrices. Bulletin of the Polish Academy of Sciences Technical Sciences 33(9–10): 473–480.
[2] M. Busłowicz (1990), "Comments on stability analysis of interval matrices: improved bounds". International Journal of Control, vol.51, no.6, 1990, 1485-1486.

 

Author Response

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Reviewer 3 Report

Comments and Suggestions for Authors

This paper presents a systematic approach for rapid assessment of the performance and robustness of linear control systems through geometric analysis in the complex plane. By combining indirect performance indices within a defined zone of desired performance in the complex s-plane, a connection is established with direct performance indices, forming a foundation for the synthesis of control algorithms that ensure root placement within this zone. It is interesting, but some minor corrections are needed before considering it.

1. Include a flowchart for a summary of the whole process from plant model → s-plane zone → z-plane mapping → performance/robustness evaluation.
2. Discuss the limitations of this study.
3. There are some typographical errors and missing punctuation in the manuscript. The authors should remove this problem.
4. Describe the application of the framework to systems with time delays, lightly nonlinear dynamics, or varying sampling periods.
5. Language and formatting issues should be addressed to enhance readability.
6. Abstract must be revised, and applications of the problem must be included.
7. Mention the main motivation of this work clearly.
8. Include a graphical abstract that highlights the entire research.
9. Some short form, like “MISO” should be defined on the first appearance.

Author Response

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Reviewer 4 Report

Comments and Suggestions for Authors

In this work, the author presents a systematic approach for rapid assessment of the performance and robustness of linear control systems through geometric analysis in the complex plane. By combining indirect performance indices within a defined zone of desired performance in the complex s-plane, the authors establishes a connection with direct performance indices, forming a foundation for the synthesis of control algorithms that ensure root placement within this zone. The author derives a analytical relationships between the complex variables s and z, thereby defining an equivalent zone of desired performance for discrete-time systems in the complex z-plane. Methods for verifying digital algorithms with respect to the desired performance zone in the z-plane are presented, along with a visual assessment of robustness through radii describing robust stability and robust performance—representing performance margins under parameter variations. Through parametric modeling of controlled processes and their projections in the complex s- and z- domains, the influence of the discretization method and sampling period - as forms of a priori uncertainty is analyzed. 

I read the manuscript in deep. The present work offers original derivations for MISO systems, facilitating the analysis, explanation, and understanding of the dynamic behavior of real-world controlled processes in both the continuous and discrete-time domains, and is aimed at integration into expert systems supporting control strategy selection. 

The introduction is well conducted (Brief overview, comparison and explanation of existing work, research Aim and specific objective).
Geometric definitions of the Gamma region are given as well as robustness.The author present consisely the parametric uncertainty Models, and apply the proposed method on several numerical examples. The work ends by discussion and conclusion. 

As minor revision, the authors must:
1- ends each equation by a corresponding punctuation sign;
2- increase the size of figures for readability;
3- Compared the results obtained by the proposed methods with the existing methods
4- Include future issue of this work in the conclusion section.

 

Author Response

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Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Ok, all my comments were correctly addressed.