Multiloop Multirate Continuous-Discrete Drone Stabilization System: An Equivalent Single-Rate Model
Round 1
Reviewer 1 Report
The paper presents the development of a Multiloop Multirate Continuous-Discrete Drone Stabilization System, the conceptual and mathematical development is complete and detailed. I suggest to reinforce the results with a simulation of the system in a software such as Matlab-Simulink.
Author Response
Response to Reviewer 1 Comments
Point 1: I suggest to reinforce the results with a simulation of the system in a software such as Matlab-Simulink..
Response 1: Corrected. In part 3 a comparative analysis of the step response of stabilization systems for the different types of mathematical models was carried out. As equivalent models we will take: a continuous system with the same transfer functions, but with continuous circuits in the feedback; a discrete single-rate system with equal sampling periods in feedback loops and an equivalent system with a sampling period T which is the greatest common divisor of the sampling periods.
Author Response File: 
Author Response.docx
Reviewer 2 Report
The presented solution seems to be interesting, but the presentation is limited to mathematical manipulations only. There is no target UAV details and parameter description presented, so the presented control system is very general. In addition, there are no implementation, testing or other validation data of the proposed method presented too, so the evaluation of the results is complicated.
The presentation of the proposed method is not clear – very similar symbols are used for different variable notation, sometimes formatting of variables in formulas and text differs. The spelling of the terms “singlerate” and “multirate” is not regular.
Author Response
Response to Reviewer 2 Comments
Point 1: There is no target UAV details and parameter description presented, so the presented control system is very general.
Response 1: Corrected. In part 1 we show the transfer functions of the stabilization system. We consider the stabilization system of the lateral movement of the air-craft-type drone, when solving, the problem of UAV landing on a non-aircraft-carrying ship.
Point 2: In addition, there are no implementation, testing or other validation data of the proposed method presented too, so the evaluation of the results is complicated.
Response 2: Corrected. In part 3 a comparative analysis of the step response of stabilization systems for the different types of mathematical models was carried out. As equivalent models we will take: a continuous system with the same transfer functions, but with continuous circuits in the feedback; a discrete single-rate system with equal sampling periods in feedback loops and an equivalent system with a sampling period T which is the greatest common divisor of the sampling periods.
Point 3: The presentation of the proposed method is not clear – very similar symbols are used for different variable notation, sometimes formatting of variables in formulas and text differs.
Response 3: The complexity of presenting the results lies in the need to designate the same variables for different sampling periods. For this, an explanation of the location of the "*" symbol has been introduced. The symbol "*" at the bottom marks the property of periodicity. The symbol "*" at the top marks denotes the execution of the discrete Laplace transform.
Also to simplify notation until the final results, the symbol in the designation of the corresponding impulse transformation will be omitted, i.e. for any function z(s), instead z*T(s) of we will write z*(s).
The designation of all variables is given in the text of the article.
Point 4: The spelling of the terms “singlerate” and “multirate” is not regular.
Response 4: These terms in various articles and books have different spellings. The term “singlerate” has been replaced by “single-rate” as the most common term.
The term “multirate” is used in various literature. So, for example, in:
- Coffey, T.C.; Williams I.J. Stability Analysis of Multiloop, Multirate, Sampled Systems AIAA Journal. 1966,4, 2178-2190 https://0-doi-org.brum.beds.ac.uk/10.2514/3.3874;
- Araki M.; Yamamoto K. Multivariable Multirate Sampled-Data Systems: State-Space Description, transfer Characteristics, and Nyquist Criterion. IEEE Trans. Auto. Contr. 1986, 31, 145-154. https://0-doi-org.brum.beds.ac.uk/10.1109/TAC.1986.1104205;
- Berg, M.C.; Amit, N.; Powell, J.D. Multirate digital control system design. IEEE Trans. Auto. Contr.1988, 33, 1139-1150. https://0-doi-org.brum.beds.ac.uk/10.1109/9.14436;
- Multirate Systems: Design and Applications by Gordana Jovanovic-Dolecek, 2012;
etc. the term “multirate” is used. Therefore, in their article, the authors use this term.
Author Response File: 
Author Response.pdf
Reviewer 3 Report
The paper deals with the problem of modeling a multidimensional (multiloop), multirate system. However, it is not clear what exactly it concerns. After showing two models from literature (fig. 1 and fig.2) (with mentioned parameters and signals but without explanation of the system), we start with equations (1) which are not explained, without explaining how they relate to the system. Do we use the system from fig.2? How do you define the output y, what is the vector x, how the transfer functions look like? Section 2 is a completely rewritten text from reference [18]; it is not clear what is new in the paper. Afterward, in the result part, we have a further derivation of the model, using the idea of the highest common divisor of the sampling period. After the final model is derived, there is no verification (e.g., by simulation) of its accuracy. Grammar of the text should be improved, see, e.g., lines 62-64, 68-71. Even the rewritten section 2 has a lot of linguistic errors (e.g., 'let u the 1xm vector'; 'let x is a vector').
Author Response
Response to Reviewer 3 Comments
Point 1: After showing two models from literature (fig. 1 and fig.2) (with mentioned parameters and signals but without explanation of the system), we start with equations (1) which are not explained, without explaining how they relate to the system. Do we use the system from fig.2? How do you define the output y, what is the vector x, how the transfer functions look like?.
Response 1: Corrected. The article provides a demonstration of a system specified using a structural diagram in the form of an input-output equation (1). All variables and transfer functions of equation (1) are added in the text of article for the system specified by the structural diagram in fig.2.
Fig. 1. demonstrates the system given in the literature [13] as confirmation that such systems are being considered.
Point 2: Section 2 is a completely rewritten text from reference [18]; it is not clear what is new in the paper.
Response 2: Corrected. As new results, we propose a complete description of a general approach to the construction of an equivalent single-rate system, which makes it possible to transform the vector-matrix model of the original MIMO multiloop multirate system into a vector-matrix model of a single-rate system in a typical form for vector-matrix models of continuous systems. The model construction includes a complete description of the equivalent T-single-rate and NT-single-rate, where T is the largest common divisor of the sampling periods of the system, N is a natural number that is the great common multiple of the numbers characterizing the sampling periods of the system. It describes the central element of the construction - a structural invariant of sampling chains - a matrix, which in the general case describes a complex picture of the mutual influence of sampling processes in digital circuits of a closed-loop system of a MIMO multirate stabilization system with different sampling periods.
The mathematical model given in Section 2 is the basic model. It is obvious that for such a model it is not possible to use the methods of analysis and synthesis due to the fact that it contains elements that depend on different sampling periods. The presented mathematical model, due to the presence of factors taking into account influencing factors in the description, is not suitable for solving problems of analysis and synthesis. It is the basic model for constructing an equivalent mathematical model, which, in turn, can be used to extend the classical methods of analysis and synthesis for multidimensional single-rate discrete systems to the class of systems under consideration.
Point 3: After the final model is derived, there is no verification (e.g., by simulation) of its accuracy.
Response 3: Corrected. In Section 3 a comparative analysis of the step response of stabilization systems for the different types of models was carried out. As equivalent models we will take: a continuous system with the same transfer functions, but with continuous circuits in the feedback; a discrete single-rate system with equal sampling periods in feedback loops and an equivalent system with a sampling period T which is the greatest common divisor of the sampling periods.
Point 4: Grammar of the text should be improved, see, e.g., lines 62-64, 68-71. Even the rewritten section 2 has a lot of linguistic errors (e.g., 'let u the 1xm vector'; 'let x is a vector').
Response 4: Corrected. Text checked with full version “Grammarly”.
Author Response File: Author Response.pdf
Round 2
Reviewer 3 Report
The contents of the paper are improved. Simulation experiments show that the derived equivalent provides a result similar to the original system. However, the presentation is still unclear. In some figures, pictures obscure other pictures. It is also not clear which model is actually considered (e.g., in fig. 2 it looks like there are two schemes of the same inputs and outputs (psi, gamma), but the first has additional inputs on the left-hand side, undefined... which of those two is considered? Are those two somehow connected and used both? In fig. 3 - why to use 'u' and 'y' and then gamma_z and gamma? Should the output and input be the same in both schemes? If you write W_0(s) you should be consistent also in the first scheme). Still, there are language errors. If the (part of the) title reads 'An Equivalent A Single-rate Model' - a should be removed (we have 'an' already for the model). In the text, we can find similar errors. E.g., in line 82 you have removed 'it is', but this way there is no adverb in the sentence.
Please check the whole paper, paragraph by paragraph, equation by equation, and improve it so it is more clear.
Author Response
Response to Reviewer 3 Comments
Point 1: «In some figures, pictures obscure other pictures».
Response 1: Corrected. In fact, in Figure 8. the step response for the constructed equivalent model and the output of the multirate system practically coincide, which confirms that the derived equivalent provides a result similar to the original system.
Point 2: «It is also not clear which model is actually considered (e.g., in fig. 2 it looks like there are two schemes of the same inputs and outputs (psi, gamma), but the first has additional inputs on the left-hand side, undefined... which of those two is considered? Are those two somehow connected and used both?».
Response 2: Corrected. In the revised version of the article, Figure 2 has been corrected. Incorrect display of pictures may have occurred. Both the new and old Figure 2 are displayed. In the current version, there is only one Figure 2. As inputs, the setting actions psi_z, gamma_z are considered.
Point 3: «In fig. 3 - why to use 'u' and 'y' and then gamma_z and gamma? Should the output and input be the same in both schemes?»
Response 3: The situation is similar to Figure 3. The revised version uses gamma_z as an input, and gamma as an output. 'u' and 'y' were used in Figure 3 in the old version of the article.
Point 4: «If you write W_0(s) you should be consistent also in the first scheme».
Response 4: Corrected. Shown is the formula for W_0(s) in accordance with Figure 2.
Point 5: «Still, there are language errors. If the (part of the) title reads 'An Equivalent A Single-rate Model' - a should be removed (we have 'an' already for the model). In the text, we can find similar errors. E.g., in line 82 you have removed 'it is', but this way there is no adverb in the sentence.»
Response 5: Corrected. Text checked by the native English-speaking colleague who made the appropriate corrections
Author Response File: Author Response.docx
