Control of PMSM Based on Switched Systems and Field-Oriented Control Strategy

Round 1

Reviewer 1 Report

This paper addresses the complexity of the algorithms of the robust type, and the use of switched systems theory as a study option is proposed. By applying these concepts to the control system of a PMSM based on the Field Oriented Control (FOC) strategy, the value of its parameters during operation are usually changed. Based on Lyapunov-Meltzer type inequalities, the stability is proof. The following suggestions should be highly recommended to the authors for improvement.

1. There exist some grammar errors in this paper

2. The quality of the figures should be highly improved.

3. In Introduction, it is suggested that the author choose some papers of high quality to expound the characteristics of the algorithm. From your Introduction, there’s no context for the algorithm.

4. The contribution of this paper is not clear.

5. It is suggested to highlight the advantages of the proposed method by comparing it with similar methods.

Author Response

Dear reviewer, thanks for your recommendations.

1. Grammatical corrections were made to this article.
2. The quality of the figures is 300dpi.
3. They were introduced in the Introduction Section, some elements referring to 4 articles that present the YALMIP toolbox facilities [30-33], among which we mention: specialized solvers for the classes of problems to which it is applied and a unitary explanation of the way to use the syntax. The 4 articles are written especially by the YALMIP toolbox developer and were chosen as examples of problem classes regarding: Automatic robust convex programming and Explicit Model Predictive control (MPC) for Linear Parameter-Varying (LPV) systems: stability and optimality. With the help of the YALMIP toolbox, Lyapunov Meltzer type inequalities can be solved, which is the way to demonstrate the stability of switched systems. The added references that provide details regarding the implementation of the algorithms in the YALMIP toolbox are the following:

[30] Löfberg, J. Automatic robust convex programming. Optimization methods and software 2022, 27, 115–129.

[31] Besselmann, T.; Lofberg, J.; Morari, M. Explicit MPC for LPV Systems: Stability and Optimality. IEEE Transactions on Automatic Control 2012, 57, 2322–2332.

[32] Chandrasekaran, V; Shah, P. Relative entropy optimization and its applications. Mathematical Programming Series A 2016, 161, 1–32.

[33] Löfberg, J. YALMIP : A Toolbox for Modeling and Optimization in MATLAB. In Proceedings of the IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508), Taipei, Taiwan, 2-4 September 2004, pp. 284–289.

 

4. The main contributions of this paper can be summarized as follows:

• Presentation of the linearization of a PMSM around a static operating point;
• Summary of the basic elements and concepts of switched systems stability;
• Application of FOC control strategy and control switched systems for the control of a PMSM under significant variation of PMSM parameters that usually change their value during operation (stator resistance R_s, stator inductances L_d and L_q, but also combined inertia of PMSM rotor and load J);
• Implementation of Matlab/Simulink programs for the calculation of the control system characteristic matrices for parametric variations, calculation of the positive definite matrices P_i from Lyapunov-Meltzer inequalities to demonstrate the system stability;
• Implementation of the Matlab program to calculate of the dwell time;
• Numerical simulations development for the switched systems of the PMSM control for the switching signal with frequency lower than the one corresponding to the dwell time;
• The qualitative study of the control system performance by presenting in phase plane and state space the evolution of state vectors: ω PMSM rotor speed, i_q current, and i_d

 

5. In terms of parametric robustness, robust control systems [19-21] have obviously been developed and implemented with the best results, but the consideration of the complexity of robust computational algorithms should not be neglected. The switched systems are characterized by the fact that at certain moments of time, under the action of a switching signal, they can change their structure or parameter values. Thus, if the system changes its parameter values in a relatively large range, the use of switched systems theory [22-28] can be an alternative on the study of parametric robustness under the circumstances of a decrease in the complexity of the implemented algorithms. Compared to other elements of qualitative analysis of the stability of systems with time-varying parameters, among which we list the Kharitonov's theorem, the Nyquist criterion stability, and the Bode characteristics with other design elements of robust controllers for PMSM presented in [34], by solving the Lyapunov-Meltzer type inequalities can be obtained information regarding the stability of the system even under the conditions of some parametric and structural changes. According to example 2 in Section 3, it is proven once again that local stability does not imply global stability, in the sense that although each subsystem is stable, the evolution of the entire switched systems can be unstable. This means that although using the classical methods of stability analysis mentioned above, each subsystem is stable, but the mode of transition between these systems is not taken into consideration, this implied that the switched systems could be unstable. This discrepancy in the analysis of the stability of the switched systems is solved by specific means, namely by introducing the notion of dwell time and solving the Lyapunov-Meltzer type inequalities.

[34] Nicola, M.; Nicola, C.-I.; Ionete, C.; Şendrescu, D.; Roman M. Improved Performance for PMSM Control Based on Robust Controller and Reinforcement Learning. In Proceedings of the 26th International Conference on System Theory, Control and Computing (ICSTCC), Sinaia, Romania, 19-21 October 2022, pp. 207–212.

Author Response File: Author Response.pdf

Reviewer 2 Report

The paper has merits and deserves to be published. Some minor corrections are required.

1. Introduction should be rewritten with more clarity.

2. Authors are advised to write novelty in the introduction part.

3. Write a conclusion with future scope related to control problems. Authors can refer (Fractal and Fractional, 6(2), p.73.,  Fractal and Fractional, 6(2), p.73.).

Author Response

Dear reviewer, thanks for your recommendations.

1 and 2. They were introduced in the Introduction Section, some elements referring to 4 articles that present the YALMIP toolbox facilities [30-33], among which we mention: specialized solvers for the classes of problems to which it is applied and a unitary explanation of the way to use the syntax. The 4 articles are written especially by the YALMIP toolbox developer and were chosen as examples of problem classes regarding: Automatic robust convex programming and Explicit Model Predictive control (MPC) for Linear Parameter-Varying (LPV) systems: stability and optimality. With the help of the YALMIP toolbox, Lyapunov Meltzer type inequalities can be solved, which is the way to demonstrate the stability of switched systems. The added references that provide details regarding the implementation of the algorithms in the YALMIP toolbox are the following:

[30] Löfberg, J. Automatic robust convex programming. Optimization methods and software 2022, 27, 115–129.

[31] Besselmann, T.; Lofberg, J.; Morari, M. Explicit MPC for LPV Systems: Stability and Optimality. IEEE Transactions on Automatic Control 2012, 57, 2322–2332.

[32] Chandrasekaran, V; Shah, P. Relative entropy optimization and its applications. Mathematical Programming Series A 2016, 161, 1–32.

[33] Löfberg, J. YALMIP : A Toolbox for Modeling and Optimization in MATLAB. In Proceedings of the IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508), Taipei, Taiwan, 2-4 September 2004, pp. 284–289.

In terms of parametric robustness, robust control systems [19-21] have obviously been developed and implemented with the best results, but the consideration of the complexity of robust computational algorithms should not be neglected. The switched systems are characterized by the fact that at certain moments of time, under the action of a switching signal, they can change their structure or parameter values. Thus, if the system changes its parameter values in a relatively large range, the use of switched systems theory [22-28] can be an alternative on the study of parametric robustness under the circumstances of a decrease in the complexity of the implemented algorithms. Compared to other elements of qualitative analysis of the stability of systems with time-varying parameters, among which we list the Kharitonov's theorem, the Nyquist criterion stability, and the Bode characteristics with other design elements of robust controllers for PMSM presented in [34], by solving the Lyapunov-Meltzer type inequalities can be obtained information regarding the stability of the system even under the conditions of some parametric and structural changes. According to example 2 in Section 3, it is proven once again that local stability does not imply global stability, in the sense that although each subsystem is stable, the evolution of the entire switched systems can be unstable. This means that although using the classical methods of stability analysis mentioned above, each subsystem is stable, but the mode of transition between these systems is not taken into consideration, this implied that the switched systems could be unstable. This discrepancy in the analysis of the stability of the switched systems is solved by specific means, namely by introducing the notion of dwell time and solving the Lyapunov-Meltzer type inequalities.

[34] Nicola, M.; Nicola, C.-I.; Ionete, C.; Şendrescu, D.; Roman M. Improved Performance for PMSM Control Based on Robust Controller and Reinforcement Learning. In Proceedings of the 26th International Conference on System Theory, Control and Computing (ICSTCC), Sinaia, Romania, 19-21 October 2022, pp. 207–212.

The main contributions of this paper can be summarized as follows:

• Presentation of the linearization of a PMSM around a static operating point;
• Summary of the basic elements and concepts of switched systems stability;
• Application of FOC control strategy and control switched systems for the control of a PMSM under significant variation of PMSM parameters that usually change their value during operation (stator resistance R_s, stator inductances L_d and L_q, but also combined inertia of PMSM rotor and load J);
• Implementation of Matlab/Simulink programs for the calculation of the control system characteristic matrices for parametric variations, calculation of the positive definite matrices P_i from Lyapunov-Meltzer inequalities to demonstrate the system stability;
• Implementation of the Matlab program to calculate of the dwell time;
• Numerical simulations development for the switched systems of the PMSM control for the switching signal with frequency lower than the one corresponding to the dwell time;
• The qualitative study of the control system performance by presenting in phase plane and state space the evolution of state vectors: ω PMSM rotor speed, i_q current, and i_d current.

3. Also, in future approaches, one of the research directions will be the study of approximate controllability of fractional integrodifferential equations using resolvent operators.

[35] Vijayakumar, V.; Nisar, K.S.; Chalishajar, D.; Shukla, A.; Malik, M.; Alsaadi, A.; Aldosary, S.F. A Note on Approximate Controllability of Fractional Semilinear Integrodifferential Control Systems via Resolvent Operators. Fractal Fract. 2022, 6, 73.

Author Response File: Author Response.pdf

Reviewer 3 Report

1, Authors should define all abbreviations at first mention, and also provide a nomenclature list.

2, Most of the MATLAB snapshots, codes and Simulink models, are redundant and makes the article unnecessarily lengthy. Rather than copying them into the study, authors should upload them in an online server and provide link for interested readers. 

3, MATLAB generated figures can be plotted better and clearer; authors can look at YouTube videos to do this.

4, Authors should provide more discussions on the attainment of steady-state conditions in their adopted PMSM control techniques.

Author Response

Dear reviewer, thanks for your recommendations.

1. We have defined all abbreviations at first mention, and also we provide a nomenclature list.

Nomenclature

PMSM          Permanent Magnet Synchronous Motor;

FOC              Field Oriented Control;

DTC              Direct Torque Control;

YALMIP      A Toolbox for Modeling and Optimization in MATLAB;

R_s                  Stator resistance of the PMSM;

R_d and R_q      Stator resistances on d-q axis;

L_d and L_q       Stator inductances on d-q axis;

u_d and u_q       Stator voltages on d-q axis;

i_d and i_q         Stator currents on d-q axis;

T_L                  Load torque;

J                    Combined inertia of PMSM rotor and load;

B                   Combined viscous friction of PMSM rotor and load;

λ₀                   Flux induced by the permanent magnets of the PMSM rotor in the stator phases;

n_p                  Pole pairs number;

ω                   PMSM rotor speed.

2 and 3. We consider that the observations from these points 2 and 3 are subjective because they refer to a manner of presentation regarding graphics and code elements, which belong to reviewer no. 3. With all due respect, as readers, authors, and Guest Editors/Academic Editors of this Special Issue "Dynamics and Intelligent Control of Complex and Switched Systems" of which this article will be a part, we ask the Editor-in-Chief's permission to leave the code fragments and graphics in the way we thought of them, given the fact that the potential readers of this article and of this Special Issue will find in a unitary and compact manner elements of theory, the concrete modeling of a system and how they can find fast "dwell times" and check the Lyapunov-Meltzer inequalities to deduce the stability of a switched system.

4. Starting from the problem of studying the parametric robustness in the case of the control of a Permanent Magnet Synchronous Motor (PMSM), although robust control systems correspond entirely to this problem, due to the complexity of the algorithms of the robust type, in this article we propose the use of switched systems theory as a study option, given the fact that these types of systems are suitable both for the study of systems with variable structure, but also systems with significant parametric variation under conditions of lower complexity of the control algorithms. For a uniform presentation, we start from the linearization of a PMSM around a static operating point and continue with a synthetic presentation of the basic elements and concepts concerning the stability of switched systems, by applying these concepts to the control system of a PMSM based on the Field Oriented Control (FOC) strategy, which usually changes the value of its parameters during operation (stator resistance R_s, stator inductances L_d and L_q, but also combined inertia of PMSM rotor and load J).

In Section 2, the mathematical model of the PMSM, the FOC control strategy and the linearization of the mathematical model of the PMSM are presented.

In Section 3, the main elements are presented, through which the stability of a switched system can be demonstrated in case of a parametric variation. Among these, the most important are the dwell time and the Lyapunov-Meltzer inequalities.

In Section 4, 3 examples are considered: the model of a PMSM controlled with FOC, the model of a PMSM in which the parametric variations contribute to the definition of 2 PMSM models, and the model of a PMSM in which the parametric variations contribute to the definition of 4 PMSM models. After calculating the dwell time and checking the Lyapunov Meltzer inequalities, the conclusion of PMSM stability can be drawn using switched systems theory. On the other hand, just to confirm the results obtained using the FOC strategy, the state space portraits are presented for the qualitative analysis of the system's behavior, confirming the stability and parametric robustness of the system. It can be concluded that, by using the switched systems theory in the presented example of PMSM control, the FOC control strategy is a control strategy that ensures parametric robustness, in the sense that in case of significant variations of the parameters in the PMSM structure, the overall performance of the control system is preserved both qualitatively and quantitatively.

In terms of parametric robustness, robust control systems [19-21] have obviously been developed and implemented with the best results, but the consideration of the complexity of robust computational algorithms should not be neglected. The switched systems are characterized by the fact that at certain moments of time, under the action of a switching signal, they can change their structure or parameter values. Thus, if the system changes its parameter values in a relatively large range, the use of switched systems theory [22-28] can be an alternative on the study of parametric robustness under the circumstances of a decrease in the complexity of the implemented algorithms. Compared to other elements of qualitative analysis of the stability of systems with time-varying parameters, among which we list the Kharitonov's theorem, the Nyquist criterion stability, and the Bode characteristics with other design elements of robust controllers for PMSM presented in [34], by solving the Lyapunov-Meltzer type inequalities can be obtained information regarding the stability of the system even under the conditions of some parametric and structural changes. According to example 2 in Section 3, it is proven once again that local stability does not imply global stability, in the sense that although each subsystem is stable, the evolution of the entire switched systems can be unstable. This means that although using the classical methods of stability analysis mentioned above, each subsystem is stable, but the mode of transition between these systems is not taken into consideration, this implied that the switched systems could be unstable. This discrepancy in the analysis of the stability of the switched systems is solved by specific means, namely by introducing the notion of dwell time and solving the Lyapunov-Meltzer type inequalities.

[34] Nicola, M.; Nicola, C.-I.; Ionete, C.; Şendrescu, D.; Roman M. Improved Performance for PMSM Control Based on Robust Controller and Reinforcement Learning. In Proceedings of the 26th International Conference on System Theory, Control and Computing (ICSTCC), Sinaia, Romania, 19-21 October 2022, pp. 207–212.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

This version is well revised. It can be accepted in present form.

Author Response

Dear reviewer, thanks for your recommendations and appreciations.

Reviewer 3 Report

Authors have responded.

Author Response

Dear reviewer, thanks for your recommendations and appreciations.