Review Reports - Iterative Feedback Tuning of Model-Free Intelligent PID Controllers

Round 1

Reviewer 1 Report

The paper introduces a novel iterative feedback tuning approach for tuning intelligent PID controllers (IFT). Using Quanser AERO laboratory equipment and an experimental test setup, the suggested technique is verified and evaluated. The study is quite fascinating. In Table 1, the author has presented some extremely intriguing findings that are useful for applications where system modeling is challenging. The author has employed a very basic lower order system to test the efficacy of proposed control scheme. The author must use an experimental setup with a higher order transfer function to test the effectiveness of the suggested control strategy.

Author Response

Dear reviewer,

Thank you for your feedback and for the aspects mentioned in you comment which helps us to improve our manuscript. Regarding the open points introduced by you in the review there are our comments:

Thank you for pointing this. From the presentation in the paper of the Aero 2 equipment in 1-DOF configuration, it follows that the model-free controller from the outer loop has as plant the inner loop transfer function (a second order system) to which a nonlinear model of the propeller is added. A more detailed scheme of the cascade control structure was added to the revised manuscript in Section 4, from which it results that the plant of the outer loop is a complex and non-linear one due to the non-linearities introduced by using a single propeller operating in both positive and negative thrust and by shifting the center of mass towards thruster 1.

Reviewer 2 Report

The manuscript entitled “ Iterative Feedback Tuning of Model-Free Intelligent PID Controllers” proposed control scheme is applied. The manuscript is well written and easy to follow, with the scientifically sound mathematical framework described in detail. The contributions of the paper are clearly stated. The stability and effectiveness of the proposed control strategy are proved for tuning the iPID controllers based on the IFT algorithm was tested and experimentally validated using a real-time application for controlling the pitch angle of the dual-motor aerospace system by QUARC. The results obtained by simulations are clearly presented and appropriately discussed. However, there are some comments I would like the authors to address before the manuscript is considered for publication:

- The authors have implemented the method in real-time and testing it on a real system to provide experimental verification, please plots same data from real robot.

- In the Conclusion section, please provide some limitations of the proposed research.

In the Conclusion section, please also provide some challenges and directions for future research.

Author Response

Dear reviewer,
Thank for the attention awarded to our manuscript. In the reviewed version we tried to clarify and improve aspects mentioned by you as follows:

Point 1: The authors have implemented the method in real-time and testing it on a real system to provide experimental verification, please plots same data from real robot.

Answer 1: Thank you for pointing this. Indeed, the methodology for tuning the model-free controllers based on IFT algorithm was tested and validated through real-time experiments performed using Quanser Aero 2 laboratory equipment in 1-DOF configuration. All the results obtained during the tests were plotted using the data collected during the real-time experiments.

Point 2: In the Conclusion section, please provide some limitations of the proposed research.

Answer 2: Additional remarks were added in the Conclusion section of the revised manuscript:

“The main advantages of the proposed tuning method based on the IFT algorithm consist of the following: the determination of both the gain parameters related to the PID algorithm and the alpha parameter; the IFT tuning of the iPID parameters does not require knowledge of the plant model; the gradients of the quadratic control criterion can be easily estimated based on the data collected after two experiments.

Among the disadvantages, it is mentioned the choosing the starting values of the iPID tuning parameters in such a way as to obtain a stable response and the performing of two experiments to determine the tuning parameters at each step. The initial choice of tuning parameters, as well as finding the related values of the step size and the penalty factor of the control effort through a trial-and-error procedure are the main design challenges.

For future research, some techniques for tuning the parameters related to the step size and the penalty factor of the control effort, as well as the extending of the proposed method to multivariable systems are considered.”

Reviewer 3 Report

The paper presents an approach for the iterative feedback tuning of model-free intelligent PID controllers. The proposed approach is verified with experimental tests. The topics of the paper are interesting. However, the following comments should be addressed to improve the overall quality and readability of the paper:

1) The main contributions of the paper should be better highlighted especially with respect to previous works on the same topic.

2) The mathematical formulation is hard to read and should be in some parts simplified, where possible. All the terms reported in the equations should be described in the text.

3) In the abstract, the parameter alpha is not defined: this point should be clarified or removed from the abstract.

4) The experimental setup should be better described. What are the models of the sensors used and their technical specifications? The software used to implement the proposed approach should be mentioned. Furthermore, all the parameters of the proposed controller should be reported in the text. Otherwise, it would be impossible for the reader to replicate the experimental tests.

5) The proposed approach should be compared with other approaches available in the literature for the tuning of PID controllers. Why the classical Ziegler Nichols method cannot be applied in this context?

6) Can the proposed approach be suitable for alternative mechanical systems, as for instance flexible multibody systems?

7) The literature review should be improved. Some suggested references are reported below:

Nonlinear control of multibody flexible mechanisms: a model-free approach. Applied Sciences, 11(3), 1082, 2021.

Kalman Filter and Variants for Estimation in 2DOF Serial Flexible Link and Joint Using Fractional Order PID Controller. Applied Sciences, 11(15), 6693, 2021.

Author Response

Dear reviewer,

Thank you for your suggestions and time offered to review our manuscript. The observation you entered was beneficial for our manuscript in order to improve it. In the reviewed version of the manuscript we tried to fulfill all of you requirements as follows:

Point 1: The main contributions of the paper should be better highlighted especially with respect to previous works on the same topic.

Answer 1: Thank you for pointing this. In the Introduction section of the revised manuscript, the following contributions were added:

“This paper proposes the following new contributions with respect to the state-of-the-art:

(i) a new method of tuning model-free iPID controllers based on the IFT algorithm

(ii) the calculation of both the tuning gains of the iPID controller and the parameter alpha

(iii) determining a fixed structure for iPID controllers based on the connections between the iPID and PID discrete-time controllers to make it possible to apply the IFT tuning algorithm

(iv) analyzing the connections between the iPID and PID controllers, the model-free control laws that correspond to some variants of the classical PID were determined and those which are uncommon in the control engineering were eliminated

(v) the tuning of the iPID controllers using the IFT algorithm was tested and validated experimentally on the Quanser AERO 2 laboratory equipment.”

Point 2: The mathematical formulation is hard to read and should be in some parts simplified, where possible. All the terms reported in the equations should be described in the text.

Answer 2: Thank you for your comment. In the revised manuscript we added a more precise description of the terms and notations used, as follows:

“The subscript of iPID₁ indicates the derivative order nu=1.” (text added after Eq. (7))

“The subscript of iPID₂ indicates the derivative order nu=2.” (text added after Eq. (17))

“The superscript (a) and (b) used in Eqs. (33-34) indicate that the signals were obtained during experiments (a) and (b).” (text added after Eq. (34))

Point 3: In the abstract, the parameter alpha is not defined: this point should be clarified or removed from the abstract.

Answer 3: Thank you for pointing this. In the abstract of the revised manuscript, we added information regarding parameter alpha:

“Among the Model-Free Control techniques, intelligent PID (iPID) algorithms based on an ultralocal model parameterized with the constant α and including a classical PID are used in many applications.”

Point 4: The experimental setup should be better described. What are the models of the sensors used and their technical specifications? The software used to implement the proposed approach should be mentioned. Furthermore, all the parameters of the proposed controller should be reported in the text. Otherwise, it would be impossible for the reader to replicate the experimental tests.

Answer 4: Thank you for your comment. A new scheme of the cascade control structure of the Quanser Aero 2 with 1-DOF configuration was added to the revised manuscript in Section 4, with a more detailed description.

Concerning the sensors, more details were added in the revised manuscript, as follows:

“Single-ended optical shaft encoders are used to measure the pitch of the Aero body and the angular speed of the DC motors [29]. The pitch encoder is 2880 counts per revolution in quadrature mode (720 lines/rev), and the angular speed encoder is a counter which returns the number of encoder counts every second.”

Regarding the software, the following text was added: “The Aero 2 system from Quanser is connected with a PC, where the control scheme from Fig. 2 is implemented using Matlab/SIMULINK software tools.”

Related to the tuning parameters of both model free controllers iP₁ and iPD₂, used to the pitch control, in the Section 4 of the revised manuscript, all the values obtained during the experiments are given.

Point 5: The proposed approach should be compared with other approaches available in the literature for the tuning of PID controllers. Why the classical Ziegler Nichols method cannot be applied in this context?

Answer 5: The IFT algorithm can be applied for tuning the classical PID controllers, as it is shown in [19], where the advantages of this data-driven tuning method compared to the classical ones, such as Ziegler-Nichols, are proven. The Ziegler-Nichols empirical methods are applied to the classical continuous-time PID controllers but cannot be applied to the iPID controller which has a different operating principle incompatible with the requirements imposed by the Ziegler-Nichols methods. An iPID controller cannot be transformed into a PID by setting some parameters, but the behavior of an iPID can be analyzed in relation to a PID.

Point 6: Can the proposed approach be suitable for alternative mechanical systems, as for instance flexible multibody systems?

Answer 6: The new data-driven method for tuning the parameters of the model-free iPID controller presented in the paper can be applied in principle to any SISO process, including for example, the mechanical or electrical systems. If the flexible multibody systems are of SISO type, the proposed tuning method for an iPID controller can be applied.

Point 7: The literature review should be improved. Some suggested references are reported below:

Nonlinear control of multibody flexible mechanisms: a model-free approach. Applied Sciences, 11(3), 1082, 2021.

Kalman Filter and Variants for Estimation in 2DOF Serial Flexible Link and Joint Using Fractional Order PID Controller. Applied Sciences, 11(15), 6693, 2021.

Answer 7: Thank you for your suggestions.

To improve the literature review, we added the following references:

Kim, B.M.; Yoo, S.J. Approximation-Based Quantized State Feedback Tracking of Uncertain Input-Saturated MIMO Nonlinear Systems with Application to 2-DOFHelicopter. Mathematics 2021, 9, 1062.

Wu, B.;Wu, J.; Zhang, J.; Tang, G.; Zhao, Z. Adaptive Neural Control of a 2DOF Helicopter with Input Saturation and Time-Varying Output Constraint. Actuators 2022, 11, 336.

Lambert, P.; Reyhanoglu, M. Observer-based sliding mode control of a 2-DoF Helicopter system. In Proceedings of the IECON 2018 44th Annual Conference of the I EEE Industrial Electronics Society, Washington, DC, USA, 21–23 October 2018.

Hjalmarsson H., Control_of nonlinear systems using iterative feedback tuning, Proceedings of the 1998 American Control Conference. ACC, Philadelphia, PA, USA, 1998.

Regarding the two references suggested, in the first one a PD controller with a nonlinear derivative gain is used. The tuning of this controller is done using a trial-and-error procedure, and hence the authors’ conclusion that it is a model-free control. In fact, the nonlinear PD controller does not fit into the model-free control techniques that use data acquired from the process for tuning the controllers.

In the second suggested reference, the controller used is a Fractional Order PID that cannot be included in the model-free category.

Reviewer 4 Report

This work proposes an original tuning approach to the more recent intelligent model-free control (MFC) structures, using a -now clasical- iterative feedback tuning approach. The results well fit the recent active research domain of data-driven control which captured widespread attention from the community.

My impression is that the work is very well written, with particular attention to details and mathematical correctness.
The papers' main strenght is the experimental validation of the proposed control. Secondly, indirect tuning for intelligent MFC controller is considered as an useful tool in control design. Indeed there are many challenges in validating such approaches, e.g. noise, experiment repeatability conditions and hyperparameter design.

My comments follow next:

1. Please motivate the approximate backward-difference discretization method selection in (3), why not use other methods.

2. The tuning effectiveness is clear from the experiments. Could authors point out how many IFT iterations did they perform and how the parameters' evolution in the iteration domain look like?

3. Could authors point out some tuning rules for the methods' hypeerparameters? I am thinking about the step-size gama in (38) and the \lambda penalty of the control effort from (25).

4. How did you preserve well-conditioning in the matrix R_i from (35) ?

5. What are the computational complexity challenges associated with the proposed method.

6. IFT tuning rules seem to work well with linear system assumption, however most real-world system are nonlinear, how to extend the results in this case?

7. How to extend the proposed design approach to multivariable systems?

8. Please describe the advantages and the disadvantages of the proposed method. Also, what were the main design challenges? This could help a lot the interested readers.

9. Minor language revision is recommended in terms of style.

A review effort from the authors is welcome.

Author Response

Dear reviewer,
We would like to thank you for your time spent for reviewing our manuscript and for giving us valuable comments and suggestions. We tried to improve our work using the aspects introduced by you and to clarify, as much as possible, the open points found in the initial version of the manuscript.

Point 1: Please motivate the approximate backward-difference discretization method selection in (3), why not use other methods

Answer 1: Thank you for your comment. To apply the IFT algorithm, the iPID controller must be put in a form with a fixed structure. In this paper, this fixed structure was built based on the connections between the discrete-time iPID and PID controllers. A simple way to obtain the discrete form of the iPID and PID controllers is to use the backward-difference discretization method, as in [9]. Of course, other discretization methods can be used, in which case the equations for computing the tuning parameters will be modified.

Point 2: The tuning effectiveness is clear from the experiments. Could authors point out how many IFT iterations did they perform and how the parameters' evolution in the iteration domain look like?

Answer 2: Thank you for your attention awarded to that topic. In the revised manuscript we added detailed information regarding iP and iPD tuning procedure. For the iP was considered 4 iterations of IFT algorithm, while for iPD 5 iterations and the evolution of the parameter values was presented in Table 2 and 3.

Point 3: Could authors point out some tuning rules for the methods' hypeerparameters? I am thinking about the step-size gama in (38) and the \lambda penalty of the control effort from (25).

Answer 3: Thank you for pointing this. The step size gamma and the penalty factor lambda were determined through a trial-and-error procedure.

Point 4: How did you preserve well-conditioning in the matrix R_i from (35) ?

Answer 4: There are several possibilities to choose the matrix R that gives the negative gradient direction. As mentioned in [6], the identity matrix is a common practice, but the matrix defined by (30) can also be used, which employs data collected from the process. In the paper we used the identity matrix, thus avoiding the cases where the matrix R calculated with (37) would become ill-conditioned

Point 5: What are the computational complexity challenges associated with the proposed method.

Answer 5: Thank you for pointing this. The proposed method does not present a huge complexity in terms of computational power and capabilities. Yes, it may exist plants which requires a high amount of data samples which can require more computational power, but from our point of view it may not represent a big challenge for MFC-IFT method.

Point 6: IFT tuning rules seem to work well with linear system assumption, however most real-world system are nonlinear, how to extend the results in this case?

Answer 6: Thank you for pointing this. Indeed, the IFT algorithm can be applied for tuning the linear time-invariant controllers with the fixed structure considering a linear time-invariant plant to be controlled. In the case of non-linear systems, the solution proposed by Hjalmarsson in [6] and [35] is to generate the true gradient using the linear time-varying system obtained by linearizing the nonlinear system around the system trajectory under normal operating conditions. In this context, the same IFT procedure as for linear systems can be applied. This information was introduced at the end of the Subsection 3.1

Point 7: How to extend the proposed design approach to multivariable systems?

Answer 7: For MIMO systems, in [H. Hjalmarsson and T. Birkeland, “Iterative Feedback Tuning of linear time-invariant MIMO systems,” Proc of CDC 98, 1998.] how to calculate the gradients of a quadratic control criterion considering a multivariable controller is given. Having in view that our proposed tuning method considers a monovariable model-free iPID controller, it cannot be extended directly to the multivariable case.

Point 8: Please describe the advantages and the disadvantages of the proposed method. Also, what were the main design challenges? This could help a lot the interested readers.

Answer 8: Thank you for your suggestions. We have introduced in the Conclusion section of the revisited manuscript the following paragraph dedicated to the mentioned aspects.

Point 9: Minor language revision is recommended in terms of style.

Answer 9: Thank you for your recommendation that we took into consideration.