An Experimental Tuning Approach of Fractional Order Controllers in the Frequency Domain

Round 1

Reviewer 1 Report

This paper presents an experimental tuning procedure of the fractional order PID controller. The advantage of the proposed tuning technique is the elimination of the need of a process mathematical model. The proposed tuning approach is validated by experiments of active vibration suppression. In my opinion, the tuning technique is very helpful for the process which can not obtain an exact model.

Comments

1. Regarding the optimization procedure (Line 111-114), the initial points of controller parameters are very important. The fractional orders of integral and differential belong to the interval (0,1), it seems to easy to choose the initial points. What about the initial points of the proportional and derivative gains (Kp and ki)? Have any guideline to choose them? For example, why Kp and Kd are chosen k_p = 15, k_d = 0.5 (line 154)?
2. It seems that the controller performance is too slower than any beam frequencies, for example, the settling time is about 1.5s with the 12.5Hz beam’s frequency (line 203). The tuned parameters are good or not?
3. The manuscript has some editing errors:

function using the method from [] ... (line 156)

Space should be added between each item in the caption of Figure 4

Author Response

Comments

1. Regarding the optimization procedure (Line 111-114), the initial points of controller parameters are very important. The fractional orders of integral and differential belong to the interval (0,1), it seems to easy to choose the initial points. What about the initial points of the proportional and derivative gains (Kp and ki)? Have any guideline to choose them? For example, why Kp and Kd are chosen kp = 15, kd = 0.5 (line 154)?

Reply: The structure of the manuscript has been modified to include a new subsection (4.1) that includes a guideline on choosing the starting points of the optimization routine.

1. It seems that the controller performance is too slower than any beam frequencies, for example, the settling time is about 1.5s with the 12.5Hz beam’s frequency (line 203). The tuned parameters are good or not?

Reply: The controller is efficient because it effectively mitigates the vibration. However, the method doesn’t guarantee that the best controller is obtained when analyzing performance such as settling time. The main advantage of the method is the simplified tuning procedure without using the process’ model. The approach based on the frequency domain phase and magnitude optimizations easily obtains a viable controller that can be successfully implemented in real-life vibration applications. The focus on the optimization procedure lies in flattening the magnitude peak which guarantees vibration suppression of the closed loop system without specific focus on the settling time.

An additional paragraph has been added at the end of Section 2 in order to include this clarification.

3. The manuscript has some editing errors: function using the method from [] ... (line 156). Space should be added between each item in the caption of Figure 4

Reply: Reference [39] has been added to reference the discretization method. The subfigures have been rearranged to include additional space between their sub-captions. In addition, several other editing and writing errors have been corrected throughout the manuscript. The figures have also been compacted and presented in a more organized manner.

We thank the reviewer for the time spent in reviewing our paper. All changes have been highlighted in the revised manuscript attached.

Author Response File: Author Response.pdf

Reviewer 2 Report

PID controller are widely used in modern engineering filed. Authors presented an effective method to determine the controller parameters experimentally, which eliminated the needs of mathematical model. Thanks for submitting this interesting work and it is potential to be very useful in certain applications, such as close-loop sensor system. However, there are some minor comments have to be addressed before publication. 

1. In the introduction section, by including practical applications of the PID controller, especially, the fractional order PID controller, can broaden the cognition of PID based technologies. For example, the utilization of fractional order PID in a specific sensor system.
2. Again, in the introduction section, a short summary of the smart beam system is recommended. 
3. Just an advice, you can draw a 3D schematic of the beam subsequent to Figure 2, which makes the beam structure clearer. 
4. It is better to have a sentence to mention the exact (numerical value) resonant frequency of the beam. 
5. In the end of page 6, "The obtained fractional order PD controller is approximated to a 6th order discrete transfer 156 function using the method from []", missing a reference.
6. In Figure 4(h), we can see the vibration amplitude falls into a low value and it might buried in the noise. Hence comes a question, what is the noise floor of your experimental set-up?
7. In page 12, "weights have been added to the free end of the beam such that the resonant frequency is moved 200 from 14Hz to 12.5Hz and 11.7Hz", it is better to clarify how much weight is added.

Author Response

Comments

1. In the introduction section, by including practical applications of the PID controller, especially, the fractional order PID controller, can broaden the cognition of PID based technologies. For example, the utilization of fractional order PID in a specific sensor system.

Reply: Two paragraphs have been added in the Introduction section that illustrate the applicability of fractional order PID control. The first one is focused on industrial applications, while the second one presents various fractional order control strategies that have been successfully applied to vibration systems. The manuscript has been enriched with references [24-31] and [32-38].

Furthermore, another paragraph has been added in the Introduction illustrating the facility of applying the developed method in industrial applicability.

2. Again, in the introduction section, a short summary of the smart beam system is recommended. 

Reply: A paragraph with a brief description of the smart beam has been added, as well as the motivation behind the experimental setup choice.

3. Just an advice, you can draw a 3D schematic of the beam subsequent to Figure 2, which makes the beam structure clearer. 

Reply: A 3D schematic of the experimental setup has been added (Figure 2) along with some additional information related to the beam experimental system.

4. It is better to have a sentence to mention the exact (numerical value) resonant frequency of the beam. 

Reply: The resonant frequency is now explicitly mentioned in a separate sentence.

5. In the end of page 6, "The obtained fractional order PD controller is approximated to a 6th order discrete transfer 156 function using the method from []", missing a reference.

Reply: The missing reference [39] has been added.

6. In Figure 4(h), we can see the vibration amplitude falls into a low value and it might buried in the noise. Hence comes a question, what is the noise floor of your experimental set-up?

Reply: Figure 5h (previously 4h) shows the response of the beam at an input sine wave of frequency 40Hz. Even if the noise is more prominent for this frequency and can be observed in the figure, an amplitude value can still be read from the acquired data. Figure 9 presents the response of the beam when applied a swept sine input between 9Hz and 90Hz. Analyzing the response of the beam to the varying frequency input, we can observe a second resonant value at frequency 83Hz. From experimental tests, we have concluded that we are unable to acquire any relevant data when applying a sine input of frequencies inside the [60, 75] interval and greater than 90Hz.

7. In page 12, "weights have been added to the free end of the beam such that the resonant frequency is moved 200 from 14Hz to 12.5Hz and 11.7Hz", it is better to clarify how much weight is added.

Reply: Two Neodymium – Iron – Boron permanent magnetic disks (10mm diameter, 5mm height, 30 grams weight and 1 kg strength) were placed on each side of the free end of the beam. The magnets were attached at a 3cm distance to the moving end, centered with respect to the upper and lower margins of the beam.

The issue has been clarified in the manuscript.

We thank the reviewer for the time spent in reviewing our paper. All changes have been highlighted in the revised manuscript attached.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

In my opinion, the revised manuscript is sufficient enough to publish in Applied Science.

Author Response

Thank you for your help in improving the manuscript.

Reviewer 2 Report

Thank you for the modified manuscript, you have addressed my comments clearly.

Author Response

Thank you for your help in improving the manuscript.