# Model-Free VRFT-Based Tuning Method for PID Controllers

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{S}) [16], using cascade control structures [18,19], and optimizing disturbance-rejection performance [20].

- -
- The tuning method requires only the measurement of the manually or automatically controlled experiment on the process, where the process changes the steady state (the process input signal is not limited to step-like signals).
- -
- The user does not need to provide any prior information about the process, such as the process transfer function, the desired control loop transfer function or time constants.
- -
- Although the closed-loop response is not defined by the user, it is implicitly defined by the open-loop response. The advantage of matching the responses is that there are no exaggerated swings of the controller output signal in response to the setpoint change, which is often preferred by plant operators in many industrial plants.
- -
- Because the tuning method discards the initial response of the process (e.g., during process time delay of the process), the method slows the closed-loop response for processes with larger time delays and therefore inherently stabilizes the closed-loop response.
- -
- Due to the regression method used, the proposed method is relatively insensitive to process measurement noise.
- -
- The proposed method is simple, does not require optimization, is not computationally intensive, and therefore can be implemented on less powerful hardware, such as lower-end PLC controllers.
- -
- All scripts (Matlab/Octave) for calculating the controller parameters are available online, allowing the user to immediately calculate the controller parameters for any selected process model or from provided process input and output signals. The scripts are open-source, can be copied and modified as needed.
- -
- The method can be extended to include an additional velocity factor that speeds up or slows down the closed-loop response compared to the open-loop response accordingly. This extension requires an additional estimate of the process time delay and process average residence time, which can be trivially determined from the open-loop response.

## 2. VRFT Method

_{CL}

^{−1}is the inverse of the desired closed-loop transfer function, and r* and e* are the fictitious reference and control error, respectively.

_{C}) can be calculated by regression from the following expression:

^{*}and Y denote the Laplace transforms of the signals u, r

^{*}and y, respectively. The bias in the estimation of the controller parameters can be reduced by additional filtering of the input and output signals of the process with the following filters [7]:

_{C}is a cutoff frequency of the low-pass filter W

_{LP}(s) and G

^{*}

_{CL}is the desired closed-loop transfer function without time delay. Therefore, the final equation used by the regression method to calculate the controller parameters is as follows:

## 3. Equalization Method

_{CR}) and feedback (G

_{CY}) transfer function blocks. Assume that the process is defined by the following unknown transfer function:

_{PR}is the steady-state process gain, T

_{delay}is the time delay, and a

_{1}, a

_{2}, …, b

_{1}, b

_{2}, … are the dynamic process parameters. The parameters of the 2-DOF controller can be calculated from expression (6) when the reference signal R is calculated from the process output signal and the desired closed-loop transfer function:

_{PR}, since the steady-state closed-loop gain must equal one. In this case, the reference signal (8) becomes

_{P}, K

_{I}, K

_{D}and T

_{F}are the proportional, integral, and derivative gains of the controller and the filter time constant of the controller, respectively. The chosen 2-DOF controller structure is one of the most common structures in process control. Considering (13), the expression (11) becomes:

_{F}depends on the desired high-frequency gain of the controller, which corresponds to K

_{D}/T

_{F}. Similar to the VRFT method, the process input and output signals can be additionally filtered (4) to reduce the bias in noisy signals, as shown in Section 4.

_{I}, K

_{P}and K

_{D}) can be calculated using a regression or optimization method. For process input and output signals, represented by n discrete measurements, the regression matrix

**Ψ**is as follows:

_{F}, e

_{F}, and y

_{F}are time equivalents of the Laplace signals I

_{F}(s), E

_{F}(s) and Y

_{F}(s), respectively, which can be derived by filtering process signals as shown in Figure 4.

**u**is the vector of the process input measurements:

_{F}(s)) even though the process input signal is constant during the time delay. To alleviate the mentioned problems, expression (14) should be considered only from the time when the process output (y) starts to change (e.g., after reaching 10% of the total steady-state change), as shown in Figure 5.

**W**is a square diagonal matrix whose diagonal elements are zero for samples 1…k − 1 (see Figure 5), where the change in process output is less than 10% of the total change in steady state, and one for the remaining diagonal elements:

_{1}…w

_{k}

_{−1}= 0 and w

_{k}…w

_{n}= 1.

_{CL}. The advantage of the proposed tuning method is that it does not require the definition of the desired closed-loop transfer function. Fortunately, for the purposes of signal filtering, the desired closed-loop transfer function need not be exact. In fact, it is sufficient to estimate the desired closed-loop dynamics by the first-order transfer function with steady-state gain 1 without time delay:

_{CL}is equal to the process time constant T

_{OL}. The latter can be estimated as the sum of the process time constants without pure time delay, which, according to expression (7), is equivalent to:

_{ar}[34], which can be easily calculated or estimated from the process step response as shown in Figure 6.

**Ψ**

_{F}contains additional filtered signals e

_{F}, i

_{F}, and y

_{F}, as shown in Figure 7.

**u**contains the measured values of the filtered signal u:

_{F}_{F}, i

_{F}and y

_{F}in the regression matrix

**Ψ**(15), including the signal vector

**u**should be additionally filtered with F(s) before calculating the controller parameters by the regression method (25). This method will be referred to as Filtered WLS (FWLS) method.

**Remark**

**1.**

_{D}= K

_{D}/K

_{P}is negative, the derivative gain must be set to K

_{D}= 0. The last column (−y

_{F}) in the regression matrices

**Ψ**(15) or

**Ψ**

_{F}(25) should be deleted. Of course, the resulting controller vector in (18) and (25) reduces only to the integrating (K

_{I}) and proportional (K

_{P}) gain.

**Remark**

**2.**

- Measure the open-loop response of the process and obtain the process input (u
_{OL}) and output (y_{OL}) signals and subtract the initial steady-state values of the signals. - Estimate the process gain K
_{PR}from the steady-states values of the input and output signals of the process. - Calculate the reference signal r from (10) or Figure 3 (can be calculated automatically).
- Find the time when the process output has risen to 10% of the final steady-state value (see Figure 5) and calculate the controller parameters from (18) or (25).

**Example**

**1.**

_{PR}is then calculated from (12) and the signals i

_{F}, e

_{F}and y

_{F}are then calculated according to Figure 4, where the filter time constant of the controller is chosen to be T

_{F}= 0.1. The open-loop process output response reaches 10% of the steady-state change at t = 2.3 s, so all diagonal elements of matrix

**W**(19) are set to zero prior to t = 2.3 s. The estimated time delay was t

_{delay}= 0.5 and the average residence time t

_{ar}= 2.5, so the desired closed-loop time was T

_{CL}≈ t

_{ar}− t

_{delay}= 2 s.

## 4. Noise Sensitivity

^{−5}and a sampling time of 0.01 s (the open-loop and closed-loop sampling time) to the process output using the Simulink block “Band-limited white noise”. The same process as in Example 1 (27) was used. The calculation of the controller parameters was repeated 100 times with different values of the random generator. All 100 open-loop responses to the input step-change are shown in Figure 9 (upper figure). As can be seen, the noise amplitude is about 0.1 (about 10% of the process steady-state change during the experiment). Therefore, the considered samples by WLS and FWLS methods were the ones when the process output reached 20% of the process steady-state change. The controller parameters are first calculated using the WLS method for each noisy open-loop response as described in the previous section. The histogram of the controller parameters for all 100 runs is shown in Figure 10. It is obvious that the calculated controller gains (with the exception of K

_{I}) are far from the values obtained with the undisturbed process signals (see red circles in the histograms). The bias caused by the process measurement noise is particularly visible in the calculation of the derivative gain K

_{D}, where all values were 0 (according to Remark 1, all originally calculated K

_{D}were negative, so the K

_{D}parameter was fixed at 0 and the remaining two controller parameters were recalculated).

## 5. Adaptive Equalization Method

_{P}should be known and the desired closed-loop transfer function G

_{CL}should be defined as in the VRFT method. Therefore, all the advantages of the proposed equalization method would be lost. On the other hand, the process transfer function could be approximated by the first-order process with estimated time delay and average residence time t

_{ar}(sum of process time constants), while the desired closed-loop transfer function G

_{CL}could be defined similarly to the process transfer function but with faster or slower time constant:

_{S}stands for speed factor (k

_{S}> 1 means that the closed-loop response is faster than the open-loop response and vice versa) and t

^{*}

_{ar}is given in (22). In this case, expression (28) simplifies to:

_{S}. However, if the desired closed-loop response is faster than the open-loop response (k

_{S}> 1), the high-frequency measurement noise in y will be amplified by an additional factor k

_{S}when y* is calculated. This can be avoided by filtering all three signals in (31) by the additional inverse filter G

_{F}

^{−1}(s) = G

_{F}

_{1}(s):

_{PR}∙u) must be additionally filtered with G

_{F}

_{1}(s), and the desired closed-loop speed can be modified according to the speed factor k

_{S}. The new signal r is shown in Figure 14. The controller parameters are then calculated by first filtering all three signals in (32) by (24), where the desired closed-loop time constant is:

^{*}

_{CL}:

_{F}(26). Increasing the closed-loop speed factor k

_{S}generally increases the difference between the u

^{*}

_{CL}and u

_{F}. Therefore, the speed factor k

_{S}should not be increased too much. The proposed method for automatically determining the speed factor k

_{S}increases it until the difference between u

^{*}

_{CL}(34) and the filtered process input signal u

_{F}(26) becomes larger than a certain threshold:

_{UR}is a relative standard deviation. The suggested value for σ

_{URmax}is 0.1.

_{y}) of the filtered process output signal y* (31) should be limited:

_{ymax}≤ 0.1.

_{S}is to increase it from the initial value k

_{S}= 1 until the relative standard deviation (35) exceeds the value 0.1 or the relative filtered overshoot y* (36) exceeds a certain value (e.g., 0.05). Of course, the upper value of speed factor k

_{S}should also be limited to a maximum value (i.e., if the closed-loop response should be up to 10-times faster than the open-loop response, the maximum value of k

_{S}can be set to 10).

_{S}should be set. In practice, the minimum value of k

_{S}should be set to 0.2. Please note that the limitation of the factor k

_{S}is not explicitly shown in the flow chart (Figure 16).

- Select the appropriate Octave (MATLAB) script (FWLS_auto_01.m if you are calculating the parameters from the process transfer function or FWLS_auto_measurements_01.m if you are calculating the parameters directly from the measured process open-loop response).
- Modify the process and the user-defined parameters.
- Press the “Save” button, and
- Press the “Run” button. The results will be displayed in the lower part of the right window.
- If the script does not finish in time (display “!!! OUT OF TIME !!!” at the bottom of the right window), click the “Run” button again and then click the “Add 15 s” link at the bottom of the right window while the script is running. Click the “Add 15 s” link several times if necessary.

## 6. Examples and Experiments

_{F}= 0.2, according to expressions (25) and Figure 16 or directly using the Matlab/Octave script [35], as mentioned in the previous section.

_{URmax}= 0.1 and o

_{ymax}= 0.05, respectively. The obtained controller parameters are listed in Table 2.

_{P}

_{1}(Figure 17) shows an ideal match between the open-loop and the closed-loop tracking responses using the FWLS method. The adaptive method automatically increased the closed-loop response by a factor of K

_{S}= 9.85 since the maximum allowable speed factor was 10.

_{P}

_{2}(Figure 18) also shows an almost ideal match between the open-loop and the closed-loop tracking response when using the FWLS method. The adaptive FWLS method increased the closed-loop response by a factor of K

_{S}= 1.61. A lower process output overshoot is observed, which is lower than the prescribed limit (5%).

_{P}

_{3}(Figure 19) are almost indistinguishable when the FWLS method is used. The adaptive FWLS method slightly increases the closed-loop response by a factor of K

_{S}= 1.33. Again, a smaller process output overshoot of less than 5% can be seen.

_{P}

_{4}(Figure 20) is slower than the open-loop response when the FWLS method is used. This is consistent with observations in Remark 2. Due to the larger difference (relative standard deviation) between the process open-loop and the closed-loop signals, the adaptive FWLS method slightly reduced the closed-loop speed by 10%.

_{P}

_{5}are shown in Figure 21. It can be seen that the closed-loop response with the FWLS method is much slower than the process open-loop response. The adaptive FWLS method significantly improves the closed-loop response without affecting the closed-loop stability.

_{S}= 0.5 s. Figure 24 shows the open-loop response.

_{S}= 1, (b) applying the adaptive FWLS method with σ

_{URmax}= 0.1 and o

_{ymax}= 0.05, and (c) applying the adaptive FWLS method with σ

_{URmax}= 0.15 and o

_{ymax}= 0.10. In the following text, cases (b) and (c) will be referred to as “adaptive FWLS 1” and “adaptive FWLS 2”, respectively. In all cases, the derivative filter T

_{F}= 2 s was chosen.

_{S}= 0.5 s. It can be seen that the closed-loop response of the original FWLS method (blue solid line) is virtually identical to the amplitude-scaled process open-loop response (cyan dash-dotted line). The adaptive algorithms provide significantly faster responses. The adaptive FWLS 1 response is slower than the adaptive FWLS 2 response. On the other hand, the adaptive FWLS 1 method exhibits a smaller overshoot (less than 5%) for a reference change, all according to the selected maximum overshoot. The adaptive FWLS 2 method has the best disturbance-rejection performance.

## 7. Comparison to Other Methods

_{F}= 0.1, according to the expressions (25) and Figure 16 or directly using the Matlab/Octave script [35]. The chosen maximum values of relative standard deviation (35) and overshoot (36) are σ

_{URmax}= 0.1 and o

_{ymax}= 0.05, respectively. The obtained controller parameters are listed in Table 4. Please note that in the Balanced method, PI controllers are used, so the derivative gain K

_{D}= 0.

_{OL}) and closed-loop (y

_{CL}) responses during the reference change (ISEy) and to measure the tracking and disturbance-rejection performance (ISEe):

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Typical process input (u) and output (y) signals during the open-loop or closed-loop steady-state change in the process, where r = K

_{PR}∙u.

**Figure 4.**Filtering process input (u) and output (y) signals to obtain the regression signals e

_{F}, i

_{F}and y

_{F}, where r = K

_{PR}u.

**Figure 6.**The average residence time t

_{ar}can be determined from the process open-loop response by means of integrating the yellow area or by estimating the position of the vertical line t

_{ar}so that the shaded areas are equal.

**Figure 7.**Filtering of input (u) and output (y) signals to obtain additionally filtered regression signals e

_{F}, i

_{F}and y

_{F}, where r = K

_{PR}u.

**Figure 8.**The comparison between the open-loop and closed-loop responses using LS, WLS and FLWS tuning methods.

**Figure 9.**One hundred process open-loop responses with measurement noise (

**top figure**) and closed-loop responses (

**bottom figure**) using the WLS tuning method without additional noise (for clarity).

**Figure 10.**Histograms of the calculated controller parameters for 100 runs with noisy open-loop responses obtained using the WLS method. The values of the controller parameters obtained with undisturbed open-loop responses without noise are represented by red circles.

**Figure 11.**Histograms of the calculated controller parameters for 100 runs in noisy open-loop responses obtained using the FWLS method. The values of the controller parameters obtained with undisturbed open-loop responses without noise are represented by red circles.

**Figure 12.**One hundred process open-loop responses with measurement noise (

**top figure**) and closed-loop responses (

**bottom figure**) using FWLS tuning method without additional noise (for clarity).

**Figure 13.**Measured process input (u) and output (y) signals with artificially calculated reference signal (r) (dashed line) represented by r = K

_{PR}∙u and filtered process input (u*) and output (y*) signals (solid green lines) during the open-loop experiment.

**Figure 14.**Measured process input (u) and output (y) signals with artificially calculated reference (r) signal (dashed line) represented by r = G

_{F}

_{1}(s)K

_{PR}∙u.

**Figure 15.**Flowchart of the applied algorithm for automatic calculation of the controller parameters.

**Figure 16.**The Octave Bucket screen which runs the script for the calculation of the controller parameters.

**Figure 23.**The electronic scheme of the laboratory thermal process connected to Arduino UNO development board.

**Figure 26.**The closed-loop tracking and control signals for process G

_{P}

_{6}when using FLWS, adaptive FLWS and Balanced method.

**Figure 27.**The closed-loop tracking and control signals for process G

_{P}

_{7}when using FLWS, adaptive FLWS and Balanced method.

**Figure 28.**The closed-loop tracking and control signals for process G

_{P}

_{8}when using FLWS, adaptive FLWS and Balanced method.

Method | K_{P} | K_{I} | K_{D} | T_{F} | T_{CL} |
---|---|---|---|---|---|

LS | 0.850 | 0.401 | 0.497 | 0.1 | 0 |

WLS | 0.762 | 0.400 | 0.325 | 0.1 | 0 |

FWLS | 0.810 | 0.399 | 0.406 | 0.1 | 2 |

Process | FWLS | Adaptive FWLS | |||||
---|---|---|---|---|---|---|---|

K_{I} | K_{P} | K_{D} | K_{I} | K_{P} | K_{D} | K_{S} | |

G_{P}_{1} | 0.125 | 1.00 | 0.03 | 1.23 | 9.85 | 0 | 9.85 |

G_{P}_{2} | 0.124 | 0.779 | 1.26 | 0.179 | 1.119 | 1.77 | 1.77 |

G_{P}_{3} | 0.125 | 0.734 | 1.02 | 0.166 | 0.918 | 1.49 | 1.33 |

G_{P}_{4} | 0.12 | 0.583 | 0.89 | 0.113 | 0.532 | 0.808 | 0.9 |

G_{P}_{5} | 0.0623 | 0.18 | 0.151 | 0.0887 | 0.286 | 0.239 | 0.39 |

Method | K_{S} | K_{P} | K_{I} | K_{D} | T_{F} |
---|---|---|---|---|---|

FWLS | 1 | 1.91 | 0.0115 | 0 | 2 |

Adaptive FWLS 1 | 3.45 | 5.11 | 0.0477 | 0 | 2 |

Adaptive FWLS 2 | 5.56 | 7.02 | 0.085 | 19.5 | 2 |

Process | FWLS | Adaptive FWLS | Balanced | ||||||
---|---|---|---|---|---|---|---|---|---|

K_{I} | K_{P} | K_{D} | K_{I} | K_{P} | K_{D} | K_{S} | K_{I} | K_{P} | |

G_{P}_{6} | 0.327 | 0.707 | 0.401 | 0.384 | 0.828 | 0.461 | 1.33 | 0.301 | 0.53 |

G_{P}_{7} | 0.249 | 0.793 | 0.752 | 0.300 | 0.92 | 0.925 | 1.21 | 0.231 | 0.6 |

G_{P}_{8} | 0.248 | 0.663 | 0.486 | 0.298 | 0.754 | 0.631 | 1.21 | 0.23 | 0.53 |

Process | FWLS | Adaptive FWLS | Balanced | ||||||
---|---|---|---|---|---|---|---|---|---|

ISEy | ISEer | ISEed | ISEy | ISEer | ISEed | ISEy | ISEer | ISEed | |

G_{P}_{6} | 0.0029 | 2.317 | 0.316 | - | 2.145 | 0.262 | 0.0325 | 2.529 | 0.410 |

G_{P}_{7} | 0.0006 | 2.957 | 0.366 | - | 2.724 | 0.292 | 0.154 | 3.220 | 0.450 |

G_{P}_{8} | 0.0036 | 3.340 | 0.492 | - | 3.184 | 0.416 | 0.221 | 3.617 | 0.598 |

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## Share and Cite

**MDPI and ACS Style**

Vrančić, D.; Moura Oliveira, P.; Bisták, P.; Huba, M.
Model-Free VRFT-Based Tuning Method for PID Controllers. *Mathematics* **2023**, *11*, 715.
https://doi.org/10.3390/math11030715

**AMA Style**

Vrančić D, Moura Oliveira P, Bisták P, Huba M.
Model-Free VRFT-Based Tuning Method for PID Controllers. *Mathematics*. 2023; 11(3):715.
https://doi.org/10.3390/math11030715

**Chicago/Turabian Style**

Vrančić, Damir, Paulo Moura Oliveira, Pavol Bisták, and Mikuláš Huba.
2023. "Model-Free VRFT-Based Tuning Method for PID Controllers" *Mathematics* 11, no. 3: 715.
https://doi.org/10.3390/math11030715