
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39232194
71261
10.1038/s41598-024-71261-1
Article
PI gain tuning for pressure-based MFCs with Gaussian mixture model
Higuchi Seiji
Ueda Takayuki
Takijiri Kotaro
Hayashi Daisuke daisuke.hayashi@horiba.com

HORIBA STEC, Co., Ltd., Research & Development Division, Kyoto, 601-8116 Japan
5 9 2024
5 9 2024
2024
14 2066027 3 2024
26 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
A vast number of mass flow controllers (MFCs) are used in semiconductor industry. For the stable supply, an efficient production method of MFC is required. The gain tuning of the proportional-integral (PI) control to realize a setting flow rate is essential for efficient mass production. The gains are tuned to meet the specifications required for evaluation indices of response time and overshoot amount in a step response waveform. The tuning is complicated especially for the case of pressure-based MFCs. In this paper, we propose a simple method for the PI gain tuning using the Gaussian mixture model (GMM) and the direct inverse analysis applicable to the pressure-based MFCs’ production. The relationship between the gains and evaluation indices for a standard unit of the MFC is modeled as the GMM. The direct inverse analysis calculates the difference between the standard and a test unit. Under the assumption that the difference can be compensated by a simple shift, gains likely to meet the specifications for the test unit are searched. We applied the method to seven test units. The result showed that the gains of all the test units were tuned within only a few iterations whose numbers were much less than the conventional manual tuning method, and there was no untunable unit.

Keywords

Semiconductor
Gaussian mixture model
PI control
Mass flow controller
Manufacturing
Subject terms

Engineering
Mathematics and computing
issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

A mass flow controller (MFC), which precisely controls fluid mass flow rate, is widely used in various industries. Especially, in the semiconductor manufacturing, many MFCs are equipped with pipelines linked to reaction chambers to feed material gas or liquid1. Since a whole process of semiconductor manufacturing consists of many reaction steps, a vast number of MFCs are required for each plant. With an expanding semiconductor market, demand for stable supply of MFCs has also been increasing. Therefore, establishing an efficient production method of MFC has become more important.

The MFC controls fluid’s flow rate by adjusting a valve opening to realize a setting flow rate. Although the design of controller including an algorithm for the adjustment is essential for the MFCs’ production, it is complicated due to the system instability or the complexity of valve-characteristics2,3. The sort of MFC is mainly classified into two types in terms of the way of measuring flow rates: a pressure-based MFC and a thermal-based MFC. The pressure-based MFC, which measures flow rate with pressure sensors, has advantages on its fast response and high accuracy. However, the control mechanism is more complicated than the thermal-based one, and the controller design is further difficult. The proportional-integral (PI) or proportional-integral-derivative (PID) controls are widely employed for the algorithm, where the PI or PID gains determine an applied voltage to the valve so that the variation between the setting and measured flow rates can be converged4–6. The gains are tuned to meet the specifications required for evaluation indices, which are a response time and an overshoot amount obtained from a step response waveform. In the pressure-based MFC case, the optimal gains that meet the specifications are configured in a narrow region compared to the thermal-based MFC. In most MFCs’ productions, the gains are tuned manually for every unit, where the respective values are independently varied, and the waveform is checked each time. Even for experienced operators, usually 10–20 times of trials are necessary. Furthermore, this method allows the tuning values to easily leave from the optimal region for the pressure-based MFC. Most of the units whose tuning values leave from the region are judged as untunable. Therefore, the PI tuning is a bottleneck against establishing the efficient production method. Tuning methods for PI or PID gains using particle swarm optimization7–16 have been well studied. The approaches by physically modeling the considered system have also been investigated17–19. More elaborated tuning methods beyond traditional PI tuning have also been researched20–30. However, considering the complexity of device characteristics installed in the pressure-based MFC, a simpler method with less numerical burden is preferable. The Gaussian mixture model (GMM) is a simple model to express a relationship among multiple variables with a superposition of Gaussian probability densities31. The machine learning approach using the GMM has attracted attention and been applied for several studies32–36. The application can provide a simple model for the relationship between the gains and resulting evaluation indices. In addition, the direct inverse analysis based on the Bayes theorem37,38 enables us to estimate the most likely gains for obtained indices, conversely. However, as devices installed in the control target system have different characteristics among individual MFC units, the relationship modeled for a single unit cannot be applied ad hoc for every unit. An elaborate application that can compensate the individual differences is required.

In this paper, we propose a simple method for the PI gain tuning using the GMM and the direct inverse analysis applicable to the pressure-based MFCs’ production. The relationship between the PI gains and evaluation indices are investigated for a standard unit. Using the GMM and the direct inverse analysis, the difference of every test unit from the standard unit is calculated. By compensating the difference, the gains are tuned to meet the specifications for every test unit. After formulating the method, we applied it to the tuning for a couple of pressure-based MFCs to confirm the applicability in mass production.

Characteristics of pressure-based MFCs

MFC structure

A schematic of a pressure-based gas MFC is shown in Fig. 1. The MFC includes a piezo valve, two pressure sensors, a temperature sensor, a flow restrictor, and a controller circuit. The inlet and outlet ports for gas are respectively connected with upstream and downstream pipelines. Gas entered from the inlet port passes through the piezo valve and flow restrictor, and then goes out from the outlet port. The flow rate of output gas (Qout) is calculated as follows:1 Qout=kp12-p22,

where k is a flow restrictor constant having a temperature dependence, and p1,p2 are outputs of pressure sensors P1,P2, respectively.Fig. 1 Internal structure of a pressure-based gas MFC.

The controller circuit adjusts the valve opening through applied voltage to realize a setting flow rate (Qset). The block diagram of PI control installed in the controller is shown in Fig. 2. Here, KP,KI mean P and I gains, respectively. The gains are tuned in accordance with characteristics of the control target including circuit delay, characteristics of piezo valve and flow restrictor, and sensor delay. The characteristics of the installed devices are individually different one by one, and the piezo valve characteristic has even a nonlinearity39. Therefore, the PI gains need to be fine tuned for each individual unit.Fig. 2 Block diagram of PI control system for the pressure-based MFC.

Evaluation indices for PI gain tuning

In this study, we consider the case that the PI gains are tuned based on the following two evaluation indices in a step response for 0→100%FS input (“%FS” means flow rate normalized by control full scale): (i) Tr (ms): response time defined as a period from the Qset input to the timing when Qout achieves 98% of Qset, and (ii) Qos (%): overshoot amount defined as,2 Qos=Qpeak-QsetQset×100,

where Qpeak is the flow rate having the maximum difference from Qset after once Qout exceeds Qset. (Qos=0 if Q never exceeds Qset.) The step response characteristics are critically important in order to achieve a high throughput and reduce waste of gas in the semiconductor manufacturing. They are emphasized more than the stability against disturbance because MFCs are used in a clean room whose environmental conditions are severely managed and sudden disturbances are not expected normally. In recent process, as various kind of gas are used while switching for a short period, the step response characteristics are especially paid attention. The total test duration for step response is fixed at 200 ms. The specifications require that Tr and Qos should be within the range of 85±5 ms and 0-0.55+0.50%, respectively. The specifications are determined from the requirement for the throughput in the recent process. Here, we denote that the optimal indices (Tr0,Qos0)=(85,0) and the tolerances (ΔTr,ΔQos)=(10,1.05).

A relationship between (KP,KI) and (Tr,Qos) were investigated for a standard unit that are arbitrarily selected from among mass produced MFCs. We independently varied (KP,KI) in the range of [0.5, 1.5] with intervals of 0.02, and the step response waveforms are acquired to get (Tr,Qos). While the intervals were determined from the minimum variation width used in the conventional manual tuning method, the range was taken more widely than a usually searched range in the manual tuning around typical optimal values of this MFC to clarify the overall trend of waveform on a KP-KI plane. For the standard unit of this study, the optimal gains that gave the indices closest to (Tr0,Qos0) were (KP0,KI0)=(0.94, 0.94). Figure 3 shows the step response waveforms at (KP,KI)=(KP0,KI0), (1.5, 0.5), and (0.5, 1.5). For (KP,KI)=(KP0,KI0), a preferable waveform with (Tr,Qos)=(85,-0.046) were obtained. For (KP,KI)=(1.5,0.5), the waveform had an oscillation with a large overshoot. For (KP,KI)=(0.5,1.5), the waveform indicated a too slow response.Fig. 3 Step response waveforms of standard unit at (KP,KI)=(KP0,KI0), (1.5, 0.5), and (0.5, 1.5).

To simply evaluate the step response, we defined the deviation index z as follows:3 z=Tr-Tr0ΔTr+QosΔQos.

The necessary condition in which either index certainly meets the specifications is z≤1.02, and the sufficient condition in which both indices certainly meet the specifications is z≤0.50. For 0.50<z≤1.02, the balance of each term in Eq. (3) determines whether the specifications are satisfied or not. The distribution of z on a KP-KI plane for the standard unit is illustrated in Fig. 4, together with regions of the necessary (white dashed line) and sufficient (white solid line) conditions and (Tr0,Qos0) location (red spot). (The approach to illustrate an overall behavior of an evaluation indices on the KP-KI plane are used in other works, for example in Ref.40) The PI gains likely to meet the specifications are configured in a narrow region diagonally around Kp≈KI. The distribution reflects the characteristics of the pressure-based MFC structure, and this is why the gains are easily to leave from the optimal region in the manual tuning method where the gains move vertically or horizontally on the plane.Fig. 4 Distribution of the deviation index z for the standard unit (See Eq. (3)).

Methods

Variables required for the formulation are summarized in Table 1. Table 1 Variables used for formulation.

Symbol	Description	
x,y	Vectors of PI gains and evaluation indices	
x0,y0	Optimal x and resulting y for the standard unit	
x^	Predictive PI gains expected to give the same waveform for the standard unit	
p(x,y)	Gaussian mixture model of (x,y)	
N	Gaussian probability density distribution	
πi	Weight of i-th Gaussian in p(x,y)	
nG	Total number of Gaussian in p(x,y)	
μν,i (ν=x,y)	Mean vector of i-th Gaussian	
Σμν,i (μ,ν=x,y)	Variance-covariance matrix of i-th Gaussian	
p(x|y)	Probability density distribution of x for given y	
wy,i	Weight of i-th Gaussian in p(x|y)	
mi (y)	Mean vector of x for given y in i-th Gaussian	
n	Iteration cycle number	

This method estimates gains likely to meet the specifications for a test using the GMM. Taking z-distribution for every test unit is not necessary. And although the test unit distribution is assumed to be similar to the standard unit distribution, it is not necessary to be completely the same. The procedure of PI gain tuning follows the next two phases: learning for the standard unit, and tuning for test units. For the formulation, the explanatory and objective variables (x,y) are defined as follows:4 x=(KP,KI),

5 y=(Tr,Qos),

and x0=(KP0,KI0), y0=(Tr0,Qos0).

Learning for standard unit

From the collected data of (x,y) for the standard unit, the probability density distribution is modeled as the form of the GMM:6 p(x,y)=∑i=1nGπiN[x,y]|[μx,i,μy,i],Σxx,iΣyx,iΣxy,iΣyy,i.

Here, πi is the weight of the i-th Gaussian, μx,i, μy,i are the mean vectors of x, y, and Σxx,i,Σxy,i,Σyx,i,Σyy,i are their variance-covariance matrices, respectively, which are determined by the expectation-maximization method41. The number of Gaussian nG is determined to minimize the square error without an overfitting. In this case, we set nG=10.

PI gain tuning for test units

A step response waveform is acquired for each test unit by setting the initial gain at x=x0 because x0 is located in the sufficient condition region with some tolerance in the standard unit z-distribution in Fig.  4. If the p(x,y) for the considering test unit is almost the same as the one for the standard unit, the resulting y is supposed to meet the specifications immediately. If the resulting y fails to meet the specifications, the direct inverse analysis of GMM37 with respect to the y is applied to get the predictive gains x^=(K^P,K^I), which is expected to give the same y for the standard unit. The probability density of x under given y can be calculated as,7 px|y=∑i=1nGwy,ipx|y,μy,i,Σyy,i,

where px|y,μy,i,Σyy,i is the probability density distribution of x in the i-th Gaussian in Eq. (6) under μy,i, Σyy,i and given y, and wy,i is the weight calculated as,8 wy,i=πipy|μy,i,Σyy,i∑j=1nGπjpy|μy,j,Σyy,j.

The mean vector of x in the i-th Gaussian under given y is calculated as,9 miy=μx,i+y-μy,iΣyy,i-1Σyx,i.

Although there are generally two options that are a “mode” and a “weighted average” as a representative value of miy, we used the mode in this study, i.e., the value having the maximum wy,i. Assuming that z-distribution for the test unit can be approximately overlapped by simply shifting the distribution for the standard unit in x-direction, the newly defined x as,10 x=x0+Δx,

is expected to give y close to y0 for the test unit. Here, the shift vector is,11 Δx=KP0-K^P,KI0-K^I.

If the new x fails again to give y meeting the specifications, the same cycle is repeated unless the number of iteration n reaches a specific maximum value nmax. If the test unit’s z-distribution largely differs from that of the standard unit, gains meeting the specifications should not be available within n≤nmax. The flow chart of this procedure is summarized in Fig. 5. Although choosing the gains of the sufficient condition region in the standard unit distribution as initial values is not essential, it can help the procedure conclude with few iterations.Fig. 5 Flow chart of PI tuning for test units.

Results

We applied the PI gain tuning method for seven test units. The results are listed in Table 2. The evaluation indices that meet the specifications were obtained for all the test units with n≤4, though we set nmax=10. The iterations were much less than the manual tuning case. Furthermore, there was no untunable unit, which means the gains are difficult to leave from the optimal region on the KP-KI plain in this method.

The required iterations were different depending on individual units as n=0-4. To consider the difference, we investigated z-distributions of units with the minimum (no. 2) and maximum (no. 4) iterations, as shown in Fig. 6. Both units indicated similar distributions to the standard unit’s distribution having the necessary and sufficient regions around KP≈KI. Thus, the assumption that a test unit’s z-distribution can be approximately overlapped by simply shifting the standard unit’s one in x-direction was appropriate. As the location and area of necessary and sufficient region in no. 4 differed more from those of the standard unit’s distribution compared to no. 2, no. 4 unit required more iterations than no. 2. However, the differences were small enough to be compensated by repeating the cycle of the direct inverse analysis and x-direction shift. Table 2 Results of PI tuning for test units.

Unit no.	Initial	Final	
KP,KI	Tr (ms)	Qos (%)	n	KP,KI	Tr (ms)	Qos (%)	
1	(0.94, 0.94)	118	-0.249	2	(0.91, 0.88)	82	0.214	
2	(0.94, 0.94)	86	-0.061	0	(0.94, 0.94)	86	-0.061	
3	(0.94, 0.94)	78	1.67	1	(0.97, 0.91)	86	-0.165	
4	(0.94, 0.94)	76	0.52	4	(0.90, 0.86)	84	-0.161	
5	(0.94, 0.94)	76	1.16	4	(0.90, 0.85)	88	-0.228	
6	(0.94, 0.94)	79	0.23	1	(0.93, 0.89)	84	-0.118	
7	(0.94, 0.94)	84	0.54	1	(0.99, 0.98)	84	-0.174	

Fig. 6 Comparison of distributions of the deviation index z (See Eq. (3)). (a) Unit no. 2 and (b) Unit no. 4.

The transitions of deviation index z and waveform with iterations for unit no.4 are shown in Fig. 7. Although the index z was finally reduced low enough and a waveform sufficiently close to the target was obtained with four iterations, the waveform once got away from the target with increase of z in the second iteration. Figure 8 shows the probability density distribution p(x|y) in Eq. (7) for each iteration. The peak in each distribution gave (K^P,K^I). The x determined from (K^p,K^I) in the distribution of n gave the index z and waveform of n+1 in Fig. 7. The distributions of n=0,2,3 were well localized, in which (K^P,K^I) can be predict with a low ambiguity. Meanwhile, the distribution of n=1 had a large variance and low peak, in which the predicted (K^P,K^I) included large ambiguity. This is because a Gaussian having large variance-covariance matrix elements in Eq. (6) was dominant for p(x|y) in the second direct inverse analysis. As the x was determined based on the prediction with large ambiguity, the resultant waveform was apart from the target. However, even if we temporarily obtain the waveform apart from the target, we can bring it close again to the target in the next iteration because the next x is determined based on the difference between the obtained (K^P,K^I) and the target. As the PI gains likely to realize the target waveform are set repeatedly, we can achieve the target with a few iterations.Fig. 7 Transition with iteration for unit no.4. (a) Deviation index z (See Eq. ( 3 )) and (b) waveforms.

Fig. 8 Transition of p(x|y) with iteration (See Eq. (7)).

Conclusion

The method for PI gain tuning using the GMM and the direct inverse analysis applicable to pressure-based MFCs’ production are proposed. The relationship between gains and evaluation indices for a standard unit is modeled as the GMM. The direct inverse analysis calculates the difference between the standard and test units. Under the assumption that the difference can be compensated by simply shifting, gains likely to meet the specifications are searched. We applied the method to seven test units. The result showed that the gains of all the test units were tuned within only a few iterations whose numbers were much less than the conventional manual tuning method. There was no untunable unit. As this method can efficiently find the optimal gains that are located at a narrow region in the KP-KI plane, it is promising for the mass production of the pressure-based MFC whose control target is significantly complicated. If the gains meeting the specifications cannot be obtained within the specific number of iterations, the result is likely to indicates that the relationship between x and y remarkably differs from that of the standard unit. Such unit should be removed from the manufacturing line, which leads to avoid excessive gain tuning. In the conventional manual tuning method, usually 10-20 times of trials are required even for experienced operators, and the tuning gains easily leaves from the optimal region, which leads to many units judged as untunable. Accordingly, we can expect time reduction of tuning process and improvement of manufacturing yields with this method. Extended applications of this method to the PID control or a system including more evaluation indices are easily available. We believe this method can simplify the manufacturing process for the complicated pressure-based MFC and contribute its stable delivery.

Acknowledgements

The authors would like to express appreciation for Mr. Maximilian Gundlach, Mr. John Dick, and Dr. Esteban Gonzalez for support and discussions.

Author contributions

S. H. and T. U. conducted numerical calculations and experimental validations. D. H. wrote the original manuscript, and K. T. improved the language. All authors collaborated and agreed on the manuscript intended for submission.

Data availability

The datasets of this article are obtainable from the corresponding author upon reasonable request.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

These authors contributed equally: Takayuki Ueda and Kotaro Takijiri.
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