C. Control experiments

Once the open-loop microfluidic system has been studied, we propose to control it to obtain a fixed desired percentage value of red. It is interesting to make this combination of microfluidics with control theory and real-time image acquisition for promising applications to biotechnology it may have. As mentioned before (cf. Sec. ), it is of great importance to control solution mixing processes around expected values in order to stimulate, for example, cell growth in microfluidic devices,19–2419. C. M. Olmos et al., “Cost-effective fabrication of photopolymer molds with multi-level microstructures for PDMS microfluidic device manufacture,” RSC Adv. 10(7), 4071–4079 (2020). https://doi.org/10.1039/C9RA07955F24. N. Bourguignon et al., “Accessible and cost-effective method of PDMS microdevices fabrication using a reusable photopolymer mold,” J. Polym. Sci. Part B: Polym. Phys. 56(21), 1433–1442 (2018). https://doi.org/10.1002/polb.24726 determination of concentration of chemical species by image processing,2525. K. Shinohara et al., “Measurement of pH field of chemically reacting flow in microfluidic devices by laser-induced fluorescence,” Meas. Sci. Technol. 15(5), 955 (2004). https://doi.org/10.1088/0957-0233/15/5/025 among others. For this reason, we propose to study two different ways of control and demonstrate the functionality of the system. The objective is to demonstrate that a microfluidic system can be controlled to reach the desired color using different approaches. Taking images in real time of the mixture in the microchip, the micropumps are controlled by means of feedback by different mathematical approaches, reaching the desired color percentage R%∣des, without previously setting the necessary flow value to reach that color. These approaches are detailed below.

1. Integrative control with variable integral gain

Integrative control has as its starting point the behavior of the open-loop system (cf. Sec. ) and the range of R% values that can be managed. The basic idea is to use an integrative control equation given bywhere vrk indicates the red flow at current time k, vrk−1 is the red flow at previous time k−1, ek is the error at current time or the difference between the desired red percentage R%∣des, and R%k is the red percentage at current time k. Finally, Ki is the integral (i) gain and it is depending on ek. By introducing a parameter f1, we calculate the estimated value of Ki under the condition Kiek≪1. Therefore, we explore two ways to obtain the integral gain Ki: the multiplicative mode where Ki=f1 and the divisive mode where Ki=1/f1.We propose a method to determine Ki using the fitting parameters of experimental saturation curves (cf. Fig. 3). First, from different experimental saturation curves, we can assume that for each red percentage of saturation, R%, outside of those obtained experimentally, corresponds a specific red flow called the theoretical red flow vrtheor. This theoretical value is calculated by solving the equation a(vrtheor)=R%∣des [cf. Eq. (4)] considering that a(vr) is the only relevant parameter in Eq. (3) for long time. In the calculation, two cases are discriminated depending on whether the initial value of the percentage of red is greater or less than the desired value, i.e., ek<0 if R%0=100, and ek>0 if R%0=0. Second, f1is calculated from Eqs. (8) and (9), solving f1 depending on the case and calculating vrtheor−vr0, with vr0=2.0μl/min if R%0=100 and vr0=0 for other case. The previous steps were repeated with other R%∣des values to obtain various f1 values, and these were plotted vs ek. Finally, the data were fitted by mathematical functions, and these are given by f1={−7.0982ek+751.48ek<0−0.4591ek+3.7109ek>0(11)for the divisive mode and f1={6×10−4ek2+0.0252ek+0.2784ek<01×10−5ek2−0.0015ek+0.0557ek>0(12)for the multiplicative mode. The correlation factors are R2 = 0.9926 for the first piece, R2 = 0.9981 for the second piece in Eq. (11), and finally R2 = 0.9999 for both pieces in Eq. (12).The control equation that we obtain by substituting the integral gain as the inverse of f1 in Eq. (11), and it is, in general, for the divisive mode, vrk=vrk−1+(1αek+β)ek,(13)where Ki1=1αek+β with constant α and β obtained from the fit explained above to determine f1. On the other hand, the control equation obtained for the multiplicative mode is given by vrk=vrk−1+(α2ek2+α1ek+α0)ek,(14)where Ki2=α2ek2+α1ek+α0 in this case and it corresponds to one generic parabolic function with constant coefficients. From Eqs. (13) and (14), we can see that the integral gain is obtained as a pair of error-dependent functions: a rational function and a polynomial of degree 2 function. The constant parameters were obtained using fits of the saturation curves. Various tests were performed using these control equations in the experimental setup, and the results are shown below.The results obtained for the control of the red percentage with variable integral gain are shown in Fig. 7 for R%∣des=80. Experiments were carried out starting from the two extreme initial conditions (R%0=100orR%0=0) to see how the system adapted in both cases. Figure 7 shows the results of the control of red percentage vs time and shows that the saturation curves that start from zero per cent (i.e.,R%0=0) are the ones that best approximate the desired value. The rate of increment of the red percentage is high, and R% passes above the desired value. However, the control action causes the percentage of red to oscillate toward the desired value with an error of approximately 10% (violet color points) and 5% (red color points) for Ki2(ek)/100 in Eq. (14). If the value of Ki1/100 is assumed in Eq. (13) (divisive mode), the system does not oscillate, but it rather approaches the desired value asymptotically and slower than the oscillatory case (see blue color curve in Fig. 5). Sometimes it is convenient to control in this way, i.e., without going over the desired value. The results indicate that the control is more effective for integral gain values reduced 100 times in both control equations because this factor is directly related to the rate of red percentage increment. It is reasonable to think that if red percentage increment slowly, there is a greater probability of reaching the desired value asymptotically without oscillating. Finally, the system is well controlled in approximately 10 min.To illustrate the behavior of integrative control without increasing the value of Ki1 in Eq. (13), the rate of change of red percentage for R%0=0 is large according to the slope at the beginning of the curve as shown in Fig. 8. As a result of this, the more difficult the stabilization of the variable to be controlled at its set point. For this reason, the Ki1 value was reduced 100 times to make the factor smaller. In these experiments, the initial flow rate was taken as vrtheor, but this choice is inconvenient for the reason mentioned above regarding the rates of change of the percentage of red. In Fig. 7, the initial flow is 0.01μl/min for all experiments performed.Figure 9 shows the control of Eq. (14) (multiplicative mode) in the cases in which Ki2 is not reduced and the case in which it is reduced 100 times. The results indicate that the rates of variation over time of the percentage of red are high; therefore, the system is difficult to stabilize, even in the case where Ki2 is reduced. Therefore, the multiplicative mode [Eq. (14)] is less successful than the divisive mode [Eq. (13)] in achieving the control of the percentage of red around the set point.Finally, the closed-loop least squares fit was applied for integrative control results in divisive and multiplicative modes. Although the tests presented in Table II are open-loop, with a step-type variation of the input (red flow), this technique has shown to be robust also in the closed-loop. The best result corresponds to the control of the red percentage starting at R%0=100 (cf. Fig. 7) for the divisive mode [Eq. (13)], and it is shown in Fig. 10. This plot shows the correct follow-up of the model to the closed-loop output data. Note that the output (red percentage) has negative values. This is because the model is incremental, that is, it represents variations above a steady state value. The robustness of First Order plus Delay Time (FODT) modeling (cf. Sec. ) makes it more reliable for the calculation of controllers in the future.

2. Integrative control with constant integral gain

The standard integrative control equation was used in the microfluidic device to control red percentage to a desired value, and it was used as a reference to compare control methods. The Ki parameter is constant, i.e., Ki≠Ki(ek), and the control equation is Eq. (8). Figure 11 shows the control of the red percentage with R%∣des=80, and it tends to reach equilibrium faster than the integrative control with variable integral gain and the margin of error is less than 5% as shown in the figure. For both cases, where R%0=0 or 100, the system tends to oscillate damped around the desired value of red.The Ki value was calculated using the Ziegler Nichols method.2626. K. Ogata, Ingeniería de Control Moderna (Pearson Educación, Ilustrada, Reimpresa, 2003). This method allows to calculate the constant parameter of the PI control formula using the fitted mathematical function of the saturation curves [cf. Eq. (3)]. The values of parameters x0, b, and a are calculated for the desired red percentage. Using the graphical or analytical method, the line that passes through the inflection point and the point where the plateau of the curve begins is determined. The intersection of the line with the abscissa axis (L) and the distance of that intersection with the point of the axis of the plateau (T) of the curve give the values required for the calculation of γ using the formula γ=0.9T/L. In this control approach, it was also necessary to reduce the value of the gamma constant 100 times to achieve convergence to the set point. This result shows us that microfluidic systems are very sensitive to flow changes and increments must be done slowly to avoid instabilities and achieve the desired control.Finally, Fig. 12 shows two cases where the control fails to reach the set point is R%jdes = 80. It can be observed for large Ki values (Ki≥103) where control fails because of these magnitude orders, the red flow variations are brusque, and the system cannot reach any equilibrium point. On the other hand, with values below the order of 10−4, the control is carried out very slowly, tripling the equilibrium time with reference to the case observed in Fig. 11 and with a set point deviation greater than 5%. Therefore, the most effective control in this microfluidic system is achieved at Ki constant of the order of 10−4.