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Design and application of a Linear Algebra Based Controller from a reduced order model for regulation and tracking of chemical processes under uncertainties María Fabiana Sardella, Mario Emanuel Serrano, Oscar Camacho, and Gustavo Juan E. Scaglia Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.9b01257 • Publication Date (Web): 29 Jul 2019 Downloaded from pubs.acs.org on August 7, 2019

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Industrial & Engineering Chemistry Research

Design and application of a Linear Algebra Based Controller from a reduced order model for regulation and tracking of chemical processes under uncertainties M. FabianaSardellaa*, M. Emanuel Serranoa, Oscar Camachob, Gustavo J.E. Scagliaa a Instituto

de Ingeniería Química, CONICET, Universidad Nacional de San Juan Av. Libertador San Martín Oeste 1109, San Juan J5400ARL, Argentina

b Departamento

de Automatización y Control Industrial, Escuela Politécnica Nacional Ladrón de Guevara, E11-253, Quito, Ecuador

[email protected]*, [email protected], [email protected], [email protected]

Abstract This work presents a simple controller, tunable by 3 parameters, capable of following variable time references. It is a Linear Algebra based Controller (LABC), developed from a First Order Plus Dead Time model (FOPDT). Two processes were selected: a high order linear process and a transesterification batch reactor from the biodiesel production process. Constant and variable time references were followed with very low tracking error. The methodology for the controller design under different conditions is described and results of simulations and experimental assays on a batch reactor are shown. Through comparison against PID and a Numerical Method Based Controller presented by other researchers, the accuracy of LABC is evidenced. Keywords: FOPDT model, Linear Algebra, Tracking control, Uncertainties 1. Introduction Rejecting persistent disturbances or tracking non-constant references with digital controllers is very difficult to achieve using classical control methods because the dynamic behavior over the whole range is usually unknown. The effectiveness of conventional feedback controllers, such as the proportional-integral-derivative (PID) or Dynamic Matrix Control (DMC), on such problems can be quite limited since their performance usually decline for processes with relatively large time delay compared to the dominant time constant. Sliding mode controllers and predictive structures, such as internal model control (IMC) and the Smith predictor, have been used to upgrade these control strategies1,2. However, it has been reported that most optimal controllers, when formulated as in the literature, lead to periodic off-sets3. Furthermore, these approaches require a process model, which is sometimes difficult to obtain, and are, therefore, sensitive to mismatches between the model and the actual plant. Thus, the controllers designed using particular models will show poor performances when implemented on the real system4.

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Controllers design using the methodology based on linear algebra and numerical methods (LABC) is a simple method that has been successfully applied to many systems. It has been used to design different kind of control systems such as trajectory tracking of mobile robots and UAV’s5-8, chemical plants and bioprocesses4,9,10. Complete nonlinear and discrete process models are used to develop these controllers4,6,7. One great advantage is that control actions are obtained by simple calculations which makes it easy to implement in a microcontroller. The development of a complete model for many industrial processes is generally difficult due to the complexity of the processes themselves, besides the important drawback that means the lack of information about process parameters. Most process models relating the controlled and the manipulated variables are of higher-order, leading to more complex controllers whose application is just for analysis11. Moreover, as the process model represents one process in particular, which is used to synthesize the mathematical controller algorithm, it can only be applied to itself, as can be seen in previous works4,6-11. First Order Plus Dead Time (FOPDT) models are a powerful tool for process control since they allow the analysis of the system with relatively simple low order linear models with dead time2,12. Their simplicity and ability to capture the essential dynamics of several industrial processes is their main advantage13. One drawback of these reduced order models is that they present uncertainties that leads to performance degradation of conventional controllers, such as PID or Smith Predictors12. There are many no lineal industrial processes whose operation under optimal conditions implies following variable profiles, requiring the use of efficient control systems. One of these cases is biodiesel production, which has gained great attention due to worldwide interest on the development of alternative renewable fuel14. Conventional biodiesel process plants involve batch reactors or continuous stirred tank reactors (CSTRs)15. Batch operation is frequently used due to its flexibility and simple operation16. It has been shown that forced periodic operations can be used to improve selectivity and yields3. Moreover, Benavides et al.17 determined the optimum conditions for biodiesel production involves a variable temperature profile. A potential drawback is that, under these conditions, operation becomes very complex and difficult to control. The literature shows that although there have been some studies exploring advanced control of the transesterification reaction, they are limited in terms of their applicability to real reactors18. This work proposes a methodology to control this kind of processes, overcoming the difficulties mentioned and showing good performance to follow variable profiles using low computing capacity. This paper presents a Linear Algebra based Controller (LABC) developed from a FOPDT model of the real process. A general controller, applicable to processes with an open loop response similar to the FOPDT model, is developed. Hence, this work summarizes the simplicity of numerical methods procedures along with a reduced order model of the process to obtain a simple and versatile controller that can be applied to


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different processes if they can be approximated by FOPDT models. This controller is tuned using only three parameters (zeros (ri) of the characteristic polynomial) ranging between 0 and 1. An interesting feature of this new controller is its ability to follow constant and variable references without overshoot, a highly desirable characteristic for most process systems. The performance of the proposed controller in this article is tested by simulations in a high order linear system and also applied to a real laboratory scale batch reactor used for biodiesel production. FOPDT models were used for controllers design and tests were driven under highly variable operation conditions and considering parametric uncertainty. The controller performance is compared with a PID controller and with a numerical method based controller (NMCr) shown by Guevara et al.19. The paper is organized as follows. Section 2 presents general concepts about First Order Plus Dead Time models. The methodology for the controller design under different conditions is described in Sections 3 and 4. The methodology applied to obtain simulated and experimental results is shown in Section 5. The performance of the new controller is contrasted against a classical PID controller and another controller based on linear algebra in Section 6. Main conclusions and remarks are summarized in the last section. 2. Process model Linear models are very common in industry, although the real dynamic behavior of many industrial processes present nonlinear characteristics. A linear model can adequately represent the real process when it works close to an operating point. Most of these models include a dead time as part of the process representation1. The First-Order Plus Dead Time Model is a widely used mathematical representation that describes the dynamics of many chemical processes. Transportation and measurement delays, analysis times, computation and communication lags all introduce dead times into control loops. Dead times are also used to compensate for model reduction where high-order systems are represented by low-order models with delays. The wide use of this model is due to its simplicity and its ability to capture the essential dynamics of several industrial processes. By a proper choice of  and to, this model can represent the dynamics of many industrial processes12. The transfer function of a first order system plus dead time is:

G1 ( s ) 

y ( s ) K e  t0 s  u (s)  s  1

(1)

It is convenient to represent the transfer function as the ratio of two polynomials. The deadtime can be approximated by a first-order Taylor series20. Thus, the dead time expression is represented by:



et0 s 

1 t0 s  1

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(2)

Replacing (2) in (1):

G2 ( s ) 

K

(3)

 s  1 t0 s  1

Note that the model used to design the controller (FOPDT model approximated by Taylor series) is a useful and simple approximation of a complex process, which still holds the required steady state and dynamic process information. The main drawback is that leads to a high uncertainty which must be assumed by the controller in order to have a good performance.

3. Controller design The LABC design involves the following steps7,10,11: 1. Get the state variables 2. Obtain the derivatives of the state variables discretized using a numerical method. 3. Set out as a linear system equation and look for the necessary condition to have exact solution. 4. Solve the equation system to calculate the control action. The development of the proposed linear algebra controller (LABC), based on a first order plus dead time model, starts from the transfer function with the dead time term replaced by Taylor approximation. G (s) 

y(s) K KB  2 u (s) s  K A s  K B

t0   t0 

KB 

(4)

where: KA 

1 t0 

From equation (4):

y  K A y  K B y  K K B u

(5)

Step 1: The state variables of the system are y 1 and y2 , defined as: (6)


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y1  y (7)

y 2  y  K K B u  K A y  K B y  y 1   0  y    K  2  B

1   y1   0   u  K A   y2   KK B 

(8)

(9)

Then, the aim is to find the values of u so that the process may follow a pre-established trajectory with a minimum error. The values of y1 (t ) , y2 (t ) and u(t ) at discrete time

t  n T , where T is the sampling period, and n  0,1, 2, will be denoted as y1,n , y2,n and un , respectively. Step 2: The derivatives of the state variables are discretized using Euler approximation, and equations (9) take the form:

 0  y1, n 1   y1, n   y    y   T   K B  2, n 1   2, n 

1   y1, n   0   un     K A   y2, n   KK B  

(10)

Here T = min(, t0)/10 Step 3: Now the tracking trajectory problem can be solved by the application of linear algebra, and the control action calculation can be done by solving the following linear equations system:

A un  b (11)

It is also desirable that the tracking error tends to zero, which can be written as follows:

e1, n 1  k1  e1, n


(12)


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y y  k (y  y )  y1, n 1  y1ref , n 1  k1e1, n , 0  k1  1  1ref ,n 1  1, n 1 1  1ref , n  1, n e1, n 1 e1, n

(13)

In order to have an exact solution for the system of equations (11), the following condition must be satisfied21:





rank(A)  rank  A b  Which implies that y2,n must fulfill:

y1ref ,n 1  k1e1,n  y1,n  T y2,n  0 

y2,n 

y1ref ,n 1  k1e1,n  y1,n T

(14)

The value of y2,n that satisfies (14) will be called y2 ez ,n and it is the necessary value the variable y 2 should take to force the tracking error to zero. Step 4: Finally, by solving system (11), the control action law is obtained

un 

 1  y2 ez ,n1  k 2 e2,n  y2,n  K A y2,n  K B y1,n   K KB  T 

Where y2, n 1  y2 ez , n 1  k2 ( y2 ez , n  y2, n ) ,

(15)

(16)

     e 2 ,n

with

e2,n  y2 ez ,n  y2,n

(17)

This last expression was obtained following an analogous procedure as used for equation (13).

Remark 2: To compute un the value of

y2 ez ,n 1 is required but what (14) allows to

calculate is y2 ez ,n However, y2 ez , n 1 can be estimated using Taylor’s formula:

dy2ez ,n d 2 y2ez ,n T 2 y2ez ,n1  y2ez ,n  T  ...  C dt dt 2


(18)

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Where,

is the complementary term. So, if the sampling time is small, y2 ez ,n 1 can be

approximated in one of following ways:

y2 ez ,n1  y2 ez ,n

y2 ez ,n 1  y2 ez ,n 

y2 ez ,n 1  y2 ez ,n 

(19)

dy2 ez ,n

T  2 y2 ez ,n  y2 ez ,n 1

dt

y2 ez ,n  y2 ez ,n 1 T

y2 ez ,n  2 y2 ez ,n 1  y2 ez ,n  2 T 2 T T2 2

(20)

(21)

Even when the first approximation, equation (19), provides excellent results for small sampling times, equation (20) was chosen for the control law calculation because it is a simple expression that gives lower deviations. Now, replacing un from (15) in (10)

y1, n  T y2, n  y1, n 1    y   y   2, n 1   2 ez , n 1  k2 ( y2 ez , n  y2 e, n )

(22)

And equations (13) and (16) in (22),

 e1, n 1  k1 T   e1, n  e       2, n 1   0 k2  e2, n 

If 0  k1 , k 2  1 

e

1,n

, e2,n   0,0 , n  

(23)

(24)

4. Controller design under uncertainty As the second order model (3) approximates the FOPDT model (1), which is also an approximation of the real system, it is important to consider the presence of uncertainty in the controller design22,23. The mathematical formulation of the uncertainty that represents model mismatches can be found as Supporting Information. Based on the results showed in section 3, an analysis for the perturbed systems is performed, assuming these as temporally disturbances as well as model mismatches. An additive uncertainty En is introduced into the model of the system and equation (10) takes the form:



 0  y1,n1   y1,n   y    y   T   K B  2,n1   2,n 

1   y1,n   0   0 un   En    K A   y2,n   KK B   1

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(25)

Which in analogous way leads to the following equation:

 e1, n 1  k1 T   e1, n  0 e        En  2, n 1   0 k2  e2, n  1

(26)

It is assumed that En is unknown. From equation (26), the influence of the uncertainty En over the tracking error is evident. Remark 3: The first order difference of En is defined as E n  E n 1  E n , the second order difference as  2 En   (En )   ( En1  En )  En 2  2 En1  En , and as a rule, the q-th order difference is defined as  q En   ( q 1 En ) . Remark 4: The q-th difference of a q-1 order polynomial is zero.

Let us consider a constant uncertainty En  const. This means: En  En1  En  0 . To reduce the influence of En , an integral term, calculated as the integral of e2, was added. The importance of the integral term is balanced by a tuning parameter K I . The control law is:

un 

Where

 1  y2 ez ,n 1  k2 ( y2 ez ,n  y2,n )  y2,n  K IU n 1  K B y1,n  K A y2,n   K KB  T 

U n1  U n  

( n 1)T

nT

e2 (t ) dt  U n  e2,n  T

(27)

(28)

For this, equation (26) is replaced in equation (24):

y2 n 1  y2,n   y2 ez ,n 1  k2 e2,n  y2,n  K I U n 1  K A y2,nT  K B y1,nT   K A y2,nT  K B y1,nT   En y2 n 1  y2 ez ,n 1  k2 e2,n  K I U n 1  En


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After some simple operations, it yields (29)

To ensure the stability of the linear system, the zeros (ri) of the characteristic polynomial of equation (29) should comply: 0  ri  1 , i = {1,2} Then

e2,n1  0

as

n   and, e1,n1  k1 e1,n  T e2,n  0 as n  

This means that the error will tend to zero despite of uncertainties, if they are constant. 5. Applications The performance of the LABC controller was tested in a linear high order system by simulation and, experimentally, on a transesterification batch reactor from biodiesel production process at laboratory scale. The controllers used were named as C1 (LABC controller) and C2 (Guevara et al. controller). 5.1 Linear high order system The system was defined by the following transfer function: G (s) 

1 e 0.5 s s  15.024 s  70.112 s 2  120.192 s  64.103 4

3

(30)

From the reaction curve procedure, a FOPDT model was obtained: G1 ( s ) 

1 e 1.15 s 1.3s  1

(31)

This is a fourth order system plus dead time with a controllability relationship close to t one ( 0  0.88 ).  The response curve of the system and the FOPDT model are shown in Figure 1.



Figure 1. Reaction curve for the system The system was tested for constant and variable reference profiles using k1 = 0.9, k2 = 0.7 and KI = 0.15. The tuning procedure used to find the controller parameters was based on a Montecarlo Experiment programmed to minimize a proposed error index9,24. In order to set certain boundaries for stability and robustness against disturbances, the system was perturbed with a magnitude 0.1 from minute 25 until the end of the simulation. Simulation results of controller C1 were compared with the ones obtained using a PID and C2.

5.2. Transesterification batch reactor Experimental assays were performed in a jacketed glass reactor with an overhead stirrer (Figure 2).

Figure 2. Experimental reactor Methanol, KOH and food grade soy vegetable were used without further refinement. Transesterification reaction was conducted under the following conditions: 6:1M ratio of methanol to soy oil,0.49 g KOH per 100g of oil, atmospheric pressure and 60°C reaction temperature.


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The response of the system was modeled by a FOPDT model, whose parameters were obtained from the reaction curve of the system to a 17% step change in the controller output (Figure 3).

Figure 3. Reaction curve of a biodiesel batch reactor

The resulting FOPDT model was: G exp ( s ) 

2.29 e 7.75 s 75.75s  1

(32)

Temperature was controlled using LABC controller (C1) with the following parameter values: k1 = 0.95; k2 = 0.94; KI = 0.0045. The tuning procedure was the same to the applied in section 5.1. The results were compared with a PID control. Constant and variable reference profiles were followed. During the assays, cold oil was introduced, in order to assess the performance of the controller against disturbances (see section 6.2.1 Nominal operation). To assess the controller performance against parametric uncertainty, one test was carried out in a reactor with different characteristics. The FOPDT for this reactor was obtained in order to show the change in model parameters. The process was tested using controller C1, designed and tuned for the nominal system (eq 32), to follow a variable profile (see section 6.2.2 Operation under uncertainty). The criterion to evaluate the performance was chosen from indexes that consider the entire close loop response. Among these, the integral of time weighted absolute value of the error (ITAE), is the best to eliminate errors that persist in the time.

ITAE   t. e1 (t ) .dt


(33)


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6. Results and discussion 6.1 Linear high order system The performance of C1 for a constant set point and persistent disturbances is shown in Figure 4. Results are compared with the ones obtained using a PID controller and the one developed in Guevara19, called C2. The best parameters values for PID controller were found applying Montecarlo algorithm9,24. Starting values were taken from Dahlin formula25 with a  20% variation. The lowest ITAE was used for the selection. It can be observed that C1 controller leads the system to the reference very fast and without overshoot, even when the disturbance added to the system appeared from minute 25. This important improvement of the controller proposed here is achieved by calculating the integral term over e2 instead e1 . The other controllers (PID and C2) uses the integral term calculated over e1 .

(a)

(b)

(c)

(d)

Figure 4. Simulation results for a fourth order system plus dead time for constant set point and disturbances. a) Response; b) Tracking error; c) Control action required by each control system; d) Cumulative tracking error


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When the integral term is defined over e1 , the expression of y2 ez ,n should be defined as:

y2 ez ,n  Now U n 1 takes the form:

y1ref ,n 1  k1e1,n  y1,n  K I U n 1 T

U n 1  U n  

( n 1)T

nT

e1 (t ) dt  U n  e1,n  T

And the expression for e1 , when there are no perturbations on the system, is:

e1n 1  k1e1,n  T e2 n  K I U n 1  0

(34)

Working under nominal conditions, e1 and e2 should tend to zero. So, to satisfy (34), the integral term should tend to zero too. To accomplish this condition, e1 will take positive and negative values, leading always to an overshoot on y1 . The integral over e2 produces the same effect on y2 , which is the derivative of y1 , but a change in the derivative sign does not imply an overshoot on y1 . This improvement allows to adjust the integral term of the controller to avoid overshoot in the response. The ITAE value obtained for C1 was the lowest, while C2 had the highest one. The tracking error, showed in Figure 4, shows the settling time is the same for C1 and PID, reaching values under 2.10-4 at approximately 15 minutes, while C2 controller took around 30 minutes to reach the same result. A remarkable feature of C1 controller is that achieves similar results than PID with a limited control action (Figure 4-c). Figure 4-d shows the cumulative tracking error is the lowest for C1 controller. The robustness of the controller was evaluated through Monte Carlo algorithm9. A 10% variation of the controller constants (k1, k2 and Ki) was tested and the results are shown in Figure 5.

Figure 5. Robustness evaluation



It was observed that even when the system response turns oscillatory for certain values, it still tends to the reference value. To improve the controller performance an anti-windup strategy was adopted. In this case, conditional integration was applied and the results, presented in Figure 6, show the robustness of the controller designed.

Figure 6. Robustness evaluation using filter

C1 controller was also tested for a variable time reference and contrasted with PID and C2 controller (Figure 7). Observe that C1 is the only one which does not present an overshoot and follows the reference profile with the lowest tracking error. The cumulative error clearly shows the best performance corresponds to the controller proposed in this work. (a)

(b) Figure 7. Simulation results for a fourth order system plus dead time for a variable reference and disturbances using C1, C2 and PID controllers. a) Response; b) Cumulative tracking error


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6.2. Transesterification batch reactor 6.2.1 Nominal operation Experimental results showed the controller proposed in this work (C1) has a good performance, better than PID and C2, when applied to a real process (Figure 8). The system follows the reference signal even when a disturbance, caused by cold oil addition, appeared at minute 230, correcting deviations faster than PID and C2. Notice the response of C2 controller showed an important overshoot.

(a) (b) Figure 8. Experimental results for C1 and PID: a) system responses against constant disturbances. b) Tracking error It has been shown that oscillatory chemical reactions may enhance the performance of continuous stirred-tank reactors3. Moreover, Benavides et al.17 determined the optimum conditions for biodiesel production involves a variable temperature profile. For this reason, two different variable time temperature profiles were proven, both with excellent results (Figures 9 and 10). In the first case (Figure 9), C1 reached the reference, without overshoot, in about one hour and followed the profile with the lowest tracking error. As seen, the systems with other two controllers were not able to track the desired oscillatory profile with a good performance. It is noteworthy C1 rejects the disturbance, caused by cold oil addition, at minute 130. The other variable time reference tested was a parabolic temperature profile. Figure 10 shows it was successfully followed using the controller proposed in this work (C1), with an outstanding performance reflected in a markedly lower tracking error. This result was contrasted with the ones obtained using a PID controller which showed it has no ability



to follow this kind of reference variability. As C2 controller showed bad performance for variable reference it was not tested for this profile.

(a) (b) Figure 9. Experimental results for C1, C2 and PID: a) system responses for variable time reference with disturbances b) Tracking error

Figure 10. Experimental results for C1 and PID: a) system responses for a parabolic temperature increase, b) Tracking error In order to show the advantages of the controller proposed in this work, called C1, the ITAE of each experimental test is shown graphically (Figure 11). It can be observed that variable profiles are very difficult to be followed using a PID, reflected in high ITAE values. Nevertheless, C1 can lead the system response through these profiles with an outstanding performance.


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Figure 11. ITAE obtained in experimental tests 6.2.2 Operation under uncertainty The results of a test driven in a different reactor showed controller C1 has an excellent performance operating under uncertainties. The FOPDT obtained for this new system is: G exp_u ( s ) 

1.11 e 8.9 s 55.6 s  1

Figure 12 shows the system response for a variable time temperature profile, when the process is under nominal operation conditions and under uncertainties. It can be seen the tracking error is very low in both cases, making evident the good performance of the controller designed.

Figure 12. Experimental results for C1 applied to the system under nominal and operation and under uncertainty.



7. Conclusions A methodology based on linear algebra to design control algorithms for process systems that can be represented by a FOPDT Model (First Order Plus Dead Time Model) has been presented. An integrator has been added on e2 to reduce the influence of uncertainties ( En ) over the tracking error. This improvement resulted in a controller that can be tuned to follow different profiles without overshoot. Different simulations and experimental tests on a batch reactor were carried out to demonstrate the effectiveness of the proposed methodology. The controller proposed in this work showed accurate results and an excellent performance against disturbances. Also, its robustness was shown through Monte Carlo simulations. From comparison against other controllers (PID and Guevara19) the superiority the controller proposed in this work was evidenced. One remarkable feature is that this new controller can be tuned to work without overshoot, a desirable characteristic for most operations. Besides its better performance, this controller has the advantage of being versatile and simple, as it has been developed for any process that can be represented by a FOPDT model. These features make its industrial application highly feasible.

Acknowledgments This work was partially funded by Universidad Nacional de San Juan and Consejo Nacional de Investigaciones Científicas y Técnicas, Argentina (CONICET-National Council for Scientific Research). OC thanks to PIMI 15-10 Project of Escuela Politécnica Nacional, for its support for the realization of this work.

Supporting Information Mathematical formulation of the uncertainty that represents model mismatches. The Supporting Information is available free of charge on ACS Publications website at http://pubs.acs.org/.


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Abstract Graphics

Persistent disturbancies

Uncertainties

k1 k2 KI

LINEAL ALGEBRA BASED CONTROLLER


Design and Application of a Linear Algebra Based ... - ACS Publications

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