processes Article
Multivariable Real-Time Control of Viscosity Curve for a Continuous Production Process of a Non-Newtonian Fluid Roberto Mei 1 , Massimiliano Grosso 1 , Francesc Corominas 2 , Roberto Baratti 1 and Stefania Tronci 1, * ID 1
2
*
Dipartimento di Ingegneria Meccanica, Chimica e dei Materiali, Università degli Studi di Cagliari, Via Marengo 3, 09123 Cagliari, Italy;
[email protected] (R.M.);
[email protected] (M.G.);
[email protected] (R.B.) Procter & Gamble Eurocor N.V., Temselaan 100, 1853 Strombeek-Bever, Belgium;
[email protected] Correspondence:
[email protected]; Tel.: +39-070-675-5050
Received: 22 December 2017; Accepted: 27 January 2018; Published: 30 January 2018
Abstract: The application of a multivariable predictive controller to the mixing process for the production of a non-Newtonian fluid is discussed in this work. A data-driven model has been developed to describe the dynamic behaviour of the rheological properties of the fluid as a function of the operating conditions using experimental data collected in a pilot plant. The developed model provides a realistic process representation and it is used to test and verify the multivariable controller, which has been designed to maintain viscosity curves of the non-Newtonian fluid within a given region of the viscosity-vs-shear rate plane in presence of process disturbances occurring in the mixing process. Keywords: non-Newtonian fluid; multivariable control system; viscosity curve
1. Introduction The industrial continuous production of complex fluids still encounters to date issues in terms of quality control of products. Parameters like shear rate dependent viscosity, which affects product quality, could easily go out of specifications if the ingredient characteristics or other variables of the process move away from the original formulation. The main issue is represented by the difficulty of controlling such processes because there aren’t commercial sensors capable of providing real-time information about the rheological properties of the fluid. Available in-line sensors can measure the viscosity only for a punctual shear rate value, therefore their use for controlling the production process of non-Newtonian fluid is not possible. Furthermore, very few in-line rheometers have been tested and few studies have been presented for non-Newtonian and opaque industrial fluids [1]. Because of the importance of the continuous monitoring of rheological parameters of industrial fluids during production, in the last years many steps have been accomplished for designing and developing sensors capable of providing rheological proprieties of complex fluids in real-time. Many recent works focus their efforts on non-invasive techniques which seem the most promising for this type of problems. Kotzé et al. [2] studied a technique based on ultrasonic velocity profiling (UVP) which measures an instantaneous one-dimensional velocity profile in a fluid containing particles across the ultrasonic beam axis or measurement line. The method combines the UVP technique with pressure difference (PD) measurements, it is non-invasive, it can be used to measure opaque and concentrated suspensions. Preliminary results obtained in concentrated cement pastes showed that UVP is a feasible and promising technique for flow characterisation in viscous fluids. Meacci et al. [1] presented an efficient, fully programmable and integrated system for in-line fluids characterisation of a wide range
Processes 2018, 6, 12; doi:10.3390/pr6020012
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of non-Newtonian and opaque fluids. This system, named Flow-Viz exploits ultrasounds to detect the velocity profile of the flow moving in the pipe, and it is designed for industrial use. Recently, Yoshida et al. [3] proposed a sensor based on ultrasonic spinning rheometry (USR), which is expected to provide details of various rheological properties. The proposed USR capabilities were assessed for three test fluids chosen as examples of thixotropic fluids, shear-thinning fluids, and multiphase fluids. The recent results on innovative sensors encourage the efforts for developing control strategies Processes 2018, 6, x FOR PEER REVIEW 2 of 14 which provide the target rheological characteristics in the production of non-Newtonian fluids. to detecttothe velocity profile of the in the pipe, and it is designed for by mixing The presentultrasounds study is aimed design a controller forflow themoving production of a detergent obtained industrial use. Recently, Yoshida et al. [3] proposed a sensor based on ultrasonic spinning rheometry different ingredients, which should lead to specific rheological and physical properties. The product is (USR), which is expected to provide details of various rheological properties. The proposed USR characterised by a complex rheological previous [4] the authors capabilities were assessed for three behavior test fluids and, choseninasa examples of work thixotropic fluids, shear- developed and multiphase a one-pointthinning controlfluids, of viscosity curvefluids. for the continuous production process, showing how the choice results on innovative sensors encourage the efforts developing of the point onThe therecent viscosity-vs-shear rate curve was crucial forforthe qualitycontrol of thestrategies product. Results which provide the target rheological characteristics in the production of non-Newtonian fluids. The also evidenced that it is possible to partly reject disturbances using only one point as controlled present study is aimed to design a controller for the production of a detergent obtained by mixing variable, but this configuration was notlead adequate the severe specifics required the plant under different ingredients, which should to specificfor rheological and physical properties. Thein product is characterised by a complex behavior and, in a previous work [4] improve the authors developed investigation. In this situation, a rheological model predictive control (MPC) can the performance of one-point control of viscosity curve for the continuous production process, showing how the choice the system.a MPC can handle non-square system, therefore the two available manipulated inputs can of the point on the viscosity-vs-shear rate curve was crucial for the quality of the product. Results be used to also control morethat than points on the viscosity-vs-shear rate one curve inasancontrolled efficient way. It is evidenced it istwo possible to partly reject disturbances using only point worth noting that but thethis plant under consideration byrequired high time if compared to variable, configuration was not adequateis forcharacterised the severe specifics in thedelay, plant under investigation. In this situation, a model predictive control (MPC) can improve the performance of thethis case [5]. the characteristic time of the mixing process, and again the MPC algorithm is effective in system. MPC can handle non-square system, therefore the two available manipulated inputs can be The multivariable control performances have been assessed by simulating the process at different used to control more than two points on the viscosity-vs-shear rate curve in an efficient way. It is operating conditions, including and measurement worth noting that the planttime underdelay consideration is characterised noise. by high time delay, if compared to the characteristic time of the mixing process, and again the MPC algorithm is effective in this case [5]. The multivariable control performances have been assessed by simulating the process at different 2. Process Description operating conditions, including time delay and measurement noise.
The production of a detergent on pilot plant scale (facilities made available by the Brussels Process Description Innovation2.Centre in Belgium, BIC) is considered in this work. The product is a compound which Theingredients production of with a detergent on pilot plant scale (facilities made obtaining available by the Brussels contains several different rheological properties, a final mixture with Innovation Centre in Belgium, BIC) is considered in this work. The product is a compound which a highly complex rheological behavior. It is important to underline that blends rheology is affected by contains several ingredients with different rheological properties, obtaining a final mixture with a factors such rheology ofrheological single components but also to temperature polydispersity [6,7]. highly complex behavior. It is important underline thatand blends rheology is affected by Details on the ingredients be reported confidentiality reasons. and polydispersity [6,7]. Details on factorscannot such rheology of single for components but also temperature the ingredients cannot1) beis reported for confidentiality reasons. The pilot plant (Figure designed as a main pipe connected to a series of tanks, each of them The pilot plant (Figure 1) is designed as a main pipe connected to a series of tanks, each of them containingcontaining one ingredient, through secondary pipes. Ingredients are pumped in the main line and one ingredient, through secondary pipes. Ingredients are pumped in the main line and a a series of static mixer between inputs a good blending of the mixture. Atthethe series of static mixer between inputs ensure ensure a good blending of the mixture. At the end of lineend the of the line product is collected in a tank for off-line rheological measurements (for further details see [8]). the product is collected in a tank for off-line rheological measurements (for further details see [8]).
Figure 1. Scheme of the pilot plant at Brussels Innovation Centre (BIC) (Patent N. US 2013/0225468 Figure 1. Scheme of the pilot plant at Brussels Innovation Centre (BIC) (Patent N. US 2013/0225468 A1). A1).
An in-line Endress-Hauser Proline Promass 83I Coriolis flowmeter, located in the main pipe after the 4 inputs of the ingredients, is used to gain information about system dynamics, along with pressure and temperature sensors. Such a flowmeter can provide only a point measurement of viscosity.
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Such measure is not exhaustive when dealing with non-Newtonian fluids but can provide a rough Processes 2018, 6, x FOR PEER REVIEW 3 of 14 estimation about dynamics. An in-line Endress-Hauser Proline Promass 83I Coriolis flowmeter, located incollected the main pipe Off-line rheological measurements were carried out on the samples in after the final tank the 4 inputs of the ingredients, is used to gain information about system dynamics, along with with a rheometer AR-2000 TA stress controlled equipped with a 40 mm cone and plate fixture, used to pressure and temperature sensors. Such a flowmeter can provide only a point measurement of obtain the viscosity dependence on shear rate. viscosity. Such measure is not exhaustive when dealing with non-Newtonian fluids but can provide a rough estimation about dynamics. Off-line rheological measurements were carried out on the samples collected in the final tank 3. Problem Statement with a rheometer AR-2000 TA stress controlled equipped with a 40 mm cone and plate fixture, used to obtain the viscosity dependence on shear The quality of the product under study is rate. determined through its rheological properties, which can be summarised by the viscosity-vs-shear rate curve. The rheological properties may deviate from 3. Problem Statement the nominal ones, because of process disturbances, and they can be adjusted by varying the ingredients The quality of the product under study is determined through its rheological properties, which flow rates which a major impact without affecting toorheological much detergent can behave summarised by the viscosity-vs-shear rate curve. The propertiescharacteristics. may deviate from As representative Figure 2 reports a typicaland viscosity for thebydetergent produced in the nominalexample, ones, because of process disturbances, they cancurve be adjusted varying the ingredients in flow rates 1), which a major impactwith without affecting too much line detergent the plant (schematised Figure withhave a target reported the green continuous and the region characteristics. (grey shaded area) of accepted viscosity values limited by the upper and lower curves. Such region As representative example, Figure 2 reports a typical viscosity curve for the detergent produced 1 ), while 20% was obtainedinby a maximum error 10%reported at low with shear than 10and s−the theconsidering plant (schematised in Figure 1), with aoftarget therate green(lower continuous line (grey shaded area)The of accepted viscositylines valuesare limited the upper and lower curves. output Such has been set region at high shear rate. red dashed twobyexamples of undesired values, region was obtained by considering a maximum error of 10% at low shear rate (lower than 10 s−1), when the process is out-of-control. It is worth noting that the target of the control problem is not while 20% has been set at high shear rate. The red dashed lines are two examples of undesired output a point valuevalues, but infinite on a one-dimensional manifold (the viscosity curve) and the when the points process islying out-of-control. It is worth noting that the target of the control problem is not a point value but infinite pointsregion lying on one-dimensional manifold (themain viscosity curve) and entire curve should stay within the grey ofa allowable viscosity. The issue is to individuate curve stay within the grey of region of allowable viscosity. The main issueplant. is to Based on the points onthe theentire curve thatshould guarantee the respect the limits at every condition in the individuate the points on the curve that guarantee the respect of the limits at every condition in the the previous plant. investigation of the authors [4], theofviscosities studied(η)compound at shear-rate Based on the previous investigation the authors(η) [4],of thethe viscosities of the studied . 1 have been selected. −1 (γ) values equal to 0.1,at1, 10 and(γ) 1100 s−equal are the most compound shear-rate values to 0.1, 1, 10 and 1100 sSuch have points been selected. Such pointsrepresentative are the mostbehavior representative for the rheological behavior for the rheological of the considered system.of the considered system.
Figure 2. Rheological curves for the system under study. The shaded region indicates the viscosity
Figure 2. Rheological curves for the system under study. The shaded region indicates the viscosity values for in-control product. The red dotted lines are examples of off-control products. values for in-control product. The red dotted lines are examples of off-control products. 4. Process Simulator A process simulator has been developed for designing and evaluating a rheological controller 4. Process Simulator
for the continuous production of a detergent. The role of this simulator is to give time responses for
thesimulator viscosities of thebeen produced compound as inputs the evaluating ingredient amounts. Because the A process has developed fortaking designing and a rheological controller for the continuous production of a detergent. The role of this simulator is to give time responses for the viscosities of the produced compound taking as inputs the ingredient amounts. Because the rheological behavior of the system is very complex, a first principle description of the process is not possible, and a data-driven model has been preferred. The experimental apparatus used in this work cannot give on-line information about the quality of the ingredients which are fed in the plant, meaning that it is only possible to describe the input-output relationship between ingredient flow rates and product’s rheological properties. For model development, several step tests were carried out at different inlet flow rates and plant setup (distance between the last mixer and the sample collection), and the products
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rheological behavior of the system is very complex, a first principle description of the process is not and aoff-line data-driven model hasrheometer been preferred. The experimental apparatus used in this work The rheological were analysed possible, analyzed with the described in the previous section. cannot give on-line information about the quality of the ingredients which are fed in the plant, information obtained with such tests can be related to steady state conditions. The transient response meaning that it is only possible to describe the input-output relationship between ingredient flow rates and product’s rheological Forcollected model development, several step tests were carried(Promass). was on the other hand observed usingproperties. the data with the in-line viscometer at different inlet flow rates and plant setup (distance between the last mixer and the sample A simple out description of the data collected in the pilot plant can be attained by means of collection), and the products were analysed analyzed off-line with the rheometer described in the a continuous-time Hammerstein model [9], where thewith nonlinear no-memory gainstate is calculated with previous section. The rheological information obtained such tests can be related to steady conditions. The transient response was on the other hand observed using the data collected with the of a nonlinear a Neural Network (NN) model [10,11]. The time invariant state space representation in-line viscometer (Promass). continuous-time Hammerstein system 3) that cascades static nonlinearity by a linear A simple description of the(Figure data collected in the pilot planta can be attained by meansfollowed of a continuous-time Hammerstein model [9], where the nonlinear no-memory gain is calculated with a dynamic system may be described by Equation (1) Neural Network (NN) model [10,11]. The time invariant state space representation of a nonlinear continuous-time Hammerstein system (Figure 3) that cascades a static nonlinearity followed by a . x( t ) = Ax(t) +(1)fNN (u), linear dynamic system may be described by Equation ( )=
( )+
( ),
(1)
(1)
where x is the n-dimensional state vector, u is the m-dimensional input vector, A is a constant matrix where x is the n-dimensional state vector, u is the m-dimensional input vector, A is a constant matrix and f NN indicates the memoryless NNNN model. and fNN indicates the memoryless model.
( )
( )
=
+ ( )
( )
Figure 3. Nonlinear continuous time system of the Hammerstein Model. Figure 3. Nonlinear continuous time system of the Hammerstein Model.
The neural network between the inputs u and the variable z on Figure 3 has been designed and tested to relate viscosities measured off-line at different shear rate values with the inputs of the pilot The neuralplant, network theMass variable on chosen Figure 3 has been designed and in terms between of ingredientthe massinputs fraction u in and the feed. fractionszwere instead of mass to avoid problems concerning the useatofdifferent different order of mass flows magnitude. Thethe outputs tested to relateflows viscosities measured off-line shear rate values with inputs of the pilot (z) are the off-line viscosities (η) of the studied compound at shear-rate ( ) values equal to 0.1, 1, 10 plant, in termsand of 1100 ingredient mass fraction in the feed. Mass fractions were chosen instead of mass s−1, as reported in the previous section. The NN structure is represented in Figure 4, where flows to avoid itproblems concerning the use of different order mass flows magnitude. The outputs is shown that the input vector consists of four units, the outputof layer has four neurons and the hidden layer has six neurons. For both the hidden and output layers a sigmoidal. activation function (z) are the off-line viscosities (η) of the studied compound at shear-rate (γ) values equal to 0.1, 1, 10 and is used. The experimental data used for model calibration were obtained varying the percentage of 1100 s−1 , as reported in the NNinstructure is represented in Figure 4, where the ingredient flow previous rates within section. the intervalThe reported Table 1. In more details, 27 different operating conditions were used in the pilot plant units, and withthe the repeated allowed it is shown that the input vector consists of four outputexperiments layer has fourthe neurons and the collection of 171 viscosity curves. The model parameter estimation was carried out using 70% of hidden layer has six neurons. For both the hidden and output layers a sigmoidal activation function is points for the training step, 15% for the validation and 15% for the test.
used. The experimental data used for model calibration were obtained varying the percentage of the ingredient flow rates within the interval reported in Table 1. In more details, 27 different operating conditions were used in the pilot plant and with the repeated experiments allowed the collection of 171 viscosity curves. The model parameter estimation was carried out using 70% of points for the training step, 15%2018, for6, xthe Processes FORvalidation PEER REVIEW and 15% for the test. 5 of 14
Ingredient A
Viscosity at = 0.1
Ingredient B
Viscosity at = 1
Ingredient C
Viscosity at = 10
Ingredient D
Viscosity at = 1100
INPUT LAYER
HIDDEN LAYER
OUTPUT LAYER
Figure 4. Structure of the neural network used for the function fNN in the continuous Hammerstein model.
Figure 4. Structure of the neural network used for the function f NN in the continuous Hammerstein model. 1. Range theingredients’ ingredients’ mass fraction. TableTable 1. Range of of the mass fraction.
MIN MIN MAX MAX
A A 0.730 0.730 0.880 0.880
B B 0.072 0.072 0.250 0.250
C C 0.000 0.000 0.039 0.039
D D 0.012 0.012 0.066 0.066
Model’s performance is evaluated by considering the coefficient R2 and the mean absolute deviation (MAD) defined in Equation (2), where yi is the measured viscosity, fi is the calculated viscosity, is the mean viscosity value and N is the number of experimental points. =1−
∑ ∑
( − ) , ( − )
=
1
−
(2)
The developed NN is capable to predict viscosities of the compound with good results, as reported in Table 2, where the performance indexes are shown for the four output variables.
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Model’s performance is evaluated by considering the coefficient R2 and the mean absolute deviation (MAD) defined in Equation (2), where yi is the measured viscosity, fi is the calculated viscosity, is the mean viscosity value and N is the number of experimental points. 2 ∑iN=1 (yi − f i )
1 R = 1− N , MAD = 2 N ∑ i =1 ( y i − y ) 2
N
yi − f i ∑ y
i =1
(2)
i
The developed NN is capable to predict viscosities of the compound with good results, as reported in Table 2, where the performance indexes are shown for the four output variables. Measured viscosities against predicted NN viscosities are reported, by way of example, in Figure 5 for shear-rate equal to 0.1 s−1 . The test dataset is evidenced in the figure to better evaluate the performance of the model, because in this case the neural model uses data which have not been used during the training/validation step. The obtained results are in the main quite good, but at = 1100 s−1 , R2 is relatively low (about 0.93) showing some predictability limits of the model. It should be considered that the viscosity variations at high shear-rate value are however rather small (from 0.1 to 0.5 Pa·s). Standard residuals (Figure 6) appear without a deterministic structure, indicating that the model captures the essential features of the data and they validate the reliability of the NN to predict the steady state values of off-line viscosities for different amounts of ingredients, at least for the investigated conditions. The second block in the Hammerstein model (Figure 3) describes the dynamic behaviour and it has been obtained using the in-line measurements. Because only a point viscosity is available with Promass, a unique characteristic time is adopted for the four output variables, in the understanding that the rheological changes in the mixture happens at the same moment for each shear rate. By the same token, the time delay is assumed equal for each calculated output. A comparison between the measured viscosity and the one obtained using a first-order plus time delay system is reported in Figure 7, with the purpose of showing the dynamics assessment. It is worth noticing that it is not possible to compare the viscosity predicted by the NN, calculated offline, and the one of the flowing detergents in the plant, because it is not possible to relate the in-line viscosity given by Promass with the off-line values. This means that a comparison between the model and plant behaviour is possible only when the in-line sensor is available. In this work, the impact of process conditions on the characteristic time is neglected and it is set equal to 20 s. Time delay has been calculated considering one configuration of the plant and it is set equal to 8 s. Table 2. Performance of the neural network. .
γ (s−1 ) 0.1 1 10 Processes 2018, 6, x FOR PEER REVIEW 1100
R2
MAD (%)
0.9757 0.9666 0.9808 0.9315
8.6 7.0 4.0 3.7
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Figure viscosities 5. Predicted viscosities measured viscosities at at 0.10.1 s−1. Training and validation: Figure 5. Predicted againstagainst measured viscosities s−1 . Training and grey validation: grey circle; test: green diamond. circle; test: green diamond.
Figure 5. Predicted viscosities against measured viscosities at 0.1 s−1. Training and validation: grey
Processes 2018, 6,circle; 12 test: green diamond.
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Figure 7, with the purpose of showing the dynamics assessment. It is worth noticing that it is not possible to compare the viscosity predicted by the NN, calculated offline, and the one of the flowing detergents in the plant, because it is not possible to relate the in-line viscosity given by Promass with the off-line values. This means that a comparison between the model and plant behaviour is possible only when the in-line sensor is available. In this work, the impact of process conditions on the Figure characteristic 6. Standardised to predicted at: 0.1 s−1 (left upper panel), time isresiduals neglected with and itrespect is set equal to 20 s. Timeviscosities delay has−1 been calculated considering −1 Figure 6. Standardised residuals with respect to predicted viscosities 0.1 s (left upper panel), 1 s −1 at: 1 s−1 (right upper panel), 10−1plant s−1 (left (right lower panel). one configuration of the and lower it is setpanel) equal toand 8−1 s. 1100 s (right upper panel), 10 s (left lower panel) and 1100 s (right lower panel).
The second block in the Hammerstein model (Figure 3) describes the dynamic behaviour and it has been obtained using the in-line measurements. Because only a point viscosity is available with Promass, a unique characteristic time is adopted for the four output variables, in the understanding that the rheological changes in the mixture happens at the same moment for each shear rate. By the same token, the time delay is assumed equal for each calculated output. A comparison between the measured viscosity and the one obtained using a first-order plus time delay system is reported in
Figure 7. Viscosity measured on line (black andvalues values calculated (blue continuous Figure 7. Viscosity measured on line (blackdots) dots) and calculated (blue continuous line) with a line) with plus time delay system (upperpanel) panel) due step change of theof ingredient D flow rate a first-orderfirst-order plus time delay system (upper duetotothe the step change the ingredient D flow rate (lower panel). (lower panel). 5. Control Algorithm The control problem is addressed using a model predictive control (MPC) in its basic formulation of dynamic matrix control (DMC), where the controlled outputs are selected among the ones given by the simulator. In more details, because the small variations at = 1100 s−1 can be challenging for the robustness of the controller [12] only the viscosities at = 0.1, 1 and 10 s−1 have been considered as controlled variable, while the ingredient B and D flow rates are the manipulated inputs. The evaluation of the control configuration has been carried out using the simulator reported in Section 4 to act as a virtual plant and a schematic representation of the control loop is reported in Figure 8.
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5. Control Algorithm The control problem is addressed using a model predictive control (MPC) in its basic formulation of dynamic matrix control (DMC), where the controlled outputs are selected among the ones given by . the simulator. In more details, because the small variations at γ = 1100 s−1 can be challenging for the . robustness of the controller [12] only the viscosities at γ = 0.1, 1 and 10 s−1 have been considered as controlled variable, while the ingredient B and D flow rates are the manipulated inputs. The evaluation of the control configuration has been carried out using the simulator reported in Section 4 to act as a virtual plant and a schematic representation of the control loop is reported in Figure 8. The objective function J used to calculate the manipulated action in the DMC is reported in Equation (3) Processes 2018, 6, x FOR PEER REVIEW 8 of 14 J = [(e(k + 1) − F∆u) T W(e(k + 1) − F∆u)] + [∆u] T K[∆u] (3) ( (p + ( ++1) ) 3xH 1) − ∆ ) + vector ∆ = ( e(k ∆representing the(3) where k denotes the time index, 1)−is ∆the dimensional difference between the desired outputs and the e(k current further control action, F is the dimensionalwithout vector representing the difference where k denotes the time index, + 1) is output the 3xHpprediction outputs and the current output without further control F is control dynamic between matrix the [5],desired Hp is the prediction horizon, u isprediction 2xHu dimensional vector of action, the future dynamic matrixhorizon, [5], Hp is the prediction horizon, u ismatrix 2xHu dimensional vector of changes the future of control moves, Hthe is the control K is a block diagonal used to penalise manipulated u moves, Hu is the control horizon, K is a block diagonal matrix used to penalise changes of inputs, W is a weighting matrix used as a tuning parameter. The prediction error is corrected by the manipulated inputs, W is a weighting matrix used as a tuning parameter. The prediction error is measuredcorrected outputs at the sampling More instant. in details, a positive (3xHp )x(3xHp ) byavailable the measured outputs availableinstant. at the sampling MoreW in is details, W is a positive block diagonal matrix which in turn is which composed three diagonal matrixes, one for (3xHp)x(3xH p) block diagonal matrix in turn of is composed of three diagonal matrixes, oneeach for output (Equationeach (4)).output (Equation (4)). ( ) = ( w1 ) (4) diag 0( ) 0 ( ) W= (4) 0 diag(w2 ) 0 The matrices diag(wi), with i = 1, 2, 3, have dimensions H p xH p and the element on the diagonal is 0 0 diag(w3 ) the positive weight for a specific output. The matrix K is a positive (2xHu)x(2xHu) block diagonal
The matrix matrices diag(w i = 1, of 2,two 3, have dimensions Hpfor xHeach the element on the diagonal which in turn composed diagonal matrixes, one input. p and i ),iswith is the positive weight for a specific output. The matrix K is a positive (2xH u )x(2xHu ) block diagonal ( ) = (5) ( ) matrix which in turn is composed of two diagonal matrixes, one for each input. The matrices diag(ki), with i = 1, 2,"have dimensions HuxHu and # the element on the diagonal is positive and it penalises a specific input. The process diag 0 (Equation (1)) provides the four points (k1 ) simulator K = to the actual situation of the process. Random noise is then (5) of the rheological curve corresponding 0 diag(k2 ) added to the outputs in order to simulate measurement noise.
Noise
MPC
+ +
Other ingredients
Ingredient B
Ingredient D
Process simulator
8. Schematic representation thecontrol control loop thethe model predictive controlcontrol (MPC). (MPC). Figure 8.Figure Schematic representation ofofthe loopforfor model predictive
A linear predictive model is required to implement the DMC in its traditional form. This task is
The addressed matrices using diag(kthe with i = to 1, carry 2, have dimensions Hu xH theFdiagonal is on step response tests from whichthe theelement dynamic on matrix is u and i ), simulator obtained [5]. The simulator model is excited by varying the input(Equation corresponding the manipulated positive and it penalises a specific input. The process simulator (1))toprovides the four points variables, starting the reference condition and considering step of different sizes [13,14]. of the rheological curvefrom corresponding to the actual situation ofchanges the process. Random noise is then It is worth noting that the simulator does not model the variations of ingredients quality (composition added to the outputs in order to simulate measurement noise. and rheological behavior). The coefficients of the dynamic matrix F are obtained by averaging the A linear predictive modelstep is required responses of the different changes. to implement the DMC in its traditional form. This task is addressed using the simulator to carry on step response tests from which the dynamic matrix F is 6. Results Various tests have been carried out with the intention of assessing the performance of the control strategy. The main purpose of the simulations is to understand how the quality of the on-line measurements (delay and noise) can affect the rheological control behavior, in view of a future
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obtained [5]. The simulator model is excited by varying the input corresponding to the manipulated variables, starting from the reference condition and considering step changes of different sizes [13,14]. It is worth noting that the simulator does not model the variations of ingredients quality (composition and rheological behavior). The coefficients of the dynamic matrix F are obtained by averaging the responses of the different step changes. 6. Results Various tests have been carried out with the intention of assessing the performance of the control strategy. The main purpose of the simulations is to understand how the quality of the on-line measurements (delay and noise) can affect the rheological control behavior, in view of a future implementation of continuous monitoring of rheological parameters in industrial plants. Ultrasound sensors are indeed candidate to monitor and control the production of materials with complex characteristic behavior (e.g., [1]), but information on their in-line response such as time required for calculation of the viscosity curve, measurement noise, reliability, repeatability is not available yet for the fluid under investigation. In particular, it could be useful to understand what happens if the time interval for the ultrasound sensor to collect measurements and calculate the velocity profile is much greater than the characteristic time of the process. 6.1. MPC for Set-Point Tracking The parameters related to the MPC development, such as prediction and control horizon, sampling time and weights, are found by analysing the dynamic response of the process and by tuning. Considering a measurement delay equal to 30 s, the following parameters have been selected in order to achieve an acceptable dynamic matrix conditioning while maintaining good controller performances: (i) the control action is applied every 10 s, and (ii) the dimension of the prediction horizon Hp is set equal to 16. The control horizon Hu is set equal to 4 for the entire control configuration, as suggested by [5], and only the first control move is applied at each sampling time. Table 3 summarises the process conditions and MPC parameters for a reference case. Table 3. Specifications for the reference case. Parameter
Value
Time interval for control action
10 s
Measurement delay
30 s
Time constants for simulator (for each controlled viscosity)
20 s, 20 s, 20 s
Time delays for simulator (for each controlled viscosity)
8 s, 8 s, 8 s
Noise in the measurements about ±2%: −0.2 Pa·s < random noise for η0.1s−1 < +0.2 Pa·s −0.05 Pa·s < random noise for η1s−1 < +0.05 Pa·s −0.02 Pa·s < random noise for η10s−1 < +0.02 Pa·s Weights of the controller (for each controlled viscosity)
w1 = 1, w2 = 12, w3 = 40
Parameters for the penalisation of the controller (Ingredients D and B respectively)
k1 = 3.5 × 104 k2 = 3.0 × 102
Figure 9 shows the system responses to step variation of the set points. The controller works well and it is capable to bring the three controlled viscosities (viscosity at shear-rates equal to 0.1, 1 and 10 s−1 ) to the desired values in a relatively low time. Actions on the two manipulated variables have been penalised to avoid excessive overshoots.
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The effects of a greater measurement delay on the controlled process is represented in Figure 10. In this case measurements are available every 60 s, and Hp was set equal to 19. The controller is capable Processes 2018, 6, x FOR PEER REVIEW 10 of 14 to bring the viscosities to the desired values but the response is slower than the reference case. 14
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Figure 9. Performance thereference referencecase: case: mass flow of ingredient (b) flow mass flow Figure 9. Performanceofofcontroller controller for for the (a)(a) mass flow of ingredient B; (b)B; mass −1 ; (d) response of the viscosity of ingredient D; (c) response of the viscosity at shear rate equal to 0.1 s −1 of ingredient D; (c) response of the viscosity at shear rate equal to 0.1 s ; (d) response of the viscosity −1. For at shear raterate equal to to 1 s1−s1−1; ;(e) of the theviscosity viscosityatat shear rate equal tos10 s−1 .(c–e) For the (c–e) the (e) response response of shear rate equal to 10 at shear equal following data areare reported: (redsolid solidline), line),measured measured value (green dashed following data reported:target target curve curve (red value (green dashed line). line).
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Figure 10. 10. Performance for the thecase casewith with time sampling equal (a) mass Figure Performanceofofcontroller controller for time sampling equal to 60tos: 60 (a) s: mass flow offlow of −1 ingredient B; (b) mass flow of ingredient D; (c) response of the viscosity at shear rate equal ingredient B; (b) mass flow of ingredient D; (c) response of the viscosity at shear rate equal to 0.1to s−10.1 ; s ; − 1 (d) response of the viscosity to11s−1 s ; (e) ; (e) response of viscosity the viscosity at shear rate equal response of the at shear rate equal (d) response of the viscosityatatshear shearrate rate equal equal to −1s.−1For to 10tos10 dataare arereported: reported: target curve measured . For(c–e) (c–e)the the following following data target curve (red(red solidsolid line),line), measured value value (green (green dashed line). dashed line).
6.2. MPC for Disturbance Rejection
6.2. MPC for Disturbance Rejection
To study the performance of the MPC controller when different batches of ingredients are used, To study the performance of the MPC controller when different batches of ingredients are used, another test has been designed and developed. More in details, a different type of ingredient B has another been designed and developed. More in details, a different type of ingredient B has been test usedhas in the process. been used in the process. Another neural network (hereafter indicated as NN_d) with the same structure shown in Figure indicated as NN_d) with same structure in Figure 3 3Another has been neural trained network using data(hereafter coming from tests where another typethe of ingredient B (typeshown 2) has been used. trained The second of from the Hammerstein model, which only related to the2)dynamic has been usinglinear data block coming tests where another type ofisingredient B (type has been used. is theblock same of ofthe theHammerstein previous case. model, The new model has been used to simulate a The behavior, second linear which is NN_d only related to the dynamic behavior, is disturbance entering the process, as explained in the following. Time intervals for process simulation, the same of the previous case. The new model NN_d has been used to simulate a disturbance entering control action and sampling are the same reported in Table 2. The magnitude of noise in the the process, as explained in the following. Time intervals for process simulation, control action and measurements is respectively ±0.2 Pa·s for η . , ±0.05 Pa·s for η and ±0.02 Pa·s for η . sampling are the same reported in Table 2. The magnitude of noise in the measurements is respectively To simulate the disturbance, at a certain point in the timeline of the simulation (when relative ±0.2time Pa·sisfor η 1 , ±0.05 Pa·s for η1 s−1 and ±0.02 Pa·s for η10 s−1 . equal0.1tos−500 s), the neural network used for the process simulator (NN) has been replaced
To simulate the disturbance, at a certain point in the timeline of the simulation (when relative time is equal to 500 s), the neural network used for the process simulator (NN) has been replaced with the other one (NN_d). It’s important to highlight that only the simulator is affected by this change, while the dynamic matrix F is the same. This action has the goal to simulate what happens to the real
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with the other one (NN_d). It’s important to highlight that only the simulator is affected by this
process when a different batch of ingredient B, which has a different rheological behavior with respect change, while the dynamic matrix F is the same. This action has the goal to simulate what happens to the previous is suddenly used batch (maybe to replace finished to the realone, process when a different of ingredient B, awhich has abatch). different rheological behavior The response ofthe theprevious systemone, to this changeused of ingredient is reported in Figure with respect to is suddenly (maybe to replace a finished batch).11. As shown in the The response the system to this change ingredientviscosities is reported in 11.The As shown the figure, the controller is of capable to maintain the of controlled toFigure target. dottedinorange line figure, controllerof is capable to maintain thechange controlled to target. The dotted orange line represents thetheresponse the process to the of viscosities ingredient B in an open loop configuration, represents the response of the process to the change of ingredient B in an open loop configuration, while the green dotted line represents the controlled process. In this case, also the behavior of the while the green dotted line represents the controlled process. In this case, also the behavior of the viscosity at high raterate is reported inorder order evaluate if controllability for the four viscosity at shear high shear is reportedininthe the figure, figure, in to to evaluate if controllability for the four − 1 states isstates satisfied. Indeed, the the viscosity atat1100 not controlled, nonetheless the difference is satisfied. Indeed, viscosity 1100ss−1 isisnot controlled, nonetheless the difference betweenbetween target and actual valuevalue is reduced indicating behavior proposed controller. target and actual is reduced indicatingaa correct correct behavior of of thethe proposed controller.
11. Performance of the controller responding responding totoa disturbance (change of ingredient B) and B) and Figure Figure 11. Performance of the controller a disturbance (change of ingredient comparison with the open loop response: (a) mass flow of ingredient B; (b) mass flow of ingredient comparison with the open loop response: (a) mass flow of ingredient B; (b) mass flow of ingredient D; −1; (d) response of the viscosity at shear rate D; (c) response of the viscosity at shear rate equal to − (c) response of the viscosity at shear rate equal to 0.1 s0.11 ;s(d) response of the viscosity at shear rate equal −1 to1 10 s−1; (f) response of the viscosity at equal to 1 s ; (e) response of the viscosity at shear rate equal − to 1 s−1 ; (e) response of the viscosity at shear rate equal to 10 s ; (f) response of the viscosity at shear rate shear rate equal to 1100 s−1. For (c–f) the following data are reported: target curve (red solid line), equal tomeasured 1100 s−1value . For (green (c–f) the following data are reported: target curve(orange (red solid line), dashed line), open loop evolution of the process dotted line).measured value (green dashed line), open loop evolution of the process (orange dotted line).
7. Conclusions
7. Conclusions The present paper was focused on the control of the rheological properties of a detergent which is produced by means of a continuous process. The control target was to maintain the product’s
The present paper was focused on the control of the rheological properties of a detergent which is produced by means of a continuous process. The control target was to maintain the product’s viscosity curve within a given region of the viscosity-vs-shear rate plane, and this issue was addressed selecting three points on the viscosity curve that were controlled manipulating two ingredient flow rates according to MPC algorithm. The paper is preparatory to the introduction of on-line innovative rheological sensors in the complex fluid production and showed that automatic control can effectively
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improve the quality of the product but it is important to reduce the time required by the sensor to compute the rheological curve. Acknowledgments: This work has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 636942. Author Contributions: Massimiliano Grosso conceived and designed the experiments; Francesc Corominas performed the experiments and contributed with reagents/materials/analysis tools; Stefania Tronci and Roberto Baratti analysed the data for modeling and control development purposes; Roberto Mei developed the simulator and controller. Stefania Tronci and Roberto Mei wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.
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