**3. Phenomenological model**

Stefanov & Hoo (2003) have developed a rigorous model with distributed parameters based on partial differential equations for a falling-film evaporator, in which the open-loop stability of the model to disturbances is verified. On the other hand, various methods have been proposed in order to obtain reduced-order models to solve such problems (Christofides, 1998; El-Farra, Armaou and Christofides, 2002; Hoo and Zheng, 2001; Zheng and Hoo, 2002). However, the models are not a general framework yet, which assure an effective implementation of a control strategy in a multiple effect evaporator.

In practice, due to a lack of measurements to characterize the distributed nature of the process and actuators to implement such a solution, the control of systems represented by partial differential equation (PDE) in the grape juice evaporator, is carried out neglecting the spatial variation of parameters and applying lumped systems methods. However, a distributed parameters model must be developed in order to be used as a real plant to test advance control strategies by simulation.

In this work, it is used the mathematical model of the evaporator developed by Ortiz *et al.*  (2006), which is constituted by mass and energy balances in each effect. The assumptions are: the main variables in the gas phase have a very fast dynamical behavior, therefore the corresponding energy and mass balances are not considered. Heat losses to surroundings are neglected and the flow regime inside each effect is considered as completely mixed.

a. Global mass balances in each effect:

92 Frontiers of Model Predictive Control

The evaporator operates in co-current. The solution to be concentrated and the steam are fed to the first effect by the bottom and by the upper section of the shell, respectively. Later on, the concentrated solution from the first effect is pumped to the bottom of the second effect, and so on until the fourth effect. On the other hand, the vapor from each effect serves as heater in the next one. Finally, the solution leaving the fourth effect attains the desired

Each effect has a baffle in the upper section that serves as a drops splitter for the solution dragged by the vapor. The vapor from the fourth effect is sent to a condenser and leaves the process as a liquid. The concentrated solution coming from the fourth effect is sent to a

Fig. 2. Photo of evaporator and scheme of effect i in the four-stage evaporator flow sheet.

Stefanov & Hoo (2003) have developed a rigorous model with distributed parameters based on partial differential equations for a falling-film evaporator, in which the open-loop stability of the model to disturbances is verified. On the other hand, various methods have been proposed in order to obtain reduced-order models to solve such problems (Christofides, 1998; El-Farra, Armaou and Christofides, 2002; Hoo and Zheng, 2001; Zheng and Hoo, 2002). However, the models are not a general framework yet, which assure an

In practice, due to a lack of measurements to characterize the distributed nature of the process and actuators to implement such a solution, the control of systems represented by partial differential equation (PDE) in the grape juice evaporator, is carried out neglecting the spatial variation of parameters and applying lumped systems methods. However, a

effective implementation of a control strategy in a multiple effect evaporator.

concentration.

storage tank.

݅ ൌ ͳǡ ڮ ǡͶǤ

**3. Phenomenological model** 

$$\frac{d\mathcal{W}\_i}{dt} = \mathcal{W}\_{i-1} - \mathcal{W}\_{si} - \mathcal{W}\_i \tag{1}$$

in this equations *W i <sup>i</sup>* , 1,...,4 are the solution mass flow rates leaving the effects 1 to 4, respectively. *W*<sup>0</sup> is the input mass flow rate that is fed to the equipment. *W i si* , 1,...,4 are the vapor mass flow rates coming from effects 1 to 4, respectively. *dMi dt i* / , 1,...,4 represent the solution mass variation with the time for each effect.

b. Solute mass balances for each effect:

$$\frac{d\langle\mathcal{W}\_i X\_i\rangle}{dt} = \mathcal{W}\_{i-1} X\_{i-1} - \mathcal{W}\_i X\_i \tag{2}$$

where, *X i <sup>i</sup>* , 1,...,4 are the concentrations of the solutions that leave the effects 1 to 4, respectively. is the concentration of the fed solution. , *Xo*

c. Energy balances:

$$\frac{d\mathcal{V}\mathcal{V}\_i h\_i}{dt} = \mathcal{V}\_{i-1} h\_{i-1} - \mathcal{V}\_i h\_i - \mathcal{V}\_{si} H\_{si} + A\_i \mathcal{U}\_i (T\_{si-1} - T\_i) \tag{3}$$

where, *<sup>i</sup> h i*, 1,...,4 are the liquid stream enthalpies that leave the corresponding effects, h0 is the feed solution enthalpy, and *H i si* , 1,...,4 are the vapor stream enthalpies that leave the corresponding effects and, *Ai* represents the heat transfer area in each effect. The model also includes algeb raic equations. The vapor flow rates for each effect are calculated neglecting the following terms: energy accumulation and the heat conduction across the tubes. Therefore:

$$\mathcal{W}\_{si} = \frac{\mathcal{U}\_i A\_i (T\_{si-1} - T\_i)}{H\_{si-1} - h\_{ci}} \tag{4}$$

For each effect, the enthalpy can be estimated as a function of temperatures and concentrations (Perry, 1997). Them:

$$H\_{si} = 2509.2888 + 1.6747 \, T\_{si} \tag{5}$$

$$h\_{ci} = 4.1868 T\_{si} \tag{6}$$

Predictive Control for the Grape Juice Concentration Process 95

∆� � � ��� And the C polynomial is chosen to be 1, from what they if C-1 can be truncated it can be

The GPC algorithm consists of applying a sequence that minimizes a multistage cost

����� ��� ��� � �∑ �����������|�� � �������� � <sup>∑</sup> ��������� � � � ��� �� �

������|�� is a sequence of (j) best predictions from the output of the system later instantly t

��������� is a sequence control signal increases to come, to be obtained from the

N1, N2 and Nu are the minimum and maximum costing horizons, and control horizon. N1 and N2 That does not necessarily coincide with the maximum prediction horizon. The meaning of them is quite intuitive, they mark the limits of the moments that criminalizes the

δ(j) and λ(j) are weighting factors they are sequences are respectively weighted tracking errors and future control efforts. Usually considered constant values or exponential

Reference trajectory: one of the benefits of predictive control is that if you know a priori the future evolution of the reference, the system can start to react before the change is actually carried out, avoiding the effects of the delay in the response of the process. On the criterion of minimizing (Bitmead et al., 1990), most of the methods often used a trajectory of reference w(t+j) which does not necessarily coincide with the actual reference. Normally it would be a soft approach from the current value of the output y (t) to the known reference, through a

α is a parameter between 0 and 1 that constitutes an adjustable value that will influence the dynamic response of the system. where α = diag( α1, α2,. . . , αn) is the diagonal soften factor

(1-α) = diag(1- α1, 1- α2,….1- αn); r(t+j) is the system's future set point sequence. By employing this cost function, the distance between the model predictive output and the

���

���� (14)

������ � ��������� � �������� � �� (15)

<sup>∆</sup> ���������� (13)

���������� � ������������ � �

**4.1 Generalized predictive control** 

with

where:

absorbed into A and B.

function of the form

first-order dynamics.

where

matrix;

The CARIMA model of the process is given by:

��

and performed with the known data to instantly t.

discrepancy of the output with the reference.

sequences. These values can be used as tuning parameters.

minimization of the cost function.

$$C\_{pi} = 0.80839 - 4.3416 \cdot 10^{-3} X\_i + 5.6063 \cdot 10^{-4} T\_i \tag{7}$$

$$h\_i = 0.80839T\_i - 4.316 \cdot 10^{-3} X\_i T\_i + 2.80315 \cdot 10^{-4} T\_i^2 \tag{8}$$

*T i <sup>i</sup>* , 1,...,4 are the solution temperatures in each effect, and *Ts*<sup>0</sup> , is the vapor temperature that enters to the first effect. *T i si* , 1,...,4 are the vapor temperatures that leave each effect.

The heat transfer coefficients are:

$$\mu U\_i = \frac{490.D^{0.57} W\_{si}^{3.6 \,\mu\text{L}}}{\mu\_i^{0.25} \Delta T\_i^{0.1}} \tag{9}$$

Once viscosity values were established at different temperatures, (apparent) flow Activation Energy values for each studied concentration were calculated using the Arrhenius equation:

$$
\mu = \mu\_{\text{co}} \exp(-\frac{E\_a}{RT}) \tag{10}
$$

$$
\mu\_{\infty} = -\exp(a\_0 + a\_1 B \text{tr}\mathbf{x} + a\_2 B \text{tr}\mathbf{x}^2) \tag{11}
$$

$$\left. \begin{aligned} ^{E}a \end{aligned} \right|\_{T} = -\exp(a\_0 + a\_1 B \dot{r} \dot{x} + a\_2 B \dot{r} \dot{x}^2) \tag{12}$$

The global heat-transfer coefficients are directly influenced by the viscosity and indirectly by the temperature and concentration in each effect. The constants ܽ, ܽଵ y ܽଶ depend on the type of product to be concentrated (Kaya, 2002; Perry, 1997; Zuritz, 2005).

Although the model could be improved, the accuracy achieved is enough to incorporate a control structure.

### **4. Standard model predictive control**

The biggest problem that arises in the implementation of conventional PID controllers, arises when there are high nonlinearities and long delays, a possible solution to these arises with the implementation of predictive controllers, in which the entry in a given time (t) will generate an output at a time (t +1), using a control action at time t.

The model-based predictive control is currently presented as an attractive management tool for incorporating operational criteria through the use of an objective function and constraints for the calculation of control actions. Furthermore, these control strategies have reached a significant level of acceptability in practical applications of industrial process control.

The model-based predictive control is mainly based on the following elements:


#### **4.1 Generalized predictive control**

The CARIMA model of the process is given by:

$$A(\mathbf{z}^{-1})\mathbf{y}(\mathbf{t}) = B(\mathbf{z}^{-1})\mathbf{u}(\mathbf{t}-\mathbf{1}) + \frac{1}{\Delta}\mathcal{C}(\mathbf{z}^{-1})\mathbf{e}(\mathbf{t})\tag{13}$$

with

94 Frontiers of Model Predictive Control

*C XT pi i i*

*T i <sup>i</sup>* , 1,...,4 are the solution temperatures in each effect, and *Ts*<sup>0</sup> , is the vapor temperature that enters to the first effect. *T i si* , 1,...,4 are the vapor temperatures that leave each effect.

*i i*

Once viscosity values were established at different temperatures, (apparent) flow Activation Energy values for each studied concentration were calculated using the Arrhenius equation:

ߤൌߤஶሺെ ாೌ

The global heat-transfer coefficients are directly influenced by the viscosity and indirectly by the temperature and concentration in each effect. The constants ܽ, ܽଵ y ܽଶ depend on the type

Although the model could be improved, the accuracy achieved is enough to incorporate a

The biggest problem that arises in the implementation of conventional PID controllers, arises when there are high nonlinearities and long delays, a possible solution to these arises with the implementation of predictive controllers, in which the entry in a given time (t) will

The model-based predictive control is currently presented as an attractive management tool for incorporating operational criteria through the use of an objective function and constraints for the calculation of control actions. Furthermore, these control strategies have reached a

The use of a mathematical model of the process used to predict the future evolution of

 The establishment of a future desired trajectory, or reference to the controlled variables. The calculations of the manipulated variables optimizing a certain objective function or

significant level of acceptability in practical applications of industrial process control.

The model-based predictive control is mainly based on the following elements:

*T* 0.57 3.6 0.25 0.1

*D W <sup>U</sup>*

490. 

*i*

ܧ ܶ

**4. Standard model predictive control** 

of product to be concentrated (Kaya, 2002; Perry, 1997; Zuritz, 2005).

generate an output at a time (t +1), using a control action at time t.

the controlled variables over a prediction horizon.

The imposition of a structure in the future manipulated variables.

The application of control following a policy of moving horizon.

The heat transfer coefficients are:

control structure.

cost function.

3 4 0.80839 4.3416 10 5.6063 10 (7)

(9)

ߤஶ ൌ െሺܽ ܽଵݎܤ݅ݔ ܽଶݎܤ݅ݔ<sup>ଶ</sup> (11)

ൗ ൌ െሺܽ ܽଵݎܤ݅ݔ ܽଶݎܤ݅ݔ<sup>ଶ</sup> (12)

ோ்ሻ (10)

*<sup>i</sup> <sup>i</sup> i i <sup>i</sup> h T X T <sup>T</sup>* <sup>3</sup> 4 2 0.80839 4.316 10 2.80315 10 (8)

*JL si*

∆� � � ���

And the C polynomial is chosen to be 1, from what they if C-1 can be truncated it can be absorbed into A and B.

The GPC algorithm consists of applying a sequence that minimizes a multistage cost function of the form

$$f(N\_1, N\_2, N\_u) = \sum\_{j=N\_1}^{N\_2} \delta(j) [\S(t+j|t) - w(t+j)]^2 + \sum\_{j=1}^{N\_1} \lambda(j) [\Delta u(t+j-1)]^2 \tag{14}$$

where:

������|�� is a sequence of (j) best predictions from the output of the system later instantly t and performed with the known data to instantly t.

��������� is a sequence control signal increases to come, to be obtained from the minimization of the cost function.

N1, N2 and Nu are the minimum and maximum costing horizons, and control horizon. N1 and N2 That does not necessarily coincide with the maximum prediction horizon. The meaning of them is quite intuitive, they mark the limits of the moments that criminalizes the discrepancy of the output with the reference.

δ(j) and λ(j) are weighting factors they are sequences are respectively weighted tracking errors and future control efforts. Usually considered constant values or exponential sequences. These values can be used as tuning parameters.

Reference trajectory: one of the benefits of predictive control is that if you know a priori the future evolution of the reference, the system can start to react before the change is actually carried out, avoiding the effects of the delay in the response of the process. On the criterion of minimizing (Bitmead et al., 1990), most of the methods often used a trajectory of reference w(t+j) which does not necessarily coincide with the actual reference. Normally it would be a soft approach from the current value of the output y (t) to the known reference, through a first-order dynamics.

$$w(t+f) = aw(t+k-1) + (1-a)r(t+f)\tag{15}$$

where

α is a parameter between 0 and 1 that constitutes an adjustable value that will influence the dynamic response of the system. where α = diag( α1, α2,. . . , αn) is the diagonal soften factor matrix;

(1-α) = diag(1- α1, 1- α2,….1- αn); r(t+j) is the system's future set point sequence. By employing this cost function, the distance between the model predictive output and the

$$\mathbf{1} = E\_{\parallel}(\mathbf{z}^{-1})\check{A}(\mathbf{z}^{-1}) + \mathbf{z}^{-1}F\_{\parallel}(\mathbf{z}^{-1})\tag{16}$$

$$
\tilde{A}(\mathbf{z}^{-1})E\_j(\mathbf{z}^{-1})\mathbf{y}(t+j) = E\_j(\mathbf{z}^{-1})B(\mathbf{z}^{-1})\Delta u(t+j-d-1) + E\_j(\mathbf{z}^{-1})e(t+j) \tag{17}
$$

$$\left(1 - \mathbf{z}^{-1} F\_{\mathbf{f}}(\mathbf{z}^{-1})\right) \mathbf{y}(t+j) = E\_{\mathbf{f}}(\mathbf{z}^{-1}) B(\mathbf{z}^{-1}) \Delta u(t+j-d-1) + E\_{\mathbf{f}}(\mathbf{z}^{-1}) e(t+j)$$

$$\mathbf{y}(t+j) = F\_{\rangle}(\mathbf{z}^{-1})\mathbf{y}(t) + E\_{\rangle}(\mathbf{z}^{-1})B(\mathbf{z}^{-1})\Delta u(t+j-d-1) + E\_{\rangle}(\mathbf{z}^{-1})e(t+j) \tag{18}$$

$$\mathfrak{H}(t+|t) = G\_{\mathfrak{f}}(\mathbf{z}^{-1})\Delta u(t+j-d-1) + F\_{\mathfrak{f}}(\mathbf{z}^{-1})\mathbf{y}(t) \tag{19}$$

$$F\_{\mathbf{j}}(\mathbf{z}^{-1}) = F\_{\mathbf{j},0} + F\_{\mathbf{j},1}(\mathbf{z}^{-1}) + F\_{\mathbf{j},2}(\mathbf{z}^{-2}) + \dots + F\_{\mathbf{j},na}(\mathbf{z}^{-na})$$

$$E\_{\mathbf{j}}(\mathbf{z}^{-1}) = E\_{\mathbf{j},0} + E\_{\mathbf{j},1}(\mathbf{z}^{-1}) + E\_{\mathbf{j},2}(\mathbf{z}^{-2}) + \dots + E\_{\mathbf{j},l-1}(\mathbf{z}^{-(j-1)})$$

$$G\_{\mathbf{j}}(\mathbf{z}^{-1}) = G\_{\mathbf{j},0} + G\_{\mathbf{j},1}(\mathbf{z}^{-1}) + G\_{\mathbf{j},2}(\mathbf{z}^{-2}) + \dots + G\_{\mathbf{j},l-1}(\mathbf{z}^{-(j-1)})$$

$$F\_{j+1}(\mathbf{z}^{-1}) = F\_{j+1,0} + F\_{j+1,1}(\mathbf{z}^{-1}) + F\_{j+1,2}(\mathbf{z}^{-2}) + \dots + F\_{j+1,na}(\mathbf{z}^{-na})$$

$$E\_{j+1}(\mathbf{z}^{-1}) = E\_{j}(\mathbf{z}^{-1}) + E\_{j+1,j}(\mathbf{z}^{-j})$$

$$G\_{j+1}(\mathbf{z}^{-1}) = G\_{j}(\mathbf{z}^{-1}) + F\_{j,0}(\mathbf{z}^{-j})B$$

$$\begin{aligned} \mathbf{N\_2} &= \mathbf{d} + \mathbf{N} \\\\ \mathbf{N\_u} &= \mathbf{N} \end{aligned}$$

$$\mathbf{y} = G\mathbf{u} + F(\mathbf{z}^{-1}) + G'(\mathbf{z}^{-1})\Delta\mathbf{u}(t-1) \tag{20}$$

$$\mathbf{y} = \begin{bmatrix} \mathbf{\hat{y}}(t+d+1|t) \\ \mathbf{\hat{y}}(t+d+2|t) \\ \vdots \\ \mathbf{\hat{y}}(t+d+N|t) \end{bmatrix} \mathbf{G} = \begin{bmatrix} G\_0 & 0 & \dots & 0 \\ G\_1 & G\_0 & \dots & 0 \\ & \vdots \\ G\_{N-1} & G\_{N-2} & G\_0 \end{bmatrix} \mathbf{u} = \begin{bmatrix} \Delta\mathbf{u}(t) \\ \Delta\mathbf{u}(t+1) \\ \vdots \\ \Delta\mathbf{u}(t+N-1) \end{bmatrix}$$

$$F(\mathbf{z}^{-1}) = \begin{bmatrix} F\_{d+1}(\mathbf{z}^{-1}) \\ F\_{d+2}(\mathbf{z}^{-1}) \\ \vdots \\ F\_{d+N}(\mathbf{z}^{-1}) \end{bmatrix} \qquad G'(\mathbf{z}^{-1}) = \begin{bmatrix} (G\_{d+1}(\mathbf{z}^{-1}) - G\_0)\mathbf{z} \\ (G\_{d+2}(\mathbf{z}^{-1}) - G\_0 - G\_1\mathbf{z}^{-1})\mathbf{z}^2 \\ \vdots \\ (G\_{d+N}(\mathbf{z}^{-1}) - G\_0 - G\_1\mathbf{z}^{-1} - \dots - G\_{N-2}\mathbf{z}^{-(N-1)})\mathbf{z}^N \end{bmatrix}$$

$$J = (Gu + f - \mathbf{w})^T (Gu + f - \mathbf{w}) + \lambda u^T u \tag{21}$$

$$f = G'(z^{-1})\Delta u(t-1)$$

$$\mathcal{W} = [\mathcal{W}(t+d+1)\mathcal{W}(t+d+2)\dots\mathcal{W}(t+d+N)]^T$$

$$J = \frac{1}{2}u^T H u + b^T u + f\_0$$

$$H = 2(G^T G + \lambda I)$$

$$b^T = 2(f - w)^T G$$

$$f\_0 = (f - w)^T (f - w)$$

$$A(\mathbf{z}^{-1})\mathbf{y}(t) = B(\mathbf{z}^{-1})\mathbf{u}(t-1) + D(\mathbf{z}^{-1})\mathbf{v}(t) + \frac{1}{\Delta}C(\mathbf{z}^{-1})e(t) \tag{22}$$

$$D(\mathbf{z}^{-1}) = d\_0 + d\_1 \mathbf{z}^{-1} + d\_2 \mathbf{z}^{-2} + \cdots + d\_{nd} \mathbf{z}^{-nd}$$

$$\begin{aligned} E\_{\not\!\!/} (\mathbf{z}^{-1}) \check{A} (\mathbf{z}^{-1}) \mathbf{y} (t+j) &= E\_{\not\!\!/} (\mathbf{z}^{-1}) B (\mathbf{z}^{-1}) \Delta u (t+j-1) \\ &+ E\_{\not\!\!/} (\mathbf{z}^{-1}) D (\mathbf{z}^{-1}) \Delta v (t+j) + E\_{\not\!\!/} (\mathbf{z}^{-1}) e (t+j) \end{aligned}$$

$$\begin{aligned} \chi(t+j) &= F\_{\mathfrak{f}}(\mathbf{z}^{-1})\chi(t) + E\_{\mathfrak{f}}(\mathbf{z}^{-1})B(\mathbf{z}^{-1})\Delta u(t+j-1) \\ &+ E\_{\mathfrak{f}}(\mathbf{z}^{-1})D(\mathbf{z}^{-1})\Delta(t+j) + E\_{\mathfrak{f}}(\mathbf{z}^{-1})e(t+j) \end{aligned}$$

$$\begin{aligned} \mathfrak{z}(t+j|t) &= E[\mathfrak{y}(t+j)] \\ = F\_{\mathfrak{f}}(\mathbf{z}^{-1})\mathfrak{y}(t) + E\_{\mathfrak{f}}(\mathbf{z}^{-1})B(\mathbf{z}^{-1})\Delta u(t+j-1) + E\_{\mathfrak{f}}(\mathbf{z}^{-1})D(\mathbf{z}^{-1})\Delta v(t+j) \end{aligned}$$

$$\mathbf{j(t+j|t)} = G\_{\mathbf{j}}(\mathbf{z}^{-1})\Delta u(\mathbf{t}+\mathbf{j}-\mathbf{1}) + H\_{\mathbf{j}}(\mathbf{z}^{-1})\Delta v(\mathbf{t}+\mathbf{j}) + G\_{\mathbf{j}}'(\mathbf{z}^{-1})\Delta u(\mathbf{t}-\mathbf{1}) + H\_{\mathbf{j}}'(\mathbf{z}^{-1})\Delta v(\mathbf{t})$$

$$+F\_{\mathbf{i}}(\mathbf{z}^{-1})\mathbf{y}(\mathbf{t})\tag{23}$$

$$\mathfrak{J}(t+j|t) = G\_{\mathfrak{f}}(\mathbf{z}^{-1})\Delta u(t+j-1) + H\_{\mathfrak{f}}(\mathbf{z}^{-1})\Delta v(t+j) + f\_{\mathfrak{f}},$$

$$f\_{\mathfrak{f}} = G\_{\mathfrak{f}}'(\mathbf{z}^{-1})\Delta u(t-1) + H\_{\mathfrak{f}}'(\mathbf{z}^{-1})\Delta v(t) + F\_{\mathfrak{f}}(\mathbf{z}^{-1})\mathbf{y}(t)$$

$$\mathfrak{H}(t+N|t) = G\_N(\mathbf{z}^{-1})\Delta u(t+N-1) + H\_{\mathfrak{f}}(\mathbf{z}^{-1})\Delta v(t+N) + f\_N$$

Predictive Control for the Grape Juice Concentration Process 101

(c) (d)

(a) (b)

 (c) (d) Fig. 6. Behavior of the temperature in the evaporator to a change of a step in the temperature

In the following figures shows the response of the open loop system, when making a step in one of the disturbance variables such as in feed concentration is one measurable disturbances; in the figure 7 is represented the concentration of output in each of the effects

of the steam supply (increase of 5% - decrease of 5% ).

and figure 8 is represented the temperature in each of the effects.

Fig. 5. Behavior of the concentration in the evaporator to a change of a step in the

temperature of the steam supply (increase of 5% - decrease of 5%).

Fig. 4. Behavior of the temperature in the evaporator to a change of a step in the flow of food (increase of 5% - decrease of 5%)

In the following figures shows the response of the open loop system, when making a disturbance in one of the manipulated variables such as steam temperature is the other manipulated variable; in the figure 5 is represented the concentration of output in each of the effects and figure 6 is represented the temperature in each of the effects.

100 Frontiers of Model Predictive Control

(a) (b)

 (c) (d) Fig. 4. Behavior of the temperature in the evaporator to a change of a step in the flow of food

In the following figures shows the response of the open loop system, when making a disturbance in one of the manipulated variables such as steam temperature is the other manipulated variable; in the figure 5 is represented the concentration of output in each of

(a) (b)

the effects and figure 6 is represented the temperature in each of the effects.

(increase of 5% - decrease of 5%)

Fig. 5. Behavior of the concentration in the evaporator to a change of a step in the temperature of the steam supply (increase of 5% - decrease of 5%).

Fig. 6. Behavior of the temperature in the evaporator to a change of a step in the temperature of the steam supply (increase of 5% - decrease of 5% ).

In the following figures shows the response of the open loop system, when making a step in one of the disturbance variables such as in feed concentration is one measurable disturbances; in the figure 7 is represented the concentration of output in each of the effects and figure 8 is represented the temperature in each of the effects.

Predictive Control for the Grape Juice Concentration Process 103

 (c) (d) Fig. 8. Behavior of the temperature in the evaporator to a step change in feed concentration

In the figures now shows the response of the open loop system, when making a disturbance in one of the disturbance variables such as in temperature of the input solution is the other measurable disturbances; in the figure 9 is represented the concentration of output in each of

(a) (b)

 (c) (d) Fig. 9. Behavior of the concentration in the evaporator to a change of a step in the temperature

of the input solution (increase of 5% - decrease of 5% ).

the effects and figure 10 is represented the temperature in each of the effects.

(increase of 5% - decrease of 5%).

Fig. 7. Behavior of the concentration in the evaporator to a step change in feed concentration (increase of 5% - decrease of 5%).

102 Frontiers of Model Predictive Control

(a) (b)

 (c) (d) Fig. 7. Behavior of the concentration in the evaporator to a step change in feed concentration

(a) (b)

(increase of 5% - decrease of 5%).

Fig. 8. Behavior of the temperature in the evaporator to a step change in feed concentration (increase of 5% - decrease of 5%).

In the figures now shows the response of the open loop system, when making a disturbance in one of the disturbance variables such as in temperature of the input solution is the other measurable disturbances; in the figure 9 is represented the concentration of output in each of the effects and figure 10 is represented the temperature in each of the effects.

Fig. 9. Behavior of the concentration in the evaporator to a change of a step in the temperature of the input solution (increase of 5% - decrease of 5% ).

Predictive Control for the Grape Juice Concentration Process 105

In analyzing the results obtained by performing perturbations in each of the four variables that enter the equipment, is considered appropriate the choice of manipulated variables chosen as the income flow of the solution to concentrate (grape juice) and the steam temperature and as measurable disturbances to the feed concentration and temperature that enters the solution concentration, this conclusion after observing emanates figures 3 to 10. We can also observe that the process of concentration has a complex dynamic, with long delays, high nonlinearity, coupling between variables, added to the reactions of

From the results shown in Figures 11 and 12 on the behavior of the controlled system verifies that the design of GPC has performed well since the variations in the controlled variable are smoother. As well as you can see the robustness of the proposed controller.

The authors gratefully acknowledge the financial support of the "Universidad de La Frontera"- Chile DIUFRO DI07-0102, "Universidad Nacional de San Juan"- Argentina, Project FI-I1018. They are also grateful for the cooperation of "Mostera Rio San Juan".

(Albertos, 1989) Albertos, P. and Ortega, R. "*On Generalized Predictive Control: Two Alternative* 

(Allidina, 1980) Allidina A. Y. and Hughes, F. M. "*Generalized Self-tuning Controller with Pole* 

(Armaou, 2002) Armaou A., Christofides P.D., "Dynamic Optimization of Dissipative PDE

Systems Using Nonlinear Order Reduction". Chemical Engineering Science 57 - 24,

deterioration of the organoleptic properties of the solution to concentrate

*Formulation"s.* Automatica, 25 (5): 753-755

*Assignment"*. Proccedings IEE, Part D, 127: 13-18.

Fig. 12. Behavior of the temperature in the first effect

**6. Conclusions** 

**7. Acknowledgments** 

pp. 5083-5114.

**8. References** 

Fig. 10. Behavior of the temperature in the evaporator to a change of a step in the temperature of the input solution (increase of 5% - decrease of 5%).

## **5.2 Close loop**

The following figures show the response of GPC controller, when conducted disturbances on the manipulated variables, ie giving an overview of the steam temperature and feed flow, one step at time 5 hours on the steam temperature and an increase to 10 hours in the feed stream**.** 

Fig. 11. Behavior of the final product concentration at the outlet of the fourth effect

Fig. 12. Behavior of the temperature in the first effect
