**Loop Transfer Recovery for the Grape Juice Concentration Process**

Nelson Aros Oñate1 and Graciela Suarez Segali2 *1Departamento de Ingeniería Eléctrica, Facultad de Ingeniería, Universidad de La Frontera, Temuco, 2Departamento de Ingeniería Química, Facultad de Ingeniería, Universidad Nacional de San Juan, San Juan, 1Chile* 

*2Argentina* 

## **1. Introduction**

280 Challenges and Paradigms in Applied Robust Control

Waller, J.B., Sågfors, M. & Waller, K.E. (1994). Ill-Conditionedness and Process

Zhou, K., Doyle, J.C. & Glover, K. (1996). *Robust and Optimal Control*. Prentice-Hall, Inc.,

*IFAC Symposium*, Kyoto, Japan, 465–470.

Upper Saddle River, New Jersey.

Directionality—The Use of Condition Numbers in Process Control. *Proceedings of* 

It is necessary to ensure the quality of concentrated, because it is highly used in the food industry in juices, drinks, sweets, etc. Its application is in full development because it can compete with any other constituent, it is a natural product, and considering that is a very important regional industry, marketing greatly affects the regional economy. Because of this, that it is extremely important to ensure quality and quantity concentrate.

Argentina is one of the principal producers and exporters of concentrated clear grape juices in the world. They are produced mainly in the provinces of San Juan and Mendoza (Argentine Republic) from virgin grape juice and in the major part from sulfited grape juices. The province of San Juan's legislation establishes that a portion of the grapes must be used for making concentrated clear grape juices. This product has reached a high level of penetration in the export market and constitutes an important and growing productive alternative.

An adequate manufacturing process, a correct design of the concentrate plants and an appropriate evaluation of their performance will facilitate optimization of the concentrated juices quality parameters. The plant efficiency is obtained from knowledge of the physics properties of the raw material and products. These properties are fundamental parameters that are used in the designing and calculations on all the equipment used and also in the control process.

The multi-step evaporation (M-SE) is the most important unit operation used in the food industry to concentrate juices of grapes and apples. Even when the main objective of this process is to produce a concentrated product, it should also possess certain organoleptic properties that are critical with respect to its quality and acceptance grade by the customers.

Product requirements and the complex characteristics of the process such as non-linear behavior, input and output constraints, time delays and loop interactions justify the use of an advanced control system.

The rheological behavior influences directly the heat transfer coefficient (Pilati, 1998; Rubio, 1998) and therefore its knowledge is essential together with the influence of temperature on

Loop Transfer Recovery for the Grape Juice Concentration Process 283

On the other hand, sensitivity theory, originally developed by Bode (Bode, 1945), has regained considerable importance, due to the recent work developed by many researchers. This research effort has made evident the fundamental role played by sensitivity theory to highlight design tradeoffs and to analyze, qualitatively, control system performance. One of the fields of active research is the analysis of different design strategies from to point of view of sensibility properties. From this perspective one of the richest strategies is the optimal linear quadratic regulator (LQR). It is well known that the sensitivity of a LQR Loop is always less than one (Anderson, 1971). However, it is also known that when the state is not directly fed back but reconstructed through an observer this property is normally lost. In fact, the situation more general, since the recovery problem appears every time a control

It has been shown that when the plant is minimum-phase, a properly design Kalman filter provides complete recovery of the input sensitivity achieved by LQR will full state feedback (Doyle, 1979). Either full or partial-order filter may be used. On the other hand it is also known that it is generally impossible to obtain LTR if we use observers for a plant with unstable zeros. An exception to this rule arises in MIMO Systems when input directions are

On the other hand, it is known that the only way to obtain full recovery for a general nonminimum-phase plant is to increase the number of independent measurements. This idea has been suggested in conjunction with the use of reduced-order Kalman filters (Friedland,

The additional independent measurements are used to modify the structure of the open loop transfer function. The standard LTR procedure is applied and it is the implemented combining the resulting full-order Kalman filter with the additional measurements optimally weighted. The idea is obviously to feed back only a subset of the state, for that reason we speak of 'partial' state feedback. The basic approach assumes that all states are available for measurement. However in this paper, it is also shown how to do ଶ optimization on the amount of recovery of the input sensitivity when a given set of measurements is available. This situation is important since in many additional situations there are limitations regarding which variables can be measured and how many additional sensors can be used. This connects the recovery theory with the issue of additional measurements raised in the context of practical ideas for control design, as illustrated in the control of the inverted pendulum; see (Middleton, 1990). The theory supporting the proposal is built on some import technical results which allow for computing the amount of

Figure 1 show the input and output streams in a vertical generic effect evaporator with long tubes. The solution to be concentrated circulates inside the tubes, while the steam, used to heat the solution, circulates inside the shell around the tubes. The evaporator operates in parallel mode. 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 vaporized solvent from each effect serve as heater in the next one. 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

design based on state feedback design is implemented through observers.

orthogonal to non-minimum phase zero directions (Zhang, 1990).

recovery, as a function of frequency (Zhang, 1990).

**2. Process description** 

1989).

its value. The juices (concentrate and intermediate products) physical properties, such as density, viscosity, boiling point elevation, specific heat and coefficient of thermal expansion, are affected by their solid content and their temperature. For this reason, it is necessary to know the physical properties values, as a function of the temperature and the solids content, during the manufacture process.

The principal solids constituents of clear grape juices are sugars and its concentration affects directly the density, viscosity and refraction index. Tables were developed to relate reducing sugar contents, refractometric values and density of pure solutions, at 20ºC, for concentrate ranges from 0% to 85% w/w and sucrose solutions for different range concentrations 0% to 70% and a temperature range from 0 to 100ºC (AOAC, 1995).

Barbieri (1980) worked with white concentrated clear grape juice in a falling film multiple effect evaporators. They obtained 18.2, 27.3, 38.6, 48.6 and 64.6ºBrix samples. They measured density, viscosity and boiling point elevation as a function of soluble solids concentration and temperature. They presented the results in plots with predictive equations for the properties which were studied.

Di Leo (1988) published density, refraction index and viscosity data for a rectified concentrated grape juice and an aqueous solution of a 1:1 glucose/levulose mixture, for a soluble solids concentrate range from 60 to 71% (in increments of 0.1%) and 20ºC.

Pandolfi, (1991) studied physical and chemical characteristics of grape juices produced in Mendoza and San Juan provinces, Argentina. They determined density at 20ºC in sulfited grape juices of 20–22ºBx and concentrated grape juices of 68–72 ºBx. They obtained no information on intermediate concentrations or other temperatures.

In general, the clarified juice concentrates have a Newtonian behavior (Ibarz 1993; Rao 1984; Saenz, 1986; Saravacos, 1970), although some authors have found a small pseudoplasticity in the flow of grape concentrates, from the variety Concord (Vitis labrusca) for concentrations above 55ºBx. It has been attributed to the presence of some soluble solids, mostly pectins and tartrates (Moressi, 1984; Saravacos, 1970). Other authors consider the juice concentrates as Newtonian, even at high soluble solids concentrations of 60–70ºBx (Barbieri, 1980; Di Leo, 1988; Rao, 1984; Schwartz, 1986).

If we analyze the temperature influence on this product's viscosity, it seems which is directly related with soluble solids concentration; the higher the concentration, the higher is the variation of the viscosity with temperature (Rao, 1984; Saravacos, 1970; Bayindirli, 1992; Crapiste, 1988; Constela, 1989).

Schwartz (1986) determined clear grape juice viscosity at 20, 30, 40 and 50ºC, for 30, 40, 50, 60 and 66% soluble solids concentration, but did not publish the experimental data. These authors presented the correlation constants values of the Arrhenius equation for temperature, a potential and an exponential model between viscosity and solids concentration for each temperature studied.

The physical property that represents density change in a material, due to an increase in its temperature at constant pressure, is called the coefficient of thermal expansion. The importance of this parameter can be seen in the effect that density change in the product can have over heat transfer during the process. There is not publish data on the coefficient of the thermal expansion for grape juices and their concentrates. The existing information did not cover all the temperature and concentration ranges that are used in the evaporation process, or else cover to pure sugar solutions, or grape juices of other varieties and/ or originating in other geographical zones.

282 Challenges and Paradigms in Applied Robust Control

its value. The juices (concentrate and intermediate products) physical properties, such as density, viscosity, boiling point elevation, specific heat and coefficient of thermal expansion, are affected by their solid content and their temperature. For this reason, it is necessary to know the physical properties values, as a function of the temperature and the solids content,

The principal solids constituents of clear grape juices are sugars and its concentration affects directly the density, viscosity and refraction index. Tables were developed to relate reducing sugar contents, refractometric values and density of pure solutions, at 20ºC, for concentrate ranges from 0% to 85% w/w and sucrose solutions for different range concentrations 0% to

Barbieri (1980) worked with white concentrated clear grape juice in a falling film multiple effect evaporators. They obtained 18.2, 27.3, 38.6, 48.6 and 64.6ºBrix samples. They measured density, viscosity and boiling point elevation as a function of soluble solids concentration and temperature. They presented the results in plots with predictive equations for the

Di Leo (1988) published density, refraction index and viscosity data for a rectified concentrated grape juice and an aqueous solution of a 1:1 glucose/levulose mixture, for a

Pandolfi, (1991) studied physical and chemical characteristics of grape juices produced in Mendoza and San Juan provinces, Argentina. They determined density at 20ºC in sulfited grape juices of 20–22ºBx and concentrated grape juices of 68–72 ºBx. They obtained no

In general, the clarified juice concentrates have a Newtonian behavior (Ibarz 1993; Rao 1984; Saenz, 1986; Saravacos, 1970), although some authors have found a small pseudoplasticity in the flow of grape concentrates, from the variety Concord (Vitis labrusca) for concentrations above 55ºBx. It has been attributed to the presence of some soluble solids, mostly pectins and tartrates (Moressi, 1984; Saravacos, 1970). Other authors consider the juice concentrates as Newtonian, even at high soluble solids concentrations of 60–70ºBx (Barbieri, 1980; Di Leo,

If we analyze the temperature influence on this product's viscosity, it seems which is directly related with soluble solids concentration; the higher the concentration, the higher is the variation of the viscosity with temperature (Rao, 1984; Saravacos, 1970; Bayindirli, 1992;

Schwartz (1986) determined clear grape juice viscosity at 20, 30, 40 and 50ºC, for 30, 40, 50, 60 and 66% soluble solids concentration, but did not publish the experimental data. These authors presented the correlation constants values of the Arrhenius equation for temperature, a potential and an exponential model between viscosity and solids

The physical property that represents density change in a material, due to an increase in its temperature at constant pressure, is called the coefficient of thermal expansion. The importance of this parameter can be seen in the effect that density change in the product can have over heat transfer during the process. There is not publish data on the coefficient of the thermal expansion for grape juices and their concentrates. The existing information did not cover all the temperature and concentration ranges that are used in the evaporation process, or else cover to pure sugar solutions, or grape juices of other varieties and/ or originating in

soluble solids concentrate range from 60 to 71% (in increments of 0.1%) and 20ºC.

information on intermediate concentrations or other temperatures.

during the manufacture process.

properties which were studied.

1988; Rao, 1984; Schwartz, 1986).

Crapiste, 1988; Constela, 1989).

other geographical zones.

concentration for each temperature studied.

70% and a temperature range from 0 to 100ºC (AOAC, 1995).

On the other hand, sensitivity theory, originally developed by Bode (Bode, 1945), has regained considerable importance, due to the recent work developed by many researchers. This research effort has made evident the fundamental role played by sensitivity theory to highlight design tradeoffs and to analyze, qualitatively, control system performance. One of the fields of active research is the analysis of different design strategies from to point of view of sensibility properties. From this perspective one of the richest strategies is the optimal linear quadratic regulator (LQR). It is well known that the sensitivity of a LQR Loop is always less than one (Anderson, 1971). However, it is also known that when the state is not directly fed back but reconstructed through an observer this property is normally lost. In fact, the situation more general, since the recovery problem appears every time a control design based on state feedback design is implemented through observers.

It has been shown that when the plant is minimum-phase, a properly design Kalman filter provides complete recovery of the input sensitivity achieved by LQR will full state feedback (Doyle, 1979). Either full or partial-order filter may be used. On the other hand it is also known that it is generally impossible to obtain LTR if we use observers for a plant with unstable zeros. An exception to this rule arises in MIMO Systems when input directions are orthogonal to non-minimum phase zero directions (Zhang, 1990).

On the other hand, it is known that the only way to obtain full recovery for a general nonminimum-phase plant is to increase the number of independent measurements. This idea has been suggested in conjunction with the use of reduced-order Kalman filters (Friedland, 1989).

The additional independent measurements are used to modify the structure of the open loop transfer function. The standard LTR procedure is applied and it is the implemented combining the resulting full-order Kalman filter with the additional measurements optimally weighted. The idea is obviously to feed back only a subset of the state, for that reason we speak of 'partial' state feedback. The basic approach assumes that all states are available for measurement. However in this paper, it is also shown how to do ଶ optimization on the amount of recovery of the input sensitivity when a given set of measurements is available. This situation is important since in many additional situations there are limitations regarding which variables can be measured and how many additional sensors can be used. This connects the recovery theory with the issue of additional measurements raised in the context of practical ideas for control design, as illustrated in the control of the inverted pendulum; see (Middleton, 1990). The theory supporting the proposal is built on some import technical results which allow for computing the amount of recovery, as a function of frequency (Zhang, 1990).

## **2. Process description**

Figure 1 show the input and output streams in a vertical generic effect evaporator with long tubes. The solution to be concentrated circulates inside the tubes, while the steam, used to heat the solution, circulates inside the shell around the tubes. The evaporator operates in parallel mode. 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 vaporized solvent from each effect serve as heater in the next one. 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

$$\frac{dM\_l}{dt} = \mathcal{W}\_{l-1} - \mathcal{W}\_{sl} - \mathcal{W}\_l \tag{1}$$

$$\frac{d(\mathcal{W}\_l X\_l)}{dt} = \mathcal{W}\_{l-1} X\_{l-1} - \mathcal{W}\_l X\_l \tag{2}$$

$$\frac{d(M\_l h\_l)}{dt} = W\_{l-1} h\_{l-1} - W\_l h\_l - W\_{sl} H\_{sl} + A\_l U\_l (T\_{sl-1} - T\_l) \tag{3}$$

$$\mathcal{W}\_{sl} = \frac{\mathcal{U}\_l A\_l (\mathbf{r}\_{sl-1} - \mathbf{r}\_l)}{H\_{sl-1} - h\_{cl}} \tag{4}$$

$$H\_{\rm sl} = 2509.2888 + 1.6747 \cdot T\_{\rm sl} \tag{5}$$

$$h\_{cl} = 4.1868 \cdot T\_{sl} \tag{6}$$

$$\mathcal{L}\_{pl} = 0.80839 - 4.3416 \cdot 10^{-3} \cdot X\_l + 5.6063 \cdot 10^{-4} \cdot T\_l \tag{7}$$

$$h\_l = 0.80839 \cdot T\_l - 4.3416 \cdot 10^{-3} \cdot X\_l T\_l + 2.80315 \cdot 10^{-4} \cdot T\_l^2 \tag{8}$$

$$U\_l = \frac{{}^{490}\!\!\!\!D^{0.87} \cdot \mathcal{W}^{3.6/L}\_{sl}}{\mu^{0.25}\_l \cdot \Delta T^{0.1}\_l} \tag{9}$$

$$
\mu\_l = \mu\_0 \cdot e^{\frac{\mathcal{A} \cdot \mathcal{X}\_l}{\mathbf{1}^{\otimes 0} - B \cdot \mathcal{X}\_l}} \tag{10}
$$

$$A = C\_1 + \frac{c\_2}{T\_\ell} \tag{11}$$

$$B = \mathcal{C}\_3 + \mathcal{C}\_4 \cdot T\_l \tag{12}$$

$$
\dot{\mathbf{x}} = \mathbf{A}\mathbf{x}(\mathbf{t}) + \mathbf{B}\mathbf{u}(\mathbf{t}) + \mathbf{v}(\mathbf{t}) \tag{13}
$$

$$\mathbf{y}(\mathbf{t}) = \mathbf{C}\mathbf{x}(\mathbf{t})\tag{14}$$

$$\mathbf{G(s)} = \mathbf{C(sI - A)^{-1}B} \tag{15}$$

$$\mathbf{u}(\mathbf{t}) = -F\mathbf{x}(\mathbf{t})\tag{16}$$

$$\mathcal{S}\_0(\mathbf{s}) = |I + H(\mathbf{s})|^{-1} \tag{17}$$

$$H(\mathbf{s}) = F(\mathbf{s}I - \mathcal{A})^{-1}\mathcal{B} \tag{18}$$

$$\mathbf{u}(\mathbf{t}) = -F\mathbf{\hat{x}}(\mathbf{t})\tag{19}$$

$$\hat{\mathbf{x}}(\mathbf{t}) = \mathbf{A}\hat{\mathbf{x}}(\mathbf{t}) + \mathbf{B}\mathbf{u}(\mathbf{t}) + L\{\mathbf{y}(\mathbf{t}) - \mathbf{C}\hat{\mathbf{x}}(\mathbf{t})\} \tag{20}$$

$$L = \sum \mathbf{C}^T \tag{21}$$

$$
\mathbf{A}\boldsymbol{\Sigma} + \boldsymbol{\Sigma}\mathbf{A}^T - \boldsymbol{\Sigma}\mathbf{C}^T\boldsymbol{C}\boldsymbol{\Sigma} + \mathbf{Q} = \mathbf{0} \tag{22}
$$

$$\mathbf{S}\_{obs}(\mathbf{s}) = \{I + [I + F(\mathbf{s}I - A + LC)^{-1}B]^{-1}F(\mathbf{s}I - A + LC)^{-1}LC(\mathbf{s}I - A)^{-1}B\}^{-1} \tag{23}$$

$$\mathcal{S}\_{obs}(\mathbf{s}) = \mathcal{S}\_o(\mathbf{s}) \{ I + F(\mathbf{s}I - \mathbf{A} + L\mathbf{C})^{-1} \mathbf{B} \} \tag{24}$$

$$\mathcal{S}\_{obs}(\mathbf{s}) = \mathcal{S}\_o(\mathbf{s}) \{ I + F\phi [B - L(sI - A + LC)^{-1} \mathcal{C}\phi B] \} \tag{25}$$

$$
\phi = (\mathbf{s}I - \mathcal{A})^{-1} \tag{26}
$$

$$E(\mathbf{s}) = \mathbf{S}\_o^{-1} \{ \mathbf{S}\_{obs} - \mathbf{S}\_o \} \tag{27}$$

$$E(\mathbf{s}) = F(\mathbf{s}I - \mathbf{A} + L\mathbf{C})^{-1}\mathbf{B} \tag{28}$$

$$\dot{\mathbf{x}} = \mathbf{A}\mathbf{x}(t) + \mathbf{B}\{\mathbf{u}(t) + \mathbf{U}\mathbf{w}(t)\} + \mathbf{v}(t) \tag{29}$$


$$\mathcal{S}\_{obs}(\mathbf{s}) \to \mathcal{S}\_o(\mathbf{s}) \tag{30}$$

$$\mathcal{G}(\mathbf{s}) = \mathcal{G}\_{\mathbf{m}}(\mathbf{s}) \mathcal{B}\_{\mathbf{z}}(\mathbf{s}) \tag{31}$$

$$\mathbf{G(s)} = \mathbf{C(sI - A)^{-1}B\_mB\_x(s)}\tag{32}$$

$$\mathbf{B}\_{\mathbf{z}}(\mathbf{s}) = \mathbf{B}\_{\mathbf{z}\_1}(\mathbf{s})\mathbf{B}\_{\mathbf{z}\_2}(\mathbf{s})\cdots\mathbf{B}\_{\mathbf{z}\_l}(\mathbf{s})\tag{33}$$

$$\mathcal{B}\_m = \mathcal{B}\_m^l \tag{34}$$

$$\mathcal{B}\_{\mathbf{z}\_l}(\mathbf{s}) = I - \frac{2\Re e(\mathbf{z}\_l)}{\mathbf{s} + \mathbf{z}\_l^\*} \eta\_l \eta\_l^H \tag{35}$$

$$B\_m^l = B\_m^{l-1} - 2\Re e\{\mathbf{z}\_l\} \xi\_l \eta\_l^H \tag{36}$$

$$
\begin{bmatrix}
\mathbf{z}\_l \mathbf{I} - \mathbf{A} & -\mathbf{B}\_m^{t-1} \\
\end{bmatrix}
\begin{bmatrix}
\xi\_l \\
\eta\_l
\end{bmatrix} = \mathbf{0} \tag{37}
$$

$$B\_m^k = \underbrace{\prod\_{l=1}^k \{I - 2\Re e(\mathbf{z}\_l)(\mathbf{z}\_l I - A)^{-1}\}}\_{\equiv \dot{M}\_k} \mathbf{B} \tag{38}$$

$$\xi\_k = (\mathbf{z}\_k I - A)^{-1} \mathbf{M}\_{k-1} \mathbf{B} \tag{39}$$

$$\mathbf{M\_0} = I \tag{40}$$

$$M\_1 = I - 2\Re e(\mathbf{z}\_1)(\mathbf{z}\_1 I - A)^{-1} \tag{41}$$

$$\mathcal{S}\_{obs}(\mathbf{s}) = \mathcal{S}\_o(\mathbf{s}) \{ I + E(\mathbf{s}) \} \tag{42}$$

$$E(\mathbf{s}) := F(\mathbf{s}I - \mathbf{A})^{-1} \{ \mathbf{B} - \mathbf{q} \mathbf{B}\_{\mathbf{m}} \mathbf{W} (I + \mathbf{q} \mathbf{C} \boldsymbol{\phi} \mathbf{B}\_{\mathbf{m}} \mathbf{W})^{-1} \mathbf{C} \boldsymbol{\phi} \mathbf{B}\_{\mathbf{m}} \mathbf{B}\_{\mathbf{z}}(\mathbf{s}) \} \tag{43}$$

$$E(\mathbf{s}) = F(\mathbf{s}I - \mathbf{A})^{-1} \{ \mathbf{B} - \mathbf{B}\_{\text{m}} \mathbf{B}\_{\text{z}}(\mathbf{s}) \} \tag{44}$$

$$E(\mathbf{s}) = E^l(\mathbf{s}) = \sum\_{k=1}^l \frac{2\Re e\langle \mathbf{z}\_k \rangle}{s + \mathbf{z}\_k^\*} F\xi\_k \eta\_k^H B\_\mathbf{z}^{k-1}(\mathbf{s}) \tag{45}$$

$$\lim\_{q \to \text{ss}} \mathbb{S}\_{obs}(\mathbf{s}) = \mathbb{S}\_o(\mathbf{s}) \left\{ \mathbf{1} + \frac{2x}{s \cdot \mathbf{z}} H(\mathbf{z}) \right\} \tag{46}$$

$$H(\mathbf{z}) = F(\mathbf{z}I - \mathcal{A})^{-1}\mathcal{B} \tag{47}$$

$$\mathcal{S}\_{\Gamma}(\mathbf{s}) = \mathcal{S}\_{o}(\mathbf{s}) \left\{ I + \sum\_{k=1}^{l} F(I - \Gamma) \xi\_{k} \eta\_{k}^{H} \mathcal{W}\_{k}(\mathbf{s}) \right\} \tag{53}$$

$$\text{The } \mathcal{L}\_{\text{int}} \text{ is the } \mathcal{L}\_{\text{int}} \text{-module.}$$

$$\mathcal{W}\_k(\mathbf{s}) = \frac{2\mathfrak{R}e(\mathbf{z}\_k)}{\mathbf{s} + \mathbf{z}\_k^\*} \mathcal{B}\_\mathbf{z}^{k-1}(\mathbf{s}) \tag{54}$$

$$\mathbf{B}\_{\mathbf{z}}^{k-1}(\mathbf{s}) = \prod\_{l=1}^{k} \mathbf{B}\_{\mathbf{z}\_l}(\mathbf{s}) \tag{55}$$

$$B\_{\mathbf{z}\_l}(\mathbf{s}) = \frac{-\mathbf{s} + \mathbf{z}\_l}{\mathbf{s} + \mathbf{z}\_l^\*} \tag{56}$$

$$\mathcal{S}\_{\Gamma}(\mathbf{s}) = \mathcal{S}\_{o}(\mathbf{s}) \left[ I + \sum\_{k=1}^{l} F(I - \Gamma) \xi\_{k} W\_{k}(\mathbf{s}) \right] \tag{57}$$


$$FT\zeta\_k = F\xi\_k \quad k = \mathbf{1}, \mathbf{2}, \cdots, l \tag{58}$$

$$
\begin{bmatrix}
\boldsymbol{\alpha}\_{1\_1} & \cdots & \boldsymbol{\alpha}\_{\lambda\_1} \\
\vdots & \ddots & \vdots \\
\boldsymbol{\alpha}\_{1\_l} & \cdots & \boldsymbol{\alpha}\_{\lambda\_l}
\end{bmatrix}
\begin{bmatrix}
\boldsymbol{\mathcal{V}}\_1 \\
\vdots \\
\boldsymbol{\mathcal{V}}\_\lambda
\end{bmatrix} = \begin{bmatrix}
\boldsymbol{\Sigma}\_{l=1}^n \boldsymbol{\alpha}\_{l\_1} \\
\vdots \\
\boldsymbol{\Sigma}\_{l=1}^n \boldsymbol{\alpha}\_{l\_l}
\end{bmatrix} \tag{59}
$$

$$\mathfrak{a}\_{\mathfrak{k}\_{\mathbf{k}}} = \mathfrak{f}\_{\mathfrak{k}} \xi\_{\mathbf{k}\_{\mathbf{k}}}; \qquad \mathfrak{i} = 1, 2, \dots, n \tag{60}$$

$$F = \begin{bmatrix} f\_1 & f\_2 & \cdots & f\_n \end{bmatrix} \tag{61}$$

$$\xi\_{\mathbf{k}} = [\xi\_{\mathbf{k}\_1} \quad \xi\_{\mathbf{k}\_2} \quad \cdots \quad \xi\_{\mathbf{k}\_n}]^{\mathsf{T}} \tag{62}$$

$$\lim\_{q \to \text{ss}} \mathbf{S}\_{\Gamma}(\mathbf{s}) = \mathbf{S}\_{o}(\mathbf{s}) \left\{ \mathbf{1} + \frac{2x}{s + \mathbf{z}^\*} H(\mathbf{z}) - \frac{2x}{s + \mathbf{z}^\*} H'(\mathbf{z}) \right\} \tag{63}$$

$$H'(\mathbf{z}) = F\Gamma(\mathbf{z}I - \mathbf{A})^{-1}\mathbf{B} \tag{64}$$

$$
\Gamma = \mathbf{y} \Gamma'\tag{65}
$$

$$
\Gamma' = \text{diag}\{\mathbf{1}, \mathbf{0}, \cdots, \mathbf{0}\} \tag{66}
$$

$$\mathbf{y} \,\, \mathbf{y} \,\, \, = \, \frac{\mathbf{F} (\mathbf{z}\mathbf{l} - \mathbf{A})^{-1} \mathbf{B}}{\mathbf{F} \mathbf{I}^{\prime} (\mathbf{z}\mathbf{l} - \mathbf{A})^{-1} \mathbf{B}} \,\, \tag{67}$$


$$\mathbf{y}^{\mathbf{o}} = \arg\min\_{\mathbf{y} \in \mathsf{R}e^{\lambda}} \mathbf{J}(\mathbf{y}) \tag{68}$$

$$\mathbf{J} = \int\_0^\infty |\mathbf{E}\_\Gamma(\mathbf{j}\mathbf{w})|^2 \,\mathbf{d}\mathbf{w} \tag{69}$$

$$\mathbf{J} = \int\_0^\infty \left\| \mathbf{G}\_1(\mathbf{j}\mathbf{w}) + \mathbf{y}^\mathbf{T} \mathbf{G}\_2(\mathbf{j}\mathbf{w}) \right\|^2 \,\mathrm{d}\mathbf{w} \tag{70}$$

$$G\_1(f\mathbf{w}) = F(f\mathbf{w}I - A + LC)^{-1}\mathbf{B} \tag{71}$$

$$\mathbf{y}^{\mathsf{T}} \mathbf{G}\_2(f\mathbf{w}) = F\Gamma(f\mathbf{w}I - A + LC)^{-1}\mathbf{B} \tag{72}$$

$$\mathbf{y}^{\mathbf{o}} = -\left[\int\_{0}^{\infty} \mathbf{G}\_{2}(\mathbf{j}\mathbf{w}) [\mathbf{G}\_{2}^{\*}(\mathbf{j}\mathbf{w})]^{\mathbf{T}} \mathbf{d}\mathbf{w}\right]^{-1} \Re\left\{\int\_{0}^{\infty} \mathbf{G}\_{1}(\mathbf{j}\mathbf{w}) \mathbf{G}\_{2}^{\*}(\mathbf{j}\mathbf{w}) \mathbf{d}\mathbf{w}\right\} \tag{73}$$

$$\mathbf{y}^{\mathbf{o}} = -\frac{\mathfrak{Re}\{\int\_{0}^{\infty} \mathbf{G}\_{1}(\|\mathbf{w}\|) \mathbf{G}\_{2}^{\*}(\|\mathbf{w}\|) \mathbf{dw}\}}{\int\_{0}^{\infty} \|\mathbf{G}\_{2}(\|\mathbf{w}\|) \|^{2} \mathbf{dw}}\tag{74}$$

Loop Transfer Recovery for the Grape Juice Concentration Process 295

Controlled system response for optimal regulator, whereas white noise disturbances, as well as step-like variation to the inlet concentration to 50, and then a step 100 is added to the feed

(a)

(b)

Fig. 8. Controlled system response to changes in the shocks in type of step and white noise

(blue for changes of +5% - green changes -5%)

**5.2 Close loop** 

temperature at the entrance.

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

Fig. 6. Behavior of the concentration in the evaporator to a change of a step in the concentration of power (increase of5% - decrease of 5%)

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

## **5.2 Close loop**

294 Challenges and Paradigms in Applied Robust Control

(a) (b)

(a) (b)

(a) (b)

Fig. 7. Behavior of the concentration in the evaporator to a change of a step in the

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

Fig. 6. Behavior of the concentration in the evaporator to a change of a step in the

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

temperature of steam power (up 5% - decrease of 5%)

concentration of power (increase of5% - decrease of 5%)

Controlled system response for optimal regulator, whereas white noise disturbances, as well as step-like variation to the inlet concentration to 50, and then a step 100 is added to the feed temperature at the entrance.

Fig. 8. Controlled system response to changes in the shocks in type of step and white noise (blue for changes of +5% - green changes -5%)

Loop Transfer Recovery for the Grape Juice Concentration Process 297

Controlled system response for optimal regulator, whereas white noise disturbances, as well as like step variation of 5% for the inlet concentration to 50 and then to 100 adds a step is 5%

(a)

(b)

Fig. 10. LQG controlled system response to changes in the type shocks of step and white

noise (blue for changes of +5% - green changes -5%)

**5.3 LQG- design** 

of the feed temperature at the entrance.

Fig. 9. Efforts to control the controlled system to changes in the shocks in type of step and white noise (blue for changes of +5% - green changes -5%)

## **5.3 LQG- design**

296 Challenges and Paradigms in Applied Robust Control

(a)

(b)

Fig. 9. Efforts to control the controlled system to changes in the shocks in type of step and

white noise (blue for changes of +5% - green changes -5%)

Controlled system response for optimal regulator, whereas white noise disturbances, as well as like step variation of 5% for the inlet concentration to 50 and then to 100 adds a step is 5% of the feed temperature at the entrance.

Fig. 10. LQG controlled system response to changes in the type shocks of step and white noise (blue for changes of +5% - green changes -5%)

Loop Transfer Recovery for the Grape Juice Concentration Process 299

(a)

(b)

Fig. 12. Controlled system response LQG / LTR to changes in the type shocks of step and

white noise (blue for changes of +5% - green changes -5%)

**5.4 LQG/LTR** 

Fig. 11. Control efforts for the LQG-controlled system to changes in the type shocks of step and white noise (blue for changes of +5% - green changes -5%)

## **5.4 LQG/LTR**

298 Challenges and Paradigms in Applied Robust Control

(a)

(b)

Fig. 11. Control efforts for the LQG-controlled system to changes in the type shocks of step

and white noise (blue for changes of +5% - green changes -5%)

(a)

Fig. 12. Controlled system response LQG / LTR to changes in the type shocks of step and white noise (blue for changes of +5% - green changes -5%)

Loop Transfer Recovery for the Grape Juice Concentration Process 301

The authors gratefully acknowledge the financial support of the "Universidad de La Frontera"- Chile DIUFRO DI07-0102, "Universidad Nacional de San Juan"- Argentina,

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(Bayindirli, 1992) Bayindirli, L. "Mathematical Analysis Of Variation Of Density And

(Bode, 1945) Bode H.W., *Network Analysis and Feedback Amplifier Design*. Van Nostrand, New York. (Camacho, 1999) Camacho E., Bordons F.C., *Model Predictive Control*. Springer-Verlag. (Crapiste, 1988) Crapiste, G.H. and Lozano, J.E. "Effect Of Concentration And Pressure On

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*básica de estadística vitivinícola argentina, Mendoza*. Varios números.

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Project FI-I900. They are also grateful for the cooperation of "Mostera Rio San Juan".

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**7. Acknowledgment** 

Engleewood Cliffs, N.J.

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*Vol. AC-24, April.* 

**8. References** 

Fig. 13. Efforts to control the controlled system LQG / LTR to changes in the type shocks of step and white noise (blue for changes of +5% - green changes -5%)

## **6. Conclusions**

Looking at the results presented in figures 4 to 7, show that it is appropriate to consider as manipulated variables steam temperature and feed rate of the solution to concentrate, and as measurable disturbance and characteristic of the system to the concentration the solution concentrated and the inlet temperature of food. You can check the analysis of these figures that the evaporation process presents a complex dynamic, high delay, coupling between the variables, high nonlinearities.

From the results shown in figures 8 to 13, on the behavior of the controlled system verifies that the design LQG/LTR has a better performance especially when control efforts are softer. Partly, it validates the robustness of the proposed control system, despite having analyzed only the rejection of disturbances, since these regulatory systems at the show a good response.

## **7. Acknowledgment**

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-I900. They are also grateful for the cooperation of "Mostera Rio San Juan".

## **8. References**

300 Challenges and Paradigms in Applied Robust Control

(a)

(b) Fig. 13. Efforts to control the controlled system LQG / LTR to changes in the type shocks of

Looking at the results presented in figures 4 to 7, show that it is appropriate to consider as manipulated variables steam temperature and feed rate of the solution to concentrate, and as measurable disturbance and characteristic of the system to the concentration the solution concentrated and the inlet temperature of food. You can check the analysis of these figures that the evaporation process presents a complex dynamic, high delay, coupling between the

From the results shown in figures 8 to 13, on the behavior of the controlled system verifies that the design LQG/LTR has a better performance especially when control efforts are softer. Partly, it validates the robustness of the proposed control system, despite having analyzed only the rejection of disturbances, since these regulatory systems at the show a good response.

step and white noise (blue for changes of +5% - green changes -5%)

**6. Conclusions** 

variables, high nonlinearities.


(Camacho, 1999) Camacho E., Bordons F.C., *Model Predictive Control*. Springer-Verlag.


**Part 4** 

**Power Plant and Power System Control** 

