2.1 Stochastic response surface models

The response surface methodology (MSR) is a collection of mathematical and statistical techniques that allows modeling, analyzing, and optimizing problems whose response variables are influenced by many variables [2]. As mentioned earlier, there is great difficulty in knowing the behavior of independent and dependent variables in a process. Thus, the response surface allows the real approximation of the process from a quadratic model. The development through a Taylor polynomial, truncated in the quadratic term, takes what we call a second-order response surface:

$$Y(\mathbf{x}) = \beta\_0 + \sum\_{i=1}^{k} \beta\_i \mathbf{x}\_i + \sum\_{i=1}^{k} \beta\_{ii} \mathbf{x}\_i^2 + \sum\_{i$$

where β represents the coefficients of the model, k is the number of independent variables considered in the study, and ε is the error term.

The fact of using the response surface in a region close to high curvature of the model, presented according to local or global maxima or minima, according to convexity, does not effectively determine the best points or operation setups. However, what can be verified is a region of space that, depending on the levels of each of the independent variables, leads to better responses.

From the color gradient shown in Figure 4, it is possible to verify regions, delimited through the Cartesian axes representing the levels of each of the factors studied, leading to better responses. Thus, the construction of models through the surface response method becomes paramount for the application of later optimization algorithms. Among several optimization algorithms, the Normal Boundary Intersection (NBI) [3] has been used in several researches, in several different fields.

#### 2.2 NBI algorithm

parameter in an effluent treatment plant, for example. The polynomial (Figure 3), which represents the Pareto frontier, can be used as a transfer function in scaling of

In order to facilitate the understanding of the possibility of implementing the NBI algorithm in controllers, let us take, for example, an industrial effluent treatment plant, which operates with a certain constant flow, due to the

residence time necessary for part of the organic load to be degraded via bacteria and protozoa in an aerobic process. As a base of the input variables, we will work with initial organic load in terms of biochemical oxygen demand (BOD) and pH. As controllable factors, we will use the air or oxygen flow (aeration) and residence time. As desired responses, we will use as an illustration the removal of the organic load in terms of biochemical oxygen demand (BOD) and chemical

Modeling a typical problem processes, we could write that both responses have a direct relationship with the two factors presented as X1 and X2. However, keeping

the process steady relative to the inputs becomes virtually impossible. By establishing, the two responses used, as an illustration of the application of the method, is it feasible to predict the aeration rate and residence time required. Certainly, the answer would be positive, if the entries were kept constant. However, if this standardization is not possible, how can we keep the responses within desirable patterns? Imagine in a situation of actual biological treatment process, where some changes can lead to periodic changes in the conditions of entry. For example, an increase in the rainfall rate may lead to the dilution of the organic matter present in the tributary and consequently the decrease of the initial BOD. The decrease of the initial BOD requires a lower concentration of dissolved oxygen so that bacteria and protozoa can decompose the organic matter in order to meet the exit standards, which would lead to the conclusion of shorter residence times required. There is a relationship as presented that can be considered a certainty. However, what is the relationship between them? What would be the best condition, to decrease aeration or increase residence time? These responses can only be met if we have this problem modeled. When working with models, we can easily predict the relationship of each of the factors to the expected response. This fact helps us reduce process costs and increase effectiveness in the targeted response. Through the use of models created from response surfaces, which have quadratic models, it is easily possible to deter-

a possible dynamic process controller for multiobjectives.

2. Description of the process NBI control

mine local or global minimum or maximum points.

oxygen demand (COD).

10

Figure 3.

Pareto border for bi-objective problem.

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The NBI algorithm is developed in terms of an array that we call the payoff matrix Φ , which represents the optimal values of the multiple objective functions

Figure 4. Counter graphic.

minimized individually. The solution vector that minimizes the i-th objective function individually fi ð Þ <sup>x</sup> is represented by <sup>x</sup><sup>∗</sup> <sup>i</sup> so that the minimum value of fi ð Þ x at this point is f <sup>∗</sup> <sup>i</sup> x<sup>∗</sup> i � �. When replacing the individual optimum point x<sup>∗</sup> <sup>i</sup> obtained in the optimization of objective function in the other functions, we have fi x<sup>∗</sup> i � � which is therefore a nonoptimal value of this function. By repeating this algorithm for all functions, we can represent the payoff matrix as

$$
\Phi = \begin{bmatrix}
f\_1^\*\left(\mathbf{x}\_1^\*\right) & \cdots & f\_1\left(\mathbf{x}\_i^\*\right) & \cdots & f\_1\left(\mathbf{x}\_m^\*\right) \\
\vdots & \ddots & & \vdots \\
f\_i\left(\mathbf{x}\_1^\*\right) & \cdots & f\_i^\*\left(\mathbf{x}\_i^\*\right) & \cdots & f\_i^\*\left(\mathbf{x}\_m^\*\right) \\
\vdots & & \ddots & \vdots \\
f\_m\left(\mathbf{x}\_1^\*\right) & \cdots & f\_m\left(\mathbf{x}\_i^\*\right) & \cdots & f\_m^\*\left(\mathbf{x}\_m^\*\right) \\
\end{bmatrix} \tag{2}
$$

Each line of Φ is composed of minimum and maximum values of fi ð Þ x . In the NBI method, these values can be used to normalize the objective functions, especially when they are represented by different scales or units. In a similar way, writing the set of individual optimums in a vector, we have

$$f^U = \begin{bmatrix} f\_1^\* \begin(\mathbf{x}\_1^\*\right) & \dots \ f\_i^\* \begin(\mathbf{x}\_i^\*\right) & \dots \ f\_m^\* \begin(\mathbf{x}\_m^\*\right) \end{bmatrix}^T \tag{3}$$

This vector is called utopia point. In the same way, by grouping the maximum (nonoptimal) values of each objective function, we have

$$f^N = \begin{bmatrix} f\_1^N & \dots & f\_i^N & \dots & f\_m^N \end{bmatrix}^T \tag{4}$$

This vector is called nadir points.

Using these two sets of extreme points, the normalization of the objective functions can be obtained as

$$\overline{f}(\mathbf{x}) = \frac{f\_i(\mathbf{x}) - f\_i^U}{f\_i^N - f\_i^U}, i = \mathbf{1}, \dots, m \tag{5}$$

2.3 Implementation of the NBI control system

smaller scale.

13

Figure 6.

Figure 5.

General scheme of the process.

Normal to intersect method (NBI).

experiments in laboratory scale:

For the process described as an example, there are two controllable factors represented by the aeration rate (x1) and residence time (x2). However, according to Figure 6, there are also two input variables that cannot be measured, mainly due to the instability of a biological treatment plant, according to initial organic charge z1 and pH z2. The first artifice presented will be the transformation of each of these variables into known values, from experiments carried out on a

Normal Boundary Intersection Applied to Controllers in Environmental Controls

DOI: http://dx.doi.org/10.5772/intechopen.83662

Thus, we will have the following factors: aeration rate (x1), residence time (x2), initial organic load (x3), and pH (x4). From a surface of response called central composite design (CCD), it is possible to construct a quadratic model, executing 31

This normalization therefore leads to the normalization of the payoff matrix, Φ. The convex combinations of each line of the payoff matrix, Φ, form the "convex hull of individual minima" (CHIM) or the utopia line.

Figure 5 illustrates the main elements associated with multiobjective optimization. The anchor points represent the individual solutions of two functions. Points a and b are calculated from the stepped payoff matrix, Φ wi. Considering a set of convex values for the weights, w, one has to Φ wi represent a point on the utopia line, making n^ denote a unit vector normal to the line at point's utopia Φ wi in the direction of origin; at the time, Φ w þ D n^ with D ∈R will represent the set of points in that normal.

The point of intersection of this normal with the boundary of the viable region that is closest to the origin will correspond to the maximization of the distance between the utopia line and the Pareto border. Thus, the NBI method can be written as a constrained nonlinear programming problem such that

$$\begin{array}{ll}\textbf{Max} & D\\ \textbf{s}(\textbf{x};\textbf{t}) & \\\\ \textbf{subject to} & \overline{\Phi}w + D\hat{n} = \overline{F}(\textbf{x})\\ & \textbf{x} \in \Omega \end{array} \tag{6}$$

Normal Boundary Intersection Applied to Controllers in Environmental Controls DOI: http://dx.doi.org/10.5772/intechopen.83662

Figure 5. Normal to intersect method (NBI).

minimized individually. The solution vector that minimizes the i-th objective func-

� �. When replacing the individual optimum point x<sup>∗</sup>

i

⋮ ⋱ ⋮

<sup>i</sup> x<sup>∗</sup> i � � ⋯ f <sup>∗</sup>

⋮ ⋱ ⋮

<sup>i</sup> x<sup>∗</sup> i � � …; f <sup>∗</sup>

> N <sup>i</sup> …, f

Using these two sets of extreme points, the normalization of the objective func-

This normalization therefore leads to the normalization of the payoff matrix, Φ. The convex combinations of each line of the payoff matrix, Φ, form the "convex

Figure 5 illustrates the main elements associated with multiobjective optimization. The anchor points represent the individual solutions of two functions. Points a and b are calculated from the stepped payoff matrix, Φ wi. Considering a set of convex values for the weights, w, one has to Φ wi represent a point on the utopia line, making n^ denote a unit vector normal to the line at point's utopia Φ wi in the direction of origin; at the time, Φ w þ D n^ with D ∈R will represent the set of

The point of intersection of this normal with the boundary of the viable region that is closest to the origin will correspond to the maximization of the distance between the utopia line and the Pareto border. Thus, the NBI method can be written

> subject to : Φw þ Dn^ ¼ Fð Þ x x∈ Ω

ð Þ� x f U i

f N <sup>i</sup> � f U i

This vector is called utopia point. In the same way, by grouping the maximum

i � � ⋯ f <sup>∗</sup>

� � <sup>⋯</sup> <sup>f</sup> <sup>1</sup> <sup>x</sup><sup>∗</sup>

is therefore a nonoptimal value of this function. By repeating this algorithm for all

the optimization of objective function in the other functions, we have fi x<sup>∗</sup>

� � <sup>⋯</sup> <sup>f</sup> <sup>1</sup> <sup>x</sup><sup>∗</sup>

� � <sup>⋯</sup> <sup>f</sup> <sup>m</sup> <sup>x</sup><sup>∗</sup>

Each line of Φ is composed of minimum and maximum values of fi

NBI method, these values can be used to normalize the objective functions, especially when they are represented by different scales or units. In a similar way,

� � ⋯ f <sup>∗</sup>

<sup>i</sup> so that the minimum value of fi

m � �

<sup>i</sup> x<sup>∗</sup> m � �

<sup>m</sup> x<sup>∗</sup> m � �

<sup>m</sup> x<sup>∗</sup> m � � � � <sup>T</sup> (3)

N m � �<sup>T</sup> (4)

, i ¼ 1, …, m (5)

ð Þ x at

(2)

(6)

ð Þ x . In the

<sup>i</sup> obtained in

i � � which

ð Þ <sup>x</sup> is represented by <sup>x</sup><sup>∗</sup>

functions, we can represent the payoff matrix as

f ∗ <sup>1</sup> x<sup>∗</sup> 1

fi x<sup>∗</sup> 1

f <sup>m</sup> x<sup>∗</sup> 1

writing the set of individual optimums in a vector, we have

(nonoptimal) values of each objective function, we have

f <sup>N</sup> <sup>¼</sup> <sup>f</sup> N <sup>1</sup> …, f

hull of individual minima" (CHIM) or the utopia line.

as a constrained nonlinear programming problem such that

Max

ð Þ <sup>x</sup>;<sup>t</sup> <sup>D</sup>

<sup>1</sup> x<sup>∗</sup> 1 � � …; f <sup>∗</sup>

f xð Þ¼ fi

Φ ¼

f <sup>U</sup> <sup>¼</sup> <sup>f</sup> <sup>∗</sup>

This vector is called nadir points.

tions can be obtained as

points in that normal.

12

tion individually fi

<sup>i</sup> x<sup>∗</sup> i

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this point is f <sup>∗</sup>

Figure 6. General scheme of the process.

#### 2.3 Implementation of the NBI control system

For the process described as an example, there are two controllable factors represented by the aeration rate (x1) and residence time (x2). However, according to Figure 6, there are also two input variables that cannot be measured, mainly due to the instability of a biological treatment plant, according to initial organic charge z1 and pH z2. The first artifice presented will be the transformation of each of these variables into known values, from experiments carried out on a smaller scale.

Thus, we will have the following factors: aeration rate (x1), residence time (x2), initial organic load (x3), and pH (x4). From a surface of response called central composite design (CCD), it is possible to construct a quadratic model, executing 31 experiments in laboratory scale:

$$\begin{aligned} Y1\_{\mathbf{x}} &= \beta\_0 + \beta\_1 \mathbf{x}\_1 + \beta\_2 \mathbf{x}\_2 + \beta\_3 \mathbf{x}\_3 + \beta\_4 \mathbf{x}\_4 + \beta\_{11} \mathbf{x}\_1^2 + \beta\_{22} \mathbf{x}\_2^2 + \beta\_{33} \mathbf{x}\_3^2 \\ &+ \beta\_{44} \mathbf{x}\_4^2 + \beta\_{12} \mathbf{x}\_1 \mathbf{x}\_2 + \beta\_{13} \mathbf{x}\_1 \mathbf{x}\_3 + \beta\_{14} \mathbf{x}\_1 \mathbf{x}\_4 + \beta\_{23} \mathbf{x}\_2 \mathbf{x}\_3 \\ &+ \beta\_{24} \mathbf{x}\_2 \mathbf{x}\_4 + \beta\_{34} \mathbf{x}\_3 \mathbf{x}\_4 + \varepsilon \end{aligned} \tag{7}$$
 
$$\begin{aligned} Y2\_{\mathbf{x}} &= \beta\_0 + \beta\_1 \mathbf{x}\_1 + \beta\_2 \mathbf{x}\_2 + \beta\_3 \mathbf{x}\_3 + \beta\_4 \mathbf{x}\_4 + \beta\_{11} \mathbf{x}\_1^2 + \beta\_{22} \mathbf{x}\_2^2 + \beta\_{33} \mathbf{x}\_3^2 \\ &+ \beta\_{44} \mathbf{x}\_4 \mathbf{x}\_4^2 + \beta\_{12} \mathbf{x}\_1 \mathbf{x}\_2 + \beta\_{13} \mathbf{x}\_1 \mathbf{x}\_3 + \beta\_{14} \mathbf{x}\_1 \mathbf{x}\_4 + \beta\_{23} \mathbf{x}\_2 \mathbf{x}\_3 \\ &+ \beta\_{24} \mathbf{x}\_2 \mathbf{x}\_4 + \beta\_{34} \mathbf{x}\_3 \mathbf{x}\_4 + \varepsilon \end{aligned} \tag{8}$$

Aeration rate ¼ x1 Residence time ¼ x2 Initial organic load ¼ x3

8 >>>>><

DOI: http://dx.doi.org/10.5772/intechopen.83662

Normal Boundary Intersection Applied to Controllers in Environmental Controls

>>>>>:

and Y2 (x).

Figure 9.

Figure 10.

15

Proposed arrangement for implementation.

Flow sheet of implementation algorithm.

according to Figure 9.

In the conditions of this chosen point, replacing in (Eqs. (6) and (7)) the response surface, we have two quadratic equations, one referring to Y1 (x)

The implementation of the transfer function in the control will be done

The two responses provided in the example, enter into a multiprocessor system according to pre-established parameters. The multiprocessing system introduces

pH ¼ x4

(9)

Each of the coefficients presented in the two equations, represented by βi, βii, and βij, is determined by the ordinary least square (OLS) regression algorithm where x1, x2, x3, and x4 are the factors already stated. With the models presented, it is possible to propose an optimization of both responses from the NBI algorithm for the four factors (Figure 7).

The Pareto frontier constructed from the optimum of both responses can now, from each of the setups assigned to each point, serve as the basis for implementation in controllers.

For each point referring to the specific response condition, a different setup is considered. For the chosen point 1 according to Figure 8, there is a BOD of 33.2 and a COD of 67, and under these conditions, we have the levels of each of the factors:

Figure 7. Modeling and optimization flowchart.

Figure 8. Pareto frontier with sample choice point.

Normal Boundary Intersection Applied to Controllers in Environmental Controls DOI: http://dx.doi.org/10.5772/intechopen.83662

> Aeration rate ¼ x1 Residence time ¼ x2 Initial organic load ¼ x3 pH ¼ x4 8 >>>>>< >>>>>: (9)

In the conditions of this chosen point, replacing in (Eqs. (6) and (7)) the response surface, we have two quadratic equations, one referring to Y1 (x) and Y2 (x).

The implementation of the transfer function in the control will be done according to Figure 9.

The two responses provided in the example, enter into a multiprocessor system according to pre-established parameters. The multiprocessing system introduces

Figure 9.

Y1<sup>x</sup> ¼ β<sup>0</sup> þ β1x<sup>1</sup> þ β2x<sup>2</sup> þ β3x<sup>3</sup> þ β4x<sup>4</sup> þ β11x<sup>1</sup>

Y2<sup>x</sup> ¼ β<sup>0</sup> þ β1x<sup>1</sup> þ β2x<sup>2</sup> þ β3x<sup>3</sup> þ β4x<sup>4</sup> þ β11x<sup>1</sup>

þ β24x2x<sup>4</sup> þ β34x3x<sup>4</sup> þ ε

þ β24x2x<sup>4</sup> þ β34x3x<sup>4</sup> þ ε

<sup>2</sup> <sup>þ</sup> <sup>β</sup>12x1x<sup>2</sup> <sup>þ</sup> <sup>β</sup>13x1x<sup>3</sup> <sup>þ</sup> <sup>β</sup>14x1x<sup>4</sup> <sup>þ</sup> <sup>β</sup>23x2x<sup>3</sup>

<sup>2</sup> <sup>þ</sup> <sup>β</sup>12x1x<sup>2</sup> <sup>þ</sup> <sup>β</sup>13x1x<sup>3</sup> <sup>þ</sup> <sup>β</sup>14x1x<sup>4</sup> <sup>þ</sup> <sup>β</sup>23x2x<sup>3</sup>

Each of the coefficients presented in the two equations, represented by βi, βii, and βij, is determined by the ordinary least square (OLS) regression algorithm where x1, x2, x3, and x4 are the factors already stated. With the models presented, it is possible to propose an optimization of both responses from the NBI algorithm for

The Pareto frontier constructed from the optimum of both responses can now, from each of the setups assigned to each point, serve as the basis for implementation

For each point referring to the specific response condition, a different setup is considered. For the chosen point 1 according to Figure 8, there is a BOD of 33.2 and a COD of 67, and under these conditions, we have the levels of each of

þ β44x<sup>4</sup>

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þ β44x<sup>4</sup>

the four factors (Figure 7).

in controllers.

the factors:

Figure 7.

Figure 8.

14

Modeling and optimization flowchart.

Pareto frontier with sample choice point.

<sup>2</sup> <sup>þ</sup> <sup>β</sup>22x<sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>β</sup>22x<sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>β</sup>33x<sup>3</sup>

<sup>2</sup> <sup>þ</sup> <sup>β</sup>33x<sup>3</sup>

2

2

(7)

(8)

Proposed arrangement for implementation.

Figure 10. Flow sheet of implementation algorithm.

polynomials referring to each different setup that consisted of the Pareto frontier, and for each setup, there are specific values of COD and BOD in mgO2L�<sup>1</sup> . From these inputs, the factors can be determined in optimized terms, X1 \*, X2 \*, and X3 \*.

References

2013

17

[1] Karna SK. An overview on Taguchi method. Society for Industrial and Applied Mathematics. 2016;1:10-17

DOI: http://dx.doi.org/10.5772/intechopen.83662

Normal Boundary Intersection Applied to Controllers in Environmental Controls

Analysis of Experiments Eighth Edition.

[3] Das I, Dennis JE. Normal-boundary intersection: A new method for generating the Pareto surface in nonlinear multicriteria optimization

Optimization. 1998;8:631-657. DOI: 10.1137/S1052623496307510

[2] Montgomery DC. Design and

problems. SIAM Journal on

An example of implementation for pH 5–9 and BOD values between 200 and 1000 mgL�<sup>1</sup> follows the flow sheet (Figure 10).

As already mentioned, one of the advantages of the method is the correction of the input parameters belonging to the Pareto frontier, consisting of innumerable set points within an optimal solution space.
