Arab J Sci Eng DOI 10.1007/s13369-015-1714-x

RESEARCH ARTICLE - CHEMICAL ENGINEERING

Genetic Algorithms Applied to PCA–Residues Optimization for Defect Localization Tawfik Najeh1 · Achraf Jabeur Telmoudi1 · Lotfi Nabli1

Received: 19 September 2013 / Accepted: 7 June 2015 © King Fahd University of Petroleum & Minerals 2015

Abstract While defect localization is vital in real-world systems, some limitations inherent to the existing techniques urge us to seek more advanced methods. This paper presents a new approach which takes advantages of genetic algorithms for optimization of non-convex objective function employed in calculating structured residues. The proposed approach so far improved the current principal component analysis (PCA) based on one of the defect localization. It has excellent impact on problem solving while dealing with optimization of residues structuring. The first part illustrates both the PCA model and the traditional residues structuring approach. The principle of optimizing a problem via genetic algorithms is explained later. A proposed objective function to be optimized is defined in the next part, and its optimization via genetic algorithms allows the structured residues computation. The new approach has been applied and proved functional for monitoring the Tennessee Eastman process. We have also proved the efficient performance of the proposed method in comparing it with some state-of-the-art methods. Keywords PCA · Defect localization · Non-convex · Genetic algorithms

B

Tawfik Najeh [email protected] Achraf Jabeur Telmoudi [email protected] Lotfi Nabli [email protected]

1

Laboratory of Automatic Control, Signal and Image Processing, National School of Engineers of Monastir, University of Monastir, 5019 Monastir, Tunisia

1 Introduction The principal component analysis (PCA) is one of the most efficient techniques for data analysis. PCA transforms the group of starting variables into new ones called principle components. The first principle components contain the important portion of variations of initial variables. The last principle components called primary residue contain measurement’s noise. Recently, the defect diagnosis put back on the PCA gained particular attention and was largely used for the surveillance of industrial processes [1–3]. One of the defect diagnosis important steps is the synthesis of residues able to prove the presence of defects, and this step is complete with localization phase to determine the defect components of the system. Hence, it is necessary to dispose of structured residues, which means being able to proceed to this localization [4]. These residues are sensitive to certain defects and insensitive to others. Several methods are proposed in order to structure the residues obtained from the PCA models [5,6]. Within all these methods, the desired choice of sensitivity for every structured residue compared to defects is indexed in a binary table, called theoretical signatures table. The calculation of structured residues must best match the already imposed sensitivity in the theoretical signatures table. After the determination of the structured residues, the diagnosis is executed starting from the measurements deducted at the present moment. An experimental signature of defects is obtained after a logical test of these residues compared to their respective threshold [7]. If this experimental signature is nullified, then the system is declared safe. On the contrary, the system is then considered failing. The identification of defect is realized by comparing the experimental signature to the columns of the theoretical signatures table. We note that our problem in relation to the proposed objective function can be considered as a large-scale problem. To

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obtain structured residues with maximized sensitivity, previous works use algebraic methods as gradient method [7]. However, this method does not guarantee the global optimum in case of large-scale problems. In fact due to the non-convex nature of the objective function, the used algebraic method may lead to a local optimum. Hence, structured residues will be less sensitive to certain defects or similar signatures to different defects [7,9]. In this case, we will approach this problem by using meta-heuristic method as GAs. In this work, we are particularly interested in the implementation of GAs to solve the problem of structured residues with maximized sensitivity. Therefore, in this paper, we apply and compare GAs and traditional structuring residues to prove the achievement of the global optimum efficiently. We note that the traditional structuring of residues with algebraic methods has been presented and discussed in [1,3,7]. There exist two categories of methods allowing to obtain structured residues. The first category looks for structured residues insensitive to certain defects without taking into account the sensitivity of the rest of defects [7]. The second category calculates the structured residues insensitive to certain defects maximizing along with it the sensitivity to others [5,6,8]. The two categories use algebraic methods to calculate the structured residues. Due to the non-convex nature of the structuring problem, the use of these methods may lead to a local optimum generating structured residues less sensitive to certain defects or similar signatures to different defects [9]. Despite the importance of the calculation cost, the second category does not allow a significant amelioration of the structured residues sensitivity with respect to some defects to be detected [3]. To offset these inconveniences, this work proposes the use of genetic algorithms (GAs) to calculate the structured residues. GAs are classified among the evolutionary optimization methods [10,11]. Both attractive and simple, they are applied to an important variety of optimization problems. The use of GAs to optimize solutions of the non-convex problems trigger the increase of interest in research working [12,13]. The originality of this technique resides in the absence of frequently used hypothesis with traditional methods in order to guarantee the convergence of the solution. In the presence of several local optima, the GAs assure the convergence of the desired solution toward the global optimum of the problem [14]. The application of this tool in the case of structuring of residues turns out to be very interesting, considering the nonconvex nature of the problem. The optimization of residues by GAs demands a new formulation of the structuring problem. Therefore, an objective function is elaborated. This function permits to obtain residues with the desired sensitivities with respect to every defect. This article is organized as follows: The second section is a recall to the PCA. The third section presents a localization

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of defects via structuring of residues. Two approaches are illustrated, the traditional approach and the sensitivity optimization one. The fourth section is devoted to the proposed method to the structuring of residues. This section elaborates, in detail, both the principle of optimization with GAs and the proposed approach to structure the residues. The fifth section presents the application of the proposed approach on two systems. The first is a linear noisy system, affected by captor defects, while the second system is an industrial one. The results and advantages of the proposed method are discussed and commented upon. The last section of this work is dedicated to the conclusion and to future perspectives. 1.1 Notations The notations used all over the paper are standard. Rm and R N ×m indicate the n-dimensional Euclidean space and the set of all N ×m matrices, respectively. For a matrix P, P t denote its transpose. Im represent the identity matrix of dimension m. rank(.) represent the rank of a matrix. Diag(λ1, λ2, . . . , λm) stands for a diagonal matrix with entries (λ1, λ2, . . . , λm) on the diagonal.

2 Modeling of Systems by PCA Let z b ∈ Rm be the vector grouping the measured variables of the systems. t  b z b (k) = z 1b · · · z m

(1)

This vector will be centered and reduced in order to obtain the vector of central and reduced variables z(k): z(k) = [z 1 · · · z m ]t

(2)

The PCA aims for a linear transformation of z(k) in a vector t (k) defined as follows: t (k) = P t z(k) = [t1 (k) · · · tm (k)]t

(3)

where P is orthogonal matrix and P t P = Im , Im ∈ Rm is the identity matrix. The variables ti (k) are called principle components. They are characterized by the null average and variations equal to λi ordered decreasingly. λi = var(ti (k))

(4)

Consider the data matrix Z N ∈ R N ×m defined as follows: Z N = [z(k) · · · z(k + N − 1)]t

(5)

Arab J Sci Eng

The matrix P and the variations λi can be obtained by diagonalization of the matrix of correlations 

=

1 Z t Z t = PΛP t N −1 N N

(6)

where Λ = diag(λ1 · · · λm ) is a diagonal matrix, containing variations of the principle components ti (k). P and λi can also be obtained by SVD decomposition of the matrix Z N Z N = USVt

(7)

with S = diag(s1 · · · sm ) diagonal matrix. In this case: 1 s2; λi = N −1 i

P=V

(8)

Starting from a certain number of components l, the variances of t j (k), j ≥ l, are quasi-null. The following partitioning is performed:   t  (9) t (k) = tˆ(k) | t˜(k) ; P = Pˆ | P˜ The vector tˆ(k) ∈ Rm−l groups the components of t (k) that provide wide variations. The vector t˜(k) ∈ Rl called residues vector contains principle components of quasi-null variations. The matrices Pˆ ∈ Rm×(m−l) and P˜ ∈ Rm×l contain vectors of P associated with the calculation of tˆ(k) and t˜(k). An estimation of z(k), noted zˆ (k) ∈ Rm , can then be calculated starting from tˆ(k). Based on the orthogonality property of P and its partitioning, we can write: Im = Pˆ Pˆ t + P˜ P˜ t

Several methods have been introduced to fix (l) in the PCA model. Most of these methods are heuristic and provide a subjective number of (l). The reader may find further details in [4]. In order to palliate against the drawbacks of heuristic methods, in [15] the author proposes to choose (l) by the minimization of an element called VRE that represents the variation of the reconstruction error of process variables. Meanwhile, it should be noted that the element underestimates the number (l) in the case of industrial process.

3 Structuring of Residues for Defects Localization The localization based on the structuring of the group of the last principle components (also called primary residues) was first presented in [7]. The method consists in finding a transformation of primary residues in a way that every transformed residue (or structured) is sensitive to certain defects and insensitive to others. The calculation of the structured is carried out by algebraic methods. 3.1 Traditional Structuring To obtain structured residues, two steps need to be fulfilled. In the first step, we define the desired sensitivity or insensitivity of the structured residues considering the defects to be (or not to be) detected. With the second step, we develop the generation of residues according to the already defined constraints. Several approaches for structuring residues have been proposed in the literature [1,3,7]. But, by utilizing only algebraic methods, they do not satisfy the isolation requirement. This is due to the calculation of residues without regard to their sensitivities to defect.

(10) 3.1.1 First Step

Multiplying this equation by z(k): z(k) = Pˆ Pˆ t z(k) + P˜ P˜ t z(k) = zˆ (k) + e(k)

(11)

e(k) denotes the error vector of the PCA model. The vectors of the matrix Pˆ form the main space of representation of data since the vectors of P˜ form the residual space of representation of data. The identification of the PCA model then consists in: – The parameters estimation is performed, so by diagonalization of the matrix Σ or by an SVD decomposition of the matrix Z N . – Determine the structural parameter (l) An incorrect choice of (l) can mask the detection of changes occurring within the process or can trigger false alarms [7].

From the monitoring specifications of systems, the desired sensitivities of structured residues are defined with respect to the defects. The sensitivities are grouped in a matrix called theoretical signatures table. Every column of this table corresponds to a defect, and every line corresponds to a structured residue. The value 1 at the intersection of the ith line and the jth column means that ith structured residue is sensitive to jth defect, while the indication of 0 means that ith structured residue is insensitive to jth defect. To assure the isolation of defects, all columns of the theoretical signatures table must be different. Once the theoretical signatures table is constructed, structured residues are generated and a detection procedure is applied on them in order to obtain their experimental signature at every instant. The value 1 is associated if the structured

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residue is superior to its threshold, if not it is the value 0. The group of these Boolean values is stored in a vector of experimental signatures. If the latter is null, then the system is declared safe. If a defect is present, at least one of the structured residues is sensitive to it and the vector of the experimental signatures becomes then non-null. Next, the localization phase consists in finding the correspondence between this vector and one of the column vectors of the theoretical signatures table.

group of defects d F , the ith line wi of W must nullify all big j vectors. The lines reduce to zero the following criteria: ϕi =

t i 2 F 1  (wi bg j ) F wit bigi 

(16)

j=1

Two conditions for the existence of the matrix W have been formulated in [7], and they are enounced as follows:

3.1.2 Second Step

rank(Bgi F ) ≤ m − 1

The structuring of the residues is characterized by a linear transformation defined by a matrix W ∈ Rq×m−l . The relation between the primary residues and structured residues is defined as follows:

This first condition implies the existence of complementary space generated by columns of the Bgi F matrix. The wi vectors are formed through a linear combination of the column vectors of the complementary space.

r (k) = W tˆ(k)

(12)

  rank Bgi F big¯ j = rank(Bgi F ) + 1

(17)

(18)

where t˜(k) ∈ Rm−l is the vector of primary residues and designs the vector of structured residues. In order to simplify the notions in the following, it is proposed that:

This second condition assumes that every big¯ j vector brings supplementary information to the last. This allows obtaining an undesired zero in one of the lines of theoretical structuring matrix.

Pˆ t = B

3.2 Structuring with Maximized Sensitivity

(13)

The structured residues may be established in function of original variables of the system r (k) = W Bz(k)

(14)

The matrix WB is called real signatures matrix. The absolute values of its coefficients must be as close as possible to those of theoretical signatures table. The structuring aims at transforming every structured residue ri (k), i ∈ {1, . . . , q} to be insensitive to a group of defects dF .  ri (k) = WB z(k) = i

W [Bgi F

Bgi¯ F ]

z ig F (k) z ig¯ F (k)

 (15)

For every ri (k), the vector z(k) is split into two parts. The first being noted as z ig F (k) which contains the variables affected by the defects d F . The second is noted z ig¯ F (k), and it contains the rest of the variables. In the same way and to each i, the columns of the B matrix are rearranged in a B i matrix that will be divided into two matrices: Bgi F and Bgi¯ F which will be respectively associated with the variables z ig F (k) and z ig¯ F (k). The matrix groups the columns of matrix Bgi F corresponding to zero in the i me line of the theoretical signature table. The Bgi F and Bgi¯ F columns are identified as big F (k) and big¯ F (k). To be insensitive to a

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Structuring the residues via the traditional approach strictly focuses on calculating wi , one by one, which give zeros in predefined positions by the matrix of theoretical signatures. In order to improve structuring by the traditional approach [5,8], authors proposed a method called structured residual approach with maximized sensitivity SRAMS. This method calculates the wi that nullify the effect of certain defects d F along with it maximizing the sensitivity to the rest of defects. The SRAMS approach maximizes an average of projections of observations in the residual space. One variant of this method is the OSR (optimal structured residual) method that maximizes independently each projection [5]. To improve the sensitivity of ri (k) to a group of defects, the scalar product of wi and the vectors bg¯ j must be as close as possible to 1. The line wi is a solution that maximizes the following criteria γi : γi =

t i 2 F 1  (wi bg¯ j ) m−F wit big¯i  j=1

(19)

Under the constraint: wit Bi = 0 Geometrically, wi is chosen orthogonal to bi while minimizing angles with the other defect directions b j ( j = i). The main difference between the conventional structuring and the method of structuring with maximized sensitivity can be illustrated geometrically in Fig. 1.

Arab J Sci Eng

The application of GAs in the case of structuring residues leads to very interesting view of the non-convex problems nature. The first section of this part presents the principles of the GAs. The second section deals with the optimization of the W matrix’s coefficients by the GAs. 4.1 Optimization by Genetic Algorithms GAs aim at calculating solutions, or individuals, which optimize an objective function. After having imposed its expression, the probable steps are implied to create an initial population of individuals [10]. The last will subdue genetic operations (such as evaluation, mutation, crossover) to converge toward the optimal solution of the problem. The optimization steps by GAs are enounced as follows:

Fig. 1 Maximization of the product w1m .b2

The SRAMS approach does not go further than the traditional one [3]. In fact, records that certain coefficients of the matrix WB do ameliorate while others degrade. This method allows obtaining better performances than the traditional one, but it is accompanied by an augmentation of false alarms rates. All existing approaches utilize algebraic methods, and there is no previous use of heuristics methods to solve the problem of structured residues with maximized sensitivity.

4 Proposed Methods for the Structuring of Residues The problem of finding the matrix W is an “under-determined” one. To find all coefficients of every line wi , the number of equations (equal to the number of zeros of the ith line in the theoretical signature table) is strictly inferior to that of unknowns (equal to the number of system variables). This problem is non-convex and has infinity of possible solutions. To resolve this type of problems, the traditional approach imposes, in an arbitrary way, the value of certain variables to succeed with determinant system. The wi coefficients obtained by this method constitute a non-optimal solution that verifies all the equations of the problem. Optimization methods based on matrix calculations have been suggested to ameliorate the value of the wi coefficients. In [3], the author noticed that despite the high calculation costs, this method does not permit a significant amelioration of the wi values causing bad sensitivities of structured residues with respect to the defects to be detected.

– Initialization It is usually random: A population of individuals is arbitrarily generated which belongs to the research space. It is usually preferable to inject the maximum knowledge upon the problem. – Evaluation Generally, this step consists in calculating the quality of individuals by attribution a positive value to each individual called aptitude or fitness. In this paper, the assignment of fitness is done by linear ranking method. Consider Nind the number of individuals in the current population, Pos the position on an individual (least fit individual has pos = 1, the fittest individual pos = Nind ) and SP the selective pressure. The aptitude is done as follows: Fitness = 2 − SP + 2(SP − 1)

Pos − 1 Nind − 1

(20)

– The selection This step selects a determined number of individuals from the current population. The selection is characterized by a parameter called pressure of selection (Ps ) that imposes the number of selected individuals. – The crossover The genetic operation of crossover creates new individuals. Starting from two arbitrarily chosen parents, the crossover produces two descendants. This step does only affect a limited number of individuals, imposed by the rate of crossover (Pc ). – The mutation The mutation consists in causing a light perturbation to a certain number (Pm ) of individuals. The effect of this step is to oppose the attraction applied by the best individuals which permits the exploration of other research areas.

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4.2 Structuring Residues with Genetic Algorithms It is illustrated in Sect. 3 that obtaining every residue ri (k) means calculating the coefficients of wi line which nullify the function ϕi and maximize the function γi . This calculation does not have unique solution, and the methods used to solve them are all deterministic. The use of these methods may lead to a local optimum, giving residues less sensitive to certain defects or with similar signatures for different defects. The resolution of such problems by the GAs permits to avoid all local convergence and to converge toward the problem global optimum. The proposed approach to structuring the residues uses GAs to calculate all the wi lines of the W matrix. The use of GAs for structuring the residues requires the elaboration of several objective functions φi . Each objective function must keep in account two criteria ϕi and γi . The minimization of the criterion ϕi (along the hypothesis that ϕi ≥ 0) permits to obtain zeros in the desired positions of wi B conforming to the theoretical signature table. The maximization of criterion γi (along the hypothesis that γi ≥ 1) assures to obtain the rest of the wi B elements of which the norm tends to 1. Structuring the residues implies the optimization of two criteria of two different natures. To unify these two criteria, a objective function Φi is proposed: Φi = α[1 − γi ] + βϕi

(21)

With α and β ∈ [0, 1], they are two used coefficients to adjust the sensitivity or insensitivity of residues. Every retained wi line for the construction of the W matrix must minimize the function Φi . The value of Φi can be negative, and in this case, the absolute value will be considered. The structured residues are then calculated by Eq. (21) which can be divided into two parts: The first part α[1 − γi ] will try to obtain the ones of the theoretical signature table which will be archived by maximizing the criteria γi given by Eq. (19) (or by minimizing the quantity 1 − γi ). The second part βϕi uses Eq. (16) and then will try to minimize the criteria ϕi to obtain the zeros of the theoretical signature table. By minimizing the objective function Φi (Eq. 21), we are going to optimize simultaneously these two criteria. We note that the first criteria γi use the product (wit big¯ j ) and the second criteria

ϕi use the product (wit big j ) In most of the research work concerning the structuring of residues, the author imposes a zero value to the function ϕi . This restrains the research space of the vectors wi . In the proposed method, the function ϕi is minimized under constraint that it would be inferior to a threshold equal to 10−4 with the condition that the maximum number of iterations (generations) of algorithm should not exceed a selected value (Gmax). This minimization permits to enlarge the space of solutions of vectors wi which favors the gain of the best

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solutions. The value of this threshold is chosen in an empirical way and can be adjusted according to the application processed. By optimizing the multi-objective function Φ, both criteria γ and ϕ are taken into consideration. As a result, the product W.B will be very close to the theoretical signature table. Therefore, structured residues are sensitive to certain defects and insensitive to others. After having determined the objective function to be minimized, an individuals’ population wi is generated arbitrarily. The aptitudes of the individuals wi are evaluated by the function Φi . The individuals with the highest aptitudes are selected to undergo different genetic operations (crossover, mutation and selection). The new GAs is constructed following the strategy outlined as follows: Algorithm 1 01: Repeat until Gmax or a threshold equal to 10−4 02: Generate at random an initial population of N vectors Wi 03: For every Wi do 04: Compute the fitness 05: End for 06: Apply crossover to produce new vectors 07: Apply mutation 08: Select better vectors Wi 09: Reinsertion of better vectors W i 10: End repeat 11: Return to 01 and repeat until obtaining all structured residues. The following flowchart (Fig. 2) describes the steps to obtain one structured residue with maximized sensitivity and can be repeated to find all other residues.

5 Applications 5.1 Linear System In order to illustrate the method previously presented, a static system is considered, along with seven outputs z i (k), i ∈ {1 . . . 7} and two inputs u j (k), j ∈ {1, 2} described by the next equations:  z b (k) =

1 1 1 0 0 3 2 0 0 0 1 1 2 1

t u(k) + ε(k)

(22)

u j (k) are square signals with amplitudes and durations that change arbitrarily. The noise εi (k) are arbitrary and uniformly arranged between +0.05 and −0.05. This group of measurements presents relations of analytic linear redundancy.

Arab J Sci Eng Table 1 The theoretical signature d1

d2

d3

d4

d5

d6

d7

r1

1

0

0

0

1

1

1

r2

1

1

0

0

0

1

1

r3

1

1

1

0

0

0

1

r4

1

1

1

1

0

0

0

r5

0

1

1

1

1

0

0

r6

0

0

1

1

1

1

0

r7

0

0

0

1

1

1

1

Table 2 Parameters of the genetic algorithm Parameter

Ps

Pc

Pm

Insertion rate

value

1.9

90 %

3.3 %

90 %

The objective function to be optimized is illustrated by Eq. (21) with α = β = 0.5. The initial population is formed of 80 individuals arbitrarily chosen, and the algorithm stops when it accomplishes a number of generations G max = 1200. The values of the genetic operations are expressed in Table 2. The matrix W issued from the traditional method and Wg issued from the genetic optimization are given by the next matrices: ⎛

Fig. 2 Proposed method to obtain structured residue with maximized sensitivity

The system is simulated with N = 500 samples. The vector z b (k) is centered and reduced to obtain a vector of reduced and central data z(k). The latter will be used to construct the data matrix Z N . The PCA model is then determined, and the choice of (l) is realized by the criterion of non-reconstructed variance. The B matrix issued from the last principle components is illustrated as follows. ⎛

−0.25 ⎜ −0.74 ⎜ B=⎜ ⎜ 0.03 ⎝ −0.35 −0.02

−0.49 0.83 0.02 0.65 0.15 0.04 −0.02 0.15 0.71 −0.29 −0.16 −0.30 −0.05 −0.04 −0.07

⎞ 0.05 −0.09 −0.08 −0.01 −0.01 −0.06 ⎟ ⎟ −0.70 −0.51 0.03 ⎟ ⎟ −0.32 0.36 0.67 ⎠ −0.15 0.80 −0.57

The theoretical signature table proposed for this example is in Table 1

−0.11 ⎜ −0.12 ⎜ ⎜ −0.83 ⎜ W =⎜ ⎜ −0.82 ⎜ −0.85 ⎜ ⎝ 0.69 0.55 ⎛ −0.13 ⎜ −0.16 ⎜ ⎜ −1.00 ⎜ Wg = ⎜ ⎜ 0.57 ⎜ 0.99 ⎜ ⎝ −0.54 0.00

⎞ 0.75 0.26 0.06 0.31 ⎟ ⎟ 0.03 0.01 ⎟ ⎟ 0.03 0.01 ⎟ ⎟ 0.02 0.01 ⎟ ⎟ 0.37 −0.35 ⎠ 0.00 0.02 ⎞ −0.68 −0.50 −1.2 −0.46 1.00 0.00 0.27 −0.67 ⎟ ⎟ −0.35 0.00 −0.11 0.00 ⎟ ⎟ −1.00 0.03 0.00 0.02 ⎟ ⎟ −0.36 0.75 0.10 0.04 ⎟ ⎟ −0.07 0.96 0.56 0.78 ⎠ 0.06 1.00 −0.12 1.00 0.14 0.71 −0.32 −0.32 −0.26 0.25 −0.48

−0.57 −0.61 −0.45 −0.45 −0.44 −0.42 −0.67

The matrix of real signatures calculated by the traditional method WB and by the genetic approach Wg B is illustrated as follows: ⎞ 0.07 0.00 0.00 0.00 −0.14 0.76 −0.62 ⎜ 0.00 0.16 0.00 0.00 0.00 −0.62 −0.76 ⎟ ⎟ ⎜ ⎜ −0.50 −0.30 0.80 0.00 0.00 0.00 0.00 ⎟ ⎟ ⎜ 0.30 −0.80 0.00 0.00 0.00 0.00 ⎟ WB = ⎜ ⎟ ⎜ 0.50 ⎜ 0.00 0.00 ⎟ 0.36 −0.39 −0.58 −0.58 0.00 ⎟ ⎜ ⎝ 0.00 0.00 −0.01 −0.71 −0.70 −0.02 0.00 ⎠ 0.00 0.00 0.00 −0.10 −0.00 0.72 −0.67 ⎛

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Fig. 3 Evolution of structured residues for different cases of defects ⎛

⎞ 0.95 0.00 0.00 0.00 0.80 −0.76 −0.51 ⎜ −0.79 0.68 0.00 0.00 0.00 −0.45 0.53 ⎟ ⎜ ⎟ ⎜ 0.56 0.30 −0.87 0.00 0.00 0.00 0.00 ⎟ ⎜ ⎟ 0.01 0.00 ⎟ Wg B = ⎜ ⎜ 0.60 −0.93 0.33 0.00 0.00 ⎟ ⎜ 0.00 −0.77 0.77 0.51 −0.52 0.00 0.00 ⎟ ⎜ ⎟ ⎝ 0.00 0.00 −0.57 0.45 −0.99 0.80 0.00 ⎠ 0.00 0.00 0.00 0.69 −0.81 0.71 −0.63

According to Table 1, the coefficients of the WB matrix are moved away from the theoretical signature table. Considering the first line of WB, the first term of this line must be close to one, but it is quasi-null in this case. This same figure case can be observed in all the lines of WB. The use of the genetic approach allowed obtaining a matrix Wg B whose coefficients lean close to those of the theoretical signature table. For example, the first term of the first line leans toward one, contrarily to what was obtained by the traditional approach. The use of genetic approach for structuring residues allowed the improvement of their sensitivity with respect to the defects to be detected. Some columns of the WB matrix, the 6th and 7th columns, for example, are quasiidentical. In this case, the obtained structured residues will behave in the same way in the presence of defects affecting the variables 6 and 7. This causes an erroneous isolation of these defects. The use of genetic approach in structuring the residues permits to avoid this problem, since all obtained columns of the Wg B matrix are different. Seven simulations are carried out on 1000 samples. Every time, a defect affects one of the variables. The defect is of constant bias type and

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has amplitude equal to 20 % of the variation range of the affected variable. It is introduced starting from the 500th sample until the end of simulation. The evolution of ri (k) in all simulations is represented in Fig. 3. Every line of this figure gives the answer of the same residue to different defects di . Every column represents the answers to the same defect of different structured residues ri (k). The columns of the Fig. 3 are different and match the sensitivities to the defined defects in the theoretical signature table. Contrarily to the traditional approach, the structured residues issued from the proposed approach permit the totality localization of the defects.

5.2 Tennessee Eastman Process To illustrate the efficiency of the proposed model, we proceed to its validation on chemical process: the Tennessee Eastman process. This process [16] is a highly nonlinear, nonminimum phase, and open-loop unstable chemical process consisting of a reactor, separator and recycle arrangement. This process produces two products G and H from four reactants A, C, D and E. Also a by-product F is present in the process. The process also produces an inert B and a by-product F. The major units of the process are: a reactor, a product condenser,a vapor, liquid separator, a recycle compressor and a product stripper. The process gives 41 measurements; in this study, we only treated the reactor unit (Fig. 4) which has 12 variables:

Arab J Sci Eng Fig. 4 Tennessee process. XA …XF feed mole fractions, F feed flow, T1,T2 temperature, P pressure, L level, Vc coolant valve position

– – – – – – –

Feed mole fractions: X A, X B, XC, X D, X E and X F Pressure: P Coolant valve position: V c Temperature: T Level: L Coolant temperature: T c Feed flow: F

All gaseous reactants are transmitted to the reactor to produce other liquid products. A catalyst dissolved in the liquid phase catalyzes the gas-phase reactions. Then, the catalyst remains in the reactor and the products leave the reactor as vapors. The input/output data, used to built the model, were generated from the model of Tennessee implemented for the program MATLAB in the toolbox Simulink [17]. In this study, the PCA model was trained with a data set of 100 samples. During the simulation, low-level noise was added to all outputs. The matrix B gives the residual subspace obtained by PCA. ⎛

−0.79 ⎜ 0.34 ⎜ B=⎜ ⎜ 0.02 ⎝ −0.07 0.00

−0.18 −0.54 −0.02 0.40 −0.02

⎞ 0.16 −0.15 −0.22 0.45 −0.20 0.06 0.27 0.17 0.24 −0.66 ⎟ ⎟ 0.15 0.71 −070 −0.51 0.03 ⎟ ⎟ 0.00 −0.53 −0.23 −0.67 −0.23 ⎠ 0.00 −0.59 −0.78 −0.19 0.00

A theoretical isolating signature is defined in Table 3: The same equation 20 with α = β = 0.5 is taken as the objective function to be optimized. The initial population

Table 3 The theoretical signature d1

d2

d3

d4

d5

d6

r1

0

1

1

0

0

1

r2

1

0

1

1

0

0

r3

1

1

0

0

1

0

r4

0

0

0

1

1

1

comprises 80 individuals randomly performed by the GA. The algorithm stops when he performs a number of generations Gmax = 600. The experimental signatures of defects computed by the traditional approach and those provided by the proposed method are described in the following matrix: ⎞ 0 −0.67 −0.10 0 0 −0.13 0.72 ⎜ 0.71 0 −0.08 0.68 0 0 −0.10 ⎟ ⎟ WB = ⎜ ⎝ −0.52 −0.49 0 0 −0.42 0 −0.10 ⎠ 0 0 0 0.60 −0.73 −0.13 0 ⎞ ⎛ 0 0.76 0.12 0 0 0.15 −0.82 ⎜ 0.99 0 −0.11 0.95 0 0 −0.15 ⎟ ⎟ ⎜ Wg B = ⎝ 0.62 0.53 0 0 0.53 0 0.61 ⎠ 0 0 0 −0.63 0.78 0.14 0 ⎛

To prove the efficiency of proposed approach, an elementby-element comparison for each previous matrix with the proposed signature table (theoretical signatures) is done, which detects a significant difference. One notices that the

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coefficients of the matrix Wg B are much closer to those in the table of theoretical signatures. For example, on the third line of WB, the last term must be as close to 1, but it is nearly zero in this case. This inconvenience can be observed in all lines of WB. The proposed approach for structuring residues provides a Wg B matrix with elements close to the table of theoretical signatures. Let us consider the second line of Wg B, the first term and fourth term of this line closer to 1. The proposed genetic approach improves significantly the sensitivity of the residues with respect to the defects to be detected. This improvement as well noticed in some of the columns of the matrix WB, for example columns 3 and 7, is almost identical. This will cause an equal behavior of residues to deal with two deferent defects that affect the variables 3 and 7. As a consequence, the isolation of these two defects is impossible. This inconvenience is avoided which proved the efficiency of proposed approach compared with traditional one.

6 Conclusion This work proposes a new method of structuring residues based on the residual space identification. The proposed method applies GAs on the residues obtained by PCA. This method calculates new residues called structured residues that are linear combinations of the old ones. These structured residues are sensitive to certain defects and insensitive to others. Obtaining structured residues is assured by optimizing objective function proposed in this work. The complexity and the non-convex nature of structuring problem imply a robust optimization method. In most cases, the traditional methods fail to find residues of the desired aptitudes of defects isolation. The use of GAs for the resolution of such problems permits to eliminate the shortcomings of traditional methods. Hence, it gives structured residues in an optimal manner. The use of GAs for structuring gives results so far improved compared to those issued from the traditional method.

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