Fault Diagnosis in Industrial Systems Using Bioinspired Cooperative
Fault Diagnosis in Industrial Systems Using Bioinspired Cooperative Strategies Lídice Camps Echevarría, Orestes Llanes-Santiago, and Antônio José da Silva Neto
Abstract. This paper explores the application of bioinspired cooperative strategies for optimization on Fault Diagnosis in industrial systems. As a first step, the Differential Evolution and Ant Colony Optimization algorithms are considered. Both algorithms have been applied to a benchmark problem, the two tanks system. The experiments have considered noisy data in order to compare the robustness of the diagnosis. The preliminary results indicate that the proposed approach, basically the combination of the two algorithms, characterizes a promising methodology for the Fault Detection and Isolation problem.
1 Introduction The increases on the complexity of the industrial systems implies that the probability of fault occurrence is more significant. The faults change the characteristic properties of the system and produce its incapacity to fulfill the intended purpose, [6]. Therefore, an automatic supervisor should be used to detect and isolate, (FDI), the faults as early as possible. This is a reason for which in the last three decades a wide variety of FDI methods have been developed. Lídice Camps Echevarría Departamento de Matemáticas, Facultad de Ingeniería Mecánica, Instituto Superior Politécnico José Antonio Echeverría (ISPJAE), Ciudad de La Habana, Cuba e-mail: [email protected] Orestes Llanes-Santiago Departamento de Automática y Computación, Facultad de Ingeniería Eléctrica, Instituto Superior Politécnico José Antonio Echeverría (ISPJAE), Ciudad de La Habana, Cuba e-mail: [email protected] Antônio José da Silva Neto Department of Mechanical Engineering and Energy, Instituto Politécnico (IPRJ), Universidade do Estado do Rio de Janeiro, UERJ, Nova Friburgo, RJ, Brazil e-mail: [email protected]
J.R. González et al. (Eds.): NICSO 2010, SCI 284, pp. 53–63, 2010. springerlink.com © Springer-Verlag Berlin Heidelberg 2010
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L.C. Echevarría, O. Llanes-Santiago, and A.J. da Silva Neto
The FDI methods are divided in two general groups, those which use a model of the process and those which do not use it. Although many approaches have been developed, [2, 6, 11], robust FDI is still considered as a problem open to further research, [14], due to the the unavoidable process disturbances and the modeling errors which make almost unfeasible the use of many FDI methods in practical application.[11]. The FDI problem approach by the model-based methods has the following structure: based on some observations and the direct model, it is necessary to establish the causes of this observed behavior. In some cases, the identification of model parameter with fault of the system allows the FDI via the parameters estimation. Recently some articles have reported applications of meta heuristics to the FDI problems via parameters estimation, [16–18]. In this sense, the FDI via parameter estimation based on the approach of the meta heuristic algorithms seems to be an adequate alternative. The simple structure of these algorithms and their robustness reported in the solution of many parameters estimation inverse problems, [12], [1, 8, 9, 13], indicate that they are a promising alternative for FDI methods which need to be fast and simple (for online process) and robust to external perturbations. Moreover, estimations based on heuristic algorithms are absolutely viable when a nonlinear model is considered, making perfectly feasible the use of non linear models in order to prevent some modeling errors when linearizing the nonlinear process. This work presents the application of two bioinspired algorithms, Differential Evolution (DE), [15], and Ant Colony Optimization (ACO), [4], to the FDI problem in order to study and compare the capabilities of both algorithms and their combination for the FDI problems . As a case of study it has been simulated the problem of the two tanks system. This system is a simplified version of the three tanks system, which was adopted as a benchmark problem for FDI and reconfigurable control [10]. In order to verify and compare the robustness of the diagnosis, several simulations were made and different fault situations were considered. In all cases noisy data were considered. The results are presented using comparative tables and figures. The structure of the paper is the following: in the next section the basis of DE and the ACO are described. The third section shows the benchmark problem of the two tanks system. The section number 4 shows the simulations, the experimental results and the analysis of these results. Finally, section 5 summarizes the contributions and achievements of the paper.
2 Differential Evolution and Ant Colony Optimization This section describes the basis of the two algorithms that are used during this paper.
2.1 Differential Evolution The Differential Evolution (DE) was proposed around 1995, for optimization problems, [15]. DE is an improved version of the Goldberg’s Genetic Algorithm (GA), [5], taking the basis of Simulated Annealing (SA), [7]. Some of the most
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important advantages of DE are: simple structure, simple computational implementation, speed and robustness, [15]. Basically, DE generates new parameter vectors by adding the weighted difference between a pair population vectors to a certain vector (the number of pair can be changed). This configuration is summarized by the notation DE/X /α /β where X denotes the vector to disturb, α the number of pair of vectors for disturbing X and β indicates the type of crossover to be used. In this case was considered DE/X jbest /1/bin. The key parameters of control in DE are the population size, N, the crossover constant, CR , and the weight applied to random differential or scaling factor, Fs . In [15] some simple rules for choosing the parameters of DE for any application are given: usually, N should be about 5 to 10 times the dimension of the variable of the problem, D and Fs lie in the range 0.4 to 1.0. Initially, D = 0.5 can be tried, and then can be increased if the population converges prematurely.
2.2 Ant Colony Optimization ACO was initially proposed, [4], for integer programming problems but it has been extended to continuous optimization problems. This algorithm is inspired on the behavior of ants seeking a path between their colony and a source of food. This behavior is due to the deposit and evaporation of pheromone. For the continuous case the idea of the ACO is to mimic this behavior with simulated ants which are identified with a feasible solution. The first step is to discretize the feasible interval of each variable of the problem in n possible values. On each iteration of the algorithm a family of N new ants are generated based on the information obtained from the previous ants. This information is saved on the pheromone probability matrix P f (dimensions m × n where m is the number of variables in the problem) which is updated at each iteration based on a evaporation factor Cevap and an incremental factor Cinc : α ∑l=1 [ fil (t)] α n ∑l=1 [ fil (t)] j
p fi j (t) =
(1)
where α = 1 and fi j is the element of the pheromone matrix which expresses the pheromone level of the discrete value j − esimo of the variable i, and it is updated on each iteration: fi j (t + 1) = (1 − Cevap) fi j (t) + δi j,best Cinc fi j (t)
(2)
3 The Two Tanks System The two tank system considered for study is represented in Fig.1. The system consists of two liquid tanks that can be filled with two similar and independent pumps acting on the tank 1 and tank 2, which have the same cross section S1 = S2 . The pumps deliver the flow rates q1 in tank 1 and q2 in tank 2. The
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L.C. Echevarría, O. Llanes-Santiago, and A.J. da Silva Neto
tanks are interconnected to each other through lower pipes. All the pipes have the same cross section S p . The liquid levels L1 and L2 in each tank are the controlled variables and they are measured with continuous valued level sensors. The variables q1 and q2 are chosen as manipulated variables to control the levels of tank 1 and tank 2. The system has two faults to be detected and isolated: • Fault 1 : Leak at the tank 1, an outflow with magnitude q f1 . • Fault 2 : Leak at the tank 2, an outflow with magnitude q f2 . The differential equations that describe the system, under the presence of faults, is derived from conservation of mass in the system of the two tanks: q1 q10 q12 q f1 L˙ 1 = − − − S1 S1 S1 S1
(3)
q2 q20 q12 q f2 − + − , L˙ 2 = S2 S2 S2 S2
(4)
and by the application of the Torricelli’s law: qi0 = μi S p 2gLi q i j = μi S p
2g Li − L j sign (Li − L j ) ,
where μi are flow coefficients and considering Ci = μi S p 2g,
Fig. 1 Two tanks system
(5)
(6)
(7)
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Table 1 Values of the constants of the two tanks system C1 , C2 S1 , S2 Sp Acceleration due to gravity, g
0.3028 m2 2.54 m2 0.1 m2 9.8 m/s2
the following system of equations is obtained: qf q1 C1 √ C1 − L1 − |L1 − L2 |sign (L1 − L2 ) − 1 L˙ 1 = S1 S1 S1 S1 √ q q2 C2 C1 f − L2 + |L1 − L2 |sign (L1 − L2 ) − 2 L˙ 2 = S2 S2 S2 S2 y1 = L1
(8)
y2 = L2 For more details see [3]. The goal here is to diagnosis the presence of the faults 1 or 2, even more, their magnitude. As a first approach it has been supposed that the leak at both tanks do not change in time and it is assumed that the magnitude of the leaks is less than 1000 ml/s. In other words, the following restrictions for the parameters q fi , i = 1, 2 have been established: q f1 , q f2 ∈ ℜ : 0 ≤ q f1 , q f2 ≤ 1 ml/s Estimation of the parameters q f1 and q f2 permit to diagnosis the system. In order to estimate these parameters, the following problem is formulated: min F (v) = min
N
∑
2 ¯ cal L¯ exp n − Ln (v)
(9)
n=1
t exp where v = q f1 , q f2 , L¯ n = (Ln1 , Ln2 )t are the observations of the liquid levels at t n n different instants of time, L¯ cal are the liquid levels computed by n = Lcal(1) , Lcal(2) the model (9) using Runge Kuta 4. The table 1 shows the values of the constants considered in the model of the two tanks system.
4 Results and Discussion The closed loop behavior of the process was simulated when no faults are present. This behavior is shown in Fig. 2.
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L.C. Echevarría, O. Llanes-Santiago, and A.J. da Silva Neto
Fig. 2 Closed loop behavior of the process when no faults are present, noise data 2-5 %
Fig. 3 Closed loop behavior of the process when leaks of 200 ml/s are present in both tanks, noise data 2-5 %
The closed loop behavior of the process when a leak of magnitude 200 ml/s in each tank (q f1 = q f2 = 0.2) is introduced at time t = 20 s is shown in Fig. 3. The closed loop behavior of the process when a leak of magnitude 50 ml/s in tank 2 (q f1 = 0, q f2 = 0.05) is introduced at time t = 20 s is shown in Fig. 4. The effect of this leak in tank 2 is graphically imperceptible. In order to diagnosis the faults, the minimization of the objective function F (v) was implemented, in the first case based on the DE algorithm. The population was considered to be 10 and the mutation mechanism is (DE/xbest j /1/bin). The second case considered the minimization of the objective function by means of ACO, described on 2.2, with 10 ants. Both algorithms stop when 100 iterations are achieved.
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Fig. 4 Closed loop behavior of the process when leaks of 50 ml/s is present in tank 2, noise data 2-5 %
Table 2 Diagnosis obtained in five runs of leaks of 200 ml/s in each tank: noisy data between 2 and 5 % error alg fault q¯ f1 q¯ f2 iter t(s) DE q f1 = q f2 = 0.2 0.1827 0.1901 93 56.5681 DE q f1 = q f2 = 0.2 0.1985 0.2034 67 29.5475 DE q f1 = q f2 = 0.2 0.1988 0.2062 65 28.3845 DE q f1 = q f2 = 0.2 0.2036 0.2044 60 25.5477 DE q f1 = q f2 = 0.2 0.1993 0.1969 56 23.8943 ACO ACO ACO ACO ACO
q f1 q f1 q f1 q f1 q f1
= q f2 = q f2 = q f2 = q f2 = q f2
= 0.2 = 0.2 = 0.2 = 0.2 = 0.2
0.1738 0.1775 0.1832 0.2087 0.1860
0.2414 0.2287 0.2198 0.2401 0.1829
75 57 54 44 41
75.0830 48.0647 45.9054 34.9798 33.9888
The tables 2 and 3 shows the results of the diagnosis of different faulty situation by both algorithms. All cases considered data with 2-5 % of noise. The abbreviations used in the tables are alg for algorithm and iter for number of iterations. The notation introduced is t for the computing time, in seconds, of the algorithm. Both algorithms detected the presence of faults but the DE algorithm is more accurate in the determination of the leak magnitudes. Both algorithms are fast, which is good for the online diagnosis, but DE is faster. In Fig. 5 are shown the evolution of both algorithms for two situation described in table 2. The figures suggest a way of combination of ACO and DE in order to
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Table 3 Diagnosis obtained in five runs of a leaks of 50 ml/s in tank 1: noisy data between 2 and 5 % error alg DE DE DE DE DE
fault q f1 = 0.05 q f2 q f1 = 0.05 q f2 q f1 = 0.05 q f2 q f1 = 0.05 q f2 q f1 = 0.05 q f2
=0 =0 =0 =0 =0
q¯ f1 0.0498 0.0515 0.0508 0.0531 0.0489
q¯ f2 0.0007 0.0000 0.0000 0.0000 0.0001
iter 79 64 64 43 42
t(s) 46.8123 35.9895 35.4267 28.7774 18.0390
ACO ACO ACO ACO ACO
q f1 q f1 q f1 q f1 q f1
= 0.05 q f2 = 0.05 q f2 = 0.05 q f2 = 0.05 q f2 = 0.05 q f2
=0 =0 =0 =0 =0
0.0622 0.0595 0.0624 0.0683 0.0814
0.0000 0.0000 0.0008 0.0007 0.0001
49 31 30 30 26
55.1990 37.0283 36.8984 36.7891 34.2564
Table 4 Comparison of the diagnosis obtained in runs of leaks of different magnitudes: noisy data between 2 and 5 % error alg ACO-DE DE ACO ACO-DE DE ACO
fault q f1 = 0.6 q f2 q f1 = 0.6 q f2 q f1 = 0.6 q f2 q f1 = 0.6 q f2 q f1 = 0.6 q f2 q f1 = 0.6 q f2
=0 =0 =0 = 0.6 = 0.6 = 0.6
mean q¯ f1 0.6081 0.5459 0.5038 0.5901 0.6068 0.6001
mean q¯ f2 0.0000 0.0000 0.0009 0.6013 0.6109 0.4683
mean iter 53 76 51 42 50 51
mean t(s) 38.2838 44.9995 57.3317 19.0167 20.0031 45.1023
Fig. 5 Evolution of the DE and ACO for a case of the table 2
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obtain better and faster diagnosis: start the minimization with ACO a few number of iterations, (no more than 30 taking in count the experimental results), and then use this solution as initial solution for DE. The table 4 shows the comparison between the hybrid algorithm ACO-DE and the diagnosis when using pure algorithms for two faults situations. Each algorithm was executed 30 times (each one starting from a different initial solution) for each fault situation and the table 4 shows the mean q¯ f1 , q¯ f2 obtained. In order to analyze the robustness of the diagnosis to unavoidable process disturbances, some numerical experiments were made with very noisy data (15- 20 % of noise). The Fig. 6 shows a simulation of the process behavior under disturbances which causes observations with noise between 15 and 20 %.
Fig. 6 Closed loop behavior of the process, noisy data 15-20 % Table 5 Comparison of the diagnosis obtained in runs of leaks of different magnitude: noisy data between 15 and 20 % error alg ACO-DE(best) DE(best) ACO(best)
q¯ f1 0 0 0.0600
q¯ f2 0 0 0.0900
iter 14 52 25
t 13.9289 16.3422 19.9327
ACO-DE(worst) 0 0 71 71.0147 DE(worst) 0.1191 0 100 67.5625 ACO(worst) 0.1900 0.0300 100 111.0318
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The table 5 shows the best and the worst diagnosis obtained for each algorithm when no faults are present but the system is under disturbances that are represented by noise on the measurable variables. The diagnosis based on the parameter estimation via DE and ACO seems to be robust.
5 Conclusions This preliminary study indicates that the application of bioinspired algorithms and their cooperative use characterize a promising methodology for the fault diagnosis problem based on a model which does not need to be linear. There are some advantages observed in the application of the two algorithms to the FDI problem: correct and fast diagnosis, easy structure, robustness to disturbances and less modeling errors due to the use of no linear model. For the detection problem some iterations of the ACO are enough, but for a correct diagnosis the DE algorithm showed better results. In general the cooperative algorithm ACO-DE shows faster diagnosis than pure DE or pure ACO. In this sense the study of a real cooperative strategies between this two nature inspired algorithms will be done: considering the influence of the parameter α in a more exploration version of the ACO algorithm and the parameter D of the DE algorithm in order to obtain a more exploitation version of DE.
References [1] Campos Knupp, D., Silva Neto, A.J., Figueiredo Sacco, W.: Estimation of radiactive properties with the particle collision algorithm. In: Inverse Problems, Design and Optimization Symposium, Miami, Florida, USA (2007) [2] Chen, J., Patton, R.J.: Robust model-based fault diagnosis for dynamic systems. Kluwer Academic Publishers, Dordrecht (1999) [3] Dolanc, G., Juricic, D., Rakar, A., Petrovcic, J., Vrancic, D.: Three-tank benchmark test. Tech. rep., Copernicus Project Report CT94-02337. J. Stefan Institute (1997) [4] Dorigo, M.: Ottimizzazione, apprendimento automático, ed algoritmi basati su metafora naturale. PhD thesis, Politécnico di Milano, Italia (1992) [5] Goldberg, D.E.: Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, MA (1989) [6] Isermann, R.: Process fault detection based on modelling and estimation methods– a survey. Automatica 30(4), 387–404 (1984) [7] Kirkpatrick, S., Gelatt Jr., C.D., Vecchi, M.P.: Optimization by simulated annealing. Science 220, 671–680 (1983) [8] Lobato, F.S., Steffen, V., Silva Neto, A.J.: Solution of inverse radiative transfer problems in two-layer participating media with differential evolution. Inverse Problems in Science and Engineering (15), 1–12 (2009) [9] Lobato, F.S., Steffen, V., Silva Neto, A.J.: Solution of the coupled inverse conductionradiation problem using multi-objective optimization differential evolution. In: 8th World Congress on Structural and Multidisciplinary Optimization, Lisboa, Portugal (2009)
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[10] Lunze, J.: Laboratory three tanks system -benchmark for the reconfiguration problem. Tech. rep., Tech. Univ. of Hamburg-Harburg, Inst. of Control. Eng., Germany (1998) [11] Patton, R.J., Frank, P.M., Clark, R.N.: Issues of fault diagnosis for dynamic systems. Springer, London (2000) [12] Sacco, W.F., Oliveira, C.R.E.: A new stochastic optimization algorithm based on particle collisions. In: 2005 ANS Annual Meeting, Transactions of the American Nuclear Society (2005) [13] Silva Neto, A.J., Moura Neto, F.D.: Problemas Inversos - Conceitos Fundamentais e Aplicações. EdUERJ (2005) [14] Simani, S., Patton, R.J.: Fault diagnosis of an industrial gas turbine prototype using a system identification approach. Control Engineering Practice 16, 769–786 (2008) [15] Storn, R., Price, K.: Differential evolution: A simple and efficient adaptive scheme for global optimization over continuous spaces. International Computer Science Institute (1995) [16] Wang, L., Niu, Q., Fei, M.: A novel quantum ant colony optimization algorithm and its application to fault diagnosis. Transactions of the Institute of Measurement and Control 30(3/4), 313–329 (2008) [17] Witczak, M.: Advances in model based fault diagnosis with evolutionary algorithms and neural networks. Int. J. Appl. Math. Comput. Sci. 16(1), 85–99 (2006) [18] Yang, E., Xiang, H., Gu, D., Zhang, Z.: A comparative study of genetic algorithm parameters for the inverse problem-based fault diagnosis of liquid rocket propulsion systems. International Journal of Automation and Computing 04(3), 255–261 (2007)
Abstract. This paper explores the application of bioinspired cooperative strategies for optimization on Fault Diagnosis in industrial systems. As a first step, the Differential Evolution and Ant Colony Optimization algorithms are considered. Both algorithms have been applied to a benchmark problem, the two tanks system. The experiments have considered noisy data in order to compare the robustness of the diagnosis. The preliminary results indicate that the proposed approach, basically the combination of the two algorithms, characterizes a promising methodology for the Fault Detection and Isolation problem.
1 Introduction The increases on the complexity of the industrial systems implies that the probability of fault occurrence is more significant. The faults change the characteristic properties of the system and produce its incapacity to fulfill the intended purpose, [6]. Therefore, an automatic supervisor should be used to detect and isolate, (FDI), the faults as early as possible. This is a reason for which in the last three decades a wide variety of FDI methods have been developed. Lídice Camps Echevarría Departamento de Matemáticas, Facultad de Ingeniería Mecánica, Instituto Superior Politécnico José Antonio Echeverría (ISPJAE), Ciudad de La Habana, Cuba e-mail: [email protected] Orestes Llanes-Santiago Departamento de Automática y Computación, Facultad de Ingeniería Eléctrica, Instituto Superior Politécnico José Antonio Echeverría (ISPJAE), Ciudad de La Habana, Cuba e-mail: [email protected] Antônio José da Silva Neto Department of Mechanical Engineering and Energy, Instituto Politécnico (IPRJ), Universidade do Estado do Rio de Janeiro, UERJ, Nova Friburgo, RJ, Brazil e-mail: [email protected]
J.R. González et al. (Eds.): NICSO 2010, SCI 284, pp. 53–63, 2010. springerlink.com © Springer-Verlag Berlin Heidelberg 2010
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L.C. Echevarría, O. Llanes-Santiago, and A.J. da Silva Neto
The FDI methods are divided in two general groups, those which use a model of the process and those which do not use it. Although many approaches have been developed, [2, 6, 11], robust FDI is still considered as a problem open to further research, [14], due to the the unavoidable process disturbances and the modeling errors which make almost unfeasible the use of many FDI methods in practical application.[11]. The FDI problem approach by the model-based methods has the following structure: based on some observations and the direct model, it is necessary to establish the causes of this observed behavior. In some cases, the identification of model parameter with fault of the system allows the FDI via the parameters estimation. Recently some articles have reported applications of meta heuristics to the FDI problems via parameters estimation, [16–18]. In this sense, the FDI via parameter estimation based on the approach of the meta heuristic algorithms seems to be an adequate alternative. The simple structure of these algorithms and their robustness reported in the solution of many parameters estimation inverse problems, [12], [1, 8, 9, 13], indicate that they are a promising alternative for FDI methods which need to be fast and simple (for online process) and robust to external perturbations. Moreover, estimations based on heuristic algorithms are absolutely viable when a nonlinear model is considered, making perfectly feasible the use of non linear models in order to prevent some modeling errors when linearizing the nonlinear process. This work presents the application of two bioinspired algorithms, Differential Evolution (DE), [15], and Ant Colony Optimization (ACO), [4], to the FDI problem in order to study and compare the capabilities of both algorithms and their combination for the FDI problems . As a case of study it has been simulated the problem of the two tanks system. This system is a simplified version of the three tanks system, which was adopted as a benchmark problem for FDI and reconfigurable control [10]. In order to verify and compare the robustness of the diagnosis, several simulations were made and different fault situations were considered. In all cases noisy data were considered. The results are presented using comparative tables and figures. The structure of the paper is the following: in the next section the basis of DE and the ACO are described. The third section shows the benchmark problem of the two tanks system. The section number 4 shows the simulations, the experimental results and the analysis of these results. Finally, section 5 summarizes the contributions and achievements of the paper.
2 Differential Evolution and Ant Colony Optimization This section describes the basis of the two algorithms that are used during this paper.
2.1 Differential Evolution The Differential Evolution (DE) was proposed around 1995, for optimization problems, [15]. DE is an improved version of the Goldberg’s Genetic Algorithm (GA), [5], taking the basis of Simulated Annealing (SA), [7]. Some of the most
Fault Diagnosis in Industrial Systems Using Bioinspired Cooperative Strategies
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important advantages of DE are: simple structure, simple computational implementation, speed and robustness, [15]. Basically, DE generates new parameter vectors by adding the weighted difference between a pair population vectors to a certain vector (the number of pair can be changed). This configuration is summarized by the notation DE/X /α /β where X denotes the vector to disturb, α the number of pair of vectors for disturbing X and β indicates the type of crossover to be used. In this case was considered DE/X jbest /1/bin. The key parameters of control in DE are the population size, N, the crossover constant, CR , and the weight applied to random differential or scaling factor, Fs . In [15] some simple rules for choosing the parameters of DE for any application are given: usually, N should be about 5 to 10 times the dimension of the variable of the problem, D and Fs lie in the range 0.4 to 1.0. Initially, D = 0.5 can be tried, and then can be increased if the population converges prematurely.
2.2 Ant Colony Optimization ACO was initially proposed, [4], for integer programming problems but it has been extended to continuous optimization problems. This algorithm is inspired on the behavior of ants seeking a path between their colony and a source of food. This behavior is due to the deposit and evaporation of pheromone. For the continuous case the idea of the ACO is to mimic this behavior with simulated ants which are identified with a feasible solution. The first step is to discretize the feasible interval of each variable of the problem in n possible values. On each iteration of the algorithm a family of N new ants are generated based on the information obtained from the previous ants. This information is saved on the pheromone probability matrix P f (dimensions m × n where m is the number of variables in the problem) which is updated at each iteration based on a evaporation factor Cevap and an incremental factor Cinc : α ∑l=1 [ fil (t)] α n ∑l=1 [ fil (t)] j
p fi j (t) =
(1)
where α = 1 and fi j is the element of the pheromone matrix which expresses the pheromone level of the discrete value j − esimo of the variable i, and it is updated on each iteration: fi j (t + 1) = (1 − Cevap) fi j (t) + δi j,best Cinc fi j (t)
(2)
3 The Two Tanks System The two tank system considered for study is represented in Fig.1. The system consists of two liquid tanks that can be filled with two similar and independent pumps acting on the tank 1 and tank 2, which have the same cross section S1 = S2 . The pumps deliver the flow rates q1 in tank 1 and q2 in tank 2. The
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L.C. Echevarría, O. Llanes-Santiago, and A.J. da Silva Neto
tanks are interconnected to each other through lower pipes. All the pipes have the same cross section S p . The liquid levels L1 and L2 in each tank are the controlled variables and they are measured with continuous valued level sensors. The variables q1 and q2 are chosen as manipulated variables to control the levels of tank 1 and tank 2. The system has two faults to be detected and isolated: • Fault 1 : Leak at the tank 1, an outflow with magnitude q f1 . • Fault 2 : Leak at the tank 2, an outflow with magnitude q f2 . The differential equations that describe the system, under the presence of faults, is derived from conservation of mass in the system of the two tanks: q1 q10 q12 q f1 L˙ 1 = − − − S1 S1 S1 S1
(3)
q2 q20 q12 q f2 − + − , L˙ 2 = S2 S2 S2 S2
(4)
and by the application of the Torricelli’s law: qi0 = μi S p 2gLi q i j = μi S p
2g Li − L j sign (Li − L j ) ,
where μi are flow coefficients and considering Ci = μi S p 2g,
Fig. 1 Two tanks system
(5)
(6)
(7)
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57
Table 1 Values of the constants of the two tanks system C1 , C2 S1 , S2 Sp Acceleration due to gravity, g
0.3028 m2 2.54 m2 0.1 m2 9.8 m/s2
the following system of equations is obtained: qf q1 C1 √ C1 − L1 − |L1 − L2 |sign (L1 − L2 ) − 1 L˙ 1 = S1 S1 S1 S1 √ q q2 C2 C1 f − L2 + |L1 − L2 |sign (L1 − L2 ) − 2 L˙ 2 = S2 S2 S2 S2 y1 = L1
(8)
y2 = L2 For more details see [3]. The goal here is to diagnosis the presence of the faults 1 or 2, even more, their magnitude. As a first approach it has been supposed that the leak at both tanks do not change in time and it is assumed that the magnitude of the leaks is less than 1000 ml/s. In other words, the following restrictions for the parameters q fi , i = 1, 2 have been established: q f1 , q f2 ∈ ℜ : 0 ≤ q f1 , q f2 ≤ 1 ml/s Estimation of the parameters q f1 and q f2 permit to diagnosis the system. In order to estimate these parameters, the following problem is formulated: min F (v) = min
N
∑
2 ¯ cal L¯ exp n − Ln (v)
(9)
n=1
t exp where v = q f1 , q f2 , L¯ n = (Ln1 , Ln2 )t are the observations of the liquid levels at t n n different instants of time, L¯ cal are the liquid levels computed by n = Lcal(1) , Lcal(2) the model (9) using Runge Kuta 4. The table 1 shows the values of the constants considered in the model of the two tanks system.
4 Results and Discussion The closed loop behavior of the process was simulated when no faults are present. This behavior is shown in Fig. 2.
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Fig. 2 Closed loop behavior of the process when no faults are present, noise data 2-5 %
Fig. 3 Closed loop behavior of the process when leaks of 200 ml/s are present in both tanks, noise data 2-5 %
The closed loop behavior of the process when a leak of magnitude 200 ml/s in each tank (q f1 = q f2 = 0.2) is introduced at time t = 20 s is shown in Fig. 3. The closed loop behavior of the process when a leak of magnitude 50 ml/s in tank 2 (q f1 = 0, q f2 = 0.05) is introduced at time t = 20 s is shown in Fig. 4. The effect of this leak in tank 2 is graphically imperceptible. In order to diagnosis the faults, the minimization of the objective function F (v) was implemented, in the first case based on the DE algorithm. The population was considered to be 10 and the mutation mechanism is (DE/xbest j /1/bin). The second case considered the minimization of the objective function by means of ACO, described on 2.2, with 10 ants. Both algorithms stop when 100 iterations are achieved.
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Fig. 4 Closed loop behavior of the process when leaks of 50 ml/s is present in tank 2, noise data 2-5 %
Table 2 Diagnosis obtained in five runs of leaks of 200 ml/s in each tank: noisy data between 2 and 5 % error alg fault q¯ f1 q¯ f2 iter t(s) DE q f1 = q f2 = 0.2 0.1827 0.1901 93 56.5681 DE q f1 = q f2 = 0.2 0.1985 0.2034 67 29.5475 DE q f1 = q f2 = 0.2 0.1988 0.2062 65 28.3845 DE q f1 = q f2 = 0.2 0.2036 0.2044 60 25.5477 DE q f1 = q f2 = 0.2 0.1993 0.1969 56 23.8943 ACO ACO ACO ACO ACO
q f1 q f1 q f1 q f1 q f1
= q f2 = q f2 = q f2 = q f2 = q f2
= 0.2 = 0.2 = 0.2 = 0.2 = 0.2
0.1738 0.1775 0.1832 0.2087 0.1860
0.2414 0.2287 0.2198 0.2401 0.1829
75 57 54 44 41
75.0830 48.0647 45.9054 34.9798 33.9888
The tables 2 and 3 shows the results of the diagnosis of different faulty situation by both algorithms. All cases considered data with 2-5 % of noise. The abbreviations used in the tables are alg for algorithm and iter for number of iterations. The notation introduced is t for the computing time, in seconds, of the algorithm. Both algorithms detected the presence of faults but the DE algorithm is more accurate in the determination of the leak magnitudes. Both algorithms are fast, which is good for the online diagnosis, but DE is faster. In Fig. 5 are shown the evolution of both algorithms for two situation described in table 2. The figures suggest a way of combination of ACO and DE in order to
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Table 3 Diagnosis obtained in five runs of a leaks of 50 ml/s in tank 1: noisy data between 2 and 5 % error alg DE DE DE DE DE
fault q f1 = 0.05 q f2 q f1 = 0.05 q f2 q f1 = 0.05 q f2 q f1 = 0.05 q f2 q f1 = 0.05 q f2
=0 =0 =0 =0 =0
q¯ f1 0.0498 0.0515 0.0508 0.0531 0.0489
q¯ f2 0.0007 0.0000 0.0000 0.0000 0.0001
iter 79 64 64 43 42
t(s) 46.8123 35.9895 35.4267 28.7774 18.0390
ACO ACO ACO ACO ACO
q f1 q f1 q f1 q f1 q f1
= 0.05 q f2 = 0.05 q f2 = 0.05 q f2 = 0.05 q f2 = 0.05 q f2
=0 =0 =0 =0 =0
0.0622 0.0595 0.0624 0.0683 0.0814
0.0000 0.0000 0.0008 0.0007 0.0001
49 31 30 30 26
55.1990 37.0283 36.8984 36.7891 34.2564
Table 4 Comparison of the diagnosis obtained in runs of leaks of different magnitudes: noisy data between 2 and 5 % error alg ACO-DE DE ACO ACO-DE DE ACO
fault q f1 = 0.6 q f2 q f1 = 0.6 q f2 q f1 = 0.6 q f2 q f1 = 0.6 q f2 q f1 = 0.6 q f2 q f1 = 0.6 q f2
=0 =0 =0 = 0.6 = 0.6 = 0.6
mean q¯ f1 0.6081 0.5459 0.5038 0.5901 0.6068 0.6001
mean q¯ f2 0.0000 0.0000 0.0009 0.6013 0.6109 0.4683
mean iter 53 76 51 42 50 51
mean t(s) 38.2838 44.9995 57.3317 19.0167 20.0031 45.1023
Fig. 5 Evolution of the DE and ACO for a case of the table 2
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obtain better and faster diagnosis: start the minimization with ACO a few number of iterations, (no more than 30 taking in count the experimental results), and then use this solution as initial solution for DE. The table 4 shows the comparison between the hybrid algorithm ACO-DE and the diagnosis when using pure algorithms for two faults situations. Each algorithm was executed 30 times (each one starting from a different initial solution) for each fault situation and the table 4 shows the mean q¯ f1 , q¯ f2 obtained. In order to analyze the robustness of the diagnosis to unavoidable process disturbances, some numerical experiments were made with very noisy data (15- 20 % of noise). The Fig. 6 shows a simulation of the process behavior under disturbances which causes observations with noise between 15 and 20 %.
Fig. 6 Closed loop behavior of the process, noisy data 15-20 % Table 5 Comparison of the diagnosis obtained in runs of leaks of different magnitude: noisy data between 15 and 20 % error alg ACO-DE(best) DE(best) ACO(best)
q¯ f1 0 0 0.0600
q¯ f2 0 0 0.0900
iter 14 52 25
t 13.9289 16.3422 19.9327
ACO-DE(worst) 0 0 71 71.0147 DE(worst) 0.1191 0 100 67.5625 ACO(worst) 0.1900 0.0300 100 111.0318
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The table 5 shows the best and the worst diagnosis obtained for each algorithm when no faults are present but the system is under disturbances that are represented by noise on the measurable variables. The diagnosis based on the parameter estimation via DE and ACO seems to be robust.
5 Conclusions This preliminary study indicates that the application of bioinspired algorithms and their cooperative use characterize a promising methodology for the fault diagnosis problem based on a model which does not need to be linear. There are some advantages observed in the application of the two algorithms to the FDI problem: correct and fast diagnosis, easy structure, robustness to disturbances and less modeling errors due to the use of no linear model. For the detection problem some iterations of the ACO are enough, but for a correct diagnosis the DE algorithm showed better results. In general the cooperative algorithm ACO-DE shows faster diagnosis than pure DE or pure ACO. In this sense the study of a real cooperative strategies between this two nature inspired algorithms will be done: considering the influence of the parameter α in a more exploration version of the ACO algorithm and the parameter D of the DE algorithm in order to obtain a more exploitation version of DE.
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