State-Space GMDH Neural Networks for Actuator Robust Fault

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Advances in Electrical and Computer Engineering

Volume 12, Number 3, 2012

State-Space GMDH Neural Networks for Actuator Robust Fault Diagnosis Marcin MRUGALSKI, Marcin WITCZAK Institute of Control and Computation Engineering, University of Zielona Góra, ul. Podgórna 50, 65-246 Zielona Góra, Poland {M.Mrugalski,M.Witczak}@issi.uz.zgora.pl 1

Abstract—Most fault diagnosis methods focus on the fault detection of the system or sensors and do not take into account the problem of the fault detection and isolation of the actuators, which are an important part of the contemporary industrial systems. To solve such a problem, the system outputs and inputs estimator based on a dynamic Group Method of Data Handling neural network in the state-space representation is proposed. In particular, the methodology of the adaptive thresholds calculation for system inputs and outputs is presented. The approach is based on the application of the Unscented Kalman Filter and Unknown Input Filter is presented. This result enables performing robust fault detection and isolation of the actuators. The final part of the paper presents an application study, which confirms the effectiveness of the proposed approach. Index Terms—Fault diagnosis, robustness, actuators, neural networks, system identification.

I. INTRODUCTION Every engineering system can malfunction and fail due to faults in its components. In technical automatic and control systems, defects may happen in sensors or actuators. Component faults can develop into a failure of the whole system. Therefore, to protect system and provide dependability several the Fault-Tolerant Control (FTC) approaches are used [4], [9], [15], [22], [23]. The key issue of the FTC is that faults are prevented from developing into a system failure. Thus, one of the crucial FTC tasks is a Fault Detection and Isolation (FDI) [2], [5], [14], [19], [31]. The effectiveness of the FTC and FDI systems mostly depends on the quality of used models. These models are often obtained in the process of system identification with the application of the Artificial Neural Networks (ANNs) [8], [21], [26]. The advantage of the application of the ANNs is the possibility of obtaining the model which describes behaviour of identified system only on the basis of measurement data. It is especially attractive in the case when laws of physical describing the system behaviour are not available or they are too complex [9], [24]. Moreover, the ANNs characterize a good generalization and approximation abilities of non-linear systems. From the other hand, the ANNs have also some disadvantages e.g. inefficient quality of the dynamic neural models, usually not available description of a neural model in the state-space representation and only rare approaches ensure the stability of the neural models during the process The work was supported by the National Science Centre of Poland under grant: 2011-2014.

of the dynamic system identification. Moreover, there are a limited number of approaches that allow to describe mathematically the neural model uncertainty, which has the main influence on the performance of the FDI and FTC systems [29]. The most desirable feature of the neural models applied to the FDI and FTC schemes is a small modeling uncertainty, which is defined as a mismatch between the model and the system being considered [3]. Therefore, it is important to use an approach reducing the contribution of the structure errors and the parameter estimation inaccuracy of the neural model uncertainty. To solve such a challenging problem, a Group Method of Data Handling (GMDH) neural network [10], [12], [20], [13], [18] have been proposed. The concept of the GMDH approach relies on replacing the complex neural model by the set of the hierarchically connected partial models, which can be chosen with the application of appropriate selection methods. Besides the reduction of the neural model inaccuracy it is important to calculate the model uncertainty in the form of mathematical description allowing performing the robust fault diagnosis. In this paper, a new approach to the actuator and sensors fault detection and isolation is proposed. To solve such a challenging problem, the system outputs and inputs estimator based on the dynamic GMDH neural network is employed. In order to achieve this goal a new structure of the dynamic neuron in the state-space representation is proposed. This description enables to obtain constraints of the parameter estimates which warranty the stability of dynamic GMDH neural model. In order to obtain the constrained parameter estimates and the neural model uncertainty description the Unscented Kalman Filter (UKF) [7], [27], [28] was used. This knowledge enables to calculate the output adaptive threshold. Moreover, the methodology of estimation of the GMDH neural model inputs and the calculation of the input adaptive threshold with the application of the Unknown Input Filter (UIF) is proposed. The obtained input and output adaptive thresholds allow to perform the robust fault diagnosis and actuators isolation of the dynamic non-linear systems. The paper consists of six sections. After the introduction, presenting the subject of this paper, in Section II a concept of robust fault detection and isolation of sensors and actuators is presented. Section III presents the process of synthesis of GMDH neural model. In section IV, the estimation of neural model inputs is realized with the UIF. The experimental results of the robust fault diagnosis are shown in section V. Finally, section VI concludes the paper.

Digital Object Identifier 10.4316/AECE.2012.03010

65 1582-7445 © 2012 AECE




II. ROBUST FAULT DETECTION AND ISOLATION LOGIC Let us consider a classical system scheme which consists of process and some amount of actuators and sensors (Fig. 1).

Figure 3. Application of adaptive threshold to robust fault detection of sensors.

Figure 1. System scheme with multiple actuators and sensors.

The model-based fault diagnosis can be defined as the detection and isolation of faults in the system based on a comparison of the system available measurements with information represented by a model [24]. The most of the fault diagnosis methods which are based on the neural networks allow performing only the fault detection with the application of the residual and the constant threshold. In this approach it is assumed that the residual ε k , defined as a difference of the system y k and the nominal model response yˆ k , should be close to zero in the fault-free case, and it should be distinguishably different from zero in the faulty case. Under such an assumption, the faults are easily detected when the residual cross arbitrary defined threshold  t . The difficulty with such a kind of symptom evaluation approach relies on the corruption of y k by noise and disturbances. Another difficulty follows from the fact that the neural model obtained during the system identification is usually uncertain [30], what results in the inappropriate work of the fault detection system e.g. the undetected faults or false alarms appears. Unfortunately, there are only some papers which describe approaches to robust fault detection with application of adaptive thresholds. The concept of the proposed approach is illustrated in Fig. 2.

In order to perform the fault isolation the method relying on the application of the bank of neural models, which generate the residuals for each fault, is usually used. In practice such an approach is difficult to realize because it is often not possible to provide the data allowing designing neural models of particular faults. In this work, a new approach of fault detection and isolation of the actuators is presented. For this aim the methodology of neural identification of the diagnosed system with its uncertainty description is proposed. This methodology enables fault detection with the application of the output adaptive threshold. Moreover, the methodology of estimation of the neural model inputs with the application of the Unknown Input Filter is shown. This method allows obtaining the adaptive thresholds for each input signal of the diagnosed system. In the consequence this approach enables to perform robust fault detection and isolation of the actuators simultaneously (Fig. 4). An occurrence of the fault for each i-th actuator is signaled when input u i , k crosses the system input uncertainty interval (Fig. 5): ˆ kM uˆ m k  uk  u

(2)

Figure 4. Scheme of robust fault detection and isolation of actuators.

Figure 2. Scheme of robust fault detection system of sensors.

The proposed technique relies on the calculation of the adaptive threshold based on the model uncertainty. This threshold should contain the real system response in the fault-free mode. An occurrence of the fault is signaled when system output y k crosses the adaptive threshold (Fig. 3):

ˆ kM yˆ m k  yk  y 66

(1)

Figure 5. Application of adaptive threshold to robust fault detection and isolation of actuators.




III. THE SYNTHESIS OF GMDH NEURAL MODEL The behaviour of each partial model (neuron) of the GMDH neural network should reflect the behaviour of the system being identified. It follows from the rule of the GMDH algorithm that the parameters of each partial model are estimated in such a way that its output is the best approximation of the real system output. In this situation the partial model should have the ability to represent the dynamics. To tackle this problem, in this paper a dynamic neuron in the state-space representation is defined. The proposed dynamic neuron consists of two submodules: the linear state-space module and the activation module. The behavior of the linear state-space part of a neuron is described by the following equation: (3) z k 1  Az k  Bri(,lk) ~s (l )  Cz k i, j, k

(4)

where ri(,lk)   nr for i  1,..., n R are the neuron input vectors

formed as the combinations of the neural model inputs ri(,lk)  [u i(,lk) ,..., u (jl,)k ]T , and sˆ i(,l )j , k   ns for j  1,..., n N are

the outputs vectors of the linear state-space submodule of the dynamic neuron formed as the combinations of the network outputs sˆ i(,l )j , k  [ yˆ i(,lk) ,..., yˆ (jl,)k ]T , and l is the number of layer the GMDH network. Moreover, A   n z  nr

ns n z

n z n z

,

B , C , z k   , where n z represents the order of the dynamics [17]. Additionally, the matrix A has an uppertriangular form, i.e. a1,1 a1, 2  a1, n z   0 a a2, n z  2, 2  A (5)         0  an z , n z   0 This means that the dynamic neuron is asymptotically stable iff: (6) ai ,i  1 , i  1,..., nz moreover: C  diag (c1 ,..., cn s , 0 ,..., 0)

nz

(7)

n z  ns

The linear state-space submodule output is used as the input for the activation module: (8) sˆ i(,l )j , k  F (~si(, lj), k ) with F ()  [ f1 (),..., f ns ()]T where f i () denotes a nonlinear activation function (e.g., a hyperbolic tangent). In order to estimate the unknown parameters of the dynamic neurons the Unscented Kalman Filter [7], [27], [28] can be applied. Moreover, an application of this algorithm to the parameter estimation process enables to obtain the uncertainty of the partial models which can be applied in the robust fault detection scheme. After the estimation, the parameters of the neurons are not modified during the further network synthesis. Let us define a state vector:

x k  [p k

z k ]T

(9)

which is composed of the parameter vector of the neuron p k as well as of the state of the neuron, which is described in a form: (10) z k 1  A (p k ) z k  B(p k )ri(,lk) ~s (l )  C(p ) z (11) i, j, k

k

k

sˆ i(,l )j , k  F( ~si(, lj), k )

(12)

The p k is composed of the diagonal elements of A , i.e. p k  [a1,1 ,..., an , n ,...]T

(13)

while the remaining elements of p k are composed of the remaining parameters of A , as well as all elements of B and C . Thus, the dimension of p k is: (n z  n z )  n z (14)  n z  nr  n s  n p 2 It should be also pointed out that instead of A , (B, C) the dim(p k ) 

notation A (p k ) , (B(p k ), C(p k )) is introduce which clearly denotes the dependence on p k . Finally, the state-space model of the dynamic neuron is: pk   x k 1   (l )    k  (15)  A(p k )z k  B(p k )ri , k 

 G(x k , ri(,lk) )   k sˆ i(,l )j , k  F(C(p k ) z k ))  ν k  H( x k )  ν k

where G     n

nr



n

(16)

and H     n

s

are the

process and observation models, respectively.  k 1   n is the process noise, and ν k   n is the measurement noise. It is assumed that the process noise and the measurement noise are uncorrelated.  (x 0 ),  ( k 1 ),  ( ν k ) are the Probability Density Function (PDF), where x 0 is the initial state vector.

Moreover, mean and covariance of  ( k 1 ) and  ( ν k ) are known and equal to zero and Q, R respectively. The profit function which is the value of the conditional PDF of the state vector x k   n given the past and present measured data s1 ,..., s k is defined as follows:

J (x k )   (x k | (s1 ,..., s k )) (17) The parameter and state estimation problem can be defined as the maximization of (17). In order to solve the following problem the UKF can be applied. UKF employs the unscented transform [11], which approximates the mean sˆ k   ns and covariance Pkss   ns ns of the random vector sk

obtained

from

the

non-linear

transformation

s k  H(x k ) , where x k is a random vector, which means xˆ k   ns and covariance Pkxx  nn are known. The task of training of dynamic neuron relies on the estimation of parameters vector x k which satisfies the following interval constraint:

 1    eTi x k  1   where:

ei  

(18) n p n

whereas

e1  [1,0,...,0]T ,

e 2  [0,1,...,0]T ,…, e n p  n  [0,0,...,1]T , and  is a small

67



positive value. These constrains follow directly from the asymptotic stability condition (6). While  is introduced in order to make the above mentioned problem tractable. The neural model has a cascade structure what follows from the fact that the neuron outputs constitute the neuron inputs in the subsequent layers. The neural model which is the result of the cascade connection of dynamic neurons is asymptotically stable, when each of neurons is asymptotically stable [16]. So, a fulfillment of (6) (being a result of (18)) for each neuron allows obtaining an asymptotically stable dynamic GMDH neural model. Thus, the objective of the interval constrained parameterestimation problem is to maximize (17) subject to (18). In order to perform the neuron training process it is necessary to truncate the probability density function at the n constraint edges given by the rows of the state interval constraint (18) such that the pseudo mean xˆ tk ,k of the truncated PDF is an interval-constrained state estimate with the truncated error covariance Pkxx,k . The probability density function truncation procedure allows avoiding the explicit on-line solution of a constrained optimization problem at each time step. Moreover, it assimilates the intervalconstraint information in the state estimate xˆ tk , k and the error covariance Pkxxt , k . The details of the PDF truncation procedure in the paper [28] can be found. At the next stage of GMDH network synthesis, a validation data set V is used to calculate a processing error of each partial model in the current l-th network layer. The evaluation of the processing errors Q for each neuron outputs is performed after the generation of the corresponding layer of neurons. Based on the defined evaluation criterion, it is possible to select the best-fitted neurons in the layer. Selection methods in the GMDH neural networks play the role of a mechanism of structural optimization at the stage of constructing a new layer of neurons. According to the chosen selection method, elements that introduce too big processing error are removed. After the selection procedure, the outputs of the selected neurons become the inputs to the neurons in the subsequent layer. During the synthesis of the GMDH neural network, the number of layers suitably increases (Fig. 6).


The synthesis of the GMDH network is completed when the network fits the data with a desired accuracy or the introduction of new neurons did not introduce a significant increase in the approximation abilities of the neural network. The application of the UKF allows obtaining the state estimates as well as the uncertainty of the each partial model and whole GMDH network in the form of matrixes P xxt which can then be applied to the calculation of the adaptive thresholds and to perform robust fault detection: yˆ im, k  Fi (c i xˆ k  t nt /2n p ˆ i ci P xxt cTi )

(19)

yˆ iM, k  Fi (ci xˆ k  t nt /2n p ˆ i ci P xxt cTi )

(20)

where c i stands for the i-th row i  1,..., ns of the matrix C of the output neuron, ˆ i is the standard deviation of the i-th fault-free residual and t nt /2n p is the t-Student distribution quantile. IV. ESTIMATION OF NEURAL MODEL INPUTS VIA THE UNKNOWN INPUT FILTER Neuron model, which was presented in previous section, can be shown in canonical observer representation [25]: x k 1  Ax k  Brk  k (21) sˆ k 1  F(Cx k 1 )  ν k 1 (22) Because F() (i.e. f ()  tanh() , i  1,..., ns are selected in this paper) is invertible, assume that: ~s  F 1 (sˆ ) (23) k k therefore (22) can be replaced by, which is linear with respect to the state: ~s  Cx  ν (24) k 1 k 1 k 1 To estimate state and input, the results of work [6] are used. The recursive filter is proposed and the filter equations are given below. The recursive part of filter consists of three steps: the estimation of the unknown input, the measurement update and time update. These three steps are given by: 1) Estimation of unknown input: ~ R k  CPkxx| k 1CT  R k ~ ~ H k  (FkT R k1Fk ) 1 FkT R k 1 rˆk  H k ( ~sk  Cxˆ k | k 1 )

(25) (26) (27)

where: Fk  CB . 2) Prediction: xˆ k 1| k  A k xˆ k | k  Brˆk Pkxx| k 1  (I n  BH k C)Pkxx1| k 1 (I n  BH k C)T  BH k R k H Tk BT

3) Update: ~ K k  ( Pkxx| k 1CT  BH k R k ) kT ( k R k | k 1 kT ) 1 k ~ ~ R k | k 1  (I p  CBH k ) R k (I p  CBH k )T

Figure 6. Synthesis of the GMDH neural network.

68

(28) (29)

(30) (31)

xˆ k | k 1  xˆ k | k 1Bs

(32)

xˆ k | k

(33)

 xˆ k | k 1  K k ( ~sk  Cxˆ k | k 1 )




(34) Pkxx| k  Pkxx| k 1  K k ( Pkxx| k 1CTk  S k | k 1 )T ~  k  [0 I r ]UTk S k1   r  n s (35) where: U is an orthogonal matrix containing the left ~ ~ singular vectors of S k1C k B k 1 in its columns, S k   ns ns ~ ~ ~ is an invertible matrix satisfying S k S Tk  R k , which can always be found by Cholesky factorization and n n

I n y   y y , I n   nn , rank (CB)  rank (B)  nr and ~ rank (R k |k 1 )  r . Note, that the time and measurement

update of the state estimate take the form of the UIF, except that the true value of the input is replaced by an optimal estimate. The equations (25)-(34) enable the estimation of the input signals of the neurons in the last layer of the neural network. Moreover, these signals represent the outputs of the neurons from the previous layer of the GMDH neural network (Fig. 7).

which can be written as follows:

d k  ziT xk  ek where: d k  ak  hiT ~ sk 1 T~ e b h s

(40) (41)

(42) In order to perform the estimation of the neuron inputs and its adaptive thresholds once again it is necessary to apply the procedure of PDF truncation [28] at the n constraint edges given by the rows of the state interval constraint (40) where xˆ k , k and Pk|xxk described by (33) and k

k

i

k 1

(34) be respectively, the pseudo-mean and pseudocovariance obtained from UIF. Knowing that the covariance ~ matrix of the input estimate is P  (F T R k 1F) 1 , the adaptive thresholds for the inputs of the GMDH neural model receive the following form: uîm, k  uî , k  t nt  n r 1 ˆ i2 Pii

(43)

uîM, k  uî , k  t nt  n r 1 ˆ i2 Pii

(44)

V. EXPERIMENTAL RESULTS

Figure 7. Estimation of the system inputs via GMDH model and UIF.

It should be underlined that the estimated signals have to be contained in the bounded interval. It follows from the fact, that the responses of the neurons from previous layer are limited by the non-linear activation function of the neuron (e.g. in the case of the tangent activation function this interval is bounded by the values a  1 and b  1 . Thus, the task of training of neuron relies on the estimation of neuron inputs which satisfies the following interval constraint: a k r k b k (36)

The objective of this section is to design a fault detection scheme based on the approaches described in the previous sections. For the modeling and fault diagnosis purpose a Maxon DC motor RE25-10W 118743 was chosen (Fig. 8) [1]. It consists of two pole permanent magnet and precious metal brushes. This brushed motor has no magnetic cogging and the ironless winding. The motor has a high acceleration thanks to a low inertia. Due to a eleventh segmented commutator there is only a small torque ripple. The winding has very specific advantages which cause that there is no magnetic detent and minimal electromagnetic interference. The efficiency of the motor is up to 90%. The nominal voltage is 12V and no load current 26mA.

where a k   n and b k   n , a j ,k b j , k for j  1,...,n r are the known minimum and maximum allowed values of signal generated by the neurons from the previous layer of the network. This signal can be defined as follows: rk  H k ~sk 1  Zx k (37) where: Z  H k CA . Let us assume that r  hT ~s  z T x i, k

i

k 1

i

k

Figure 8. Brushed Maxon DC motor RE25-10W 118743.

(38)

where: hiT and z iT denote i-th rows of matrixes H and Z , respectively. Substituting (38) into (36) the following interval constraint can be obtained: a  hT ~ s  z T x  b  hT ~ s (39) k

i

k 1

i

k

k

i

The model of the brushed DC motor which has a connection to a load implemented in the Matlab Simulink [1] was used to generate the data sets applied to modelling purpose (Fig. 9).

k 1

69




Figures 10 and 11 show the rotor angle  r, k and the rotor's angular velocity r, k of the DC motor and the output adaptive thresholds obtained with the application of equations (19-20) for the validation data set (no fault case).

Figure 9. Scheme of the brushed DC motor with load implemented in the Matlab Simulink [1].

At the beginning of the experiment the model of DC motor with the application of the GMDH network was built. The modeled motor can be considered as a one-input and five-output system (i k ,  r , k , r , k ,  l , k , l , k )  r (u k ) , where u k is the input voltage to the motor, ik represents the motor

current,  r, k denotes the rotor angle, r, k is the rotor's angular velocity, l, k represents the load angle and l, k denotes the load angular velocity. The nominal parameters of the DC motor have the following values: R  2.18  - motor resistance,  Lm  0.000238 H - motor inductance,  

kb  0.0235 Vs / rad - back-emf constant,



kt  0.0235 Nm / A - torque constant,



J m  1.07e  6 kgm2 - rotor inertia,

Figure 10. Rotor angle  r , k of the DC motor and the corresponding output adaptive threshold obtained with the GMDH neural model.

bm  12e  7 - mechanical damping (linear model of friction), whereas the assumed nominal parameters of the load are: J l  10  J m - load inertia,  



bm  12e  6 - load damping,

K  100 - spring constant for connection rotor/load,  b  0.0001 - spring damping for connection rotor/load. The assumption of nominal values of parameters of motor implemented in Matlab allowed to generate the two 1000-th samples data sets used for the identification and validation purpose. Moreover, it was assumed that the disturbances affecting data were generated according to Gaussian distribution and had value 5% of motor input and outputs. The parameters of the dynamic GMDH neural model were estimated with application of the UKF algorithm, which was explained in section III. The selection of best performing neurons in terms of their processing accuracy was realized with the application of the soft selection method [13] based on the sum of squared error. The architecture of the GMDH neural network obtained in the process of synthesis consisted of the three layers of neurons. 

70

Figure 11. Rotor's angular velocity  r , k of the DC motor and the corresponding output adaptive threshold obtained with the GMDH neural model.

After the synthesis of the GMDH model, it was possible to employ it for robust fault detection of the DC motor. In order to achieve this goal the model of the DC motor implemented in the Matlab Simulink was also applied to generate the data containing the faults in the electrical and mechanical subsystems of the motor. The first fault, in the electrical subsystem, relied on changing of the motor resistance up to 1000% nominal values i.e. R  21.8  . Figure 12 presents the detection of the resistance change, when the rotor's angular velocity r,k crosses (for k  750 ) the output adaptive threshold obtained with the application of the dynamic GMDH neural model.



Figure 12. Detection of the resistance change with the rotor's angular velocity  r , k and its output adaptive threshold.


Figure 14. Detection of the load bumping change with the application of the motor current ik and its output adaptive threshold.

The second fault simulated in the mechanical part of the motor, relied on change of the rotor inertia from the nominal value J m  1.07e  6 kgm2 to J m  1.07e  5 kgm2 . Such type of damage can be caused by changing of the mechanical parameters of rotor or bearings. As it can be seen in Fig. 13, the fault was detected for k  750 when value of the motor current ik crossed its output adaptive threshold.

Figure 15. Detection of the load interia change with the application of the load angle  l , k output signal and its output adaptive threshold.

In the next stage of the experiment the input signals of the GMDH neural model were estimated with the application of the approach presented in the section IV. The input voltage u1 and the corresponding adaptive threshold for the validation data are given in Fig. 16.

Figure 13. Detection of the rotor interia change with the application of the motor current ik and its output adaptive threshold.

The third fault simulated in the load of the motor, relied on change of the load dumping from the nominal value bm  12e  6 to bm  24e  6 . Once again for the fault detection purpose the current output adaptive threshold was successively used. As it can be seen at figure 14, fault was detected for k  750 when value of the motor current ik crossed its output adaptive threshold. The fourth fault simulated in the load of the motor, relied on change of the load inertia from the nominal value J l  10  J m to J l  1000  J m . In order to detect this fault the load angle l, k output signal and its output adaptive threshold were applied (Fig. 15).

Figure 16. Input signal of the DC motor and the input adaptive threshold obtained with the application of the GMDH neural model.

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Figure 17 presents the detection of turning down voltage by the corresponding input adaptive threshold. As it can be seen the fault was detected for k = 750 when the value of the voltage crossed the input adaptive threshold.

Volume 12, Number 3, 2012 [7] [8] [9] [10] [11] [12]

[13] [14] [15]

Figure 17. Detection of turning down voltage with application of the input adaptive threshold.

[16]

VI. CONCLUSION

[17]

The objective of this paper was concerned with designing the robust fault detection system based on the dynamic GMDH neural network. The state-space representation of the neurons and application of the unscented Kalman filter to its parameters estimation allows to obtain the stable nonlinear dynamic GMDH neural model. Moreover, the application of the unscented Kalman filter enables to calculate the outputs adaptive thresholds of the GMDH neural model. Moreover, in the paper the GMDH neural model inputs estimation approach using the unknown input filter was developed. This algorithm allows to calculate the adaptive thresholds for the input signals of the diagnosed system. The developed approach enables to perform the robust fault detection and isolation of the actuators. Finally, in the experimental part of the paper the results of application of the proposed approach to the identification and robust fault detection of the brushed DC motor are presented.

[18]

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