Modelling for fault detection and isolation versus modelling for control

Mathematics and Computers in Simulation 53 (2000) 259–271

Modelling for fault detection and isolation versus modelling for control P.M. Frank∗ , E. Alcorta Garc´ıa, B. Köppen-Seliger Department of Measurement and Control, Gerhard-Mercator-Universität Duisburg, Bismarckstr 81 BB, D-47048 Duisburg, Germany Accepted 28 July 2000

Abstract The goal of this paper is to emphasize both the particularities of models needed for model-based fault detection and isolation (FDI) and the differences with respect to the models used in control. Of special interest is the question of complexity. This depends basically on the given situation such as the kind of plant, the kind and number of faults to be detected, the demands for fault isolation, robustness and the measurements available. However, in contrast to the wide-spread opinion that models for FDI have always to be more complex than those for control, the paper shows that diagnostic models for controllable and observable plants comprise only a partial description of the input/output model and are therefore less complex than those for control. This issue is discussed in terms of different model-based FDI approaches — analytical, data- and knowledge-based. As for the analytical approaches the necessary order of the diagnostic model is that of the transfer operator from the fault vector to the system output. © 2000 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: Modelling; Fault detection and isolation; Residuals

1. Introduction A key issue in the design of fault tolerant control systems that aim at increased reliability and safety is fault detection and isolation (FDI), which has received much attention in the last two decades. A number of different FDI approaches making use of either output signal processing or of a model of the process have been proposed over the years. The most powerful approaches are those using a process model, where either quantitative, qualitative, knowledge-based or data-based models or combinations of them are applied. The basic idea behind the model-based FDI approach is to take advantage of the nominal model of the system to generate residuals that contain information about the faults. Evidently, the quality of the ∗ Corresponding author. Tel.: +49-203-379-3386; fax: +49-203-379-2928. E-mail address: [email protected] (P.M. Frank).

0378-4754/00/$20.00 © 2000 IMACS. Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 4 7 5 4 ( 0 0 ) 0 0 2 1 2 - 3

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model is of fundamental importance for both fault detectability and isolability and the avoidance of false alarms. Since the early work of [3], system models common in control theory, which represent the dynamical behavior between the system inputs and outputs (or states), have been taken for the design of FDI systems, although it is well known, that different kinds of applications require different types of models. For control system analysis and design, the system model has to represent the dynamical input-output behavior of the system and should be as simple as possible. Hence, the model used is often drastically simplified and linearized ignoring many of the attributes of the physical nature of the system and only retaining the attributes that are deemed relevant for the behavior of the resulting control system. Not so in FDI: here one needs a representative model of high fidelity and in some cases high preciseness which is in general of higher complexity than the one for control. But under certain circumstances, models for FDI can also be simpler than those for control, which has often been overseen in the FDI society. The key point is that for FDI one needs only that part of the model which reflects the faults of interest and, with respect to robustness, is not or only weakly affected by disturbances and modelling uncertainty. Clearly, the resulting submodel highly depends upon several factors such as the kind and number of faults to be detected, the disturbances, uncertain parameters, and available measurements. Hence, the first step of determining the reduced model for FDI is to find out that input/output measurable part of the system which contains the sources of the faults. Then, in a second step, this model can be reduced by confining on that part of it which is most strongly affected by the faults and most weakly influenced by the modelling uncertainties and unmodelled disturbances. In this paper we focus our attention on the second step. For the different types of quantitative, qualitative and data-based models it is shown that for systems of minimal order (controllable and observable) and a specific set of faults to be monitored, the required diagnostic model comprises in general only a partial description of the original system. This issue is discussed in terms of different model-based FDI approaches. Note that the principal basic ideas of this paper are on the line with those discussed in [4], where the modelling for FDI in the context of fault isolation is the main concern. In addition to the ideas related to fault isolability, this paper focuses on the complexity of the models.

2. Features of models for residual generation The topic addressed in this section is the analysis of the features of the representative models for FDI compared to the models of control. Of special concern is the question of complexity. To the best knowledge of the authors, this issue has not been studied systematically so far. It has been considered previously only indirectly as a by-product of the results of specific approaches. Some general considerations are found in [4] and, for specific approaches some examples are given in [13] for the observer-based approach and in [6] for the parity space approach. Our aim is to show from a more general point of view in what sense representative models for FDI are not coincident with the input/output models for control. First, it is evidenced that the diagnostic model can be of higher order than the input/output model for control if the system is not of minimal order (not controllable). Then it is demonstrated that in the case of minimal order systems the different model-based FDI approaches lead in general to partial descriptions of the system, i.e. to reductions of the input/output models. This will be discussed in terms of different types of modelling: analytical, data-based and knowledge-based.

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Fig. 1. Controllable and uncontrollable subsystems under the influence of faults.

2.1. Controllability versus fault detectability As pointed out by [4], the fault isolation task can only be realized if the fault to be isolated has been previously taken into account in the model. In particular situations, the model used for control may not have all the information required for FDI. This is the case, if the system is not controllable and the fault occurs in the uncontrollable part. Then the model for FDI must evidently be of higher order than the one for control in order to detect the faults. To give an example, consider the following scenario: a system model which is not completely controllable with respect to the system input but observable has to be controlled and monitored. For control purposes a minimal realization based on Kalman canonical decomposition should be used. But then the uncontrollable modes of the original model are no longer present in the reduced model. The lost information (uncontrollable modes), however, is needed for FDI if a fault occurs in the eliminated (uncontrollable) states, see Fig. 1. Remark. The ideas of this subsection are not new. Under the name Natural Redundancy a residual generation approach taking advantage of the above ideas has been proposed for linear systems in [18]. A generalization for non-linear systems was considered in [25]. 2.2. Analytical approaches The approaches considered in this subsection make use of analytical (quantitative) models of the system. It will be shown that a reduced model for FDI compared to that for control is needed when we wish to detect only a limited set of faults, obtain robustness with respect to unknown inputs (disturbances, modelling uncertainty) and aim at selective detectability for the purpose of fault isolation. Then we need structured residual vectors, where each component should reflect only a subset of the fault vector; for this purpose only a part of the overall model of the process is needed. 2.2.1. Observer-based approach There is a number of different approaches for the design of diagnostic observers. These are the geometric methods [17], spectral theory [22], frequency domain [8] and algebraic methods [7,13,23]. In this contribution we follow the approach of [13] (see also [1]). The basic idea is to find a state transformation of the given system such that the state can be divided into two parts: one independent of unknown inputs and the second one which is affected by the unknown inputs. Under the assumption that the set of states affected by the unknown inputs is obtainable from the output measurement of the system, the subsystem defined by the states which are not affected is obtained. Faults in this subsystem can be detected (isolated) by the use of an observer-based residual generator. The design is simple, because a Luenberger-like observer can be used.

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The subsystem can be constructed as follows. First the vector fk is partitioned into f¯k and f˜k . The vector f¯k contains the set of faults whose effect on the subsystem is undesired and the vector f˜k contains the set of faults that will be monitored from this subsystem. Without loss of generality and to keep the analysis simple, the matrix D is neglected and no sensor  faults are considered. The matrix Ea is partitioned into f¯ Ea , [E¯ E] and the fault vector into fk , ˜k . fk   E¯ 1 ¯ Considering a non-singular transformation matrix T such that T E = and applying it as a state 0 transformation zk = Txk to the system xk+1 = Axk + Buk + Ea fk ;

yk = Cxk + Duk + Es fk ,

(1)

we have zk+1 = TAT−1 zk + TBuk + TE¯ f¯k + TEf˜k ;

yk = CT−1 zk

(2)

With the partitions     A11 A12 z1k −1 ; TAT = ; zk = Txk = z2k A21 A22       B1 E1 −1 TB = ; CT = C1 C2 ; TE = B2 E2 the transformed system can be written as z1k+1 = A11 z1k + A12 z2k + B1 uk + E¯ 1 f¯k + E1 f˜k

(3)

z2k+1 = A21 z1k + A22 z2k + B2 uk + E2 f˜k

(4)

yk = C1 z1k + C2 z2k

(5)

Note that if rank(C1 ) = m − 1 the disturbed state x1k is eliminated from (4) and with this the desired subsystem can be obtained. Otherwise consider the singular value decomposition of C1   Σ1 0 T (6) C1 = USV ; S = 0 0 Define an output transformation T1 as T1 = U T ; applying it to yk results in   Σ1 0 ∗ V T z1k + T1 C2 z2k yk = T1 yk = 0 0

(7)  yk∗

 y1k ; V T C2 = y2k

In order to eliminate a part of the unknown state z1k in Eq. (4), consider =     C21 z ; V T z1k = 11k , and z11k can be obtained from the equation of y1k as z11k = Σ −1 (y1k −C21 z2k ). C22 z12k Substituting it into (4) results in z2k+1 = A211 Σ −1 y1k + (A22 − A211 Σ −1 C21 )z2k + B2 uk + A212 z12k

y2k = C22 z2k

(8)

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Fig. 2. Decoupling of unknown inputs in observer-based residual generation.

where A21 V , [A211 A212 ]. In order to have (8) insensitive to f¯k , the elimination of z12k from (8) is required. To do so, the above procedure has to be applied to (8) once more assigning z12k → f¯k . The procedure is continued until the undesired faults have lost their effect on the calculated subsystem. The procedure is illustrated in Fig. 2. After the subsystem sensitive to the desired group of faults has been found, a residual generator is obtained by designing an output observer for the subsystem (8). The residual is defined as the difference between the output of the system and the estimated output, see Fig. 3. 2.3. Parity space The residual in the parity space approach is either defined based on the input output operator representation of the system (1) as rk = W (z)(yk − Gu (z)uk ) = W (z)Gf (z)fk

(9)

with W(z) representing a filter yet free to select, or based on the state space representation as rk = v1 z−s |zI − A| · Gf (z)fk | {z } ,W (z)

Fig. 3. Observer-based residual generator for decoupled subsystem.

(10)

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The transformation matrix v1 can be used to reduce the set of the parity equations to those that only contain information about the faults and/or are only little or not affected by the uncertain parameters. As can be seen from Eqs. (9) and (10), the residual in the parity space can be expressed as a relationship of inputs, nominal (fault free) model and outputs. The supervision of faults can be carried out in a direct way by selecting an adequate parity matrix (vector). If multiple faults have to be detected and isolated, the use of structured residuals is mandatory. Structured residuals for parity space have for example been studied by [9] for the I − O approach and, for example, by [24] for the state space approach. The basic idea is to design a set of residuals, each of them sensitive to a different set of faults. As can be seen from Eqs. (9) and (10), this can be achieved by a proper choice of W(z) or v1 , respectively. Hence, for the generation of the residual rk we need only a model of the form W(z)Gu or v1 zs |zI − a|Gu , respectively instead of Gu . This leads to a model reduction of the original nominal model Gu . An example taken from [6] may illustrate this result. Consider the system       1 0 a11 a12 0 x(k) (11) u(k) y(k) = x(k + 1) = x(k) + 0 1 1 0 a22 and suppose that both the actuator and the component faults have to be supervised. Following [6], there are three linearly independent parity equations y1 (k) − (a11 + a22 )y1 (k − 1) + a11 a22 y1 (k − 2) − a12 u(k − 2) = 0

(12)

y1 (k) − a11 y1 (k − 1) − a12 y2 (k − 1) = 0

(13)

y2 (k) − a22 y2 (k − 1) − u(k − 1) = 0

(14)

Note that for the supervision of a22 and of the actuator only the parity Eq. (14) is required, which represents only a partial description of the original system (11). In a similar way, for the supervision of a11 and a12 the parity Eq. (13) can be used, which is another partial description of (11). To select the proper equations means to find v1 in (10), for which task a number of methods have been developed, see, e.g. [6]. Remark. An easy and systematic way to design structured residuals is by applying the same procedure used to get a robust subsystem in the observer-based approach and building a parity space model for the resulting subsystem. 2.4. Parameter estimation Parameter estimation (PE) as developed for and widely used in modelling, signal processing and control can also be applied to FDI [14]. For control applications, both the model structure and the parameters of the plant are identified either off-line (for controller design) or on-line (for controller adaptation). The original possibly complex non-linear physical system is usually approximated by linear differential or difference equations valid in a certain range of operation. The physical meaning and the internal relationships of the original system parameters are not important, i.e. the mathematical parameters are sufficient for this task. The basic idea behind the application of parameter estimation to FDI is the on-line estimation of the parameters of the actual system model and the comparison to their nominal values. The resulting deviations are the residuals used for FDI [14]. For the interpretation of the faults we need in general the

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deviations in terms of the physical parameters. This is why the model should be as detailed as possible. In other words, it is not preferable to simplify the non-linear functions into linear ones etc. Most systems in engineering are continuous in time. The mathematical parameters are here closer related to the physical parameters than in a discrete time model. Hence, the parameter estimation for continuous time models bears special findings for FDI. The linearization of non-linear functions and the discretization of the differential equations make the relationship between mathematical parameters and physical parameters more complicated. It can be concluded that in this case the model for FDI must be more complex than that for control system design. In return, these arduous efforts bring along the following advantages. 1. This approach can provide a deeper insight in the system. With a bit more efforts (e.g. adaptive control), the effects of the faults can be compensated in order to gain fault-tolerance. 2. When the relationship between the model parameters and physical parameters is unique, the fault isolation is easy to implement. 3. This method also provides direct fault identification because the physical parameter deviations stand directly for the severity of the faults. The main limitation of this approach is that the estimated parameters must be persistently excited by an input signal and that for the isolation of the faults the relationship between the physical and mathematical parameters must be unique. It is, however, also possible to utilize directly the mathematical parameters (equation coefficients or zeros and poles of the transfer functions) [10,11]. In this case, simplified models can be used, because only those parameters which are related to the possible defect parts in the system need to be estimated, whereas the other parameters can be taken as constant. This means that only a partial model is required. In practice, the original system is divided into subsystems as small as possible, and with only few parameters to be estimated. An extra bonus of the reduction and partition of the original model is that the requirements for the excitation signals can be relaxed. This also speeds up the estimation convergence.

3. Data-based approaches Data-based models constitute an alternative to the analytical approach to FDI when the latter is either not available or not feasible. In many practical applications large archives of process data exist which can be used to set up a data-based model. The given set of input-output data of the possibly non-linear system can be used to train a properly prestructured non-linear model. The learning can be achieved by adaptation due to a given performance index. Data-based models designed for FDI principally aim at the (best possible) estimation of those output measurements which are influenced by the faults of interest. This means that the resulting models have a different input and output measurement space compared to the functional models for control. This can lead to less complexity. However, the data-based models are usually non-linear in contrast to the functional analytical models which are often linear and hence less complex. Let us consider the two most common types of data-based models are neural networks and fuzzy relational models. 3.1. The neural network approach Basically, the artificial neural net (ANN) represents a non-linear system refereing to its input-output behavior. The non-linear transformation results from its inner structure. In general the ANN consists

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of neurons, simple processing elements, which are activated as soon as their inputs exceed a certain threshold. The neurons are arranged in layers which are connected such that the signals at the input are propagated through the network to the output. The choice of the transfer function of each neuron (e.g. sigmoidal function) yields the non-linear overall behavior of the network. During a training period a set of parameters of the neural network is learned from a given set of data aiming at the “best” approximation of the behavior of the system. Since the late eighties artificial neural networks have been studied for FDI. First only slowly varying processes ([12,20,21]) were considered, but due to recent efforts to model non-linear dynamic systems [5], FDI can greatly benefit from this. The training is generally performed using measurements from the fault free process. For residual generation purposes the neural network simply replaces the analytical model describing the process under normal operation [16]. Employing a non-linear input/output description y (k) = g(y (k − 1), . . . , y (k − q), u(k), . . . , u(k − p)) (15) ¯ ¯ ¯ ¯ ¯ the neural network approximates the non-linear vector function g(·). This general pattern may be substantially compressed considering that • it is useful to estimate each system output with a seperate neural network • not all of the system outputs may be influenced by the faults under consideration and need to be estimated. The first point implies that for each output the neural network’s input space is chosen differently, each leading to the best possible approximation of the respective output. This choice includes the different inputs and outputs from the available ones and the number of tapped delays for each signal (Fig. 4). As an example the estimates for a system with two inputs and three outputs can look as follows: yˆ1 (k) = g1 (y1 (k − 1), y2 (k − 1), u1 (k)) yˆ2 (k) = g2 (y2 (k − 1), y2 (k − 2), y3 (k − 1), u2 (k)) yˆ3 (k) = g3 (y1(k − 1), y2 (k − 1), y3 (k − 1)) The appropriate choice of the input space is one of the most difficult tasks when configuring the neural network. One either needs to have enough process knowledge and then use a trial and error strategy or has to apply an optimization algorithm such as a genetic algorithm [16].

Fig. 4. Separate estimation of output signals by ANN.

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Keeping in mind that for FDI only those outputs need to be estimated which are affected by faults one can conclude that neural models for FDI may be of reduced order and hence of less complexity due to an adequate input space in comparison to functional models used for control. 3.2. Residual generation based on fuzzy relational models The approach considered here has been described in [2,19]. The fuzzy observer is founded on a fuzzy relational model of the process, which is formed by the composition operator (T-conorm/T-norm operator) applied to a fuzzy relational matrix R defining the relation between process input and output, and the fuzzy cartesian product of the fuzzified input–output signals and its delays on a time window. A mathematical representation of the fuzzy observer is given by Yˆi = R0 ◦ X

(16)

where Yˆi and X are given in the fuzzy space. X is defined by the fuzzy cartesian product of the input-output measurements and its delays, i.e. X = U (k) × · · · × U (k − n)Yi (k) × · · · × Yi (k − n)

(17)

and R0 is the relational matrix of the nominal (fault free) model. The fuzzy residual generator as shown in Fig. 5 determines the difference between the measured and the estimated output using a fuzzy output observer. In order to show that models for FDI used in the fuzzy output observer can be different from functional ones, we consider a (non-linear) system, whose inputs and outputs are represented by uk and yk , respectively. Suppose, we are interested to detect a fault in the jth output sensor. For this we need a residual generator based on fuzzy relational models which is sensitive to faults in the jth sensor. This can be constructed as follows. Consider the vector X, X = Uk × · · · × Uk−µ1 × Yjk × Yjk−µ2 ,

Fig. 5. Fuzzy observer-based residual generation.

(18)

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which represents a combination of all premise variables appearing in the premises of the rules that are represented by the relational matrix R. The variables Uk and Yk are the fuzzified values of uk and yk , respectively. A relational fuzzy model for the jth output is given by Yjk = R0 ◦ X

(19)

To obtain the relational matrix R0 , a fuzzy relational equation has to be solved. However, in general, the set of solutions is empty [2]. The alternative approach is to find a solution Ra such that the output of the relational model (19) when Ra is used is an approximation Yˆjk of Yj k in the sense of a given criterion J(Ra ), e.g. the quadratic error in the real valued space N

J (Ra ) =

1X (yjk − yˆjk )2 , 2 k=1

(20)

where N is the number of elements in the learning set and the quantities used in the criterion are the defuzzified corresponding values of the outputs. The above procedure shows that the representative model used for the design of a residual generator based on relational models is a partial description of the original system. Note that the structure of the residual generator based on fuzzy relational models is similar to the one based on neural nets. A basic difference is that the signals used for the fuzzy relational model have been previously fuzzified, for the neural net approach the measured signals are utilized directly. 4. Knowledge-based approaches Another alternative to the analytical FDI approach is the knowledge-based approach which makes use of the knowledge available to derive either a qualitative description of the system in the form of a qualitative model or a rule-based representation. In this paper, we restrict ourselves to qualitative models on the basis of qualitative differential equations following [26]. Other approaches, as for example the important concept of fuzzy rule-based modelling, are not considered here. Before discussing details of the qualitative models some general remarks on qualitative modelling strategies should be made. In order to perform totally reliable fault diagnosis and to avoid false alarms an accurate model is of absolute importance. However, an accurate model does neither mean the highest precision in the description nor the highest complexity of the model [15]. Developing a model on a higher level of abstraction can still lead to an accurate model though less complex. On the other hand preciseness of measurements is only of critical importance for diagnostic tasks if the models are very precise in their description as well. An increased imprecision of the measurements can be tolerated by models on a higher level of abstraction still leading to a correct fault decision. In exchange with this advantage in modelling effort one has to put up with the drawback of less sensitivity to smaller faults. 4.1. Qualitative residual generation When complete information about an industrial process is not available, the quantitative model-based techniques for FDI can be replaced by qualitative ones that make use of the available incomplete in-

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Fig. 6. Graphical description of behaviors.

formation by building a qualitative model, in terms of which the analysis and reasoning can be carried out. Different from the other model-based approaches, a qualitative model may be based on qualitative differential equations (QDE). A QDE for a specific system has the same structure as the corresponding ordinary differential equation that models the dynamic system in continuous time. However, the information about the parameters is only of “semi-quantitative” nature being frequently only partially known or uncertain. As a result, a constrained model is obtained which consists of qualitative variables representing the physical parameters of the system and a set of constraints of how those parameters are related to each other. The behavior can be described by a graph consisting of the possible future states of the system. Due to the inherent ambiguity of the qualitative representation and calculus, the simulated possible behavior is in general not unique, but could take any path through the graph starting at the initial state, as shown in Fig. 6. A qualitative observer (QOB) used for residual generation makes use of qualitative simulation on the basis of conventional filtering techniques to perform an observation filtering [26]. The principle of observation filtering is that the simulated qualitative behavior of a variable must cover its counterpart of the measurement obtained from the system itself, otherwise the simulated behavioral path is inconsistent and can be eliminated [26]. A scheme of the qualitative observer is shown in Fig. 7. The idea behind the qualitative observer-based FDI is that a fault causes a deviation of the system output in such a way that its counterpart of the estimated output is no more consistent, i.e. a fault will produce an empty set of qualitative estimated states, which is impossible in a fault free case. A model for FDI can now be derived following the basic idea of using fuzzy relational models. To show this, consider a non-linear system described by the QDE [x] ˙ = f ([x], [u]);

[y] = C[x]

(21)

Fig. 7. Qualitative output observer.

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where [x] ∈ Rn is the system state, [u] ∈ Rp is the system input and [y] ∈ Rm is the system output. If we are only interested in the detection of a single sensor fault, i.e. in the ith sensor, a qualitative observer can be designed obeying the equation [x] ˙ = f ([x], [u]);

[yi ] = Ci [x]

(22)

where [yi ] is the ith element of [y] and Ci is the ith row of C. A more detailed description of the design of qualitative observers can be found in [26].

5. Conclusions The most common model-based FDI approaches have been examined for their modelling efforts, accentuating the differences between models used for FDI and those used for control. Firstly, we have studied the role of the transfer operator from the fault vector to the system output and its decisive impact on the residual generation with respect to fault isolation. The similarities of the different analytical FDI approaches have been underlined, and the dependence of the residual upon the transfer operator from the fault to the output has been discussed in terms of complexity. As a main result it has been shown that for robust FDI of controllable systems and making use of quantitative, data-based or qualitative models, the model needed comprises only a partial description of the system and is thus not simply identical with the model for control. Mostly it is less complicated. Even though the most common model-based approaches have been taken into account in this paper, we do not claim completeness of our study. Some important concepts such as the continuous-time parity space approach, the continuous-time dead-beat observer approach, analytical non-linear FDI methods and stochastic FDI approaches have not been included into this consideration. However, it seems to be evident that they do not require fundamental revision of the message given in this paper.

Acknowledgements The authors want to thank the Deutscher Akademischer Austauschdienst (DAAD) for the financial support of this work. Moreover, the authors are grateful to Z. Han, Z. Zhuang and P. Amann for helpful discussions and support in the preparation of the paper.

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