Proceedings of the 3rd International Conference on Systems and Control, Algiers, Algeria, October 29-31, 2013
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Residual structuration based on a new observer scheme for sensor fault detection and isolation Abdelghani DJEDDI
1
and Mohamed Faouzi HARKAT
Abstract— In observer-based approach for fault detection and isolation, two schemes are generally considered, namely the dedicated observer scheme (DOS) and the generalized observer scheme (GOS). DOS is a bank of observers sensitive to only one fault while GOS is composed of observers sensitive to all faults except one. In this paper a new sensor fault diagnosis approach named Reconstruction Observer Scheme (ROS) is proposed, which does not need any bank of observer, only one observer is used. The proposed method based on reconstruction of variables is used to generate a structured residuals for fault isolation. After the fault detection, the reconstruction is carried of all the variables. Reconstruction of a variable consists on the replacement of this variable to the input of the observer by its estimation. This operation eliminates fault effect when a faulty variable is reconstructed. The proposed approach is illustrated by an academic example.
I. I NTRODUCTION Over the last three decades, the growing demand for safety, reliability, maintainability, and survivability in technical systems has drawn significant research in Fault Detection and Diagnosis (FDD). Such efforts have led to the development of many FDD techniques, for example survey papers [2], [3]. The diagnosis of faults can be done using observers. One great advantage of the diagnosis schemes based on observers is that in comparison with other methods they are very large schemes. In many publications about non linear observers for the design of FDI systems, the residuals are based in the error of the estimation obtained by the observer. Diagnosis schemes based on observers can be classified according to the type of fault detected: sensor faults (Instrument Fault Detection or IFD), actuator faults (Actuator Fault Detection or AFD), and component faults (Component Fault Detection or CFD). Diagnosis schemes can also be classified according the number of observers that are used. There are schemes with one observer: a Direct scheme is a scheme of just one observer of complete order. The Simplified Observer Scheme (SOS), is a scheme of one observer of reduced order. For sensor faults isolation only observer in this scheme uses all the inputs and one output, which only provides simple redundancy and only allows the isolation of faults in one sensor. In AFD, the only observer uses all the outputs and just one input. When several observers constitute a bank of observers of reduced order we have a Dedicated Observer Scheme (DOS). For faults in sensors (IFD), each observer uses all the inputs and just one output. The number 1 Djeddi A. is with Department of Electronics, Kasdi Merbah - Ouargla University, Algeria
[email protected] 2 Harkat M-F. is a Professor with Department of Electronics, Badji Mokhtar Annaba University, P.O. Box 12, Annaba 23000, Algeria
[email protected]
2
of observers equals the number of outputs (sensors). For actuator faults (AFD) each observer uses one input and all the outputs. It should be mentioned that the DOS scheme allows the localization of multiple faults, either in sensors (IFD) or in actuators (AFD). The Generalized Observer Scheme (GOS) is formed by a bank of observers of reduced order. For faults in sensors (IFD), each observer uses all the inputs and m-1 outputs, where m is the number of outputs. For actuator faults (AFD), each observer uses all the outputs and n-1 inputs, n being the number of inputs. There is no doubt that the theory (and practice, as a consequence) of fault diagnosis and control is well-developed and mature for linear systems only. Observers are commonly used in both control and fault diagnosis schemes of non-linear systems [1], [5], [7]. We propose, in this paper the design of new approach for sensors faults diagnosis without unknown inputs observers design. Reconstruction observer scheme (ROS) is a new approach which uses one observer of full order. For sensor faults (IFD), the only observer in this scheme uses measurable inputs and all the outputs, which provides simple redundancy and allows sensors fault isolation. The paper is organized as follows. Section 2 presents the reconstruction observer scheme in the linear case and the link between the unknown inputs observer and reconstruction approaches. In section 3 application of ROS for nonlinear model represented by multiple model. Section 4 presents numerical examples with discussion about the performances of the new sensors diagnosis method. Finally, this note is ending with conclusions and perspectives. The idea is based on two steps: the first step consists to detect faults by using SP E index. The second step is to isolate faulty variable by reconstruction of outputs and calculated SP E index to define the faulty variable. II. R ECONSTRUCTION APPROACH To present the reconstruction approach, we consider the class of linear systems which can be described a follows: x (k + 1) = Ax(k) + Bu(k) y(k) = Cx(k)
(1)
Where x ∈ Rn denotes the state vector, u ∈ Rm is the control input vector, y ∈ Rp is the measurable output vector. A, B and C are known parameter matrices with proper dimensions. We consider the outputs vector y = [yN (k) y¯(k)]T , where y ¯(k) represents variable to be reconstructed and yN (k) the
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remaining variables of outputs vector y. Then equation 1 can be rewritten as: x(k + 1) = Ax(k) + Bu(k) yN (k) CN = x(k) y¯(k) C¯
(2)
Where y¯(k) is the variable to be reconstructed and C¯ its corresponding row of the matrix C, yN is the remaining component of vector y and CN represents its corresponding rows of matrix C. The full order observer for the system 2 can be defined as: ˆ (k + 1) = Aˆ ˆ (k)) x x(k) + Bu(k) + L(y(k) − y ˆ N (k) y CN ˆ (k) = x yˆ¯(k) C¯
(3)
SP E(k) = rT (k)r(k)
To improve fault detection by reducing the rate of false alarms (due to noise), EWMA (ExponentiallyWeighted Moving Average) filter can be applied to the residuals. The filtered residuals ¯r(k) are thus obtained: ¯r(k) = (1 − β)r(k − 1) + β¯r(k)
SP E(k) = ¯r(k)T ¯r(k)
ˆ N (k) yN (k) − y y¯(k) − yˆ¯(k)
¯r(k) and SP E(k) respectively define the filtered residual and the SP E. III. R ECONSTRUCTION OBSERVER SCHEME (ROS) FOR
(4) (5)
The main idea of reconstruction consists of the replacement of the variable to be reconstructed [6], y¯(k), by its estimated value, yˆ¯(k), given by the observer. The estimation error is then given by: ˆ N (k) yN (k) − y r(k) = (6) yˆ¯(k) − yˆ¯(k) And we obtaine: r(k) =
ˆ N (k) yN (k) − y 0
(7)
From equation 7, equation 3 can be rewritten as: ˆ (k + 1) = Aˆ x x(k) + Bu(k) + L ˆ N (k) y CN ˆ (k) = x yˆ¯(k) C¯
(11)
SENSOR FAULT ISOLATION
ˆ (k) r(k) = y(k) − y
(10)
Where β is a forgetting factor for residuals and r(0) = 0. The filtered SP Eis given by:
The output error estimation is given by:
r(k) =
(9)
ˆ N (k) yN (k) − y 0
(8) From the expression of the estimation error, the effect of the reconstructed variable is eliminated. This operation of reconstruction can be used for the other variables in output vector y(k) and several estimation errors can be generated. Each estimation error vector is insensitive to its corresponding reconstructed variable. This property is very useful for fault isolation. A. Observer based sensor fault detection Fault detection is performed by using the statistical SP E (Squared Prediction Error), also known as Q statistics, is an index obtained from the estimation error of the outputs, it depends on all the variables to be monitored. The SP E index is given by:
To present sensor fault isolation based on reconstruction approach, let us consider the linear system (1) affected by a sensor fault and its corresponding observer (3). From equations (1) and (3) we have : ˆ (k) = Aˆ x x(k−1)+Bu(k−1)+L(y(k−1)−ˆ y(k−1)) (12) and ˆ (k) = C [Aˆ y x(k − 1) + Bu(k − 1)+ ˆ (k − 1)) L (y(k − 1) − y
(13)
ˆ (k) depend only on From this equation we can see that y variables measurement at time (k −1). If fault affect a sensor at time k (ith component of y(k) for example), it is clear that ˆ (k) will not be affected by this fault since it its estimation y depend only on variable measurement at time (k − 1). So, if we try to reconstruct the faulty variable (yi (k)) which mean that yi (k) (which is affected by fault) will be replaced by its estimation yˆi (k) (which is fault free), the fault is eliminated. It is also clear that if we replace yj (k) (another component of vector y(k)) by its estimation yˆj (k) the fault will not be eliminated. To prevent the fault propagation at time (k + 1), ˆ (k + 1) will the vector y(k) which is used to estimate y consist of all its component except the ith component which is replaced by yˆi (k) (the reconstructed value at time k). When a fault is detected, it is necessary to identify the faulty variable. Sensor fault isolation approach based on the reconstruction principle assumes that each sensor may be faulty (in the case of single fault) and suggests to reconstruct the assumed faulty sensor using the observer model from the remaining measurements. By examining the residuals given by the model and observer before and after reconstruction, we can determine the faulty sensor. The isolation is performed by comparing the detection index before and after reconstruction. The faulty sensor is identified as the sensor for which the detection index calculated after reconstruction, is below its control limit (fault is eliminated). This approach can be illustrated by the following scheme:
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y1 (k)
The goal is to design a bank of unknown input observer to generate a structured residuals sensitive to faults that affect outputs yN (k) and insensitive to y¯. To design a bank of observer, unknown input observers are used for decoupling. We use the same approach as that used for actuator fault isolation. It is possible after a transformation of the system to rewriting the sensor fault in the form of actuator fault, this method is developed by [8]. For this, we define a stable filter as follows:
u(k)
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y2 (k) y3 (k)
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wf (k + 1) = Af wf (k) + Bf y¯(k) Fig. 1.
Reconstruction based observer principal
Let us consider the SP E j (k) which denotes the index SP E(k) calculated after reconstruction of the j th variable. This computation has to be performed successively for all the variables (j = 1, ..., M ). Therefore, if the ith faulty variable is reconstructed (j = i), the index SP E j is in the control limit because the fault is eliminated by reconstruction. If the reconstructed variable is not faulty, the index SP E j being always affected by the fault, SP E j is outside its control limit. In summary, when a fault has been detected, all the indices SP E j are computed, and if SP E j is under its control limit, the j th sensor is considered as a faulty one. Table (I) gives sensor fault signatures [4], each residual is insensitive to one variable or sensor (the reconstructed variable): Residuals SP E1 SP E2 . . . SP Ei . . . SP EM
y1 0 × . . . × . . . ×
y2 × 0 . . . × . . . ×
... ... ... . . . ... . . . ...
yi × × . . . 0 . . . ×
... ... ... . . . ... . . . ...
yM × × . . . × . . . 0
TABLE I S ENSOR FAULT SIGNATURES
IV. L INK BETWEEN UNKNOWN INPUT OBSERVER AND
(14)
Where wf (k) ∈ Rp−l . The matrix Af ∈ R(p−l)×(p−1) and Bf ∈ R(p−l)×(p−1) are known and 1 ≤ l ≤ m. The T augmented state is defined by x ˜(k) = x(k)T wf (k)T . The dynamics is given by the following equations:
x(k + 1) wf (k + 1)
A 0 x(k) = + 0 Af wf (k) Bj 0n×l u(k) + y¯(k) 0(p−l)×m Bf (15)
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˜x(k) + Bu(k) ˜ ˜f y¯(k) ˜ (k + 1) = A˜ x +B yN (k) = CN x(k)
(16)
(17)
Then considering y¯(k) as the vector of unknown inputs, an unknown input observer can be developed under the structural conditions on the matrix of influence of y¯(k) allowing the decoupling, to detect output faults only on yN (k) . Then, the approach based of observers bank for actuator fault isolation also can be applied to the isolation of sensor fault. z(k + 1) = Ni z(k) + Gi u(k) + Li yN (k) ˆ (k) = z(k) − HyN (k) x ˆ N (k) = CN x ˆ (k) y
(18)
Each observer of the observer bank generates residuals formed by the output error between each observer and the measured output of the system:
RECONSTRUCTION
The main idea of sensor fault isolation based on observer is based on the generation of a set of structured residuals by using a bank of unknown inputs observers. The proposed reconstruction approach (Reconstruction Observer Scheme) is based on the same idea but by using a single full order observer. To make a link between those two approaches let us consider the system given by equation (2) where y¯ is considered us an unknown input of the observer and C¯ its corresponding row of matrix C, yN is a vector containing the remaining components of y and CN its its corresponding rows of matrix C.
ˆ N (k) r(k) = yN (k) − y
(19)
From the vector y¯, different combinations of outputs used yN and ignored y¯ lead to the generation of structured residuals for sensor fault detection and isolation. And residuals used for fault detection and isolation are given by : By comparing equations (7) and (19) we can conclude that residuals obtained by the two approaches (reconstruction approach and the input observer approach) are the same. In conclusion, the two approaches are equivalent and give the same structured residuals.
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Proceedings of the 3rd International Conference on V. R ECONSTRUCTION OBSERVER SCHEME FOR NONLINEAR SYSTEM
The reconstruction observer scheme can be easily extended to nonlinear systems represented by multiple linear models. A multi-model is based on the decomposition of the dynamic behavior of the system in a number M of operating zones, each zone being characterized by a linear sub-model [1]. It was shown that the multiple model is a universal approximation tool. In fact, any nonlinear system can be approximated with an accuracy imposed by increasing the number of sub-models and optimizing the weighting functions. Two essential structures of multiple model can be distinguished, one where the sub-models share the same state vector (Takagi-Sugeno), where the other sub-models are decoupled, with each sub-model has its own state vector (decoupled). The Takagi-Sugeno is, at present, the most commonly used. VI. ACADEMIC EXAMPLE
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VII. C ONCLUSION In this paper a new approach for sensor fault detection and isolation using observer is proposed. The proposed approach
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Sensor are now assumed to be affected by faults and noise measurements. In this case study, a fault is simulated on variable y2 between samples 200 and 300. Figure 3 shows the time evolution of residuals and the SP E. Fault is easily detected on the SP E. After fault detection, it is important to isolate the faulty variable. Figure 4 shows the residuals SP Ei calculated after the reconstruction of the ith variable (i = 1, 2, 3). The SP E2 is under its control limit which indicate that the faulty variable is y2 . It is important to note that the residual corresponding to the reconstracted variable is equal to zero (figure 4, r1 for the reconstruction of y1 , r2 for the reconstruction of y2 and r3 for the reconstruction of y3 ).
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Consider the linear discrete system, in which the parameters are given as follows: 0.137 0.199 0.284 A = 0.0118 0.299 0.47 0.894 0.661 0.065
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Proceedings of the 3rd International Conference on is based on the reconstruction method and uses only one observer. After the presence of fault has been detected, it is important to identify the fault and apply the necessary corrective actions to eliminate the abnormal data. The reconstruction approach assumes that each sensor or variable may be faulty and suggests to reconstruct the assumed faulty variable using its estimation given by the observer. By examining the residuals before and after reconstruction, we can determine the faulty variable. R EFERENCES [1] Akhenak A., Chadli M., Ragot J., an Maquin D. (2004), Estimation of state and unknown inputs of a nonlinear system represented by a multiple model. In 11h IFAC Symposium on Automation in Mineral and Metal processing, Nancy, France. [2] Frank P. M. (1996), Analytical and qualitative model-based fault diagnosis.- A survey and some new results. European Journal of Control, 2(1), 6-28. [3] Frank P. M., and Ding X. (1997), Survey of robust residual generating and evaluation methods in observer-based fault detection systems. Journal of Process Control, 7(6), 403-424. [4] Gertler J. (1993), Residual generation in model-based fault diagnosis. Control Theory and Advanced Technology, 9(1), 259-285. [5] Chen J. and Patton R.J. (1999), Robust Model-based Fault Diagnosis for Dynamic Systems. Kluwer Academic Publishers, London. [6] R. Dunia, S. J, Qin, T. F. Edgar and T. J. McAvoy, Identification of faulty sensors using principal component analysis,AIChE.J., vol. 42,1996, pp 2797-2812. [7] L. H. Chiang and E. L. Ressell and R. D. Braatz, Fault Detection and Diagnosis in industrial systems,Springer, 2000. [8] Tan, C. and Edwards, C. ”Sliding mode observers for robust detection and reconstruction of actuator and sensor faults. International journal of robust and nonlinear control”, 13: 443-463, 2003.
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