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Fault prediction for nonlinear stochastic system with incipient faults based on particle filter and nonlinear regression
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Research article
Fault prediction for nonlinear stochastic system with incipient faults based on particle filter and nonlinear regression Bo Ding, Huajing Fang n School of Automation, Key Laboratory of Image Processing and Intelligent Control, Ministry of Education, Huazhong University of Science and Technology, Wuhan 430074, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 18 July 2016 Received in revised form 25 January 2017 Accepted 22 March 2017
This paper is concerned with the fault prediction for the nonlinear stochastic system with incipient faults. Based on the particle filter and the reasonable assumption about the incipient faults, the modified fault estimation algorithm is proposed, and the system state is estimated simultaneously. According to the modified fault estimation, an intuitive fault detection strategy is introduced. Once each of the incipient fault is detected, the parameters of which are identified by a nonlinear regression method. Then, based on the estimated parameters, the future fault signal can be predicted. Finally, the effectiveness of the proposed method is verified by the simulations of the Three-tank system. & 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Fault detection Fault prediction Nonlinear stochastic system Incipient fault Particle filter Nonlinear regression
1. Introduction In order to keep the safety and reliability of the modern complex system, and with the development of digital computers and the system identification techniques, the design and analysis of the model-based fault detection (FD) algorithms has received considerable attention in recent decades [1–3]. As an important branch of the FD technology, FD based on stochastic models has also been well studied. For linear stochastic system (LSS), the mainly FD algorithm is by using the innovation (residual) generated by Kalman filter (KF), where the fault are detected by the statistic testing on whiteness of the residual [4,5] or by the generalized likelihood ratio (GLR) testing on the residual [6]. In order to isolate the fault, the multiple KF method is proposed in [7], where each KF is designed for detecting a specific fault. Once a fault does occur, all filters except the one has the correct hypothesis will generate the large estimation error, hence the specific fault can be isolated. For nonlinear stochastic system (NSS), the FD algorithm is similar to that of the LSS, the difference is that the KF is replaced by extended Kalman filter (EKF), unscented Kalman filter (UKF) or particle filter (PF) according to the difference assumption of the system and noise [8–10]. With the development of the FD technology, researchers begin to explore the algorithm of incipient fault detection. In [11], the n
Corresponding author. E-mail address: [email protected] (H. Fang).
on-line approximation method is proposed to detect the incipient fault, where the evolution of the incipient fault is expressed as a nonlinear function with unknown parameters. This expression about incipient fault has been cited by many literatures, such as [12–14]. In [15], a sliding-mode observer is proposed to reconstruct the incipient sensor fault. The similar method can also be used in [16] to detect and isolate the incipient sensor fault for the uncertain nonlinear system and in [17] for the quadrotor helicopter attitude control system. Owing to its wide application in fault tolerant control, fault estimation (FE) has also been noticed and some meaningful results have been achieved [18]. In [19], a multi-constrained full-order fault estimation observer (FFEO) design with finite frequency specifications was proposed. Later, this result was extended to the discrete-time systems [20], and an reduced-order fault estimation observer (RFEO) design was also obtained. It should be noted that, the incipient fault often changes slowly and it is almost unnoticeable during its early stage, hence the research on detection and estimation of the incipient fault is still a challenge problem, especially in the nonlinear systems with stochastic noise. Meanwhile, with the improvement of requirement of the safety and reliability, researchers not only want to be able to detect the incipient fault after it occurs, but even more hope to predict the trend of the fault. If the incipient fault is detected in time and then its trend can be predicted as early as possible, then the manager will have more time to take action to avoid the possible serious damage. Moreover, fault prediction can also provide more support
http://dx.doi.org/10.1016/j.isatra.2017.03.018 0019-0578/& 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Ding B, Fang H. Fault prediction for nonlinear stochastic system with incipient faults based on particle filter and nonlinear regression. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.018i
B. Ding, H. Fang / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
for system management and maintenance [21]. Therefore, the study about fault prediction technology has an important theoretical significance and application values. However, very few results are related to the issue of fault prediction since it is difficult to detect the incipient fault in time. More recently, for a class of nonlinear stochastic system with additive faults, based on modified PF (MPF), a FD algorithm which can simultaneously estimate the fault and state is proposed [22]. The error of the fault estimation is closely related to the system noise and measurement noise. Theoretically, if the intensity of the system noise and measurement noise are small enough, then the accuracy of the estimated fault will be high enough. Once the estimated fault with enough accuracy is obtained, it can be used to estimate the fault's parameters, then the future fault can be predicted. However, in practice, the intensity of the system noise or measurement noise may not always sufficiently small, instead in most cases we should enlarge the system noise to compensate for modeling errors, so the MPF algorithm can not provide the estimated fault with enough accuracy. And in the case of large error of the fault estimation, it is hard to detect the fault in time, not to mention to predict the fault. In this work, an improved method is introduced to enhance the accuracy of fault estimation in the mild case, where the intensity of the noise is not exceptionally small, and the parameters of each fault is successfully estimated through the nonlinear regression, thus realize the fault prediction. As in previous literatures, the evolution function of each incipient fault is also expressed as a nonlinear function with unknown parameters. Obviously, this nonlinear function can not be directly used in the filter design because of the unknown parameters. However, as is known, the incipient fault often changes slowly, then based on this feature, a reasonable assumption about the incipient fault is introduced, i.e., the change of each fault signal between any two adjacent time steps is always bounded and small. Combine this assumption with the PF, the modified fault estimation (MFE) algorithm is derived, and then the state estimation can also be obtained. The results show that the fault estimation precision is effectively improved. Based on the MFE, an intuitive fault detection strategy is introduced. Once the fault is detected, the parameters of each component fault are identified by an effective nonlinear regression method: Gauss–Newton method (GNM). After that, the future fault signal can be predicted in light of the expression of the incipient fault and the estimated parameters. Lastly, the effectiveness of the proposed method is illustrated by the simulation of the Three-tank system. The rest of this paper is organized as follows. Section 2 illustrates the formulation of the nonlinear dynamic system and the process of the incipient fault. Section 3 gives the details of modified fault estimation and state estimation based on particle filter. Section 4 discusses the fault detection and fault prediction algorithm. Section 5 illustrates the proposed approach by the simulation of Three tank system. Finally, some conclusions are drawn in Section 6.
2. Problem formulation Consider the following non-linear system with incipient faults:
⎧ + wk ⎪ xk + 1 = g (xk ) + Γ (xk )f k ⎨ ⎪ y C x v = + ⎩ k k k k n
p
(1)
where xk ∈ and yk ∈ are the state vector and measurement vector respectively. g (xk ) is the system function. fk ∈ q represents the incipient fault vector, Γ(xk ) is a known distribution matrix function, Ck is the measurement matrix with compatible
dimensions. The process noise wk ∈ n and measurement noise vk ∈ p are mutually uncorrelated white noise sequences, with known covariance matrices Q k ≥ 0 and Rk > 0 respectively. The initial state x0 is independent of wk and vk with the known mean x^ and covariance P . In order to simplify the discussion and 0
0
without loss of generality, the known input vector uk is not included in the system model. Remark 1. In (1), Γ(xk ) is a matrix function about xk, hence the fault is multiplicative. This type of fault often represents the degradation of the component. If Γ(xk ) is not related to xk, then the fault is additive, and additive fault usually represents the offset in actuator. Therefore, model (1) has a wide applicability. Let fi denotes the ith component of the fault vector, and the incipient fault fi can be described as
⎧ ⎪ 0, 0 ≤ k < θ2, i fi, k = ⎨ −θ (k − θ 2, i) ⎪ ), θ2, i ≤ k ⎩ ai(1 − e 1, i
(2)
where ai is a known constant that represents the maximum magnitude of the ith fault; θ1, i > 0 is an unknown constant that represents the rate at which the fault in state xi evolves; θ2, i is the occurring time of the ith fault. The abrupt fault can also be represented by Eq. (2). For a large value of the growth rate θ1, i , the profile of (2) is approximate to a step function, which often models abrupt fault. However, for fault prediction, we only consider the incipient fault. Actually, the Eq. (2) represent the evolution of the incipient fault fi , but we do not use it to improve the accuracy of the fault and state estimation due to the unknown parameter θ1, i and θ2, i . In order to obtain the useful feature of the fault signal, it is reasonable to analyze (2). Before the fault occur, the amplitude of the fault signal fi is always a constant value 0. Once the incipient fault occurs, then the fault signal will changes slowly. Hence it is reasonable to assume that the fault signal always changes slowly. That is to say, the fault signal at present step are often related to their previous step. In detail, the amplitude of each component of the fault vector fk almost keeps unchanged or changes slowly. Then the characteristic of the fault signal can be expressed as the following assumption and we can use it to improve the fault estimation and state estimation in Section 3. Assumption 1.
fk = fk − 1 + wkf− 1
(3)
where wkf− 1 ∈ q is a virtual Gaussian white noise with covariance matrices Q kf− 1 > 0, which represents the noise intensity. The initial fault f−1 is 0 and the covariance is P −f 1. f
Remark 2. The noise wk is virtual, hence it is not related to wk and vk. For convenient, we can choose a fixed covariance matrices, denote as Q f . Because the fault does not always occur and the fault is incipient, it is reasonable to considered that the fault does not change or changes slowly, then a small Q f should be choose. It is also assumed that rank Ck + 1Γ (xk ) = rank Γ (xk ) = q for all k and xk. This assumption is a necessary condition to ensure that the state estimation is unbiased. Our aim is as follows: (1) Effectively estimate the system state and the fault signal. (2) Detect the fault and estimate the parameters of the incipient fault. (3) Predict the trend of the fault based on the estimated parameters.
Please cite this article as: Ding B, Fang H. Fault prediction for nonlinear stochastic system with incipient faults based on particle filter and nonlinear regression. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.018i
B. Ding, H. Fang / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
3. Modified fault estimation and state estimation
3
Once the estimation of fault vector is obtained by (11) and (7), then each of the a priori particles can be revised by
3.1. The existing results
( )
^ xki k − 1 = g xki − 1 + wki − 1 + Γk − 1 fk − 1
In this subsection, we give some results about the fault estimation and state estimation, which can also be shown in [22]. For the dynamic system (1), the fault act as the unknown input of the system. In order to obtain the accurate state estimation, the unknown fault should be estimated firstly at every time step. To do this, we take the following steps. Firstly, based on the best estimation of the mean and covariance of xk 1, i.e., x^k − 1 and Pk 1, generate randomly N initial particles xki − 1(i = 1, … , N ). The parameter N is chosen as a tradeoff between computational effort and estimation accuracy. Using the nonlinear system equation g (·), we can obtain a priori particle vectors xki | k − 1 as
( )
xki | k − 1 = g xki − 1 + wki − 1 (i = 1, …, N )
(14)
Now according to the principle of PF, the unbiased state estimation can be obtained. For clarity, the entire algorithm can be briefly listed in Algorithm 1. Algorithm 1. Fault estimation and state estimation. Step 1 Initialization: The initial state is x^0 with covariance P0 . Step 2 Fault estimation: The a priori particle vectors xki | k − 1 are obtained by (4), the a priori state estimation and its covariance are calculated by (5) and (6), respectively.
R˜ k = CkPk k − 1CkT + Rk
(15)
(4)
where each noise vector wki − 1 is randomly generated on the basis of the known probability distribution function (PDF) of wk − 1. Then the a priori state estimate x^ and the a priori error covariance
(
−1
Lk = FkT R˜ k Fk
−1
)
−1
FkT R˜ k
(16)
k|k−1
Pk | k − 1 are calculated by
(
∑ xki | k − 1
(17)
(5)
i=1 N
1 Pk | k − 1 = N
)
^ fk − 1 = Lk yk − Ckx^k | k − 1
N
1 x^k | k − 1 = N
T
∑ ( xki | k − 1 − x^k | k − 1)( xki | k − 1 − x^k | k − 1) i=1
(6)
Step 3 State estimation Based on the fault estimation, the revised a priori particles are obtained by (14). (i) Importance sampling: Compute the relative likelihood qi of each particle xki k − 1 conditioned on the measurement yk, i.e.,
The estimation of fault fk − 1 can be calculated by
^ fk − 1 = Lky˜k
(
(
)
evaluate the PDF p yk xki k − 1 based on the measurement
(7)
equation and the PDF of the measurement noise. Then normalize the relative likelihoods by
)
where y˜k = yk − Ckx^k k − 1 , and Lk has still to be determined. It is easy to show that
y˜k = CkΓk − 1 fk − 1 + ek
(8)
qi = qi
N
∑ j = 1 qj
(18)
where
(
)
ek = Ck g (xk − 1) + wk − 1 − x^k k − 1 + vk
(9)
( )
and Γk − 1 = Γ x^k − 1 . The expectation and variance of ek can be calculated as follows:
ek ] = 0 R˜ k = ⎡⎣ ek ekT ⎤⎦ = CkPk k − 1CkT + Rk
(10)
Using the similar method as in ([22] Theorem 3.1), the optimal matrix Lk can be obtained by
(
−1
Lk = FkT R˜ k Fk
−1
)
i=1
(19)
(11)
(12)
The variance of the fault estimation error is given by
Rkf− 1 ≜ [(Lkek )(Lkek )T ] = LkR˜ kLkT −1
N
∑ xki k
3.2. Modified fault estimation
^ fk − 1 = fk − 1 + Lkek
(
1 x^k = N
−1
FkT R˜ k
where Fk = CkΓk − 1. Then the unbiased estimation of fk − 1 in the sense of linear least squares is given by (7). Substituting (11) and (8) in (7) yields
= FkT R˜ k F
Now the sum of all the likelihoods is equal to one. (ii) Re-sampling: Generate a set of a posteriori particles x i on k|k the basis of the relative likelihoods qi. (iii) A posteriori state estimation: The state estimation at time k can be calculated by
−1
)
(13)
In the above subsection, the unbiased estimation of the fault fk 1 is obtained, it is optimal under the assumption that their are no prior knowledge about fk 1 is available. However, the estimate of the fault always exists estimation error, which is related to the system noise and measurement noise. Therefore, the precision of the estimated fault is impaired, and further affect the accuracy of state estimation, especially in the case of large intensity of the system or measurement noise. In order to improve the accuracy of the fault estimation, recall that the fault fk 1 always keeps unchanged or changes slowly in most time step. Hence, we can treat (3) as the dynamic equation of
Please cite this article as: Ding B, Fang H. Fault prediction for nonlinear stochastic system with incipient faults based on particle filter and nonlinear regression. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.018i
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the fault signal and (13) as the measurement equation of the fault signal. Then, according to the principle of Kalman filter [23,24], the MFE can be obtained immediately as follows:
φk − 1 | k − 2 = φk − 2
Pkf− 1 | k − 2 = Pkf− 2 + Q kf− 2
(27)
(20)
(
Pkf− 1 | k − 2 = Pkf− 2 + Q kf− 2
Kkf− 1
=
(
Pkf− 1 | k − 2
)
+
)
q
f k−1
(
)
k−2
k−2
^ + Kkf− 1 fk − 1
)
Pkf− 1 = Iq − Kkf− 1 Pkf− 1 | k − 2
^ fk − 1 − φk − 2
( I − K )φ
f k−1
q
(22)
(
=
( I − K )φ
φk − 1 =
−1
Rkf− 1
^ φk − 1 = φk − 1 | k − 2 + Kkf− 1 fk − 1 − φk − 1 | k − 2
(
Step 4 State estimation: The revised a priori particles is given by
^ + Kkf− 1 fk − 1
(23)
( )
)
Pkf− 1 = Iq − Kkf− 1 Pkf− 1 | k − 2 where φk − 1 | k − 2 and
(29)
(30)
xki k − 1 = g xki − 1 + wki − 1 + Γk − 1φk − 1
(
(28)
(21)
Pkf− 1 | k − 2
= φk − 2 + Kkf− 1
−1
)
Kkf− 1 = Pkf− 1 | k − 2 Pkf− 1 | k − 2 + Rkf− 1
(24)
(31)
and the following process are the same as that of in Algorithm 1.
φk 1 represent a priori and a posteriori esti-
mation of the fault fk 1, respectively; Pkf− 1 | k − 2 and Pkf− 1 are the corresponding covariance of the estimation error, Kkf− 1 is the Kalman gain, Iq is the identity matrix with order q. Denoting the MFE error as φ˜ k − 1 ≜ fk − 1 − φk − 1, then it follows from (3), (11) and (24) that
−
(φ
−
Kkf− 1
k−1
+ Lkek − φk − 2
= φ˜ k − 2 + wkf− 2 − Kkf− 1
k−2
+ wkf− 2 + Lkek − φk − 2
φ˜ k − 1 = fk − 2 +
wkf− 2
= φ˜ k − 2 +
wkf− 2
=
k−2
( I − K )( φ˜ q
f k−1
k−2
+
(f (f
(
Kkf− 1
)) ))
^ fk − 1 − φk − 2
))
)
+ wkf− 2 − Kkf− 1Lkek
(25)
Remark 3. It can be shown that the error of MFE (25) is smaller that of fault estimation in Algorithm 1. Therefore, in the same intensity of the system and measurement noise case, the MFE can improve the estimation precision of the fault signal.
4. Fault detection and fault prediction 4.1. Fault detection In Algorithm 2, the fault signal and the system state are both estimated, and the corresponding estimation errors are also obtained. If there are no fault in the system, i.e., each fault signal is zero, then the value of the estimated fault will be very close to zero. Once the fault occurs, i.e., one or some component of the fault vector is not zero, then the corresponding estimated fault signal will deviate from zero. Since the MFE can effectively track the fault signal, an intuitive FD strategy is introduced. The threshold value can be obtained by Monte-carlo methods in the case of no fault. Denote the threshold value as γi (i = 1, … , q). Then the threshold value is given by
γi = max φi, k
^ Now that the MFE φk 1 is obtained, we can use it instead of fk − 1 to estimate the state. To do this, a priori particles should be replaced by
( )
xki k − 1 = g xki − 1 + wki − 1 + Γk − 1φk − 1
(26)
Then, the following processes are similarly as that of in Algorithm 1. Up to now, the modified recursive estimation of fault and state are obtained. In order to make it easier to understand, the complete modified fault estimation and state estimation is listed in Algorithm 2. Algorithm 2. The modified fault estimation and state estimation. Step 1 Initialization: The initial state is x^ with covariance P , 0 0 and the initial fault signal is 0 with covariance P −f 1. Step 2 Fault estimation: This step is the same as that of in Algorithm 1. Step 3 Modified fault estimation:
(32)
k
3.3. State estimation
where φi, k is the ith component of the estimated fault signal If
φk.
φi, k > γi
(33)
then it will be considered that the fault occurs in the ith component at time step k. 4.2. Parameter estimation and fault prediction In order to realize the fault prediction, we should determine the parameters of the incipient fault, i.e., determine the values of θ1 and θ2. To do this, and notice that Eq. (2) is a nonlinear function with two unknown parameters, the nonlinear regression method is adopted. Firstly, once the ith fault is detected, we select l MFEs, denote as Y, i.e.,
Y=
{φ
i, k d
, φi, k
, …, φi, k
d +1
}
d +l−1
(34)
Please cite this article as: Ding B, Fang H. Fault prediction for nonlinear stochastic system with incipient faults based on particle filter and nonlinear regression. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.018i
B. Ding, H. Fang / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
where kd is the detection time, and for convenience the subscript i will be omitted form now on. By (2) and the definition of MFE error, it follows that
(
)
φj = a 1 − e−θ1(j − θ 2) − φ˜j
( j = k , …, k d
)
(35)
d + l− 1
Denote the nonlinear function a(1 − e−θ1(j − θ2)) as h(j, θ ), where θ represents the parameter set, i.e., θ = {θ1, θ2}, then (35) can be rewritten as
φj = h( j, θ ) − φ˜j
( j = kd, …, kd + l − 1)
(36)
This expression is a typical nonlinear regression model. The optimal parameter will be that minimize the residual sum of squares, i.e., minimize the expression kd + l− 1
∑
S(θ ) =
(37)
An effective technique to accomplish this minimization is GNM, which uses a linear approximation to the function h to improve the initial parameter θ0 for θ and keep improving the estimates until there is no change. For comprehension, a brief process about Gauss-Newton method is introduced as follows, and more details can be seen in [25]. Expending the nonlinear function h in a first order Taylor series about θ0, we have
(
)
(
)
(
)
h j, θ ≈ h j, θ 0 + d1, j θ1 − θ10 + d2, j θ2 − θ20
(38)
where
d1, j =
∂h( j, θ ) ∂θ1
d2, j = θ= θ
0
∂h( j, θ ) ∂θ2
)
(44)
θ = θ^
Remark 4. It should be noted that, the approximate inference interval (42) is valid under the assumption that φ˜j( j = kd, … , kd + l − 1) obey the Gaussian distribution. If the nonlinear system (1) are time invariant or nearly time invariant, then this Assumption is tenable according to (25). Actually, the more MFE we choose, i.e. the bigger l, the more accurate of the estimated parameter. However, the bigger l means that we should let the system go on running in the faulty-case. Usually, the l is selected between 30 and 60 according to the actual situation.
(
)
(
In this section, the well known Three-tank system DTS200 is considered to illustrate the effectiveness of the proposed method. 5.1. Plant description The Three-tank system is produced by the Amira Automation Corporation in Germany and widely serves as a benchmark in laboratory for process control and fault detection methods [2]. The structure and principle of the DTS200 is shown in Fig. 1. The discrete state equations of the Three-tank system is described as
⎧h 1, k + 1 = h1, k + T /A(Q 1 − Q 13) + w1, k ⎪ ⎪ ⎨ h2, k + 1 = h2, k + T /A(Q 2 + Q 32 − Q 20) + w2, k ⎪ ⎪ = h + T /A(Q − Q ) + w ⎩h 3, k + 1
θ= θ
(39)
0
In matrix notation we have
(
∂h( j, θ ) ∂θ
5. Simulations
(φj − h( j, θ ))2
j = kd
( )
d=
5
)
h K, θ ≈ h K, θ0 + V 0 θ − θ0
(40)
0
where V is the l × 2 matrix of partial derivatives evaluated at θ = θ 0 , and K = (kd, … , kd + l − 1). Then the first estimates beyond the initial values is obtained as θ1 = θ 0 + b0 , where b0 = (V 0T V 0)−1V 0T (Y − h(K, θ 0)). This process is then repeated with θ0 replaced by θ1 (and V0 by V1), and this produces a new set of estimates. The iterative processes continues until convergence is achieved. Once the optimal parameter is obtained, denoted as θ^ = {θ^ , θ^ }, 1
2
we can use it to predict the future magnitude of the fault. In detail,
3, k
13
32
3, k
(45)
where T is the sampling interval of the system; A is the cross section area of tanks; hi (i = 1, 2, 3) are the water level of each tank; Q1 and Q2 are incoming mass flow; wk is a vector of process noise having three components and a covariance of Qk; Qij is the mass flow from the ith tank to the jth tank, and given as
⎧ ⎪ Q 13 = az1 ssgn(h1 − h3) 2g h1 − h3 ⎪ ⎨ Q 32 = az3 ssgn(h3 − h2) 2g h3 − h2 ⎪ ⎪ Q = az s 2gh 2 ⎩ 20 2
(46)
where s is the cross section area of pipe; azi(i = 1, 2, 3) are the flow coefficient of each pipe; g represent the acceleration of gravity. The values of parameters are listed in Table 1, which is adopted from [26].
the magnitude of the fault at the jth ( j > kd + l − 1) step can be calculated by
( ) (
^ ^ f¯j = h j, θ^ = a 1 − e−θ1( j − θ 2)
)
(41)
furthermore, according to the theory of the Nonlinear Regression Inference, the (1 − α ) approximate inference interval is
h( j, θ^) ± s dT (V T V )−1d t (l − 1; α/2)
(42)
where
s=
Y − h(K , θ^) l−1
(43)
is the residual mean square based on (l − 1) degree of freedom, t (l − 1; α /2) is the upper α/2 quantile for Student's T distribution with (l − 1) degree of freedom, and d is the derivative vector Fig. 1. DTS200 setup.
Please cite this article as: Ding B, Fang H. Fault prediction for nonlinear stochastic system with incipient faults based on particle filter and nonlinear regression. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.018i
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Table 1 The value of parameters.
A = 0.0154 m2 az1 = 0.49
g = 9.81 m/s2 az2 = 0.61
s = 5 × 10−5 m2 az3 = 0.45
Let x i, k = hi, k for i = 1, 2, 3, T = 0.5 s, Q1 = 4 × 10−5 m3/s and
Q 2 = 1.4 × 10−5 m3/s. The initial state vector is assumed to be (0.455, 0.159, 0.320)T . Each component of system noise vector is assumed to be a non-Gaussian distribution as follows:
wi ∼ N (0, σ ) + U[ − σ , σ ] (i = 1, 2, 3)
(47)
where N (0, σ ) denote the Gaussian distribution with mean 0 and variance s; U ( − σ , σ ) denote the uniform distribution from −σ to s; σ = 1 × 10−7. In order to show the validity of the proposed method, two simulation cases are conducted.
Fig. 2. State estimation.
5.2. The first case: Simulation with two faults In this subsection, two typical incipient faults are considered, i.e., leakage fault and jamming fault. These two faults are described respectively as follows:
⎧ 0, 0 ≤ k < 100 f1, k = ⎨ ⎩ 3.5 × 10−5(1 − e−0.013(k − 100)), 100 ≤ k ≤ 300
(48)
⎧ 0, 0 ≤ k < 120 f2, k = ⎨ ⎩ 3 × 10−5(1 − e−0.01(k − 120)), 120 ≤ k ≤ 300
(49)
⎪
⎪
⎪
⎪
where f1, k represents that the area of the leakage hole growing slowly under the first tank at the 100th second, and f2, k means that the jamming area of the outflow pipe occurs at the 120th second. Then the matrix function Γ(xk ) and the fault vector fk can be written as T ⎡ −T 2gx /A ⎡f ⎤ 0 0⎤ 1, k ⎢ ⎥ ⎢ 1, k ⎥ , f = Γ(xk ) = ⎢ ⎥ k ⎢f ⎥ 0 az T 2 gx / A 0 ⎣ 2, k ⎦ 2 2, k ⎣ ⎦
Fig. 3. Fault estimation.
(50)
The measurement equation is given as T yk = Ck⎡⎣ x1, k x2, k x3, k ⎤⎦ + vk
(51)
where
⎡ 1 0 0⎤ Ck = ⎢ ⎣ 0 1 0⎥⎦
(52)
The measurement noise vector is assumed to be Gaussian vector with the mean 0, and covariance matrix 0.25 × 10−6I2. The variance of the virtual noise is chosen as Qd = 0.5 × 10−12I2. In the case of no fault, running (Algorithm 1) 2500 s and according to (32), the threshold is obtained and the value as 4.5 × 10−6 , a very small value, which also means that the accuracy of fault detection is satisfactory. Let the total simulation time is 300 s, then the results are shown in Figs. 2–4. Fig. 2 gives the results of state estimation. It can be seen that the states of the system are estimated accurately both in fault-free case and faulty case. Fig. 3 shows the time profile of the actual incipient fault, the estimated fault and the modified fault estimation, respectively. It can be seen that the estimated fault has deviated seriously from the original fault signal due to the large estimation error, hence it
Fig. 4. Parameter estimation and fault prediction.
is hard to use it to detect the fault, not to mention to predict the fault. On the other hand, the modified fault estimation has a good approximation to the original fault signal. This indicate that the MFE is indeed enhance the estimated accuracy of the fault.
Please cite this article as: Ding B, Fang H. Fault prediction for nonlinear stochastic system with incipient faults based on particle filter and nonlinear regression. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.018i
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In Fig. 4, the first fault is detected at the 110 s and the second fault is detected at 139.5 s. Let l ¼50, the first initial parameter θ10 is 0.01 and the second initial parameter θ02 equals to the corresponding detection time. Then according to GNM, the first fault parameters are estimated as θ^ = 0.01359 and θ^ = 102.147, the 1,1
2,1
second fault parameters are θ^1,2 = 0.00872 and θ^2,2 = 114.863. It is shown that the results are well consistent with the true fault parameters. Based on the estimated fault parameter, the predicted fault amplitudes are easy obtained and the results can also be shown in Fig. 4. 5.3. The second case: Simulation with three faults In this section, a more complex situation is implemented: assume that there are three incipient faults will occur in the Threetank system. These faults are described respectively as follows:
f1, k
⎧ 0, 0 ≤ k < 180 =⎨ ⎩ 3 × 10−5(1 − e−0.013(k − 180)), 180 ≤ k ≤ 300
Fig. 5. State estimation.
⎪
⎪
(53)
⎧ 0, 0 ≤ k < 130 f2, k = ⎨ ⎩ 3.5 × 10−5(1 − e−0.01(k − 130)), 130 ≤ k ≤ 300
(54)
⎧ 0, 0 ≤ k < 100 f3, k = ⎨ ⎩ 4 × 10−5(1 − e−0.012(k − 100)), 100 ≤ k ≤ 300
(55)
⎪
⎪
⎪
⎪
where f1, k denotes that the area of the leakage hole growing slowly under the first tank at the 180th second; f2, k means that the jamming area of the outflow pipe occurs at the 130th second; f3, k represents that the jamming area between the first tank and the third tank occurs at the 100th second. the matrix function Γ(xk ) and the fault vector fk can be written as
⎡ −T 2gx /A ⎡f ⎤ 0 Fak ⎤ 1, k ⎥ ⎢ ⎢ 1, k ⎥ ⎥, f = ⎢ f ⎥ Γ(xk ) = ⎢ 0 az T 2 gx / A 0 k 2 2, k ⎥ ⎢ ⎢ 2, k ⎥ ⎥ ⎢ ⎢⎣ f3, k ⎥⎦ 0 0 −Fak ⎦ ⎣
(56)
az T
where Fak = A1 sgn(x1, k − x3, k ) 2g x1, k − x3, k . Assume that all the states are directly measured with Gaussian noises, then the measurement equation is shown as follows:
yk = Ck[x1, k x2, k x3, k ]T + vk
(57)
where
⎡ 1 0 0⎤ ⎢ ⎥ Ck = ⎢ 0 1 0⎥ ⎣ 0 0 1⎦
(58)
The measurement noise vector is assumed to be Gaussian vector with the mean 0, and covariance matrix 0.25 × 10−6I3. The variance of the virtual noise is chosen as Qd = 0.5 × 10−12I3. In the case of no fault, running (Algorithm 1) 2500 s and according to (32), the threshold is obtained and the values are 4.5 × 10−6 , 4.6 × 10−6 , 4.7 × 10−6 , respectively. Let the total simulation time is 300 s, the results can be shown in Figs. 5 and 6. Fig. 5 gives the results of state estimation in this situation. Fig. 6 gives the time profile of the actual incipient fault and the modified fault estimation. Three faults are detected at the 192 s,
Fig. 6. Parameter estimation and fault prediction.
the 147 s and 123.5 s, respectively. Let l ¼50, based on GNM, the first fault parameters are estimated as θ^1,1 = 0.01161 and θ^ = 180.452; the second fault parameters are θ^ = 0.00918 and 2,1
1,2
θ^2,2 = 130.155; the third fault parameters are θ^1,3 = 0.01392 and
Please cite this article as: Ding B, Fang H. Fault prediction for nonlinear stochastic system with incipient faults based on particle filter and nonlinear regression. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.018i
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θ^2,3 = 104.939. It can be shown that in this complex situation, the results are also well consistent with the true fault parameters. According to the estimated fault parameter, the predicted fault amplitudes can be obtained and the results can also be shown in Fig. 6. Once the future fault signal is obtained, we can evaluate the system performance in advance and take appropriate action to recondition the system.
6. Conclusions In this paper, we have considered the fault prediction problem for nonlinear stochastic system with incipient faults, where each of the incipient fault is expressed by a nonlinear function with unknown parameters. The estimated fault with higher accuracy is obtained by the MFE algorithm, which is based on the PF and the reasonable assumption about the incipient fault process. Meanwhile, the state estimation is also obtained. According to the estimated fault, an intuitive fault detection strategy is introduced. Once the fault is detected, the parameters of each component fault are identified by the GNM. Then, the future fault signal can be predicted based on the expression incipient fault and the estimated parameters, thus realize the fault prediction. Simulations of the Three-tank system show the effectiveness of the proposed method.
Acknowledgments This work was supported by the National Natural Science Foundation of China under Grant 61473127.
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shock and vibration. Vol. 5. Springer; New York, 2011. p. 399–404. [6] Summerer L. Generalized likelihood ratio approach for fault detection in linear dynamic stochastic systems with unknown inputs. Int J Syst Sci 1996;27 (12):1231–41. [7] Kobayashi Takahisa, Simon Donald L. Application of a bank of Kalman filters for aircraft engine fault diagnostics. In: ASME Turbo Expo 2003, collocated with the 2003 International joint power generation conference. American Society of Mechanical Engineers, Atlanta, Georgia, 2003. p. 461–470. [8] Luo Zhen, Fang Huajing. Fault detection for nonlinear systems with unknown input. Asian J Control 2013;15(5):1503–9. [9] Li Ping, Kadirkamanathan Visakan. Fault detection and isolation in non-linear stochastic systems a combined adaptive Monte Carlo filtering and likelihood ratio approach. Int J Control 2004;77(12):1101–14. [10] Compare Michele, Baraldi Piero, Turati Pietro, Zio Enrico. Interacting multiplemodels, state augmented particle filtering for fault diagnostics. Probab Eng Mech 2015;40:12–24. [11] Demetriou Michael A, Polycarpou Marios M. Incipient fault diagnosis of dynamical systems using online approximators. IEEE Trans Autom Control 1998;43(11):1612–7. [12] Zhang Xiaodong, Polycarpou Marios M, Parisini Thomas. A robust detection and isolation scheme for abrupt and incipient faults in nonlinear systems. IEEE Trans Autom Control 2002;47(4):576–93. [13] Wang Wenhui, Li Linglai, Zhou Donghua, Liu Kaidi. Robust state estimation and fault diagnosis for uncertain hybrid nonlinear systems. Nonlinear Anal: Hybrid Syst 2007;1(1):2–15. [14] Thumati Balaje T, Feinstein Miles A, Jagannathan Sarangapani. A model-based fault detection and prognostics scheme for takagi-sugeno fuzzy systems. IEEE Trans Fuzzy Syst 2014;22(4):736–48. [15] Alwi Halim, Edwards Christopher, Tan Chee Pin. Sliding mode estimation schemes for incipient sensor faults. Automatica 2009;45(7):1679–85. [16] Zhang Juyong, Swain Ayas Kanta, Nguang Sing Kiong. Detection and isolation of incipient sensor faults for a class of uncertain non-linear systems. IET Control Theory Appl 2012;6(12):1870–80. [17] Liu Chun, Jiang Bin, Zhang Ke. Incipient fault detection using an associated adaptive and sliding-mode observer for quadrotor helicopter attitude control systems. Circuits Syst Signal Process 2016:1–20. [18] Zhang Ke, Jiang Bin, Shi Peng. Observer-based fault estimation and accomodation for dynamic systems. Springer, Berlin, Heidelberg, 2013;436. [19] Zhang Ke, Jiang Bin, Shi Peng, Xu Jinfa. Multi-constrained fault estimation observer design with finite frequency specifications for continuous-time systems. Int J Control 2014;87(8):1635–45. [20] Zhang Ke, Jiang Bin, Shi Peng, Xu Jingfa. Fault estimation observer design for discrete?time systems in finite?frequency domain Int J Robust Nonlin Control 2015;25(9):1379–98. [21] Xu Zhengguo, Ji Yindong, Zhou Donghua. A new real-time reliability prediction method for dynamic systems based on on-line fault prediction. IEEE Trans Reliab 2009;58(3):523–38. [22] Ding Bo, Fang Huajing. Multi-faults detection and estimation for nonlinear stochastic system based on particle filter and hypothesis test. Int J Syst Sci 2016;47(16):3812–21. [23] Kailath Thomas, Sayed Ali H, Hassibi Babak. Linear estimation. Upper Saddle River, NJ: Prentice Hall; 2000. [24] Simon Dan. Optimal state estimation: Kalman, H infinity, and nonlinear approaches. John Wiley & Sons, Hoboken, New Jersey; 2006. [25] Bates Douglas M, Watts Donald G. Nonlinear regression analysis and its applications; 1988. [26] Xu Zhengguo, Ji Yindong, Zhou Donghua. Real-time reliability prediction for a dynamic system based on the hidden degradation process identification. IEEE Trans Reliab 2008;57(2):230–42.
Please cite this article as: Ding B, Fang H. Fault prediction for nonlinear stochastic system with incipient faults based on particle filter and nonlinear regression. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.018i
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Research article
Fault prediction for nonlinear stochastic system with incipient faults based on particle filter and nonlinear regression Bo Ding, Huajing Fang n School of Automation, Key Laboratory of Image Processing and Intelligent Control, Ministry of Education, Huazhong University of Science and Technology, Wuhan 430074, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 18 July 2016 Received in revised form 25 January 2017 Accepted 22 March 2017
This paper is concerned with the fault prediction for the nonlinear stochastic system with incipient faults. Based on the particle filter and the reasonable assumption about the incipient faults, the modified fault estimation algorithm is proposed, and the system state is estimated simultaneously. According to the modified fault estimation, an intuitive fault detection strategy is introduced. Once each of the incipient fault is detected, the parameters of which are identified by a nonlinear regression method. Then, based on the estimated parameters, the future fault signal can be predicted. Finally, the effectiveness of the proposed method is verified by the simulations of the Three-tank system. & 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Fault detection Fault prediction Nonlinear stochastic system Incipient fault Particle filter Nonlinear regression
1. Introduction In order to keep the safety and reliability of the modern complex system, and with the development of digital computers and the system identification techniques, the design and analysis of the model-based fault detection (FD) algorithms has received considerable attention in recent decades [1–3]. As an important branch of the FD technology, FD based on stochastic models has also been well studied. For linear stochastic system (LSS), the mainly FD algorithm is by using the innovation (residual) generated by Kalman filter (KF), where the fault are detected by the statistic testing on whiteness of the residual [4,5] or by the generalized likelihood ratio (GLR) testing on the residual [6]. In order to isolate the fault, the multiple KF method is proposed in [7], where each KF is designed for detecting a specific fault. Once a fault does occur, all filters except the one has the correct hypothesis will generate the large estimation error, hence the specific fault can be isolated. For nonlinear stochastic system (NSS), the FD algorithm is similar to that of the LSS, the difference is that the KF is replaced by extended Kalman filter (EKF), unscented Kalman filter (UKF) or particle filter (PF) according to the difference assumption of the system and noise [8–10]. With the development of the FD technology, researchers begin to explore the algorithm of incipient fault detection. In [11], the n
Corresponding author. E-mail address: [email protected] (H. Fang).
on-line approximation method is proposed to detect the incipient fault, where the evolution of the incipient fault is expressed as a nonlinear function with unknown parameters. This expression about incipient fault has been cited by many literatures, such as [12–14]. In [15], a sliding-mode observer is proposed to reconstruct the incipient sensor fault. The similar method can also be used in [16] to detect and isolate the incipient sensor fault for the uncertain nonlinear system and in [17] for the quadrotor helicopter attitude control system. Owing to its wide application in fault tolerant control, fault estimation (FE) has also been noticed and some meaningful results have been achieved [18]. In [19], a multi-constrained full-order fault estimation observer (FFEO) design with finite frequency specifications was proposed. Later, this result was extended to the discrete-time systems [20], and an reduced-order fault estimation observer (RFEO) design was also obtained. It should be noted that, the incipient fault often changes slowly and it is almost unnoticeable during its early stage, hence the research on detection and estimation of the incipient fault is still a challenge problem, especially in the nonlinear systems with stochastic noise. Meanwhile, with the improvement of requirement of the safety and reliability, researchers not only want to be able to detect the incipient fault after it occurs, but even more hope to predict the trend of the fault. If the incipient fault is detected in time and then its trend can be predicted as early as possible, then the manager will have more time to take action to avoid the possible serious damage. Moreover, fault prediction can also provide more support
http://dx.doi.org/10.1016/j.isatra.2017.03.018 0019-0578/& 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Ding B, Fang H. Fault prediction for nonlinear stochastic system with incipient faults based on particle filter and nonlinear regression. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.018i
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for system management and maintenance [21]. Therefore, the study about fault prediction technology has an important theoretical significance and application values. However, very few results are related to the issue of fault prediction since it is difficult to detect the incipient fault in time. More recently, for a class of nonlinear stochastic system with additive faults, based on modified PF (MPF), a FD algorithm which can simultaneously estimate the fault and state is proposed [22]. The error of the fault estimation is closely related to the system noise and measurement noise. Theoretically, if the intensity of the system noise and measurement noise are small enough, then the accuracy of the estimated fault will be high enough. Once the estimated fault with enough accuracy is obtained, it can be used to estimate the fault's parameters, then the future fault can be predicted. However, in practice, the intensity of the system noise or measurement noise may not always sufficiently small, instead in most cases we should enlarge the system noise to compensate for modeling errors, so the MPF algorithm can not provide the estimated fault with enough accuracy. And in the case of large error of the fault estimation, it is hard to detect the fault in time, not to mention to predict the fault. In this work, an improved method is introduced to enhance the accuracy of fault estimation in the mild case, where the intensity of the noise is not exceptionally small, and the parameters of each fault is successfully estimated through the nonlinear regression, thus realize the fault prediction. As in previous literatures, the evolution function of each incipient fault is also expressed as a nonlinear function with unknown parameters. Obviously, this nonlinear function can not be directly used in the filter design because of the unknown parameters. However, as is known, the incipient fault often changes slowly, then based on this feature, a reasonable assumption about the incipient fault is introduced, i.e., the change of each fault signal between any two adjacent time steps is always bounded and small. Combine this assumption with the PF, the modified fault estimation (MFE) algorithm is derived, and then the state estimation can also be obtained. The results show that the fault estimation precision is effectively improved. Based on the MFE, an intuitive fault detection strategy is introduced. Once the fault is detected, the parameters of each component fault are identified by an effective nonlinear regression method: Gauss–Newton method (GNM). After that, the future fault signal can be predicted in light of the expression of the incipient fault and the estimated parameters. Lastly, the effectiveness of the proposed method is illustrated by the simulation of the Three-tank system. The rest of this paper is organized as follows. Section 2 illustrates the formulation of the nonlinear dynamic system and the process of the incipient fault. Section 3 gives the details of modified fault estimation and state estimation based on particle filter. Section 4 discusses the fault detection and fault prediction algorithm. Section 5 illustrates the proposed approach by the simulation of Three tank system. Finally, some conclusions are drawn in Section 6.
2. Problem formulation Consider the following non-linear system with incipient faults:
⎧ + wk ⎪ xk + 1 = g (xk ) + Γ (xk )f k ⎨ ⎪ y C x v = + ⎩ k k k k n
p
(1)
where xk ∈ and yk ∈ are the state vector and measurement vector respectively. g (xk ) is the system function. fk ∈ q represents the incipient fault vector, Γ(xk ) is a known distribution matrix function, Ck is the measurement matrix with compatible
dimensions. The process noise wk ∈ n and measurement noise vk ∈ p are mutually uncorrelated white noise sequences, with known covariance matrices Q k ≥ 0 and Rk > 0 respectively. The initial state x0 is independent of wk and vk with the known mean x^ and covariance P . In order to simplify the discussion and 0
0
without loss of generality, the known input vector uk is not included in the system model. Remark 1. In (1), Γ(xk ) is a matrix function about xk, hence the fault is multiplicative. This type of fault often represents the degradation of the component. If Γ(xk ) is not related to xk, then the fault is additive, and additive fault usually represents the offset in actuator. Therefore, model (1) has a wide applicability. Let fi denotes the ith component of the fault vector, and the incipient fault fi can be described as
⎧ ⎪ 0, 0 ≤ k < θ2, i fi, k = ⎨ −θ (k − θ 2, i) ⎪ ), θ2, i ≤ k ⎩ ai(1 − e 1, i
(2)
where ai is a known constant that represents the maximum magnitude of the ith fault; θ1, i > 0 is an unknown constant that represents the rate at which the fault in state xi evolves; θ2, i is the occurring time of the ith fault. The abrupt fault can also be represented by Eq. (2). For a large value of the growth rate θ1, i , the profile of (2) is approximate to a step function, which often models abrupt fault. However, for fault prediction, we only consider the incipient fault. Actually, the Eq. (2) represent the evolution of the incipient fault fi , but we do not use it to improve the accuracy of the fault and state estimation due to the unknown parameter θ1, i and θ2, i . In order to obtain the useful feature of the fault signal, it is reasonable to analyze (2). Before the fault occur, the amplitude of the fault signal fi is always a constant value 0. Once the incipient fault occurs, then the fault signal will changes slowly. Hence it is reasonable to assume that the fault signal always changes slowly. That is to say, the fault signal at present step are often related to their previous step. In detail, the amplitude of each component of the fault vector fk almost keeps unchanged or changes slowly. Then the characteristic of the fault signal can be expressed as the following assumption and we can use it to improve the fault estimation and state estimation in Section 3. Assumption 1.
fk = fk − 1 + wkf− 1
(3)
where wkf− 1 ∈ q is a virtual Gaussian white noise with covariance matrices Q kf− 1 > 0, which represents the noise intensity. The initial fault f−1 is 0 and the covariance is P −f 1. f
Remark 2. The noise wk is virtual, hence it is not related to wk and vk. For convenient, we can choose a fixed covariance matrices, denote as Q f . Because the fault does not always occur and the fault is incipient, it is reasonable to considered that the fault does not change or changes slowly, then a small Q f should be choose. It is also assumed that rank Ck + 1Γ (xk ) = rank Γ (xk ) = q for all k and xk. This assumption is a necessary condition to ensure that the state estimation is unbiased. Our aim is as follows: (1) Effectively estimate the system state and the fault signal. (2) Detect the fault and estimate the parameters of the incipient fault. (3) Predict the trend of the fault based on the estimated parameters.
Please cite this article as: Ding B, Fang H. Fault prediction for nonlinear stochastic system with incipient faults based on particle filter and nonlinear regression. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.018i
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3. Modified fault estimation and state estimation
3
Once the estimation of fault vector is obtained by (11) and (7), then each of the a priori particles can be revised by
3.1. The existing results
( )
^ xki k − 1 = g xki − 1 + wki − 1 + Γk − 1 fk − 1
In this subsection, we give some results about the fault estimation and state estimation, which can also be shown in [22]. For the dynamic system (1), the fault act as the unknown input of the system. In order to obtain the accurate state estimation, the unknown fault should be estimated firstly at every time step. To do this, we take the following steps. Firstly, based on the best estimation of the mean and covariance of xk 1, i.e., x^k − 1 and Pk 1, generate randomly N initial particles xki − 1(i = 1, … , N ). The parameter N is chosen as a tradeoff between computational effort and estimation accuracy. Using the nonlinear system equation g (·), we can obtain a priori particle vectors xki | k − 1 as
( )
xki | k − 1 = g xki − 1 + wki − 1 (i = 1, …, N )
(14)
Now according to the principle of PF, the unbiased state estimation can be obtained. For clarity, the entire algorithm can be briefly listed in Algorithm 1. Algorithm 1. Fault estimation and state estimation. Step 1 Initialization: The initial state is x^0 with covariance P0 . Step 2 Fault estimation: The a priori particle vectors xki | k − 1 are obtained by (4), the a priori state estimation and its covariance are calculated by (5) and (6), respectively.
R˜ k = CkPk k − 1CkT + Rk
(15)
(4)
where each noise vector wki − 1 is randomly generated on the basis of the known probability distribution function (PDF) of wk − 1. Then the a priori state estimate x^ and the a priori error covariance
(
−1
Lk = FkT R˜ k Fk
−1
)
−1
FkT R˜ k
(16)
k|k−1
Pk | k − 1 are calculated by
(
∑ xki | k − 1
(17)
(5)
i=1 N
1 Pk | k − 1 = N
)
^ fk − 1 = Lk yk − Ckx^k | k − 1
N
1 x^k | k − 1 = N
T
∑ ( xki | k − 1 − x^k | k − 1)( xki | k − 1 − x^k | k − 1) i=1
(6)
Step 3 State estimation Based on the fault estimation, the revised a priori particles are obtained by (14). (i) Importance sampling: Compute the relative likelihood qi of each particle xki k − 1 conditioned on the measurement yk, i.e.,
The estimation of fault fk − 1 can be calculated by
^ fk − 1 = Lky˜k
(
(
)
evaluate the PDF p yk xki k − 1 based on the measurement
(7)
equation and the PDF of the measurement noise. Then normalize the relative likelihoods by
)
where y˜k = yk − Ckx^k k − 1 , and Lk has still to be determined. It is easy to show that
y˜k = CkΓk − 1 fk − 1 + ek
(8)
qi = qi
N
∑ j = 1 qj
(18)
where
(
)
ek = Ck g (xk − 1) + wk − 1 − x^k k − 1 + vk
(9)
( )
and Γk − 1 = Γ x^k − 1 . The expectation and variance of ek can be calculated as follows:
ek ] = 0 R˜ k = ⎡⎣ ek ekT ⎤⎦ = CkPk k − 1CkT + Rk
(10)
Using the similar method as in ([22] Theorem 3.1), the optimal matrix Lk can be obtained by
(
−1
Lk = FkT R˜ k Fk
−1
)
i=1
(19)
(11)
(12)
The variance of the fault estimation error is given by
Rkf− 1 ≜ [(Lkek )(Lkek )T ] = LkR˜ kLkT −1
N
∑ xki k
3.2. Modified fault estimation
^ fk − 1 = fk − 1 + Lkek
(
1 x^k = N
−1
FkT R˜ k
where Fk = CkΓk − 1. Then the unbiased estimation of fk − 1 in the sense of linear least squares is given by (7). Substituting (11) and (8) in (7) yields
= FkT R˜ k F
Now the sum of all the likelihoods is equal to one. (ii) Re-sampling: Generate a set of a posteriori particles x i on k|k the basis of the relative likelihoods qi. (iii) A posteriori state estimation: The state estimation at time k can be calculated by
−1
)
(13)
In the above subsection, the unbiased estimation of the fault fk 1 is obtained, it is optimal under the assumption that their are no prior knowledge about fk 1 is available. However, the estimate of the fault always exists estimation error, which is related to the system noise and measurement noise. Therefore, the precision of the estimated fault is impaired, and further affect the accuracy of state estimation, especially in the case of large intensity of the system or measurement noise. In order to improve the accuracy of the fault estimation, recall that the fault fk 1 always keeps unchanged or changes slowly in most time step. Hence, we can treat (3) as the dynamic equation of
Please cite this article as: Ding B, Fang H. Fault prediction for nonlinear stochastic system with incipient faults based on particle filter and nonlinear regression. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.018i
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the fault signal and (13) as the measurement equation of the fault signal. Then, according to the principle of Kalman filter [23,24], the MFE can be obtained immediately as follows:
φk − 1 | k − 2 = φk − 2
Pkf− 1 | k − 2 = Pkf− 2 + Q kf− 2
(27)
(20)
(
Pkf− 1 | k − 2 = Pkf− 2 + Q kf− 2
Kkf− 1
=
(
Pkf− 1 | k − 2
)
+
)
q
f k−1
(
)
k−2
k−2
^ + Kkf− 1 fk − 1
)
Pkf− 1 = Iq − Kkf− 1 Pkf− 1 | k − 2
^ fk − 1 − φk − 2
( I − K )φ
f k−1
q
(22)
(
=
( I − K )φ
φk − 1 =
−1
Rkf− 1
^ φk − 1 = φk − 1 | k − 2 + Kkf− 1 fk − 1 − φk − 1 | k − 2
(
Step 4 State estimation: The revised a priori particles is given by
^ + Kkf− 1 fk − 1
(23)
( )
)
Pkf− 1 = Iq − Kkf− 1 Pkf− 1 | k − 2 where φk − 1 | k − 2 and
(29)
(30)
xki k − 1 = g xki − 1 + wki − 1 + Γk − 1φk − 1
(
(28)
(21)
Pkf− 1 | k − 2
= φk − 2 + Kkf− 1
−1
)
Kkf− 1 = Pkf− 1 | k − 2 Pkf− 1 | k − 2 + Rkf− 1
(24)
(31)
and the following process are the same as that of in Algorithm 1.
φk 1 represent a priori and a posteriori esti-
mation of the fault fk 1, respectively; Pkf− 1 | k − 2 and Pkf− 1 are the corresponding covariance of the estimation error, Kkf− 1 is the Kalman gain, Iq is the identity matrix with order q. Denoting the MFE error as φ˜ k − 1 ≜ fk − 1 − φk − 1, then it follows from (3), (11) and (24) that
−
(φ
−
Kkf− 1
k−1
+ Lkek − φk − 2
= φ˜ k − 2 + wkf− 2 − Kkf− 1
k−2
+ wkf− 2 + Lkek − φk − 2
φ˜ k − 1 = fk − 2 +
wkf− 2
= φ˜ k − 2 +
wkf− 2
=
k−2
( I − K )( φ˜ q
f k−1
k−2
+
(f (f
(
Kkf− 1
)) ))
^ fk − 1 − φk − 2
))
)
+ wkf− 2 − Kkf− 1Lkek
(25)
Remark 3. It can be shown that the error of MFE (25) is smaller that of fault estimation in Algorithm 1. Therefore, in the same intensity of the system and measurement noise case, the MFE can improve the estimation precision of the fault signal.
4. Fault detection and fault prediction 4.1. Fault detection In Algorithm 2, the fault signal and the system state are both estimated, and the corresponding estimation errors are also obtained. If there are no fault in the system, i.e., each fault signal is zero, then the value of the estimated fault will be very close to zero. Once the fault occurs, i.e., one or some component of the fault vector is not zero, then the corresponding estimated fault signal will deviate from zero. Since the MFE can effectively track the fault signal, an intuitive FD strategy is introduced. The threshold value can be obtained by Monte-carlo methods in the case of no fault. Denote the threshold value as γi (i = 1, … , q). Then the threshold value is given by
γi = max φi, k
^ Now that the MFE φk 1 is obtained, we can use it instead of fk − 1 to estimate the state. To do this, a priori particles should be replaced by
( )
xki k − 1 = g xki − 1 + wki − 1 + Γk − 1φk − 1
(26)
Then, the following processes are similarly as that of in Algorithm 1. Up to now, the modified recursive estimation of fault and state are obtained. In order to make it easier to understand, the complete modified fault estimation and state estimation is listed in Algorithm 2. Algorithm 2. The modified fault estimation and state estimation. Step 1 Initialization: The initial state is x^ with covariance P , 0 0 and the initial fault signal is 0 with covariance P −f 1. Step 2 Fault estimation: This step is the same as that of in Algorithm 1. Step 3 Modified fault estimation:
(32)
k
3.3. State estimation
where φi, k is the ith component of the estimated fault signal If
φk.
φi, k > γi
(33)
then it will be considered that the fault occurs in the ith component at time step k. 4.2. Parameter estimation and fault prediction In order to realize the fault prediction, we should determine the parameters of the incipient fault, i.e., determine the values of θ1 and θ2. To do this, and notice that Eq. (2) is a nonlinear function with two unknown parameters, the nonlinear regression method is adopted. Firstly, once the ith fault is detected, we select l MFEs, denote as Y, i.e.,
Y=
{φ
i, k d
, φi, k
, …, φi, k
d +1
}
d +l−1
(34)
Please cite this article as: Ding B, Fang H. Fault prediction for nonlinear stochastic system with incipient faults based on particle filter and nonlinear regression. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.018i
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where kd is the detection time, and for convenience the subscript i will be omitted form now on. By (2) and the definition of MFE error, it follows that
(
)
φj = a 1 − e−θ1(j − θ 2) − φ˜j
( j = k , …, k d
)
(35)
d + l− 1
Denote the nonlinear function a(1 − e−θ1(j − θ2)) as h(j, θ ), where θ represents the parameter set, i.e., θ = {θ1, θ2}, then (35) can be rewritten as
φj = h( j, θ ) − φ˜j
( j = kd, …, kd + l − 1)
(36)
This expression is a typical nonlinear regression model. The optimal parameter will be that minimize the residual sum of squares, i.e., minimize the expression kd + l− 1
∑
S(θ ) =
(37)
An effective technique to accomplish this minimization is GNM, which uses a linear approximation to the function h to improve the initial parameter θ0 for θ and keep improving the estimates until there is no change. For comprehension, a brief process about Gauss-Newton method is introduced as follows, and more details can be seen in [25]. Expending the nonlinear function h in a first order Taylor series about θ0, we have
(
)
(
)
(
)
h j, θ ≈ h j, θ 0 + d1, j θ1 − θ10 + d2, j θ2 − θ20
(38)
where
d1, j =
∂h( j, θ ) ∂θ1
d2, j = θ= θ
0
∂h( j, θ ) ∂θ2
)
(44)
θ = θ^
Remark 4. It should be noted that, the approximate inference interval (42) is valid under the assumption that φ˜j( j = kd, … , kd + l − 1) obey the Gaussian distribution. If the nonlinear system (1) are time invariant or nearly time invariant, then this Assumption is tenable according to (25). Actually, the more MFE we choose, i.e. the bigger l, the more accurate of the estimated parameter. However, the bigger l means that we should let the system go on running in the faulty-case. Usually, the l is selected between 30 and 60 according to the actual situation.
(
)
(
In this section, the well known Three-tank system DTS200 is considered to illustrate the effectiveness of the proposed method. 5.1. Plant description The Three-tank system is produced by the Amira Automation Corporation in Germany and widely serves as a benchmark in laboratory for process control and fault detection methods [2]. The structure and principle of the DTS200 is shown in Fig. 1. The discrete state equations of the Three-tank system is described as
⎧h 1, k + 1 = h1, k + T /A(Q 1 − Q 13) + w1, k ⎪ ⎪ ⎨ h2, k + 1 = h2, k + T /A(Q 2 + Q 32 − Q 20) + w2, k ⎪ ⎪ = h + T /A(Q − Q ) + w ⎩h 3, k + 1
θ= θ
(39)
0
In matrix notation we have
(
∂h( j, θ ) ∂θ
5. Simulations
(φj − h( j, θ ))2
j = kd
( )
d=
5
)
h K, θ ≈ h K, θ0 + V 0 θ − θ0
(40)
0
where V is the l × 2 matrix of partial derivatives evaluated at θ = θ 0 , and K = (kd, … , kd + l − 1). Then the first estimates beyond the initial values is obtained as θ1 = θ 0 + b0 , where b0 = (V 0T V 0)−1V 0T (Y − h(K, θ 0)). This process is then repeated with θ0 replaced by θ1 (and V0 by V1), and this produces a new set of estimates. The iterative processes continues until convergence is achieved. Once the optimal parameter is obtained, denoted as θ^ = {θ^ , θ^ }, 1
2
we can use it to predict the future magnitude of the fault. In detail,
3, k
13
32
3, k
(45)
where T is the sampling interval of the system; A is the cross section area of tanks; hi (i = 1, 2, 3) are the water level of each tank; Q1 and Q2 are incoming mass flow; wk is a vector of process noise having three components and a covariance of Qk; Qij is the mass flow from the ith tank to the jth tank, and given as
⎧ ⎪ Q 13 = az1 ssgn(h1 − h3) 2g h1 − h3 ⎪ ⎨ Q 32 = az3 ssgn(h3 − h2) 2g h3 − h2 ⎪ ⎪ Q = az s 2gh 2 ⎩ 20 2
(46)
where s is the cross section area of pipe; azi(i = 1, 2, 3) are the flow coefficient of each pipe; g represent the acceleration of gravity. The values of parameters are listed in Table 1, which is adopted from [26].
the magnitude of the fault at the jth ( j > kd + l − 1) step can be calculated by
( ) (
^ ^ f¯j = h j, θ^ = a 1 − e−θ1( j − θ 2)
)
(41)
furthermore, according to the theory of the Nonlinear Regression Inference, the (1 − α ) approximate inference interval is
h( j, θ^) ± s dT (V T V )−1d t (l − 1; α/2)
(42)
where
s=
Y − h(K , θ^) l−1
(43)
is the residual mean square based on (l − 1) degree of freedom, t (l − 1; α /2) is the upper α/2 quantile for Student's T distribution with (l − 1) degree of freedom, and d is the derivative vector Fig. 1. DTS200 setup.
Please cite this article as: Ding B, Fang H. Fault prediction for nonlinear stochastic system with incipient faults based on particle filter and nonlinear regression. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.018i
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Table 1 The value of parameters.
A = 0.0154 m2 az1 = 0.49
g = 9.81 m/s2 az2 = 0.61
s = 5 × 10−5 m2 az3 = 0.45
Let x i, k = hi, k for i = 1, 2, 3, T = 0.5 s, Q1 = 4 × 10−5 m3/s and
Q 2 = 1.4 × 10−5 m3/s. The initial state vector is assumed to be (0.455, 0.159, 0.320)T . Each component of system noise vector is assumed to be a non-Gaussian distribution as follows:
wi ∼ N (0, σ ) + U[ − σ , σ ] (i = 1, 2, 3)
(47)
where N (0, σ ) denote the Gaussian distribution with mean 0 and variance s; U ( − σ , σ ) denote the uniform distribution from −σ to s; σ = 1 × 10−7. In order to show the validity of the proposed method, two simulation cases are conducted.
Fig. 2. State estimation.
5.2. The first case: Simulation with two faults In this subsection, two typical incipient faults are considered, i.e., leakage fault and jamming fault. These two faults are described respectively as follows:
⎧ 0, 0 ≤ k < 100 f1, k = ⎨ ⎩ 3.5 × 10−5(1 − e−0.013(k − 100)), 100 ≤ k ≤ 300
(48)
⎧ 0, 0 ≤ k < 120 f2, k = ⎨ ⎩ 3 × 10−5(1 − e−0.01(k − 120)), 120 ≤ k ≤ 300
(49)
⎪
⎪
⎪
⎪
where f1, k represents that the area of the leakage hole growing slowly under the first tank at the 100th second, and f2, k means that the jamming area of the outflow pipe occurs at the 120th second. Then the matrix function Γ(xk ) and the fault vector fk can be written as T ⎡ −T 2gx /A ⎡f ⎤ 0 0⎤ 1, k ⎢ ⎥ ⎢ 1, k ⎥ , f = Γ(xk ) = ⎢ ⎥ k ⎢f ⎥ 0 az T 2 gx / A 0 ⎣ 2, k ⎦ 2 2, k ⎣ ⎦
Fig. 3. Fault estimation.
(50)
The measurement equation is given as T yk = Ck⎡⎣ x1, k x2, k x3, k ⎤⎦ + vk
(51)
where
⎡ 1 0 0⎤ Ck = ⎢ ⎣ 0 1 0⎥⎦
(52)
The measurement noise vector is assumed to be Gaussian vector with the mean 0, and covariance matrix 0.25 × 10−6I2. The variance of the virtual noise is chosen as Qd = 0.5 × 10−12I2. In the case of no fault, running (Algorithm 1) 2500 s and according to (32), the threshold is obtained and the value as 4.5 × 10−6 , a very small value, which also means that the accuracy of fault detection is satisfactory. Let the total simulation time is 300 s, then the results are shown in Figs. 2–4. Fig. 2 gives the results of state estimation. It can be seen that the states of the system are estimated accurately both in fault-free case and faulty case. Fig. 3 shows the time profile of the actual incipient fault, the estimated fault and the modified fault estimation, respectively. It can be seen that the estimated fault has deviated seriously from the original fault signal due to the large estimation error, hence it
Fig. 4. Parameter estimation and fault prediction.
is hard to use it to detect the fault, not to mention to predict the fault. On the other hand, the modified fault estimation has a good approximation to the original fault signal. This indicate that the MFE is indeed enhance the estimated accuracy of the fault.
Please cite this article as: Ding B, Fang H. Fault prediction for nonlinear stochastic system with incipient faults based on particle filter and nonlinear regression. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.018i
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In Fig. 4, the first fault is detected at the 110 s and the second fault is detected at 139.5 s. Let l ¼50, the first initial parameter θ10 is 0.01 and the second initial parameter θ02 equals to the corresponding detection time. Then according to GNM, the first fault parameters are estimated as θ^ = 0.01359 and θ^ = 102.147, the 1,1
2,1
second fault parameters are θ^1,2 = 0.00872 and θ^2,2 = 114.863. It is shown that the results are well consistent with the true fault parameters. Based on the estimated fault parameter, the predicted fault amplitudes are easy obtained and the results can also be shown in Fig. 4. 5.3. The second case: Simulation with three faults In this section, a more complex situation is implemented: assume that there are three incipient faults will occur in the Threetank system. These faults are described respectively as follows:
f1, k
⎧ 0, 0 ≤ k < 180 =⎨ ⎩ 3 × 10−5(1 − e−0.013(k − 180)), 180 ≤ k ≤ 300
Fig. 5. State estimation.
⎪
⎪
(53)
⎧ 0, 0 ≤ k < 130 f2, k = ⎨ ⎩ 3.5 × 10−5(1 − e−0.01(k − 130)), 130 ≤ k ≤ 300
(54)
⎧ 0, 0 ≤ k < 100 f3, k = ⎨ ⎩ 4 × 10−5(1 − e−0.012(k − 100)), 100 ≤ k ≤ 300
(55)
⎪
⎪
⎪
⎪
where f1, k denotes that the area of the leakage hole growing slowly under the first tank at the 180th second; f2, k means that the jamming area of the outflow pipe occurs at the 130th second; f3, k represents that the jamming area between the first tank and the third tank occurs at the 100th second. the matrix function Γ(xk ) and the fault vector fk can be written as
⎡ −T 2gx /A ⎡f ⎤ 0 Fak ⎤ 1, k ⎥ ⎢ ⎢ 1, k ⎥ ⎥, f = ⎢ f ⎥ Γ(xk ) = ⎢ 0 az T 2 gx / A 0 k 2 2, k ⎥ ⎢ ⎢ 2, k ⎥ ⎥ ⎢ ⎢⎣ f3, k ⎥⎦ 0 0 −Fak ⎦ ⎣
(56)
az T
where Fak = A1 sgn(x1, k − x3, k ) 2g x1, k − x3, k . Assume that all the states are directly measured with Gaussian noises, then the measurement equation is shown as follows:
yk = Ck[x1, k x2, k x3, k ]T + vk
(57)
where
⎡ 1 0 0⎤ ⎢ ⎥ Ck = ⎢ 0 1 0⎥ ⎣ 0 0 1⎦
(58)
The measurement noise vector is assumed to be Gaussian vector with the mean 0, and covariance matrix 0.25 × 10−6I3. The variance of the virtual noise is chosen as Qd = 0.5 × 10−12I3. In the case of no fault, running (Algorithm 1) 2500 s and according to (32), the threshold is obtained and the values are 4.5 × 10−6 , 4.6 × 10−6 , 4.7 × 10−6 , respectively. Let the total simulation time is 300 s, the results can be shown in Figs. 5 and 6. Fig. 5 gives the results of state estimation in this situation. Fig. 6 gives the time profile of the actual incipient fault and the modified fault estimation. Three faults are detected at the 192 s,
Fig. 6. Parameter estimation and fault prediction.
the 147 s and 123.5 s, respectively. Let l ¼50, based on GNM, the first fault parameters are estimated as θ^1,1 = 0.01161 and θ^ = 180.452; the second fault parameters are θ^ = 0.00918 and 2,1
1,2
θ^2,2 = 130.155; the third fault parameters are θ^1,3 = 0.01392 and
Please cite this article as: Ding B, Fang H. Fault prediction for nonlinear stochastic system with incipient faults based on particle filter and nonlinear regression. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.018i
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θ^2,3 = 104.939. It can be shown that in this complex situation, the results are also well consistent with the true fault parameters. According to the estimated fault parameter, the predicted fault amplitudes can be obtained and the results can also be shown in Fig. 6. Once the future fault signal is obtained, we can evaluate the system performance in advance and take appropriate action to recondition the system.
6. Conclusions In this paper, we have considered the fault prediction problem for nonlinear stochastic system with incipient faults, where each of the incipient fault is expressed by a nonlinear function with unknown parameters. The estimated fault with higher accuracy is obtained by the MFE algorithm, which is based on the PF and the reasonable assumption about the incipient fault process. Meanwhile, the state estimation is also obtained. According to the estimated fault, an intuitive fault detection strategy is introduced. Once the fault is detected, the parameters of each component fault are identified by the GNM. Then, the future fault signal can be predicted based on the expression incipient fault and the estimated parameters, thus realize the fault prediction. Simulations of the Three-tank system show the effectiveness of the proposed method.
Acknowledgments This work was supported by the National Natural Science Foundation of China under Grant 61473127.
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Please cite this article as: Ding B, Fang H. Fault prediction for nonlinear stochastic system with incipient faults based on particle filter and nonlinear regression. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.018i