Proceedings of the IEEE International Conference on Automation and Logistics Qingdao, China September 2008
A Fault Detection and Diagnosis Scheme for Discrete Nonlinear System Using Output Probability Density Estimation Yumin Zhang Temasek Laboratories, National University of Singapore
Qing-Guo Wang Department of Electrical & Computer Engineering,National University of Singapore
Kai-Yew Lum Temasek Laboratories, National University of Singapore
[email protected]
[email protected]
[email protected]
tribution processing [15]. Through image processing, some useful features can be obtained. For an instance, the flame contour reflects its temperature distribution information, and a series monitor snapshots reflect the varying in time of flame. Image processing and statistic technique can help us to grasp the rule both in space and in time, so a space-time function is obtained to characterize the distribution in space and the varying in time. In addition, distributed statistic method can be used to large quantum or batch output processing, for an example, in the particle making process, distributed statistic information reflects different quality information at different part of particle making field, its united distribution function is certain a space-time function. Motivated by such typical processes, a new group of strategies of PDF shape control for stochastic systems have been developed in the past decade. In references [17-22], [26], a class of general stochastic systems has been investigated, an output PDF approach via B-splines expansion technique has been presented, where the B-spline bases represent the space information of an output distribution while the weighting functions reflect its time varying information. The output PDF approach transfers the corresponding stochastic process to a deterministic dynamical system, and hence the corresponding stochastic problem transfers a deterministic one. Since the stochastic system can be modeled as a deterministic one by using PDF approach, the conventional linear or nonlinear filter can tackle the corresponding FDD problem. Wang and Lin [17] present a linear spline functional approach and a fault detection threshold. Guo et al. [19, 20, 26] put up a nonlinear functional approach named square root B-spline functional approach to further investigate FDD filter problems, where time delays, nonlinearity and modeling uncertainties are considered and some optimization performances such as or are applied [19], [26]. Their filter of FDD scheme consists of two parts, one is conventional systems filter and the other is fault filter. The fault filter is designed to perform faults mapping and faults measurement, which is the key of FDD and FTC schemes. For discrete systems, our present FDD research [21, 22] is only about fault detection problem, which needs further investigation on fault diagnosis problem. This paper will study the FDD problem for discrete nonlinear stochastic system. The research includes both fault detection and fault diagnosis. By constructing a time-domain
Abstract— In this paper, a fault detection and diagnosis (FDD) scheme for a class of discrete nonlinear system fault using output probability density estimation is presented. Unlike classical FDD problems, the measured output of the system is viewed as a stochastic process and its square root probability density function (PDF) is modeled with B-spline functions, which leads to a deterministic space-time dynamic model including nonlinearities, uncertainties. A weighted average function is given as an integral form of the square root PDF along space direction, which leads a function only about time and can be used to construct residual signal. Thus, the classical nonlinear filter approach can be used to detect and diagnose the fault in system. A feasible detection criterion is obtained at first, and a new adaptive fault diagnosis algorithm is further investigated to estimate the fault. The simulation example given demonstrates the effectiveness of the proposed approaches. Index Terms—Fault detection, fault diagnosis, probability density function
I. I NTRODUCTION As an important aspect for practical processes, such as large-scale chemical engineering processes, oil refining processes and aeronautical engineering processes, safety and reliability problem of control systems has long been considered [1-26]. For stochastic systems, the methodologies of the fault detection and diagnosis (FDD) mainly include filter- or observer-based approaches, identification-based approaches, et al. [3-4], [17-22], [26]. Generally, the filter- or observer-based approaches suit systems with unknown input while identification-based approaches suit systems with unknown parameters [3-4], [17-22], [26] or their unpredictable change [31]. Up to now, most approaches focus on Gaussian systems. In fact, some processes exhibit asymmetric nonGaussian distributions [17], the expectation/variance of traditional Kalman filtering approach is obviously insufficient for characterizing such processes and hence probability density functions (PDF) approach is needed. The PDF approach is actual a shape control method. To approximate a kind of distributions, one way is to use statistic-based approach, such as Monte-Carlo approach or particle filter approach, where Bayesian lemma and likelihood method are used [29-30]. Another way is to use function or functional approach, such as spline approach [17-22], [26], where B-spline expansions technique is used. The image monitor techniques are widely used in modern life. In industrial field, the typical examples include the retention of paper-making processing and flame grey-level dis-
978-1-4244-2503-7/08/$20.00 © 2008 IEEE
45
T
where β (k) = [ β1 (u, F ) β2 (u, F ) · · · βq−1 (u, F ) ] , T Φ b(z) =T [ φ1 (z) φ2 (z) ·b· · T φq−1 (z) ] , Λ1 = Φ (z) Φ (z) dz, Λ2 = Φ (z) φq (z) dz, Λ3 = a ab 2 φ (z) dz = 0, Λ0 = Λ1 Λ3 − ΛT2 Λ2 . Obviously, h (β (k)) a q satisfies Lipschitz condition, that is, there exists a matrix U2 such that
reference output function, the residual signal is given naturally and some results on fault detection and fault diagnosis are obtained. As a popular tool, linear matrix inequality (LMI) is used for its computing convenience. The remainder of this paper is organized as follows: In Section 2, some preliminaries on output PDF and related nonlinear system are introduced. In section 3, a fault detection criterion is given. In section 4, a fault diagnosis problem is studied, where an adaptive filtering algorithm is provided. A simulation example in section 5 shows the effectiveness of the proposed presented and the conclusion is given in the last section.
h (β1 (k)) − h (β2 (k)) ≤ U2 (β1 (k) − β2 (k))
for any β1 (k) and β2 (k). The output PDF model is set up if β(k) is modeled. Assume that β(k) satisfies β (k) = Ex (k) ,
= Ax (k) + Ad x (k − d) + Gg (x (k)) +Hu (k) + JF (k) ,
y (k) = f (x (k) , x (k − d) , u (k) , ξ (k) , F (k)) ,
(1)
¯ T (z) β¯ (k) + ω (z, u, F ) . γ (z, u, F ) = Φ
(2)
(3)
ρ (k) = Γ1 x (k) + Γ2 h (Ex (k)) + Δ (k) ,
for any x1 (k) and x2 (k). Suppose the measured output y (k) ∈ [a, b]. Based on the statistic information of sample data, the distribution function of the output sample can be obtained, and the corresponding probability density function (PDF) can be further studied. Of course, the output distribution law is usually complicated, which often results in the complexity of the output PDF. To obtain the output PDF, the B-spline approximation technique is often used. Under conditions of u (k), ξ (k) μ and F (k), P {a ≤ y (k) < μ} = a p (z, u, F ) dz defines a conditional probability on [a, b], where p (z, u (k) , F ) ≥ 0 is the corresponding conditional PDF. Let γ (z, u (k) , F ) = p (z, u (k) , F ), then p (z, u (k) , F ) ≥ 0 is known if γ (z, u (k) , F ) is modeled. Assume that γ (z, u, F ) =
q
βi (u, F ) φi (z) + ω (z, u, F ) ,
(8)
Deterministic equations (1), (7) and (8) describe a stochastic process together, (7) and (8) characterize the probability feature of the output. The PDF is space-time function on z and k, respectively, while system (1) is only a function about the time series k. To study conveniently, we design a weighted average function b ρ (k) = a σ (z) γ (z, u (k) , F ) dz (σ (z) = γ (z, u, F )), where σ (z) is a pre-specified weighted function defined on [a, b]. It is certainly that σ (z) = γ (z, u, F ) is necessary because the probability constraint. It can be seen that
where x (k) ∈ Rn is the system state, u (k) ∈ Rm is the control input, y (k) ∈ R is measured output, ξ (k) is output noise or exogenous disturbance, F (k) is fault to be detected and diagnosed with F (k) ≤ b ( b > 0 is a constant number ). A, Ad , G, H and J are constant matrices of appropriate dimensions. g (x (k)) is a continuous nonlinear function on x satisfying Lipschitz condition, that is, there exists a known matrix U1 such that g (x1 (k)) − g (x2 (k)) ≤ U1 (x1 (k) − x2 (k))
(7)
T where E is a known matrix. Let β¯ (k) = [ β T (k) βq (k) ] , T T ¯ Φ (z) = [ Φ (z) φq (z) ] , equation (4) can be further written as
II. S YSTEM D ESCRIPTION Consider a dynamic system x (k + 1)
(6)
where Γ1 Γ2 Δ (k)
b = a σ (z) ΦT (z) Edz, b = a σ (z) φq (z) dz, b = a σ (z) ω (z, u, F ) dz.
(9)
⎫ ⎪ ⎬ ⎪ ⎭
(10)
In (10), Γ1 and Γ1 are constant numbers or constant vectors, Δ (k) is a function on k, so the reference vector (9) is a time function. Since |ω (z, u (k) , F )| ≤ δ, we have b ˜ (11) Δ (k) = σ (z) ω (z, u, F ) dz ≤ δ, a b where δ˜ = δ a σ (z) dz . The system of (1), (7-9) is a deterministic formulation, the standard filter-based approach can be used to FDD scheme. In the following sections, we will design a nonlinear filter and a residual signal to detect the fault of system and design an adaptive fault diagnosis filter to estimate the fault.
(4)
i=1
where φi (z),i = 1, 2, · · · , q are pre-specified basis functions, βi (u, F ) i = 1, 2, · · · , q are corresponding dynamic weighting functions. The model error ω (z, u (k) , F ) satisfies |ω (z, u (k) , F )| ≤ δ for all {z, u (k) , F }, where δ is a known b positive constant. Due to P {a ≤ y (k) ≤ b} = 1 ⇒ a γ 2 (z, u, F ) dz = 1, φi (z) i = 1, 2, · · · , q are not independent. Assume that φq (z) can be described by φi (z) i = 1, 2, · · · , q − 1, φq (z) is a function of βi (u, F ) i = 1, 2, · · · , q − 1, that is βq (u, F ) = Λ−1 −Λ2 β + Λ3 + β T Λ0 β = h (β) , (5) 3
III. FAULT D ETECTION To detect the fault in system, the following filter is applied x ˆ (k + 1)
46
ˆ (k) + Gg (ˆ x (k)) = Aˆ x (k) + Ad x +Hu (k) + Lε (k) ,
(12)
ε (k) = ρ (k) − ρˆ (k) ,
(13)
ρˆ (k) = Γ1 x ˆ (k) + Γ2 h (E x ˆ (k))
(14)
ˆ¯ ¯ T (z)β(k), γˆ (z, u) = Φ
(15)
βˆ (k) = E x ˆ (k)
(16)
Corollary 1: The solution of error system (17) with no fault is asymptotic stable, if ω(z, u, F ) = 0 and satisfies ˜ x (k) ≤ α := max {α0 , α2 } , if there exist matrices P > 0, Q > 0, R and constant η > 0 satisfying LMI (20) and L = P −1 R for all k > −d for some λi with λi > 0, i = 1, 2.
where x ˆ (k) is the estimation of the state, L ∈ Rn×s is the filter gain to be determined, the residual signal ε (t) is the difference of two reference outputs ρ (k) and ρˆ (k). In fact, ε (t) can be regarded as a generalized distance or difference of two PDFs. γˆ (z, u) and βˆ (k) are the square root PDF and dynamic weight of the filter. Let x ˜ (k) = x (k) − x ˆ (k) , g˜ (k) = g (x (k)) − g (ˆ x (k)) , ˜ (k) = h (Ex (k)) − h (E x h ˆ (k)) the estimation error system of (1) and (12) is formulated as (17), ˜ k + JF (k) − LΔ (k) , x ˜ (k + 1) = AX where
A˜ = [ A − LΓ1 XkT = [ x ˜T (k)
Ad
x ˜T (k − d)
G
Theorem 1 provides a sufficient criterion for the stability of the error system in the absence of the fault, which actually gives a necessary condition to detect . In order to detect more sensitively, such a bound should be made as small as possible. Unlike the conventional filtering problem, for the concerned objective, the distance between the measured PDFs and estimated PDFs should be minimized. The following result is an extensive one of Theorem 2 in [20].
(17)
−LΓ2 ] ,
˜ T (k) ] . g˜T (k) h
Theorem 2: For the error system (17), if there exist matrices P > 0, Q > 0, R and constant η > 0 satisfying LMI (20), then the fault F can be detected by the following criterion
In addition, ˜ (k) + Δ (k) . ε (k) = Γ1 x ˜ (k) + Γ2 h
(18)
˜ ε (k) > β = α (Γ1 + Γ2 U2 E) + δ,
For the sample time {−d, −d + 1, · · · − 1, 0}, we define two useful parameters
2 2 α0 = max {˜ x (j)} , α1 = 2 + U1 + U2 E .
where α is defined in Theorem 1. Remark 2: From Theorem 1 and 2, the solvability conditions can be regarded as extensions of the well-known Lyapunov condition in LMI context for augmented plants, the threshold β can be tuned by α, which is further dependent on η as well as P , Q and R. This problem can be dealt with as an optimization procedure for α subjected to LMIs. On the other hand, modeling error Δ(k) and fault F (k) can be viewed as a disturbance and the H∞ index can be applied to justify their impaction to the system. Once fault is detected, it needs to be estimated, which follows an adaptive fault diagnosis algorithm in the next section.
−d≤j≤0
(19) The following result shows that the error system is bounded within a small field, which can be tuned by the solutions of LMI-based conditions. Theorem 1: The solution of error system (17) with no fault is bounded and satisfies ˜ x (k) ≤ α := max {α0 , α2 } , if there exist matrices P > 0, Q > 0, R and constant η > 0 satisfying Π1 ΠT2 Π= −d, where δ˜ is denoted by (11), 2 −1 T ˜ α1 A˜ R + α2 A˜T R + η RT P −1 R , α2 := δη 1
Π1
=
(21)
IV. A DAPTIVE FAULT D IAGNOSIS To estimate the fault, the following filter is applied x ˆ(k + 1)
= Aˆ x (k) + Ad x ˆ (k − d) + Gg (ˆ x (k)) +J Fˆ (k) + Hu (k) + Lε (k) ,
(22)
(23) Fˆ (k + 1) = Υ1 Fˆ (k) + Υ2 ε (k) , diag −P + Q + λ21 U1TU1 + λ22 E T U2T U2 E + ηI, 2 2 −Q + ηI, −λ1 I, −λ2 I , ˆ where x ˆ (k) is the estimation of the state, F (k) = T ˆ Π2 = [ P A − RΓ1 P Ad P G −RΓ2 ] is the estimation of F (k), F1 (k) , Fˆ2 (k) , · · · , Fˆm (k) L ∈ Rn×s is the filter gain to be determined, Υ1 ( −I < Υ1 < I) and Υ2 are learning operators to be determined. ˆ ε(k) is defined by (13), ρˆ(k), γˆ (z, u) and β(k) are defined by (14), (15) and (16), respectively. Let F˜ (k) = F (k) − Fˆ (k). The error system of (1) and (22) is formulated as
for some λi with λi > 0, i = 1, 2. Remark 1: Theorem 1 shows that if LMI (20) holds for P > 0, Q > 0, R and constant η > 0, state error ˜ x(k) is bounded,and the bound can be adjusted by adjusting η (as well as other matrices) to a proper extent. In another aspect, if modeling error Δ(k) is sufficient small, δ˜ can be set to be 0, which leads the asymptotical stability of the error system.
x ˜(k + 1)
47
= (A − LΓ1 ) x ˜ (k) + Ad x ˜ (k − d) ˜ +G˜ g (k) − LΓ2 h (k) + J F˜ (k) − LΔ (k) ¯ ¯ k − LΔ (k) , = AX (24)
0.0163 0.0065 , Λ2 = [ 0.0003 0.0065 ], Λ3 = 0.0163, 0.0065 0.0177 Γ1 = [ 0.0104 0.0134 ], Γ2 = 0.0211. The fault in system is F = 0.5 + 0.02 sin (0.05πk) ( k ≥ 20 ). To detect the fault in system, the filter of (13-16) is designed, the initial values are x ˆ(k) = [0, 0]T (k = −2, −1, 0). Choose λ1 = 1 and λ2 = 0.1850. It follows from Theorem 1 that L = [4.3213, 3.1809]T , η = 0.0664. By Theorem 2, the residual signal should satisfy |ε(k)| > β = 0.0021, where β is the threshold determined by (21). In this example, the time delay is d = 2. With such a filter, Fig. 1 shows the 3-D mash plot of the plant system output PDF, it has an obvious change at k = 20. Fig. 2 shows the threshold and residual signal tendency, it is clear that threshold can alarm the occurrence of fault during fault detection filter working in 20 ≤ k ≤ 60. This shows the effectiveness of the proposed fault detection algorithm. To diagnose the fault in system, the filter of (1416,22,23) is designed, the initial values are given as x ˆ(k) = [0, 0]T (k = −2, −1, 0). Choose λ1 = 0.1, λ2 = 6 ζ1 = 1.0044, ζ2 = 1.0025. It follows from Theorem 3, L = [−4.6544, −25.4588], η = 7.6758e − 005, Υ1 = 0.9534, Υ2 = 15.0096. Fig. 2 shows that the residual signal tends to zero rapidly after fault diagnosis filter working. Fig. 3 shows that the fault is well estimated. The estimated error of fault and its estimate is less than 4.58%. This also shows the effectiveness of the proposed fault diagnosis algorithm.
¯ T = [ X T F˜ T (k) ]. For F˜ (k), we where A¯ = [ A˜ J ], X k k have F˜ (k + 1) = F (k + 1) − Υ1 Fˆ (k) − Υ2 ε (k) (25) ¯ k + ΔF (k) − Υ2 Δ (k) , = TX where T = [ T1 Υ1 ], T1 = [ −Υ2 Γ1 0 0 −Υ2 Γ2 ], ΔF (k) = F (k + 1) − Υ1 F (k). Suppose x ˆ(0) = 0, Fˆ (0) = 0, it can be shown that the initial condition is represented by x ˜(k) = ϕ(k), k ≤ 0. The following result provides an algorithm to estimate fault. Theorem 3: The solution of error system (24) and (25) is bounded and satisfies max{˜ x(k), F˜ (k)} ≤ M = max{α0 , α3 }, if there exist matrices P > 0, Q > 0, R and constant η > 0 satisfying ⎡ ⎤ Π1 0 ζ1 ΠT2 ζ2 T1 −(1 + η)I ζ1 J T P ζ2 ΥT1 ⎥ ⎢ 0 Ω=⎣ ⎦ < 0, (26) ζ1 Π2 ζ1 P J −ζ1 P 0 ζ2 T1 ζ2 Υ1 0 −ζ2 I and L = P −1 R for all k ≥ −d, where δ˜ is defined by (11), Π1 and Π2 are defined in Theorem 1, T1 = [ −Υ2 Γ1 0 0 −Υ 2 Γ2 ], −1 α3 = [η −1 ((1 + (ζ1 − 1) ) RT P −1 R δ˜2 + −1 ˜ 2 )] 12 for some (1 + (ζ2 − 1) )((1 + Υ1 ) b + Υ2 δ) λi , ζi with λi > 0 and ζi > 1, i = 1, 2. Remark 3: Theorem 3 gives a fault diagnosis algorithm via LMI. Noted that the bound M characterizes the tracking efficiency of filter while the bound M depends on the modeling error ω(z, u, F ), the tracking efficiency can be improved if the modeling precision is improved. Furthermore, the error system of (24) and (25) is stable provide δ˜ small enough. Since the filter gain L and adaptive learning candidates Υ1 and Υ2 can be designed by Theorem 3, we need to very the effectiveness of the given algorithm in the following section.
3
Square root PDF
2.5 2 1.5 1 0.5 0 60 50
V. S IMULATION E XAMPLE
200 40
150
30 z
100
20 50
10
A linear system on particle making is considered here, the corresponding parameters are given as 0.6 0 0.1 0 A= , Ad = , 0.005 0.4 0 0.1 0.1 0.6 0.36 J= ,E = . 0.3 −0.12 0.36
0
Fig. 1.
time
0
3-D Plot of output square root PDF of plant
−3
8
x 10
Residual Threshold
7 6
The initial value is −0.03 + exp(k − 5) ]
4 residual
x(k) = [ 0.02 + exp(k − 5)
5
T
for k = −2, −1, 0. The square root output PDF can be approximated by 3 base functions, that is, for i = 1, 2, 3, 2 φi (z) = exp −0.5 (z − μi ) σi−2 , where z ∈ [0; 0.0.6], μi = 0.01 + 0.02(i − 1), σi = 0.02. Assume that the modeling error satisfies |ω (z, u, F )| ≤ 0.001. The bound of modeling error satisfies δ˜ = 0.00005 for σ (z) = 1. It is −3 0.2666 0.1041 easy to compute that Λ0 = 10 , Λ1 = 0.1041 0.2471
3 2 1 0 −1 −2
0
50
100
150
time
Fig. 2.
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Threshold and Residual signal
200
0.8
[10] Zhong M.Y., Ding S. X., J. Lam, and Zhang C.H., Fault detection filter design for LTI system with time-delay. Proceedings of the 42nd IEEE CDC, Manui, Hawaii, USA, 1467-1472, 2003. [11] Isermann R., Balle P., Trends in the application of model based fault detection and diagnosis of technical process. Proceedings of the IFAC World Congress, N, 1-12, 1996. [12] Krny M., Nagy I, Novoviov J., Mixed-data multi-modelling for fault detection and isolation. International Journal of Adaptive Control and Signal Processing, Vol.16: 61-83, 2002. [13] Nikiforv I., Staroswiecki M., and Vozel B., Duality of analytical redundancy and statistical approach in fault diagnosis. Proceedings of the IFAC World Congress, N, 19-24, 1996. [14] Patton R. J., Chen J. Control and dynamic systems: Robust fault detection and isolation (FDI) systems. London: Academic Press, 1996, 74: 171-224. [15] Zhang X., Polycarpou X. X., and Parisini T., A Robust Detection and Isolation Scheme for Abrupt and Incipient Faults in Nonlinear Systems. IEEE Trans. on Automatic Control, Vol. 47, 4:576-594, 2002. [16] Stoustrup J. and H. H. Niemann, Fault estimation-a standard problem approach, Int. J. Robust Nonlinear Control, 12:649-673, 2002. [17] Wang H., Lin W., Applying observer based FDI techniques to detect faults in dynamic and bounded stochastic distributions. Int. J. Control, 73(15): 1424-1436, 2000. [18] Guo L., Wang H., Applying Constrained Nonlinear Generalized PI Strategy to PDF Tracking Control through Square Root B-Spline Models. Int. J. Control, VOL. 77, NO. 17, 1481-492, 2004. [19] Guo L., Zhang Y. M., Wang H., etc., Observer-based optimal fault detection and diagnosis using conditional probability distributions. IEEE Trans. Signal Processing, Vol. 54, No. 10, 3712-3718, 2006. [20] Zhang Y. M., Guo L. and Wang H., Filter-Based FDD Using PDFs for stochastic systems with time delays. Int. J. Adaptive Control & Signal Processing, Vol. 20, 185-204, 2006. [21] Guo L, Zhang Y M, Liu C L,etc. Optimal Actuator Fault Detection via MLP Neural Network for PDFs. In ISNN 2005, LNCS 3498, SpringerVerlag Berlin Heidelberg, pp. 550.555, 2005. [22] Zhang Y M, Yu H S, Guo L., Using Guaranteed Cost Filters for Fault Detection of Discrete-Time Stochastic Distribution Systems with Time Delays. Proceedings of the 6th World Congress on Control and Automation, pp. 5521-5525, Dalian, China, 2006. [23] Chen W., Saif M., Fault Detection and accommodation in nonlinear time-delay sytems. Proceedings of the ACC, Denver, Colorado, 42554260, 2003. [24] Mufeed M., Jiang J., Zhang Y. M., Stochastic stability analysis of fault-tolerant control systems in the presence of noise. IEEE Trans. on Automatic Control, 46(11): 1810-1813, 2001. [25] Liu J., Wang J. L. and Yang G. H. Reliable Guaranteed Variance Filtering Against Sensor Failures, IEEE Trans. on Signal Processing, VOL. 51, NO. 5: 1403-1411. 2003. [26] Zhang Y. M., Guo L., Wang H., Robust filtering for fault tolerant control using output PDFs of non-Gaussian systems. IET Control Theory Appl., Vol. 1, No. 3: 636-645, 2007. [27] Wang Y., Zhou D, Gao F., Robust fault-tolerant control of a class of non-minimum phase nonlinear processes. Journal of Process Control, Vol 17: 523-537, 2007. [28] Jiang B., Staroswiechi M. and Cocquempot V., Fault Accommodation for nonlinear dynamic systems. IEEE Trans. Auto. Contr., Vol. 51, No. 9: 1578-1583, 2006. [29] Gordon NJ, Salmond DJ, Smith AFM. Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proc-F, Vol. 140:107-13, 1993. [30] Musso C, Oudjane N, LeGland F. Improving regularised particle filters. In: Doucet A, de Freitas JFG, Gordon NJ, editors. Sequential Monte Carlo methods in practice. New York: Springer-Verlag, 2001. [31] Wang Q. G., Lin C., Ye Z., etc., A quasi-LMI approach to computing stabilizing parameter ranges of multi-loop PID controllers. Journal of Process Control, Vol. 17: 59-72, 2007. [32] Liao X. X., Theory and Applications of Stability for dynamical Systems. Defensive Technology Press, Beijing, 2000.
Fault Estimate 0.7
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Fig. 3.
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Fault and its Estimation
VI. C ONCLUSIONS In this paper, a fault detection and diagnosis (FDD) scheme for a class of discrete nonlinear system fault using output probability density estimation is presented. Unlike classical FDD problems, the measured output of the system is viewed as a stochastic process and its square root probability density function (PDF) is modeled with B-spline functions, which leads to a deterministic space-time dynamic model including nonlinearities, uncertainties. A weighted average function is given as an integral form of the square root PDF along space direction, which leads a function only about time and can be used to construct the residual signal. Thus, the classical nonlinear filter approach can be used to detect and diagnose the fault in system. A feasible detection criterion is obtained at first, and a new adaptive fault diagnosis algorithm is further investigated to estimate the fault. Simulation example is given to demonstrate the effectiveness of the proposed approaches. It is noted that only the modeling error of PDF is concerned in this paper, the following research will include exogenous disturbances and some optimization techniques such as H2 /H∞ will be used. R EFERENCES [1] Basseville M., Nikiforov I., Detection of Abrupt Changes: Theory & Applications. Prentice-Hall: Englewood Clils, NJ, 1993. [2] Frank P. M., Ding S. X., Survey of robust residual generation and evaluation methods in observer-based fault detection systems. Journal of Process Control, Vol. 7, No. 6: 403-424,1997. [3] Wang H., Bounded Dynamic Stochastic Systems: Modelling and Control. Springer-Verlag, London, 2000. [4] Chen R. H., Mingori D. L., and Speyer J. L., Optimal stochastic fault detection filter. Automatica, Vol. 39:377-390, 2003. [5] Wang H. B., Wang J. L., Liu J., and Lam J., Iterative LMI approach for robust fault detection observer design. Proceedings of the 42nd IEEE CDC, Manui, Hawaii USA, 1794-1799, 2003. [6] Stoorvogel, A. A., H. H. Niemann, A. Saberi and P. Sannuti, Optimal fault signal estimation, Int. J. Robust Nonlinear Control, 12: 697-727, 2002. [7] Rambeaux F., Hamelin F. and Sauter D., Optimal threshold for robust fault detection of uncertain systems Int. J. Robust Nonlinear Control, 10:1155-1173, 2000. [8] Ding S. X., Zhang P. and Frank P. M., Threshold calculation using LMI-technique and its integration in the design of fault detection systems. Proceedings of the 42nd IEEE CDC, Manui, Hawaii USA, 469-474, 2003. [9] Ding S. X., Jeinsch T., Frank P. M., Ding E. L., A unified approach to the optimization of fault detection systems. International Journal of Adaptive Control and Signal Processing, Vol. 14: 725-745, 2000.
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