Fault detection and estimation for non-Gaussian stochastic systems with time varying delay

Abstract

In this paper, fault detection and estimation problem is studied for non-Gaussian stochastic systems with time varying delay. A new approach based on the output probability density function (PDF) and observers technique to detect and estimate time varying faults is presented. Some slack variables and scalars are introduced to design observersâ[euro](TM) parameters, which can provide more degrees of freedom. A particle distribution example is given to illustrate the design procedures, and the simulation results show the performance of the proposed approaches.[PUBLICATION ABSTRACT]

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R E S E A R C H Open Access

Fault detection and estimation for non-Gaussian stochastic systems with time varying delay

Kai Hu1,2, AiGuo Song2*, WeiLiang Wang1, Yingchao Zhang1 and Zhiyong Fan1

*Correspondence: mailto:[email protected]

Web End [email protected]

2School of Instrument Science and Engineering, Southeast University, Nanjing, 210096, ChinaFull list of author information is available at the end of the article

Abstract

In this paper, fault detection and estimation problem is studied for non-Gaussian

stochastic systems with time varying delay. A new approach based on the output probability density function (PDF) and observers technique to detect and estimate

time varying faults is presented. Some slack variables and scalars are introduced to

design observers parameters, which can provide more degrees of freedom. A particle

distribution example is given to illustrate the design procedures, and the simulation

results show the performance of the proposed approaches.

Keywords: fault detection; fault estimation; observer; PDF

1 Introduction

Automatic control systems are widely applied to many industrial processes. However, unexpected faults may destroy the stability of the systems. For such reasons, fault detection and estimation for dynamical systems has received much attention []. In past two decades, many signicant approaches have been presented and applied to practical processes successfully []. In general, the fault detection (FD) results can be classied into three types: lter- or observer-based approaches []; the identication-based FD scheme [, ]; and statistic approach []. For the dynamic stochastic systems, the lter-based FD approach has been shown as an eective way where generally the variables are supposed to be Gaussian in [] and []. It has been shown that in systems where either the system variables or not, the noise are not Gaussian in [, ]. Existing methods may not be sucient to characterize the closed loop system behavior. As a result, the output PDF rather than the mean variance was proposed []. Here, we rstly introduce the output PDF denition. For a dynamic stochastic system, suppose that the random process y [a, b] is the output of the stochastic system, its output PDFs are dened by (z, u(t)),

where u(t) Rm is control input. In output PDFs shape control, the B-spline expansion

technique has been introduced in the output PDF modeling in [], i.e., the following square root B-spline expansion model has been used to approximate (z, u(t)):

z, u(t) =

n

i=vi(u)bi(z), ()

2013 Hu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu

tion License (http://creativecommons.org/licenses/by/2.0

Web End =http://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any

medium, provided the original work is properly cited.

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where bi(z) (i = , , . . . , n) are pre-specied basis functions dened on [a, b], and vi(u(t)) (i = , , . . . , n) are the corresponding weights of such an expansion. Denote

B(z) = b(z) b(z) bn(z)

,

()

V(t) := V

u(t) =

v v vn

.

a bn(z) dz = , =

. Furthermore, it can be veried that () can be rewritten as:

r

z, u(t) = B (z)V(t) + h

V(t)

And let =

b

a B(z)bn(z) dz, =

b

b

a B(z)B(z) dz, =

bn(z), ()

where

B (z) = B(z)

V (t) V (t)

, ()

where h(V(t)) satises h(V(t))h(V(t)) U(V(t)V(t)) for any V(t) and V(t),

and U is a known matrix.

The motivation of fault detection and estimation via the output PDFs from the retention system in papermaking was rst studied in [], where the weight dynamical system was supposed to be a precise linear model. However, linear mappings cannot change the shape of output PDFs, which implies that the fault cannot be detected through the shape change of the PDFs. To meet the requirement in complex processes, nonlinearity should be considered in the weighting dynamic behavior [, ]. For example, the following nonlinear dynamic model was considered in []:

V(t) =

bn(z), h

x(t) = Ax(t) + Adx(t d(t)) + Gg(x(t)) + Hu(t) + Jf (t), V(t) = Ex(t),

()

where x(t) Rm is the unmeasured state, f (t) is the fault to be detected and be assumed f (t) and f(t) . A, Ad, G, H, D and E represent the known parametric matrices

of the dynamic part of the weight system. d(t) is time varying delay and satises < d(t)

h and d(t) . The nonlinear function g(x(t)) is assumed to be Lipschitz with respect to

the state x, i.e., g(x(t)) g(x(t)) U(x(t) x(t)) , where U is a known matrix.

Recently, a fault detection algorithm has been established by using the output PDFs in [, , ]. However, the algorithms in [] did not consider time delay information in the designed fault detection observer and the threshold. The method in [] provides less conservative fault detection algorithms than [] by designing delay-dependent observer and minimizing the threshold. To further improve the previous results, in this paper, a new delay-dependent observer design is presented such that the estimation error system is stable, and the fault can be detected and estimated through a threshold by introducing the tuning parameter and slack variable. Finally, particle distribution process example is given to demonstrate the applicability of the proposed approach.

Notation Throughout this paper, for a vector (t), its Euclidean norm is dened by (t) = (t)(t). A real symmetric matrix P > ( ) denotes P being a positive

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Figure 1 Fault-detection system.

denite (positive semi-denite) matrix, and A > ()B means A B > (). I is used to

denote an identity matrix with proper dimension. Matrices, if not explicitly stated, are assumed to have compatible dimensions. The symmetric terms in a symmetric matrix are denoted by .

2 Fault detection

Generally speaking, a fault-detection system consists of a residual generator and a residual evaluator including an evaluation function and a threshold as in Figure []. We will consider two parts of fault detection systems by using the information of PDF in the following section.

2.1 Residual generator

For the purpose of residual generation, we construct the following nonlinear observer:

x(t) = Ax(t) + Ad x

t d(t) + Gg

+ Hu(t) + L(t), ()

where x(t) is the estimated state, L Rmp is the gain to be determined, (t) is output

PDFs estimation error dened as

(t) =

b a

x(t)

(z)

z, u(t), f (t)

z, u(t) dz

and

z, u(t) = B (z)Ex(t) + h

Ex(t)

bn(z).

Dene a state estimation error as e(t) = x(t) x(t) and (t), it can be shown that

e(t) = (A L )e(t) + Ade

t d(t) + G

g

x(t) g

x(t)

L

h

Ex(t) h

Ex(t) + Jf (t), ()

(t) = e(t) +

h

Ex(t) h

Ex(t)

,

()

where =

b

a (z)bn(z) dz.

Thus, the problem of designing an observer-based fault detection can be described as designing a matrix L such that the error system () is asymptotically stable and the fault can be detected.

In order to formulate some practically computable criteria to check the stability of the error system described by () and provide a feasible observer design method, the following lemma is needed.

b

a (z)B (z)E dz, =

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Lemma [] For any matrix M > , scalars b > a and c < d , if there exists a Lebesgue

vector function (s), then the following inequalities hold:

b a

[intercal](s)M(s) ds

b a

[intercal](s)M

(s), ()

d c

t t+

[intercal](s)M(s) ds d

c d

[intercal](s)M

(s), ()

where

(s) =

b

a [intercal](s) ds,

(s) =

d

c

t

t+ (s) ds d.

Based on the above lemma, a new delay-dependent fault detection observer can be designed by using the following result.

Theorem Given the scalars i > (i = h and ), if there exist matrices P > , Q > , R > , R > , any matrices Z and N, satisfying

=

h R NAd

h R NG Z NAd NG Z

h R hR

h R

I

I

< , ()

where

=

hR R + Q +

E U UE +

U U

NA Z + A[intercal]N[intercal] [intercal]Z[intercal]

,

= P A[intercal]N[intercal] + [intercal]Z[intercal] + N,

= N + N[intercal] + hR + h

R,

=

hR (

)Q,

then in the absence of the fault f (t), the error system () with gain L = NZ is stable.

Proof Dene g := g(x(s)) g(x(s)), h := h(Ex(s) h(Ex(s))) and denote the Lyapunov func

tion candidate as follows:

V(t) = e (t)Pe(t) +

h

tt+ e (s)Re(s) ds d

+

h

tt+v e (s)Re(s) ds dv d

+

ttd(t) e (s)Qe(s) ds

+

t

U

Ee(s)

ds +

t

ds ()

with P > , T > , Q > . Then following () and () gives V(t) . Along the trajectories of

() in the absence of f (t) and by using the completion-of-square method, it can be shown

h

U

e(s)

g

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that

V(t) e (t)Pe(t) + e (t)

hR + h

R e

(t)

tth e (s)Re(s) ds

h

tt+ e (s)Re(s) ds d

+ e (t)Qe(t) ( )e

t d(t)

Qe

t d(t)

+

e (t) E

U UE

e(t) + e (t) U U

e(t) h h g g

+ (t). ()

It is noted that (t) = e(t)(AL )e(t)Ade(t d(t))Gg +L h = in the absence of

f (t). According to the free weighting matrix method in [], for any matrix N, the following equality holds:

e (t)N + e (t)N e(t

) (A L )e(t) Ade

t d(t) Gg + L h

= . ()

From Lemma , it is easily shown that

tth e (s)Re(s) ds =

td(t)th e (s)Re(s) ds

ttd(t) e (s)Re(s) ds

e[intercal](t) e[intercal](t d(t)) e[intercal](t h)

Rh

R

h

e(t)e(t d(t)) e(t h)

, ()

R

h

Rh

Rh

h

tt+ e (s)Re(s) ds d

h

h

tt+ e(s) ds d

[intercal]

R

h

tt+ e(s) ds d

=

e[intercal](t)

t

th e[intercal](s) ds

R

h R h R

e(t)

. ()

From () and (), we can have < , which implies V(t) [intercal](t) (t) < , where

(t) = [e[intercal](t) e[intercal](t) e[intercal](t d(t)) e[intercal](t h)

t

t

th e(s) ds

th e[intercal](s) ds g[intercal] h[intercal]][intercal] and the error system () is

asymptotically stable. This completes the proof.

Compared with the result in [], time varying delay is considered and a new method in [] to deal with time delay is also used in Theorem . Meanwhile, to reduce complex computations, some free weighting matrices Y, W in [] are not introduced in this paper.

2.2 Residual evaluator

After the fault detection observer is designed, the next important task for fault detection is the evaluation of the generated residual, including a threshold and a decision logic unit

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[]. In this case, we choose

Jr =

t+t

to

[intercal](s)(s) ds ()

as the residual evaluation function, where t denotes the initial evaluation time instant and t stands for the evaluation time, and (t) is dened in (). Let

Jth = sup

f (t)= Jr () be the threshold. Based on this, the following logical relationship is used for fault detection:

Jr > Jth alarm fault, Jr < Jth no fault.

3 Fault estimation

For the purpose of estimation, we construct the following nonlinear observer:

x(t) = Ax(t) + Ad x

t d(t) + Gg

x(t)

+ Hu(t) + J f(t) + L(t),

f (t) = f(t) + (t),

()

where x(t) and f(t) are estimation of x(t) and f (t). L, and are the gain parameters

to be determined. (t) has been denoted in ().By using e(t) = x(t) x(t) and ef (t) = f (t) f(t), the estimation error system can be for

mulated to give

e(t) = (A L )e(t) + Ade

t d(t) + G

g

x(t) g

x(t)

L

h

Ex(t) h

Ex(t) + Jef (t). ()

Theorem Given the scalars i > (i = , ), h, and , if there exist scalars ui > (i = , , . . . , ), matrices P > , P > , Q > , R > , R > , and any matrices Z, N, W and W satisfying

[intercal]

[intercal] +

+

diag{uI, uI, uI, uI, uI, uI, uI, uI} < , ()

where

=

( + J[intercal]N[intercal])J[intercal]N[intercal]

()

and is dened in (). When ef (t) |P|+|W|u , the error system () with gain L =

NZ, = PW and = PW is asymptotically stable in the presence of f (t).

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Proof Denote the Lyapunov function candidate as follows:

V(t) = V(t) + e[intercal]f(t)ef (t) ()

with > . It can be shown that

V(t) = V(t) + e[intercal]f(t)ef (t) + (t). ()

It is noted that (t) = e(t) (A L )e(t) Ade(t d(t)) Gg + L h Jef (t) = .

According to the free weighting matrix method in [], for any matrix N, the following equality holds:

e[intercal](t)N + e (t)N e(t

) (A L )e(t) Ade

t d(t) Gg + L h Jef (t) = . ()

Then we have

V(t) [intercal](t) (t) + e[intercal](t)NJef (t) + e[intercal](t)NJef (t) + e[intercal]f(t)ef (t)= [intercal](t) (t) + e[intercal](t)NJef (t) + e[intercal](t)NJef (t) + e[intercal]f(t)f(t) e[intercal]f(t)f (t)

+ e[intercal]f(t)ef (t) e[intercal]f(t) e(t) e[intercal]f(t) h

= [intercal](t)

[intercal]

W[intercal] + W

(t) + e[intercal]f(t)f(t) e[intercal]f(t)f (t)

u

e(t)

u

e(t)

u

e t

d(t)

u

e(t

h)

u

tth e(s) ds

u g u h u

e

f (t)

+

e

f (t)

+

+

e

f (t)

e

f (t)

|W

|

f(t)

+

e

f (t)

f

(t)

u

e

f (t)

=

u

e

f (t)

+ + |W|

e

f (t)

,

()

where (t) = [[intercal](t) e[intercal]f(t)][intercal], if ef (t) +|W|u , then the above () has the form of V(t) < . That is to say, the estimation error of the fault is asymptotically stable.

In Theorem , some parameters ui (i = , , . . . , ) and are introduced. These parameters may provide more degrees of freedom in fault estimation observer design and estimation performance.

4 Simulations

In this section, we consider a simple example related to the particle distribution control problems, where the shapes of measured output PDF usually have two or three peaks (see []). Suppose these output PDFs can be approximated using a square root B-spline model as

(z, u(t)) =

i= vi(u(t), F)bi(z), where z is dened in [, .] and

bi =

| sin z|, z [.(i ), .i], , others.

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Figure 2 The 3-D mesh plot of the output PDF when the fault occurs.

For i = , , , it can be veried that = diag{., .}, = [, ], = .. It is

assumed that the identied weighting system is formulated by () with the following coecient matrices:

A =

.

, Ad =

. .

, G =

,

H =

, E =

, J =

.

.

.

The upper bounds of nonlinearity are denoted by U = diag{., .}, U = diag{., .}.

It can be tested that = [. .], = . for (z) = . In the simulation, the initial condition of the system state and its estimation are selected as

x(t) =

. + exp(t ) . + exp(t )

, t [., ], x() =

, t [., ]

with the parameters being given as = , = , = . The fault is supposed as

f (t) =

, t < ,. + . sin(t), t , , t > .

By using Theorem and Theorem , we can obtain Figures , , , , , the three-dimensional (-D) mesh plot shows the changes of the measured output PDFs and Figure demonstrates the responses of residual signal, Figure shows the threshold and the evaluation function. Figures and demonstrate the response of the error system and fault estimation, when the fault occurs at seconds to .

5 Conclusion

In this paper, a new fault detection and estimation scheme has been developed for the stochastic dynamic systems with time varying delay by using stochastic distribution of

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Figure 3 The response of the residual signal.

Figure 4 Threshold and the evaluation function.

system output. Based on LMI techniques and by using the slack variables, a new delay-dependent fault detection observer is designed to detect the system fault with a threshold. Furthermore, an observer-based fault estimation method is provided to estimate the size of the fault. Particle distribution example is to show the eciency of the proposed approach.

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Figure 5 The response of the error system when the fault occurs.

Figure 6 The fault and its estimation.

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Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors drafted the manuscript, read and approved the nal manuscript.

Author details

1School of Information and Control, Nanjing University of Information Science and Technology, Nanjing, 210044, China.

2School of Instrument Science and Engineering, Southeast University, Nanjing, 210096, China.

Acknowledgements

This paper was funded under a grant from China National Nature Science (No. 61105075, 61104206), Nanjing University of Information Science & Technology University Student Renovation Project (N1885012179), Open Project (KDX1102) of Jiangsu Key Laboratory of Meteorological Observation and Information Processing, as well as under National Department Public Benet Research Foundation (GYHY200806017).

Received: 8 November 2012 Accepted: 11 January 2013 Published: 24 January 2013

References

1. Feng, ZG, Lam, J: Stability and dissipativity analysis of distributed delay cellular neural networks. IEEE Trans. Neural Netw. 22(6), 976-981 (2011)

2. Chen, RH, Mingori, DL, Speyer, JL: Optimal stochastic fault detection lter. Automatica 39, 377-390 (2003)3. Shao, HY, Han, QL: New stability criteria for linear discrete-time systems with interval-like time varying delays. IEEE Trans. Autom. Control 56, 619-625 (2011)

4. Chen, WT, Saif, M: Fault detection and isolation based on novel unknown input observer design. In: Proc. of American Control Conference, Minneapolis, Minnesota, USA, pp. 5129-5234 (2006)

5. Li, T, Yao, X, Wu, L, Li, J: Improved delay-dependent stability results of recurrent neural networks. Appl. Math. Comput. 19, 9983-9991 (2012)

6. Li, T, Zheng, W, Lin, C: Delay-slope-dependent stability results of recurrent neural networks. IEEE Trans. Neural Netw. 12, 2138-2143 (2011)

7. Cen, ZH, Wei, JL, Rui, J: Fault diagnosis based on grey-box neural network identication model. CAS2010, 249-254 (2010)

8. Guo, L, Wang, H: Applying constrained nonlinear generalized PI strategy to PDF tracking control through square root b-spline models. Int. J. Control 77, 1481-1492 (2004)

9. Guo, L, Wang, H: PID controller design for output PDFs of stochastic systems using linear matrix inequalities. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 35, 65-71 (2005)

10. Guo, L, Wang, H: Fault detection and diagnosis for general stochastic systems using B-spline expansions and nonlinear lter. IEEE Trans. Circuits Syst. I 52, 1644-1652 (2005)

11. Jiang, B, Chowdhury, FN: Fault estimation and accommodation for linear MIMO discrete time systems. IEEE Trans. Control Syst. Technol. 13, 493-499 (2005)

12. Jiang, B, Chowdhury, FN: Parameter fault detection and estimation of a class of nonlinear systems using observers.J. Franklin Inst. 342, 725-736 (2005)13. Li, P, Kadirkamanathan, V: Particle ltering based likelihood ratio approach to fault diagnosis in nonlinear stochastic systems. IEEE Trans. Syst. Man Cybern., Part C, Appl. Rev. 31, 337-343 (2001)

14. Liu, J, Wang, JL, Yang, GH: Residual guaranteed variance ltering against senor failures. IEEE Trans. Signal Process. 51, 1403-1411 (2003)

15. Stoustrup, J, Niemann, NN: Fault estimation - a standard problem approach. Int. J. Robust Nonlinear Control 12, 649-673 (2002)

16. Wang, H: Bounded Dynamic Stochastic Systems: Modelling and Control. Springer, London (2000)17. Wang, H, Lin, W: Applying observer based FDI techniques to detect faults in dynamic and bounded stochastic distributions. Int. J. Control 73, 1424-1436 (2000)

18. Zhang, YM, Guo, L, Wang, H: Filter-based fault detection and diagnosis using output PDFs for stochastic systems with time delays. Int. J. Adapt. Control Signal Process. 20, 175-194 (2006)

19. Zhao, Q, Xu, Z: Design of a novel knowledge-based fault detection and isolation scheme. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 34, 1089-1095 (2004)

20. Li, T, Guo, L, Wu, LY: Observer-based optimal fault detection using PDFs for time-delay stochastic systems. Nonlinear Anal., Real World Appl. 9, 2337-2349 (2008)

21. Li, T, Guo, L: Optimal fault-detection ltering for non-Gaussian systems via output PDFs. IEEE Trans. Syst. Man Cybern., Part A, Syst. Hum. 39, 476-481 (2009)

22. Li, T, Zhang, YC: Fault detection and diagnosis for stochastic systems via output PDFs. J. Franklin Inst. 348, 1140-1152 (2011)

23. Zheng, B-C, Yang, G-H: Further results on quantized feedback sliding mode control of linear uncertain systems. In: Control and Decision Conference (CCDC), 2012 24th Chinese, 23-25 May 2012, pp. 4249-4253 (2012)

24. Weng, L, Xia, M, Liu, Q, Wang, W: An immunology-inspired fault detection and identication system. Int. J. Adv. Robot. Syst. 9, 64 (2012)

25. Li, T, Ye, X: Improved stability criteria of neural networks with time-varying delays: an augmented LKF approach. Neurocomputing 73, 1038-1047 (2010)

26. Wang, W, Song, G, Nonami, K, Hirata, M, Miyazawa, O: Autonomous control for micro-ying robot and small wireless helicopter X.R.B. In: IEEE/RSJ International Conference on Intelligent Robots and Systems, 2006, pp. 2906-2911 (2006)

27. Li, T, Guo, L, Sun, C, Lin, C: Further results on delay dependent stability criteria of neural networks with time-varying delays. IEEE Trans. Neural Netw. 19(4), 426-430 (2008)

28. Li, T, Guo, L, Lin, C: Stability criteria with less LMI variables for neural networks with time-varying delay. IEEE Trans. Circuits Syst. II, Express Briefs 55(11), 1188-1192 (2008)

Hu et al. Advances in Dierence Equations 2013, 2013:22 Page 12 of 12 http://www.advancesindifferenceequations.com/content/2013/1/22

Web End =http://www.advancesindifferenceequations.com/content/2013/1/22

29. Xia, M, Zhang, Y, Weng, L, Ye, X: Fashion retailing forecasting based on extreme learning machine with adaptive metrics of inputs. Knowl.-Based Syst. (2012). doi:http://dx.doi.org/10.1016/j.knosys.2012.07.002

Web End =10.1016/j.knosys.2012.07.002

30. Wang, W, Song, YZ, Nonami, K, Cheng, Y, Zhou, Y, Wang, F: Attitude controller design for a six-rotor type MAV. Key Eng. Mater. 480-481, 1155-1160 (2011)

31. Wang, W, Wang, F, Zhou, Y, Cheng, Y, Song, YZ, Nonami, K: Modeling and embedded autonomous control for quad-rotor MAV. Appl. Mech. Mater. 130-134, 2461-2464 (2011)

32. Zhu, J, Park, J, Lee, K-S, Spiryagin, M: Switching controller design for a class of Markovian jump nonlinear systems using stochastic small-gain theorem. Adv. Dier. Equ. 2009, 896218 (2009)

33. Xia, M, Wang, Z, Fang, J: Temporal association based on dynamic depression synapses and chaotic neurons. Neurocomputing 74, 3242-3247 (2011)

34. Xia, M, Weng, L, Ye, X: Sequence memory based on ordered pattern interrelation. Adv. Sci. Lett. 5, 547-551 (2012)35. Xia, M, Fang, J, Tang, Y, Wang, Z: Dynamic depression control of chaotic neural networks for associative memory. Neurocomputing 73, 776-783 (2010)

36. Xia, M, Fang, J, Tang, Y: Ecient multi-sequence memory with controllable steady-state period and high sequence storage capacity. Neural Comput. Appl. 20, 17-24 (2011)

37. Li, T, Sun, N, Lin, CQ, Li, J: Improved criterion for the elimination of overow oscillations in digital lters with external disturbance. Adv. Dier. Equ. 2012, 197 (2012)

38. Kaslik, E: Stability results for a class of dierence systems with delay. Adv. Dier. Equ. 2009, 938492 (2009)39. Hou, C, Han, L, Cheng, SS: Complete asymptotic and bifurcation analysis for a dierence equation with piecewise constant control. Adv. Dier. Equ. 2010, 542073 (2010)

40. Zang, Q, Zhou, Y: Asymptotic stabilization of nonlinear DAE subsystems using articial neural networks with application to power systems. Adv. Int. Syst. 138, 125-134 (2012)

41. Qi, H, Zhu, L, Yang, A, Zang, Q: The design of thermal generating unit controller based on new energy balance. Adv. Mater. Res. 516-517, 463-466 (2012)

42. Ying, Z, Qiang, Z: Output feedback adaptive maneuvering for nonlinear MIMO systems with high frequency gain matrix Hurwitz. Adv. Mater. Res. 383-390, 2417-2422 (2012)

43. Alonso-Quesada, S, De la Sen, M, Agarwal, RP, Ibeas, A: An observer-based vaccination control law for a SEIR epidemic model based on feedback linearization techniques for nonlinear systems. Adv. Dier. Equ. 2012, 161 (2012)

44. Razminia, A, Majd, V, Baleanu, D: Chaotic incommensurate fractional order Rssler system: active control and synchronization. Adv. Dier. Equ. 2011, 15 (2011)

45. Haddad, WM, Chellaboina, VS, Hui, Q, Hayakawa, T: Neural network adaptive control for discrete-time nonlinear nonnegative dynamical systems. Adv. Dier. Equ. 2008, 868425 (2008)

46. Zang, Q, Zhou, Y, Hu, K, Sun, N, Zhang, K, Dai, X: Initialized high gain observer design for a class of nonlinear dierential-algebraic equation subsystems. In: The 31st Chinese Control Conference, pp. 916-920 (2012)

doi:10.1186/1687-1847-2013-22Cite this article as: Hu et al.: Fault detection and estimation for non-Gaussian stochastic systems with time varying delay. Advances in Dierence Equations 2013 2013:22.