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.
Similar content being viewed by others
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 [1–5]. In past two decades, many significant approaches have been presented and applied to practical processes successfully [4]. In general, the fault detection (FD) results can be classified into three types: filter- or observer-based approaches [6–8]; the identification-based FD scheme [9, 10]; and statistic approach [11]. For the dynamic stochastic systems, the filter-based FD approach has been shown as an effective way where generally the variables are supposed to be Gaussian in [12] and [13]. It has been shown that in systems where either the system variables or not, the noise are not Gaussian in [14, 15]. Existing methods may not be sufficient to characterize the closed loop system behavior. As a result, the output PDF rather than the mean variance was proposed [16–18]. Here, we firstly introduce the output PDF definition. For a dynamic stochastic system, suppose that the random process is the output of the stochastic system, its output PDFs are defined by , where is control input. In output PDFs shape control, the B-spline expansion technique has been introduced in the output PDF modeling in [18–20], i.e., the following square root B-spline expansion model has been used to approximate :
where () are pre-specified basis functions defined on , and () are the corresponding weights of such an expansion. Denote
And let , , , . Furthermore, it can be verified that (1) can be rewritten as:
where
where satisfies for any and , and is a known matrix.
The motivation of fault detection and estimation via the output PDFs from the retention system in papermaking was first studied in [20–25], 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 [18, 26–31]. For example, the following nonlinear dynamic model was considered in [18]:
where is the unmeasured state, is the fault to be detected and be assumed and . A, , G, H, D and E represent the known parametric matrices of the dynamic part of the weight system. is time varying delay and satisfies and . The nonlinear function is assumed to be Lipschitz with respect to the state x, i.e., , where is a known matrix.
Recently, a fault detection algorithm has been established by using the output PDFs in [16, 18, 32–36]. However, the algorithms in [16] did not consider time delay information in the designed fault detection observer and the threshold. The method in [18] provides less conservative fault detection algorithms than [16] 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 1 Throughout this paper, for a vector , its Euclidean norm is defined by . A real symmetric matrix (≥0) denotes P being a positive definite (positive semi-definite) matrix, and means . 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 1 [33–42]. 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:
where is the estimated state, is the gain to be determined, is output PDF’s estimation error defined as
and
Define a state estimation error as and , it can be shown that
where , .
Thus, the problem of designing an observer-based fault detection can be described as designing a matrix L such that the error system (7) 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 (7) and provide a feasible observer design method, the following lemma is needed.
Lemma 1 [1]
For any matrix , scalars and , if there exists a Lebesgue vector function , then the following inequalities hold:
where , .
Based on the above lemma, a new delay-dependent fault detection observer can be designed by using the following result.
Theorem 1 Given the scalars (), if there exist matrices , , , , any matrices Z and N, satisfying
where
then in the absence of the fault , the error system (7) with gain is stable.
Proof Define , and denote the Lyapunov function candidate as follows:
with , , . Then following (5) and (6) gives . Along the trajectories of (8) in the absence of and by using the completion-of-square method, it can be shown that
It is noted that in the absence of . According to the free weighting matrix method in [3], for any matrix N, the following equality holds:
From Lemma 1, it is easily shown that
From (13) and (15), we can have , which implies , where and the error system (7) is asymptotically stable. This completes the proof. □
Compared with the result in [18], time varying delay is considered and a new method in [1] to deal with time delay is also used in Theorem 1. Meanwhile, to reduce complex computations, some free weighting matrices Y, W in [18] 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 [43–46]. In this case, we choose
as the residual evaluation function, where denotes the initial evaluation time instant and t stands for the evaluation time, and is defined in (8). Let
be the threshold. Based on this, the following logical relationship is used for fault detection:
3 Fault estimation
For the purpose of estimation, we construct the following nonlinear observer:
where and are estimation of and . , and are the gain parameters to be determined. has been denoted in (8).
By using and , the estimation error system can be formulated to give
Theorem 2 Given the scalars (), h, μ and γ, if there exist scalars (), matrices , , , , , and any matrices Z, N, and satisfying
where
and Ξ is defined in (11). When , the error system (20) with gain , and is asymptotically stable in the presence of .
Proof
Denote the Lyapunov function candidate as follows:
with . It can be shown that
It is noted that . According to the free weighting matrix method in [3], for any matrix N, the following equality holds:
Then we have
where , if , then the above (26) has the form of . That is to say, the estimation error of the fault is asymptotically stable. □
In Theorem 2, some parameters () and γ 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 [18–22]). Suppose these output PDFs can be approximated using a square root B-spline model as , where z is defined in and
For , it can be verified that , , . It is assumed that the identified weighting system is formulated by (5) with the following coefficient matrices:
The upper bounds of nonlinearity are denoted by , . It can be tested that , for . In the simulation, the initial condition of the system state and its estimation are selected as
with the parameters being given as , , The fault is supposed as
By using Theorem 1 and Theorem 2, we can obtain Figures 2, 3, 4, 5, 6, the three-dimensional (3-D) mesh plot shows the changes of the measured output PDFs and Figure 3 demonstrates the responses of residual signal, Figure 4 shows the threshold and the evaluation function. Figures 5 and 6 demonstrate the response of the error system and fault estimation, when the fault occurs at 5 seconds to 15.
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 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 efficiency of the proposed approach.
References
Feng ZG, Lam J: Stability and dissipativity analysis of distributed delay cellular neural networks. IEEE Trans. Neural Netw. 2011, 22(6):976-981.
Chen RH, Mingori DL, Speyer JL: Optimal stochastic fault detection filter. Automatica 2003, 39: 377-390. 10.1016/S0005-1098(02)00245-5
Shao HY, Han QL: New stability criteria for linear discrete-time systems with interval-like time varying delays. IEEE Trans. Autom. Control 2011, 56: 619-625.
Chen WT, Saif M: Fault detection and isolation based on novel unknown input observer design. Proc. of American Control Conference, Minneapolis, Minnesota, USA 2006, 5129-5234.
Li T, Yao X, Wu L, Li J: Improved delay-dependent stability results of recurrent neural networks. Appl. Math. Comput. 2012, 19: 9983-9991.
Li T, Zheng W, Lin C: Delay-slope-dependent stability results of recurrent neural networks. IEEE Trans. Neural Netw. 2011, 12: 2138-2143.
Cen, ZH, Wei, JL, Rui, J: Fault diagnosis based on grey-box neural network identification model. CAS2010, 249-254 (2010)
Guo L, Wang H: Applying constrained nonlinear generalized PI strategy to PDF tracking control through square root b -spline models. Int. J. Control 2004, 77: 1481-1492. 10.1080/00207170412331326972
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. 2005, 35: 65-71. 10.1109/TSMCB.2004.839906
Guo L, Wang H: Fault detection and diagnosis for general stochastic systems using B-spline expansions and nonlinear filter. IEEE Trans. Circuits Syst. I 2005, 52: 1644-1652.
Jiang B, Chowdhury FN: Fault estimation and accommodation for linear MIMO discrete time systems. IEEE Trans. Control Syst. Technol. 2005, 13: 493-499.
Jiang B, Chowdhury FN: Parameter fault detection and estimation of a class of nonlinear systems using observers. J. Franklin Inst. 2005, 342: 725-736. 10.1016/j.jfranklin.2005.04.007
Li P, Kadirkamanathan V: Particle filtering based likelihood ratio approach to fault diagnosis in nonlinear stochastic systems. IEEE Trans. Syst. Man Cybern., Part C, Appl. Rev. 2001, 31: 337-343. 10.1109/5326.971661
Liu J, Wang JL, Yang GH: Residual guaranteed variance filtering against senor failures. IEEE Trans. Signal Process. 2003, 51: 1403-1411. 10.1109/TSP.2003.810303
Stoustrup J, Niemann NN: Fault estimation - a standard problem approach. Int. J. Robust Nonlinear Control 2002, 12: 649-673. 10.1002/rnc.716
Wang H: Bounded Dynamic Stochastic Systems: Modelling and Control. Springer, London; 2000.
Wang H, Lin W: Applying observer based FDI techniques to detect faults in dynamic and bounded stochastic distributions. Int. J. Control 2000, 73: 1424-1436. 10.1080/002071700445433
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. 2006, 20: 175-194. 10.1002/acs.894
Zhao Q, Xu Z: Design of a novel knowledge-based fault detection and isolation scheme. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 2004, 34: 1089-1095. 10.1109/TSMCB.2003.820595
Li T, Guo L, Wu LY: Observer-based optimal fault detection using PDFs for time-delay stochastic systems. Nonlinear Anal., Real World Appl. 2008, 9: 2337-2349. 10.1016/j.nonrwa.2007.06.010
Li T, Guo L: Optimal fault-detection filtering for non-Gaussian systems via output PDFs. IEEE Trans. Syst. Man Cybern., Part A, Syst. Hum. 2009, 39: 476-481.
Li T, Zhang YC: Fault detection and diagnosis for stochastic systems via output PDFs. J. Franklin Inst. 2011, 348: 1140-1152. 10.1016/j.jfranklin.2011.04.005
Zheng B-C, Yang G-H: Further results on quantized feedback sliding mode control of linear uncertain systems. Control and Decision Conference (CCDC), 2012 24th Chinese, 23-25 May 2012, 4249-4253 (2012).
Weng L, Xia M, Liu Q, Wang W: An immunology-inspired fault detection and identification system. Int. J. Adv. Robot. Syst. 2012., 9: Article ID 64
Li T, Ye X: Improved stability criteria of neural networks with time-varying delays: an augmented LKF approach. Neurocomputing 2010, 73: 1038-1047. 10.1016/j.neucom.2009.10.001
Wang W, Song G, Nonami K, Hirata M, Miyazawa O: Autonomous control for micro-flying robot and small wireless helicopter X.R.B. IEEE/RSJ International Conference on Intelligent Robots and Systems 2006, 2906-2911 (2006).
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. 2008, 19(4):426-430.
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 2008, 55(11):1188-1192.
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:10.1016/j.knosys.2012.07.002
Wang W, Song YZ, Nonami K, Cheng Y, Zhou Y, Wang F: Attitude controller design for a six-rotor type MAV. Key Eng. Mater. 2011, 480-481: 1155-1160.
Wang W, Wang F, Zhou Y, Cheng Y, Song YZ, Nonami K: Modeling and embedded autonomous control for quad-rotor MAV. Appl. Mech. Mater. 2011, 130-134: 2461-2464.
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. Differ. Equ. 2009., 2009: Article ID 896218
Xia M, Wang Z, Fang J: Temporal association based on dynamic depression synapses and chaotic neurons. Neurocomputing 2011, 74: 3242-3247. 10.1016/j.neucom.2011.05.009
Xia M, Weng L, Ye X: Sequence memory based on ordered pattern interrelation. Adv. Sci. Lett. 2012, 5: 547-551. 10.1166/asl.2012.1997
Xia M, Fang J, Tang Y, Wang Z: Dynamic depression control of chaotic neural networks for associative memory. Neurocomputing 2010, 73: 776-783. 10.1016/j.neucom.2009.10.015
Xia M, Fang J, Tang Y: Efficient multi-sequence memory with controllable steady-state period and high sequence storage capacity. Neural Comput. Appl. 2011, 20: 17-24. 10.1007/s00521-010-0453-x
Li T, Sun N, Lin CQ, Li J: Improved criterion for the elimination of overflow oscillations in digital filters with external disturbance. Adv. Differ. Equ. 2012., 2012: Article ID 197
Kaslik E: Stability results for a class of difference systems with delay. Adv. Differ. Equ. 2009., 2009: Article ID 938492
Hou C, Han L, Cheng SS: Complete asymptotic and bifurcation analysis for a difference equation with piecewise constant control. Adv. Differ. Equ. 2010., 2010: Article ID 542073
Zang Q, Zhou Y: Asymptotic stabilization of nonlinear DAE subsystems using artificial neural networks with application to power systems. Adv. Int. Syst. 2012, 138: 125-134. 10.1007/978-3-642-27869-3_16
Qi H, Zhu L, Yang A, Zang Q: The design of thermal generating unit controller based on new energy balance. Adv. Mater. Res. 2012, 516-517: 463-466.
Ying Z, Qiang Z: Output feedback adaptive maneuvering for nonlinear MIMO systems with high frequency gain matrix Hurwitz. Adv. Mater. Res. 2012, 383-390: 2417-2422.
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. Differ. Equ. 2012., 2012: Article ID 161
Razminia A, Majd V, Baleanu D: Chaotic incommensurate fractional order Rössler system: active control and synchronization. Adv. Differ. Equ. 2011., 2011: Article ID 15
Haddad WM, Chellaboina VS, Hui Q, Hayakawa T: Neural network adaptive control for discrete-time nonlinear nonnegative dynamical systems. Adv. Differ. Equ. 2008., 2008: Article ID 868425
Zang Q, Zhou Y, Hu K, Sun N, Zhang K, Dai X: Initialized high gain observer design for a class of nonlinear differential-algebraic equation subsystems. The 31st Chinese Control Conference 2012, 916-920.
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 Benefit Research Foundation (GYHY200806017).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors drafted the manuscript, read and approved the final manuscript.
Authors’ original submitted files for images
Below are the links to the authors’ original submitted files for images.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Hu, K., Song, A., Wang, W. et al. Fault detection and estimation for non-Gaussian stochastic systems with time varying delay. Adv Differ Equ 2013, 22 (2013). https://doi.org/10.1186/1687-1847-2013-22
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1847-2013-22