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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 141867, 27 pages
Fault Detection of Markov Jumping Linear Systems
School of Mechatronic Engineering, China University of Mining and Technology, Xuzhou 221116, China
Received 10 December 2011; Accepted 1 February 2012
Academic Editor: Tadeusz Kaczorek
Copyright © 2012 Wei Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this paper, the fault detection (FD) problems of discrete-time Markov jumping linear systems (MJLSs) are studied. We first focus on the stationary MJLS. The proposed FD system consists of two steps: residual generation and residual evaluation. A new reference model strategy is applied to construct a residual generator, such that it is robust against disturbances and sensitive to system faults. The generated residual signals are then evaluated according to their stochastic properties, and a threshold is computed for detecting the occurrences of faults. The upper bound of the corresponding false alarm rate (FAR) is also given. For the nonstationary MJLS, similar results are also obtained. All the solutions are presented in the form of linear matrix inequalities (LMIs). Finally, a numerical example is used to illustrate the results.
The complexity and automation degree of modern technical systems are continuously increasing. In order to guarantee the system safety and reliability, the model-based fault detection and isolation (FDI) technology has been developed since the early 1970s [1–3]. It is fully integrated into many industrial processes and automatic control systems, which can provide valuable information about system faults. The basic idea of model-based FDI is to generate a residual signal based on the system model and measurements and to determine a residual evaluation function to compare with a threshold regarding to all possible model uncertainties and unknown inputs. Exceeding the threshold indicates a fault in the system. Hence the FDI technology consists of two steps: residual generation and residual evaluation. As mentioned in the survey papers [4, 5] and book , the robustness issues have been studied in the purpose of designing of the FDI system under a cost function that expresses a tradeoff between the system robustness against unknown inputs as well as model uncertainties and the system sensitivity to faults. A unified solution for the fault detection system design was presented in [5, 6] for continuous and discrete time linear systems, respectively. In [7, 8], an -filtering formulation of the robust FDI design for uncertain system has been proposed and widely accepted, which tries to make the difference between residual and so-called weighted fault as small as possible by considering model uncertainties and unknown inputs.
Markov jump linear systems have also attracted a great deal of attention. The MJLS is one of the hybrid systems in which a state takes values in a countably finite set, referred to as the state. It can be used to represent a class of linear systems subjects to abrupt changes in their structures due to random components failures, repairs, sudden environment disturbance, change of the operation point of a linearized model of nonlinear systems, for example, electric power systems, aircraft flight control and especially networked control systems (for reasonable model of the packet delivery characteristic in communication channels). There are many theoretical works contributed in the field of MJLS. In [9, 10] the results on stability of MJLS were presented. The linear quadratic Gaussian control problem was studied in [11, 12]. The bounded real lemma for MJLS has been fully developed by  in the form of LMIs. The -control problems were discussed in [14, 15], where a controller stabilizing a linear system ensures that the induced norm from unknown inputs to the outputs is bounded. In [16, 17], the filtering for MJLS was studied as the dual problem of control. For networked systems, recently there were also many new results obtained by applying the MJLS theorem, for example, in [18–20].
Although there is intensive research in FD and MJLS, the design of FD system for MJLS has just begun. In , the packet loss in networked control systems was modeled as Markov process, and an FD system has been designed in terms of LMIs. In , the fault detection system over noisy communication channels was also described via a MJLS. In , the design of FD for MJLS was formulated as a -filtering problem. In those works, observer-based fault detection filters were designed to minimize the influence of unknown inputs [21, 22] or to minimize the difference between residual signals and (weighted) faults , and the threshold is simply computed based on the expectation of the norm of residual signals. However, by only considering the unknown inputs or faults in the design, the residual generator usually cannot achieve an optimal performance in sense of the tradeoff between the system robustness and the fault sensitivity . Besides, further statistic properties of residual signals were not analyzed and considered in the system design, and a proper residual evaluation scheme for MJLS is still missing to the best of the authors’ knowledge. We will show that the usual norm-based residual evaluation method, in which the norm of residual signals is used as evaluation function and compared with the threshold, cannot be directly applied in FD of MJLS. Without taking the variance of residual signals into account, the evaluation will result in a possible high FAR.
In this paper, the fault detection system design of MJLS is formulated as a set of optimization problems. The residual generator is designed to stochastically match an deterministic reference residual model which can achieve an optimal tradeoff between the system robustness and the fault sensitivity. The reference residual model is selected according to the statistic properties of the MJLS. For the stationary MJLS, in which the distributions of Markov state at different time instances remain the same, a new bounded real lemma is also derived, and the constraints of the expectation and variance of residual signals are considered in the residual generator design. The stochastic model matching problem is then solved by optimizing the -norm of an MJLS in terms of LMIs. In the residual evaluation, the absolute value of each residual signals is selected as the evaluation functions, and the corresponding threshold is computed by considering their expectations and variances. Those statistic properties of evaluated residual signals are calculated with the help of LMIs based on convex optimization problems. An upper bound of the FAR is also derived for this evaluation scheme. The FD problem of nonstationary MJLS is then addressed in a similar way. Finally a numerical example is given to illustrate the feasibility and effectiveness of the proposed design approach.
The rest of this paper is organized as follows. The problem of fault detection of MJLS is formulated in Section 2. Section 3 presents the design of residual generator and the new residual evaluation approach. In Section 4 the FD system design for nonstationary MJLS is discussed. An example is given in Section 5 and we conclude the paper in Section 6.
Notation. The notation used throughout the paper is fairly standard. The superscript “T” stands for transposition. denotes the -dimensional Euclidean space and the notation means that is real symmetric and positive definite. diag stands for a block-diagonal matrix. means the expectation of . For convenience we denote . The notation is used to represent the th row of a matrix or a vector. When space, its norm is given by and if is a stochastic value and its expectation is in space, then we denote
2. Problem Formulation
The considered MJLS is defined as follows: where denotes the state vector, denotes the control inputs, denotes the measured output vector, denotes the unknown inputs, and is the fault to be detected. is a discrete homogeneous Markov chain taking values in a finite state space with transition probability matrix , and is defined as which are subjected to the restriction , for any . For notation, define as a sample of possible sequences of Markov states and as the vector of state probabilities
In this paper we consider the Markov chain with the following assumption.(A1) is homogeneous and .
This assumption means that, in such a Markov chain, it is possible to get to any state from any state.
2.1. Residual Generator Design
Residual generation is the first step of FD. We propose the following residual generator for the system (2.1): where and are the estimated state vector and output vector, respectively. is the residual vector. The matrices and are to be designed. For the convenience, we denote the matrices associated with by
With , the residual dynamics of (2.4) can be written as with
The objective of the design is to generate the residual signals which is robust to unknown inputs and sensitive to faults. It is clear that (2.7) itself can be an MJLS instead of a deterministic one. The Markov state is not known as a perquisite, while only its probability is known. Hence we propose a reference residual model which achieves an optimal tradeoff between system robustness and fault sensitivity, and then the residual generator is designed to match the reference residual model. In this approach the matrices and in (2.4) should be selected such that subject to where and denotes the residual vector of a reference residual model in the form of Here and are chosen by applying the unified solution proposed in , such that Since is stochastic vector, usually the expectation, , is not enough to characterize its behavior. Hence the constraint (2.9) is applied to ensure that the summation of variances of each residual signal is bounded with an expected value . In this approach the expectation and variance of are both considered in the model matching problem for MJLS, such that the stochastic could approach a deterministic .
Remark 2.1. A significant difference between the reference model for the purpose of FD adopted here and the one in most of the literatures is that unknown inputs are included in our model, such that an optimal tradeoff between system robustness against unknown inputs and sensitivity to faults can be achieved. Simply reducing the influence of or increasing the sensitivity to system faults does not automatically lead to an optimal tradeoff between system robustness and fault sensitivity. For instance, with a residual generator which decouples from , some may also be decoupled from and thus cannot be detected. Hence it is necessary to take unknown inputs into account in the reference model.
It is well known that where is a unique constant vector called the stationary state distribution of a Markov chain with the assumption (A1), and can be computed according to This means that as time goes by, the Markov chain forgets its initial condition and converges to its stationary distribution. Hence it is reasonable to set the following matrices: so that the reference residual model describes the optimal stationary expected behavior of the MJLS (2.7).
In fact, the MJLS (2.7) consists of two groups of system state: and . In the robust control system design, the initial condition of system state is usually assumed to be zeros (so-called zero initial condition). Comparably, we make three assumptions in MJLS for the rest of the paper:(A2),(A3) is deterministic and ,(A4) is independent of .
Assumption (A2) can be regarded as the “zeros” initial condition of . It is worthy to mention that, with assumption (A2), we have . That means the state probabilities is independent of time, that is, the Markov chain is in the stationary state. We call MJLSs with assumption (2.4) as stationary MJLSs. For the FD purpose, we can generally assume that the MJLS under consideration is operating in its stationary state before a fault occurs. We denote for all for later use.
The problem of residual generator design of a stationary MJLS is summarized as follows.
Problem RGFD [Residual Generator for Fault Detection]
Consider system (2.1). Given the reference residual model (2.10), determine the matrices and , of the residual generator in the form of (2.4) under assumption (A1)–(A4), such that the residual dynamics described by (2.7) satisfy (2.8) and (2.9).
2.2. Residual Evaluation Design
The second step in FD is the evaluation of residual signals, where is evaluated and compared with a threshold. The following logic is applied to detect the occurrences of faults where denotes the evaluated residual signals and denotes a threshold. The residual evaluation problem of deterministic systems has been extensively studied. One important evaluation strategy is the so-called norm-based residual evaluation . Assume the dynamics of a residual generator for some deterministic systems are governed by the following time invariant system where is unknown inputs, and the system faults. Then the residual evaluation function is chosen as with being the length of the evaluation window. It is clear that the residual signals are corrupted with . Hence threshold is set to distinguish the faults from the unknown inputs. As widely accepted in the literatures, the threshold is set as such that false fault alarms can be prevented and meanwhile missing detection of faults can be reduced as much as possible.
The residual signals of MJLS are stochastic values. Their statistic properties are associated with the Markov process, and they are determined by the dynamics of the system (2.4). It is possible to compute such that in a similar way as stated in  for deterministic systems and to set as in many literatures. In this case only the expectation of the -norm of is considered for the computation of . Due to the variance of , there could be false fault alarms and the probability of those false alarms (also called false alarm rate (FAR)) is not known. Even a fault is detected, it is difficult to say with how large percentages the fault alarm is correct. Therefore, we apply the residual evaluation approach proposed in . Define a set of residual evaluation functions as follows: where and is the th measurement. That means the absolute value of each residual signal is selected as the evaluation function. For the evaluation function (2.20), we suggest the following threshold: where is the absolute value of the expectation of and is its variance, where is some constant.
The residual evaluation problem is summarized as follows.
Problem REFD [Residual Evaluation for Fault Detection]
Consider system (2.1). Given the residual generator (2.4) and assumptions (A1)–(A4), determine the threshold , for each residual signal, that is, , and , such that FAR is smaller than a given constant.
3. Main Results
3.1. Residual Generation
In this section the dynamics of are described at first. Then the existing bounded real lemma (BRL) for MJLS is reviewed, and a new BRL is derived for MJLS under assumption (A2). Based on those lemmas, the solution of RGFD is presented.
Before giving the solution, we first introduce the following useful lemmas. The first one is the standard BRL for MJLS given in .
Lemma 3.1. Consider the system for , where , , , , , , and are defined as in (2.1), is the -norm bounded input sequence. Given a constant , and any , then if there exist , satisfying the following LMIs: where
Proof. Here we give a short proof which is slightly different from the one in . Define the function
Given , for any initial mode , we have
With , for any and , we have .
Recall that is any possible sequences of Markov state. By taking expectation over , we can obtain (3.6) which implies for each . Thus the lemma is proved.
Remark 3.2. Lemma 3.1 assumed that is deterministic. However, it can also be used when there is no assumption made on as shown in the proof.
The second lemma gives an equivalent expression of (3.6).
By observing that is independent of under assumption (A2), we can obtain the following bounded real lemma.
Lemma 3.4. Consider the system (3.4), where , , , , , , and are defined as in (2.1), is the -norm bounded input sequence. Given a constant , and under assumption (A2), then (3.5) is satisfied, if there exist satisfying the following LMIs:
Proof. Under the assumption (A2), the expectation terms in (3.10) are
Hence in (3.10) can be rewritten as
If , then for any and , we have . Applying Shur complement and congruence transformation with , can be formulated as (3.13).
Remark 3.5. When the number of modes of is 1, Lemmas 3.1 and 3.4 reduce to the standard bounded real lemma for deterministic systems . It is clear that inequality (3.6) in Lemma 3.1 implies (3.13). But Lemma 3.4 not only requires less computational efforts, but also provides less conservative results as shown in the following example.
The fourth lemma is given to compute the bound of variances for a stationary MJLS.
Lemma 3.7. Consider the system (3.4), where , , , , , , and are defined as in (2.1), is the -norm bounded input sequence. Given a constant , , and under assumption (A2), then if there exist satisfying the following LMI: with
Now we are in the position to give the theorem for the solving RGFD.
Theorem 3.8. Given the system (2.1), a constant and under assumptions (A1)–(A4), the optimal and in the residual generator (2.4) in the sense of minimizing (2.8) and satisfying (2.9) can be obtained by solving the following optimization problem subject to where for and The optimal is then given by .
3.2. Residual Evaluation
In this section, the solution of REFD is given. The bounds of the expectation and variance of (2.20) are firstly computed by using the -norm and -norm of unknown inputs. Then the threshold is determined, and the upper bound of the guaranteed FAR is obtained.
The following lemma gives the computation of in terms of norm.
Lemma 3.9. Given system (2.7), a constant and assumption (A1)–(A4), then if there exist , and satisfying the following LMIs:
Proof. The expected behavior of (2.7) is just described by with The expected residual dynamics are governed by (3.30), which is time invariant system. The results can be easily obtained following the idea in . Thus the rest of the proof is omitted.
The variance of can be computed in terms of -norm using the following lemma.
Lemma 3.10. Given system (2.7), a constant , and assumption (A1)–(A4), then if there exist , satisfying the following LMIs: with
Proof. From (2.7) and (3.30), we have Define for some and assume that for and . Note that satisfying is bounded by (3.38). By denoting , the inequality with ensures (3.39) and thus (3.38). Noticing that with and , then the inequality can be obtained which implies Applying Shur complement and congruence transformation, (3.41) and (3.46) can be reformulated as (3.33) and (3.34), respectively.
We propose also methods to compute the expectation and variance of based on -norm. The following lemma gives the computation of .
Lemma 3.11. Given system (2.7), , and under assumption (A1)–(A4), then if there exist , satisfying the following LMIs:
Proof. The proof is similar with Lemma 3.9.
The computation of based on -norm is given in the following lemma.
Proof. The dynamics of are governed by (3.36). Define for some , . Consider that implies By denoting , the inequality (3.57) is equivalent with with Then (3.52) is guaranteed, if Applying Shur complement and congruence transformation, (3.59)–(3.62) can be reformulated as (3.53)–(3.55), respectively.
Based on above results, the following theorem gives the solution of REFD.
Theorem 3.13. Given system (2.7), a constant , assumption (A1)–(A4),(i)and , then the threshold can be set as where , are the optimum of the constrained optimization problem (ii), , then the threshold can be set as where , , and are the optimum of the constrained optimization problem: The false alarm rate is upper bounded as
Proof. The theorem can be proved in a similar way as in .
Now we have the threshold for each residual signal. The fault can be detected if one evaluated residual exceeds its threshold, for example, A false alarm occurs when there is no fault but . Its probability is upper bounded by (3.67).
Remark 3.14. If the number of Markov state is 1, then the computed will be zero. The proposed approach reduced to the standard norm-based residual evaluation methods .
Remark 3.15. The evaluated residual is a stochastic variable. By using the expectation and variance of for the computation of the threshold, an upper bound of FAR is obtained. Such a bound is very useful in practice, as it can provide confidential information for a rising fault alarm. Without considering the variance in the residual evaluation, the FAR can be very high and no bounds of FAR can be established. Since there is unknown inputs in the system, we can only derive the upper bounds of the expectation and variance. Its higher order moments are difficult to obtain, which is lack of physical means.
4. Discussion on FD of Nonstationary MJLS
Without the assumption (A2), the MJLS could be in nonstationary state, that is, . In this case the expectation of is difficult to obtain, and (2.9) cannot be established. Furthermore, the residual evaluation approach presented in last section cannot be applied. Hence we propose also an FD system for nonstationary MJLS. Firstly, the residual generator is designed to satisfy (2.8), and the solution is given in the following theorem.
Theorem 4.1. Given the system (2.1) under assumptions (A1), (A3)–(A4), the optimals and in the residual generator (2.4) in the sense of minimizing (2.8) can be obtained by solving the following optimization problem for all : subject to where the nonzero elements of are where The optimal is then given by .
Proof. By applying Lemmas 3.1 and 3.3 and setting , the LMI (4.2) can be easily obtained for system (3.1). Notice that implies that is nonsingular. Therefore, the feasibility of (4.2) always ensure the existence of optimal and .
Then (2.17) is selected as the evaluation functions, and the threshold is suggested as where can be easily obtained by applying Lemma 3.1. According to Markov inequality , we have which yields Hence the FAR is bounded by with the threshold (4.7). The result is summarized as the following theorem.
Remark 4.3. The evaluation method suggested in Theorem 4.2 can also be used for the MJLS in stationary state. The variance of , which is difficult to obtain, is not involved in the computation of the threshold. When the variance of is very small, this evaluation method can be fairly conservative.
5. Numerical Example
To illustrate the proposed method, the following two-mode discrete-time MJLS is considered: The transition probability matrix is given by then Its reference residual model has By Theorem 3.8, we have the following: During the simulation, assume that the unknown inputs are discrete-time random noises with power 0.5, the fault appears at the discrete time as a step function of amplitude 15. The residual signals are evaluated as in (2.20) and the thresholds are computed based on -norm and -norm as in (2.20). We take the second residual signal of the system as an example. The thresholds are computed according to Theorem 4.1 with , that is, . The results are shown in Tables 1 and 2, respectively. The simulation results are given in the corresponding figures, where is the threshold calculated only based on the expectation of residual signals. Figures show that many false alarms arise in the first 500 time steps with , and the number of false alarms is significantly reduced by considering the variances in the threshold computation. The fault is detected by all four thresholds.
In this paper a complete design approach of FD system for stationary and nonstationary MJLS was proposed. The main focus was how to deal with the stochastic properties of MJLS. First the residual generator was designed to stochastically match an optimal reference residual model in order to achieve the best tradeoff between robustness against system unknown inputs and sensitivity to faults, where the reference residual model was selected based on the stationary expected behavior of the MJLS. Then novel residual evaluation methods based on -norm and -norm were presented for MJLS, which can guarantee an expected FAR and meanwhile reduce missing detection of faults. In those evaluation methods, not only the expectation of evaluated residuals but also their variances were taken into account for the computation of thresholds. The proposed FD system can provide confidential information of occurring faults, which allows a practical application in real physical systems.
The work was supported by the National Natural Science Foundation of China (61004071), Jiangsu Key Technologies R&D Program (BE2011046), the China Postdoctoral Science Foundation (20100471409), and the Priority Academic Program Development of Jiangsu Higher Education Institutions.
- M. Blanke, M. Kinnaert, J. Lunze, and M. Staroswiecki, Diagnosis and Fault-Tolerant Control, Springer, 2003.
- S. X. Ding, Model-Based Fault Diagnosis Techniques: Design Schemes, Algorithms, and Tools, Springer, 2008.
- R. Patton, P. M. Frank, and R. N. Clark, Eds., Issues of Fault Diagnosis for Dynamic Systems, Springer, 2002.
- P. M. Frank, S. X. Ding, and T. Marcu, “Model-based fault diagnosis in technical processes,” Transactions of the Institute of Measurement and Control, vol. 22, no. 1, pp. 57–101, 2000.
- P. Zhang and S. X. Ding, “On fault detection in linear discrete-time, periodic, and sampled-data systems,” Journal of Control Science and Engineering, vol. 2008, Article ID 849546, 19 pages, 2008.
- S. X. Ding, T. Jeinsch, P. M. Frank, and E. L. Ding, “Unified approach to the optimization of fault detection systems,” International Journal of Adaptive Control and Signal Processing, vol. 14, no. 7, pp. 725–745, 2000.
- H. Gao, T. Chen, and L. Wang, “Robust fault detection with missing measurements,” International Journal of Control, vol. 81, no. 5, pp. 804–819, 2008.
- M. Zhong, S. X. Ding, J. Lam, and H. Wang, “An LMI approach to design robust fault detection filter for uncertain LTI systems,” Automatica, vol. 39, no. 3, pp. 543–550, 2003.
- O. L. V. Costa and M. D. Fragoso, “Stability results for discrete-time linear systems with Markovian jumping parameters,” Journal of Mathematical Analysis and Applications, vol. 179, no. 1, pp. 154–178, 1993.
- Y. Ji, H. J. Chizeck, X. Feng, and K. A. Loparo, “Stability and control of discrete-time jump linear systems,” Control Theory and Advanced Technology, vol. 7, no. 2, pp. 247–270, 1991.
- H. J. Chizeck, A. S. Willsky, and D. Castañon, “Discrete-time Markovian-jump linear quadratic optimal control,” International Journal of Control, vol. 43, no. 1, pp. 213–231, 1986.
- Y. Ji and H. J. Chizeck, “Jump linear quadratic Gaussian control: steady-state solution and testable conditions,” Control Theory and Advanced Technology, vol. 6, no. 3, pp. 289–319, 1990.
- P. Seiler and R. Sengupta, “A bounded real lemma for jump systems,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 48, no. 9, pp. 1651–1654, 2003.
- O. L. V. Costa and J. B. R. do Val, “Full information -control for discrete-time infinite Markov jump parameter systems,” Journal of Mathematical Analysis and Applications, vol. 202, no. 2, pp. 578–603, 1996.
- M. D. Fragoso, J. B. Ribeiro do Val, and D. L. Pinto, Jr., “Jump linear control: the discrete-time case,” Control Theory and Advanced Technology, vol. 10, no. 4, part 3, pp. 1459–1474, 1995.
- O. L. V. Costa and S. Guerra, “Robust linear filtering for discrete-time hybrid Markov linear systems,” International Journal of Control, vol. 75, no. 10, pp. 712–727, 2002.
- C. E. de Souza and M. D. Fragoso, “ filtering for Markovian jump linear systems,” International Journal of Systems Science, vol. 33, no. 11, pp. 909–915, 2002.
- Z. Jin, V. Gupta, and R. M. Murray, “State estimation over packet dropping networks using multiple description coding,” Automatica, vol. 42, no. 9, pp. 1441–1452, 2006.
- J. Wu and T. Chen, “Design of networked control systems with packet dropouts,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 52, no. 7, pp. 1314–1319, 2007.
- J. Xiong and J. Lam, “Stabilization of linear systems over networks with bounded packet loss,” Automatica, vol. 43, no. 1, pp. 80–87, 2007.
- P. Zhang, S. X. Ding, P. M. Frank, and M. Sader, “Fault detection of networked control systems with missing measurements,” in Proceedings of the 5th Asian Control Conference, pp. 1258–1263, Melbourne, Australia, July 2004.
- W. Li, P. Zhang, S. X. Ding, C. Chihaia, and O. Bredtmann, “Fault detection over wireless channels,” in Proceedings of the 47th IEEE Conference on Decision and Control, New Orleans, La, USA, 2007.
- M. Zhong, H. Ye, P. Shi, and G. Wang, “Fault detection for Markovian jump systems,” IEE Proceedings: Control Theory and Applications, vol. 152, no. 4, pp. 397–402, 2005.
- W. Li, Z. Zhu, and S. X. Ding, “Fault detection design of networked control systems,” IET Control Theory & Applications, vol. 5, no. 12, pp. 1493–1449, 2011.
- C. Scherer, P. Gahinet, and M. Chilali, “Multiobjective output-feedback control via LMI optimization,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 42, no. 7, pp. 896–911, 1997.
- A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, New York, NY, USA, 1965.