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Hybrid filter design of fault detection for networked linear systems with variable packet dropout rate

2018; Institution of Engineering and Technology; Volume: 13; Issue: 9 Linguagem: Inglês

10.1049/iet-cta.2018.5626

1751-8652

Dongzhe Wang, Xueheng Mei, Rui Weng, Zhenshen Qu, Lixian Zhang,

Control Systems and Identification

IET Control Theory & ApplicationsVolume 13, Issue 9 p. 1239-1245 Special Issue: Recent Advances in Control and Verification for Hybrid SystemsFree Access Hybrid filter design of fault detection for networked linear systems with variable packet dropout rate Dongzhe Wang, Dongzhe Wang School of Astronautics, Harbin Institute of Technology, Harbin, People's Republic of ChinaSearch for more papers by this authorXueheng Mei, Xueheng Mei School of Astronautics, Harbin Institute of Technology, Harbin, People's Republic of ChinaSearch for more papers by this authorRui Weng, Rui Weng IC Design Department, School of Software & Microelectronics, Harbin University of Science and Technology, Harbin, People's Republic of ChinaSearch for more papers by this authorZhenshen Qu, Zhenshen Qu Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin, People's Republic of ChinaSearch for more papers by this authorLixian Zhang, Corresponding Author Lixian Zhang lixianzhang@hit.edu.cn School of Astronautics, Harbin Institute of Technology, Harbin, People's Republic of ChinaSearch for more papers by this author Dongzhe Wang, Dongzhe Wang School of Astronautics, Harbin Institute of Technology, Harbin, People's Republic of ChinaSearch for more papers by this authorXueheng Mei, Xueheng Mei School of Astronautics, Harbin Institute of Technology, Harbin, People's Republic of ChinaSearch for more papers by this authorRui Weng, Rui Weng IC Design Department, School of Software & Microelectronics, Harbin University of Science and Technology, Harbin, People's Republic of ChinaSearch for more papers by this authorZhenshen Qu, Zhenshen Qu Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin, People's Republic of ChinaSearch for more papers by this authorLixian Zhang, Corresponding Author Lixian Zhang lixianzhang@hit.edu.cn School of Astronautics, Harbin Institute of Technology, Harbin, People's Republic of ChinaSearch for more papers by this author First published: 01 June 2019 https://doi.org/10.1049/iet-cta.2018.5626Citations: 9AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract The study focuses on the fault detection filter design problem for a class of networked systems with intermittent measurements. The fault detection filter, which is used as residual generator, is formulated as an filtering form. The random packet dropouts are governed by a Bernoulli distributed sequence, and the packet dropout rate is uncertain and variable, which is described by a semi-Markov stochastic process. A more general class of Lyapunov functions that not only depend on the system modes, but also on the staying time during the current system mode is utilised. Afterwards, numerically testable sufficient conditions on the existence of a desired fault detection filter are established such that the filtering error system is -error mean square stable with a prescribed disturbance attenuation level. Finally, an illustrative example is provided not only to demonstrate the effectiveness of the designed filter and the superiority of the utilisation of semi-Markov chain, but also the necessity of considering the variation of packet dropout rate in the design phase. 1 Introduction Owing to the advantages of the low installation and implementation costs, high reliability, and decreased hard-wiring, networked control systems (NCSs) have received significant attention, see, e.g. [1–3]. However, due to the limited communication capacity of network media [2], data missing, and transmission delay may inevitably occur between the physical plants and controllers or filters in reality, which may lead to performance deterioration or even instability of NCSs. So far, many studies have been devoted to overcoming the effect of packet dropouts, see, e.g. [1, 4–7], and the references therein. During the past decades, allowing for the demands for higher performance, higher safety, and reliability standards, the issue of fault detection of dynamic systems has been intensively investigated see, e.g. [8–12] and so on. As is well known, a fault detection process comprises constructing a residual signal, which is sensitive to faults but robust to disturbances. Then a residual evaluation function is computed to be compared with a predefined threshold [4, 13]. So far, various kinds of fault detection techniques, such as fault detection filters, system identification approach, parity relations approach, and artificial intelligence technique, have been extensively employed in chemical engineering, aerospace engineering and modern manufacturing process, see, e.g. [10, 14, 15]. Due to the wide application of networks, fault detection problem for NCSs with data missing has received increasing attention [4, 16, 17]. Nevertheless, it is worth mentioning that most of the existing results on fault detection problems of NCSs with packet dropout mainly focus on fixed packet dropout rates [4, 16, 17]. In practical systems, sudden changes in parameters, operational environment and inaccurate measurements can cause variable and uncertain PDRs. As has been demonstrated in [3] that controllers or filters designed under fixed PDR cannot reach the required performance. In practice, the dynamics may experience abrupt changes in their structures and parameters. Such systems can be modelled as Markov jump systems (MJSs). However, Markov process requires that sojourn time of each mode is subject to the exponential or geometric distribution for the continuous- or discrete-time case, respectively. Thus the mode switchings are memoryless in an MJS, which is not applicable in many scenarios. To relax the restriction, the concepts of semi-Markov chain, where the transition probabilities (TPs) among different modes are dependent on sojourn time and time-varying, have been introduced into control community, see e.g. [18, 19] and the references therein. Meanwhile, the studies on semi-MJSs (S-MJSs) have been confined to the use of several special probability density functions (PDFs) of sojourn time [20]. It should be pointed out that among these studies, only a single type of distribution of the sojourn time is utilised to describe the switchings among different modes. However, the types of distributions or the parameters of the sojourn-time PDFs for each mode are likely to be different and dependent on the next mode. Therefore, in [21, 22], the so-called discrete-time semi-Markov kernel (SMK) approach is developed and the PDFs of the sojourn time depend on both the current and the next system mode, which is more general in contrast with other investigations on S-MJSs. Motivated by the aforementioned discussions, in this paper, the fault detection problem for a class of stochastic switching systems with variable communication capability is investigated. The variable communication capability refers to variable PDR, which is modelled as mode switchings among several different fixed rates. At the same time, considering the imprecise measurements, each fixed PDR possesses both upper and lower bounds. Semi-Markov chain (SMC) is introduced to describe the random jumps in different nominal PDRs. The superiority of introducing the concept of SMC and the necessity of considering the variation of the PDR are demonstrated in the illustrative example part. The parameters or the types of sojourn time PDFs for each mode can be different via the SMK approach. Then the concept of -error mean square stability ( -MSS) is introduced and performance is guaranteed. Finally, a numerical example is used to show the effectiveness of the proposed fault diagnosis strategy and the necessity of considering the variation of PDR in the design phase. The main contributions of this paper are summarised as follows. (i) The practical phenomenon of variable rate of packet losses is taken into consideration for the fault detection problem of networked linear systems, and the dynamics of the packet dropout rate are considered to be subject to a semi-Markov stochastic process for the first time. (ii) A new form of Lyapunov function, which depends not only on the mode of packet dropout rate but also on the time for which the system has been operating in the current mode since the last switching occurs, is adopted to reduce the conservatism. Notation: The superscripts ' ' and ' ' stand for matrix transposition and inverse, respectively. denotes the n -dimensional Euclidean space. The notation () means that P is symmetric and positive (semi-positive) definite. and stand for, respectively, the expectation of the stochastic variable x and the expectation of x conditional on y. Prob means the occurrence probability of the event ' '. In symmetric block matrices expressions, we use the symbol ' ' as an ellipsis. stands for a block-diagonal matrix. is used to refer to the Euclidean vector norm. stands for the usual norm, and represents the -induced norm of a transfer function matrix. and denote the set of non-negative real numbers and set of non-negative integers, respectively; denotes the space of continuously differentiable functions. 2 Problem formulation and preliminaries 2.1 Physical plant Fix the complete probability space and consider the following discrete-time linear system with disturbance and possible faults after sampling: (1)where is the state vector, represents the measurement output, is the exogenous disturbance that is assumed to be arbitrary signal in and denotes the fault to be detected with bounded energy. A, B, C, D and G are known as real-valued matrices with appropriate dimensions. 2.2 Fault detection filter The following fault detection filter is adopted: (2)where and are the state vector and input vector of the filter, respectively. is the so-called residual that is used to be compared with the fault vector . , , and are appropriately-dimensioned filter gains depending on the semi-Markov process and the staying time during the current system mode, where is a discrete-time semi-Markov process, governing the switchings among different system modes and taking values in the finite set , , . In the following, for notational simplicity, , , and are denoted as , , and , respectively, when . In order to introduce the SMC more formally, the following definitions are needed. Definition 1.The stochastic process is said to be a discrete-time homogeneous Markov renewal chain (MRC) if , and , , where , denotes the sojourn time of mode between the th jump and th jump. Then, let , , , the matrix is called discrete-time semi-Markov kernel (SMK), where and with . In addition, is called the embedded Markov chain (EMC) of MRC , and the TPs matrix of is defined by , with [20]. According to the above concepts, the definition of SMC is introduced as below. Definition 2.Consider a MRC . The chain is said to be a SMC associated with MRC , if , where [20]. Remark 1.SMK is dependent on sojourn time . In this paper, the PDF depending on both the current and next system modes is taken into account, which is defined as , . Then, it can be inferred that . 2.3 Communication links In this paper, there exists a communication channel between the physical plant and the filter, and the phenomenon of packet dropouts emerges randomly. Hence, the filter inputs are unequal to the system measurement outputs, i.e. . This random data loss phenomenon is described as (3)where is a Bernoulli distributed white sequence. When there occur data losses, , , and when the transmission is successful, , . It is straightforward that the expectation of holds as below: (4) In (4), is the nominal expectation of packet arrivals, and is the norm-bounded uncertainty of satisfying , where and are the upper and lower bounds of the uncertainty, respectively. Therefore, it is clear that (5) Remark 2.In fact, the packet arriving rate may fluctuate because of the complex network environment. To guarantee the performance of fault detection filter in case of a large range of variation in , the concept of variable communication capability is proposed, which means that the PDR varies among different nominal expectations randomly, and is governed by a semi-Markov process in this paper. The types of sojourn-time PDFs for each mode of uncertainties can be different with the aid of SMK approach. 2.4 Filtering error system From (1)–(3), the filtering error system can be expressed by (6)where (7a) (7b) It should be pointed out that there appears a stochastic variable in the filtering error system (6), which differs from the traditional deterministic systems without data losses. To obtain the numerically testable sufficient conditions on the existence of a desired fault detection filter with given performance, the following definition is introduced. Definition 3.System (6) with , is said to be -MSS, if for any initial conditions and the upper bound of sojourn-time , the following holds: (8) Further, is defined as (9)where , which is called the cumulative density function of sojourn time for mode i and it is set that . It can be noticed that only varies with ( decreases if all increases). Specially, when , we have , then, Then, and the -MSS reduces to MSS if and only if . For system (1), a fault detection filter is designed in the form of (2), in such a way that the filtering error system (6) is -MSS with the following performance constraint under zero initial conditions: (10)where and is a prescribed scalar, which is made as small as possible in the feasibility of (10). In this paper, the evaluation function and threshold are adopted the following form: (11) Based on (11), the faults can be detected by comparing with according to the following logics: It can be directly seen that when the residual evaluation function exceeds the threshold, the fault is detected, and an alarm of fault is generated. 3 Main results In this section, the filter analysis and design problems are investigated. The sufficient criteria for the -MSS with a prescribed performance index of the system (6) are established based on the linear matrix inequality (LMI) technique. 3.1 Stability and performance In the derivation of our main results, the following lemmas are needed. Lemma 1.Consider a discrete-time stochastic switching non-linear system , where and denote the system state and mode index, respectively. The switching instants are denoted by with . Then the system is mean square stable [21], if there exists a set of functions and functions , , , such that for any initial conditions and a given finite , (12) (13) (14) Lemma 2.Given the infinite series , and , . It is obvious that Therefore, the necessary condition that is convergent can be described as (15) In the following theorem, a sufficient condition is established for -MSS of the filtering error system (6) with a given performance index . Theorem 1.Consider discrete-time system (1) and suppose that the filter parameters , are given. The filtering error system (6) is -MSS with a given disturbance attenuation level , if there exist , , satisfying (16) (17)where , , and Proof.We first prove the -MSS of the system (6). A Lyapunov function is chosen as , , where , . Assuming that and , along the trajectory of system (6), for , it yields that From (16), it can be inferred that . Therefore, , , it holds that . Because , , (16) holds, we can obtain that , , . Then, it can be derived that Considering (17), we have (18)If there exists , then Taking , , into account, therefore, there exists a constant , and from (18), we can further get (19)Summing (18) from to , when , it can be deduced that which yields On account of , , it can be obtained that via Lemma 2. Since , one can know that equals to . According to the above analysis, the system is -MSS.Next, we are devoted to analysing the performance when . From (16), we can get (20)Then, multiplying the left side and right side of (20) by and , respectively, it is obvious that (21)Setting , , and summing (21) from to , it can be inferred that Due to , and , then it leads to (22)From (22), it yields that Then let , and hence Through the above analysis, the stability and performance (10) are satisfied, which completes the proof. □ 3.2 Fault detection filter design In this subsection, the sufficient condition is presented on the existence of the desired fault detection filter in the form of (2), such that the filtering error system (6) is -MSS with a guaranteed disturbance attenuation level . Theorem 2.Considering a discrete-time system (1) with a given constant , system (6) is -MSS with a guaranteed disturbance attenuation level , if there exist and , , , , satisfying (23) (24)where Furthermore, if the aforementioned conditions are satisfied, then, a feasible solution for the filter (2) is given by (25)where and can be obtained by the decomposition on . Proof.Assuming that there exist matrices and satisfying (23). From (23), we get . One can find square and non-singular matrices , and . We denote (26)and . By (7) and (26), it can be verified that The filter in (2) can be calculated by With the congruence transformation by I, I, and for (16) and noting , the following inequality is obtained: (27)where One can readily obtain from (27) that (23) is equal to where , and together with (24), it implies that for , Theorem 1 holds. The proof is completed. □ Remark 3.As can be seen, the obtained criteria in this paper are sufficient but not necessary, there still exists some space to reduce the conservatism. One future work is the further reduction of the conservatism in the developed methodology. 4 Illustrative example To illustrate the effectiveness and applicability of the fault detection filter design approaches, a numerical example is provided in this section. The parameters of the discrete-time system (1) are given as follows: (28)The observable matrix of the system (28) is supposed to be which is full rank, thus the system (1) is observable. The TP matrix of EMC is given as In this paper, the PDFs of sojourn time in mode 1 and mode 3 are considered as Bernoulli distribution and Weibull distribution, respectively, and the two types of distributions are supposed to coexist in mode 2, which is given as below: The variable packet arriving nominal rates are chosen as , and the lower bound of the uncertainty of the PDR is chosen to be , with . Besides, the upper bound of sojourn time is set as , and the minimal -error is calculated to be . The -errors depending on the parameters , (time-varying case) and (time-invariant case) are calculated out and listed in Table 1. It can be observed that is larger than , which indicates the time-varying case is less conservative than the time-invariant one. Furthermore, to show the superiority of introducing the concept of SMC and the necessity of considering the variation of PDR, (SMC case), (MC case) and (fixed PDR case) are calculated as listed in Table 2. It is obvious that the SMC case owns less conservativeness than the Markov case and the variable PDR is much more robust than the fixed one. Fig. 1 shows the interrelation of , and . By fixing , becomes larger when increases. On the contrary, with the increase of with fixed , is smaller (in the form of small increments). The time-varying filter gain-scheduling is presented in Fig. 2. For , the disturbance input is simulated as and the fault signals are given as Table 1. performance index of different scenarios and for different (2,2,2) (4,4,4) (6,4,4) (6,6,4) (10,10,10) 1.0149 1.0152 1.0154 1.0150 1.0156 1.0190 1.0185 1.0188 1.0190 1.0187 4.7604 1.7996 0.9470 0.4714 0.0027 Table 2. performance index of different scenarios for 0.05 0.06 0.07 0.08 0.09 1.0371 1.0444 1.0516 1.0588 1.0659 1.0410 1.0482 1.0555 1.0589 1.0694 1.4020 1.0480 1.0530 1.0626 infeasible Fig. 1Open in figure viewerPowerPoint Variation of with and Fig. 2Open in figure viewerPowerPoint Time-varying filter gain-scheduling Consider the initial conditions , Figs. 3 and 4 show the realisations of residual evaluation function varying with time k when . From Fig. 3, it can be observed that, when for the first time, and the designed fault detection filter can alarm the fault within six steps after the fault occurs at time . From Fig. 4, it can be seen that an alarm is generated at . By solving the LMIs in Theorem 2, the guaranteed performance defined in (10) is . The resulting disturbance attenuation ratio is computed as which shows the effectiveness of the fault detection filter. Fig. 3Open in figure viewerPowerPoint Evolution of residual evaluation function in the case of Fig. 4Open in figure viewerPowerPoint Evolution of residual evaluation function in the case of 5 Conclusion The paper focuses on the fault detection filter design problem for a class of networked systems with random packet dropouts that are depicted by a Bernoulli distributed sequence. The PDR is considered to uncertain and variable and supposed to be governed by a semi-Markov stochastic process. A more general class of Lyapunov functions depending not only on the system modes, but also on the staying time during the current system mode, is introduced to analysing the stability of the filtering error system via the SMK approach. The parameters or the types of distributions of the sojourn time for each mode can be different for different targeted modes. Then, sufficient conditions on the existence of a desired fault detection filter are presented such that the filtering error system is -MSS with the given performance. Finally, an illustrative example is presented to demonstrate the effectiveness of the filter design methodology as well as the necessity of considering the variation of PDR and the superiority of using an SMC to model the time-varying PDR. 6 References 1Wang Z. Yang F., and Ho D.W. et al.: 'Robust control for networked systems with random packet losses', IEEE Trans. Syst. Man Cybern. B, Cybern., 2007, 37, (4), pp. 916– 924 2Zhang L. Gao H., and Kaynak O.: 'Network-induced constraints in networked control systems' survey, IEEE Trans. Ind. Inf., 2013, 9, (1), pp. 403– 416 3Lu Q. 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