Title: fault detection for linear discrete‐time descriptor systems
Abstract: IET Control Theory & ApplicationsVolume 12, Issue 15 p. 2156-2163 Brief PaperFree Access fault detection for linear discrete-time descriptor systems Weixin Han, Weixin Han School of Astronautics, Harbin Institute of Technology, Harbin, 150001 People's Republic of ChinaSearch for more papers by this authorZhenhua Wang, Corresponding Author Zhenhua Wang [email protected] School of Astronautics, Harbin Institute of Technology, Harbin, 150001 People's Republic of ChinaSearch for more papers by this authorYi Shen, Yi Shen School of Astronautics, Harbin Institute of Technology, Harbin, 150001 People's Republic of ChinaSearch for more papers by this authorYipeng Liu, Yipeng Liu College of Information Engineering, Zhejiang University of Technology, Hangzhou, 310023 People's Republic of ChinaSearch for more papers by this author Weixin Han, Weixin Han School of Astronautics, Harbin Institute of Technology, Harbin, 150001 People's Republic of ChinaSearch for more papers by this authorZhenhua Wang, Corresponding Author Zhenhua Wang [email protected] School of Astronautics, Harbin Institute of Technology, Harbin, 150001 People's Republic of ChinaSearch for more papers by this authorYi Shen, Yi Shen School of Astronautics, Harbin Institute of Technology, Harbin, 150001 People's Republic of ChinaSearch for more papers by this authorYipeng Liu, Yipeng Liu College of Information Engineering, Zhejiang University of Technology, Hangzhou, 310023 People's Republic of ChinaSearch for more papers by this author First published: 01 October 2018 https://doi.org/10.1049/iet-cta.2017.1408Citations: 11AboutSectionsPDF 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 Sensor fault detection of discrete-time descriptor systems with bounded disturbances is studied. The authors propose a new structure of a fault detection observer by augmenting the output, which can broaden the application scope. A criterion is used to achieve the fault sensitivity and disturbance attenuation ability of the residual simultaneously. The residual generation and time-varying threshold calculation are integrated together based on analysis. Furthermore, sufficient conditions for the performance of the proposed observer are formulated in terms of linear matrix inequalities. Numerical simulations of a direct current motor and a flight vehicle are conducted to verify the applicability of the proposed fault detection observer design method. 1 Introduction To improve the safety of control systems, model-based fault diagnosis has been intensively studied in the past three decades, see [1–4] and the references therein. The observer-based fault detection method is one of the most widely-used approaches among the existing model-based fault diagnosis techniques. The observer-based fault detection scheme may suffer from false alarm since disturbances and noises are inevitable. The fault detection observer is designed to achieve the fault sensitivity and disturbance attenuation ability simultaneously. As a result, fault detection has attracted considerable attention in the past two decades [5–8]. The concept of fault detection was first proposed in [5], which was formulated as a special type of constrained optimisation problem. After the pioneering work of Hou and Patton [5], many results on the fault detection observer design have been proposed in the literature, e.g. [7–13]. The main idea of observer-based fault detection methods is to generate residuals using an observer and then detect the fault occurrence by residual evaluation. In the past few decades, fault detection observer design has been widely studied to achieve robust residual generation. The residual evaluation method, however, is ignored in most of the existing results. As pointed out in [14], performance is used to measure the energy-to-energy gain of systems, while the performance describes the peak-to-peak gain of systems. Compared with performance, the peak-to-peak gain is more reasonable and useful in residual evaluation [2, 15]. Most recently, the concept of fault detection was first proposed for continuous-time systems in [16] and later was extended to a discrete-time case in [17]. Based on analysis, time-varying thresholds were obtained in [16, 17]. A descriptor system, which is also called singular system or differential-algebraic system, can describe both the dynamic property and algebraic relationship of a system. Since the descriptor system representation is able to characterise a more general class of systems than the regular state-space system model, it has been used in energy systems, mechanical systems, and network analysis [18–21]. In recent years, several papers on fault diagnosis of descriptor systems have been reported in [22–27]. In [24], actuator fault diagnosis was studied for descriptor systems by using augmented observer technique. A robust fault detection filter is designed for a class of non-linear descriptor systems using the criterion in [23]. Fault detection and identification are studied for cyber-physical systems, which can be modelled as descriptor systems [28]. For descriptor systems, most of the existing fault detection methods rely on the assumption of C-observability [23, 29–31], which restricts the scope of application. Moreover, much attention has been paid to actuator fault diagnosis, while few papers on sensor fault diagnosis have been reported, especially for discrete-time descriptor systems. We also note that most of the existing fault detection schemes mainly focus on residual generation, while the threshold computation in residual evaluation is seldom studied. Motivated by the above discussions, this work studies a sensor fault detection observer for discrete-time descriptor systems using the concept of fault detection proposed in [16, 17]. The main contributions are the following aspects. We propose a novel structure of the fault detection observer for discrete-time descriptor systems. The proposed method relaxes the assumptions used in many existing papers such as [23, 29–31] and hence broaden the application scope. fault detection of the descriptor system is studied for the first time in this study. Different from the fault detection observer, the fault detection observer can generate an adaptive threshold used in residual evaluation. Therefore, the proposed method can achieve not only residual generation but also residual evaluation. We propose a new design method for the fault detection observer. In the proposed design, extra parameter matrices are introduced to provide more design degrees of freedom and then reduce design conservatism. Moreover, the design conditions are converted into linear matrix inequalities (LMIs), which can be solved efficiently. 2 Preliminaries and problem formulation 2.1 Preliminaries Notation : represents the set of all real matrices. The superscript and denote the transposition and inverse of a matrix, respectively. 0 and I denote the zero and identity matrices with appropriate dimensions, respectively. The rank of a given matrix M is denoted by . If M has a full column rank then denotes its Moore–Penrose inverse. is a matrix satisfying . denotes the sum of matrix and its transpose, i.e. . The represents a term that is induced by symmetry in a symmetric matrix. For a symmetric matrix P, means that P is positive (negative) definite. For a signal , its norm is defined as where denotes the Euclidean norm of , i.e. . The following lemma is used in the sequel. Lemma 1.[32] Given matrices and , there exists a matrix such that if and only if (1)Moreover, the general solution to is given by (2)where is a freely chosen matrix with an appropriate dimension. 2.2 Problem formulation In this study, we consider the following discrete-time descriptor system: (3)where , , and are the state, input, measurement output and sensor fault, respectively, is the unknown input including process disturbance and measurement noise, , , , , , and are known constant matrices with appropriate dimensions. In system (3), E may be a singular matrix, i.e. . The following assumptions are considered in this study. Assumption 1.The unknown vector is bounded as , where is a known constant. Assumption 2.The descriptor system in (3) is S -observable, i.e. the system simultaneously is R -observable (4)and I -observable (5) Remark 1.It is known from [33] that the I -observability (5) in Assumption 2 is equivalent to (6)and is more general than the following assumption: (7)which is widely used in observer-based fault diagnosis for descriptor systems [23, 29–31]. Note that these methods study fault detection under the assumption of C-observability, which is stricter than S -observability. Therefore, the proposed method has a broader scope of application than the approaches presented in [23, 29–31]. To design a fault detection observer, we augment the measurement output as follows: (8)where with , and are as follows: Using , we propose the following fault detection observer: (9)where denotes the state estimation, is the intermediate state, is the residual vector, is the gain matrices to be synthesised, and matrices and should be designed to satisfy (10) Remark 2.In fault detection observer (9), L is a gain matrix while T and N can be viewed as weighted matrices. T and N are weighted information associated with models and measurements, respectively. Thanks to the equation constraint in (10), the information from models and measurements is fused together for state estimation. By using Lemma 1, it can be verified that the solution of (10) is (11)where is a freely chosen matrix and To design fault detection observer (9), we define the following error vector: Based on (3), (9) and (10), the error dynamic system can be obtained as (12)where , . Next, we will analyse the effects of faults and disturbances on residual, respectively. Inspired by [27], we split error dynamic system (12) into three subsystems as follows: (13) (14) (15)where with , , and , Based on the subsystem in (13), we give the following definition. Definition 1.The error system in (12) is said to have an performance index , if its subsystem (13) satisfies the following inequality for all (16)where with to be specified, is a given positive scalar. Remark 3.The term in (16) will converge to zero as . The peak-to-peak gain from d to is . This is why the performance index is . Note that under assumptions of and , the performance (16) reduced to the following peak-to-peak performance defined in [2] This constant performance index is used to detect the fault in [2]. In this study, the time-varying performance (16) is used to set the required threshold for the fault detection scheme, which can reduce fault-alarm. Similar performance definitions were used in [16, 17] to achieve fault detection. The following definition of performance is used to describe fault sensitivity based on subsystem (14). Definition 2.The error system in (12) is said to have an performance index , if its subsystem (14) satisfies the following inequality: (17) In this study, we use performance index and (peak-to-peak) performance index to measure fault sensitivity and disturbance attenuation ability, respectively. Based on the above definitions, we present the following fault detection observer design problem. 2.2.1 Design objective. For the discrete-time descriptor system in (3), we design a fault detection observer (9) such that the generated residual is sensitive to the fault and robust against disturbances in an sense or with an performance index. Furthermore, the bound of the residual norm is obtained for fault detection. Specifically, (12) is designed such that the following conditions are satisfied: (i) Error system (12) is asymptotically stable. (ii) Error system (12) has an performance. (iii) Error system (12) has an performance index . 3 Main results In this section, we will design a fault detection observer such that the residual is sensitive to a fault and robust against disturbances. Then the residual evaluation will also be given. 3.1 Disturbance attenuation condition The following theorem is given to ensure that the error system (12) is asymptotically stable and has an performance. Theorem 1.Given a scalar , the error system (12) satisfies (i) and (ii) if there exist positive scalars and , a positive definite matrix , and matrices , , and such that (18) (19)where Proof.Note that under fault-free condition. Therefore, in the analysis of the robustness against disturbances, we use instead of for simplicity.For the error subsystem (13), we choose a Lyapunov function as (20)Then the difference of the Lyapunov function is where (21)Note that T,N are given in (11). The inequality (18) is equivalent to (22)By pre- and post- multiplying (22) with and its transpose, we have (23)By pre-multiplying and post-multiplying inequality (23) with and its transpose, we have (24)When disturbance d of error system (12) is zero, the difference of the Lyapunov function is (25)Hence the error system (12) is asymptotically stable.Inequality (24) follows that which implies (26)In addition, inequality (19) implies that which follows By setting , performance index (16) is satisfied. □ 3.2 Fault sensitivity condition Based on Definition 2, the following theorem will provide the condition for the error system (12) satisfying the performance (17). Theorem 2.Given a scalar , the asymptotically stable error dynamic system (12) has an performance if there exist a symmetric matrix , and matrices , , , and such that (27)where Proof.For the asymptotically stable error subsystem (14), there exists a Lyapunov function (28) Note that T,N are given in (11). Pre- and post-multiplying (27) by (29)and its transpose, we have (30)where Pre- and post-multiplying (30) by and its transpose, it is obtained that Taking the summation from to , we have (31)Hence, the condition of Definition 2 is satisfied. The error system (12) has an performance index . □ Note that the conditions in Theorems 1 and 2 are non-LMIs and cannot be easily solved. Therefore, to facilitate the design, we need to convert (18), (19) and (27) into a set of LMIs. 3.3 Fault detection observer design To design a fault detection observer, we need to find parameter matrices T,N,L such that the error system (12) satisfies conditions (i)–(iii) simultaneously. To this end, we give the following theorem, which provides sufficient conditions of Theorems 1 and 2. Theorem 3.Given scalars , , , and matrix , the error system (12) satisfies (i)–(iii) if there exist positive scalars , , symmetric matrices , and matrices , and such that (32) (33) (34)where Moreover, the parameter matrices in fault detection observer (9) are given by (35) Proof.Considering (35), we substitute , , and into (32) and (34). It can be verified that LMIs (32)–(34) provide sufficient conditions of Theorems 1 and 2. Hence the error system (12) satisfies conditions (i)–(iii). □ Remark 4.Note that (32)–(34) are LMIs which can be easily solved by using some available toolboxes in MATLAB. To improve the fault detection performance, the required parameter matrices T,N,L can be designed by solving the following optimisation problem: (36)where is a weighted coefficient. Remark 5.Note that in the existing observer-based fault detection approaches [23, 27, 29, 31], the parameter matrices are solved according to condition (10), then the gain matrix L is designed based on the value of . In our proposed method, the parameter matrices can be solved together by introducing the matrix Y into Theorem 3, which increases the design freedom and reduces some conservatism of the gain matrix. 3.4 Residual evaluation For the residual evaluation, one of the commonly used approaches is the so-called threshold method [2]. In this study, we adopt the following fault detection scheme: where the residual evaluation function is defined as and is a time-varying threshold as the following: (37)where denotes the upper bound of , is the maximum eigenvalue of , , , which are obtained by Theorem 3. 4 Simulations In this section, two simulation examples are used to illustrate the effectiveness of our proposed method. 4.1 Example 1 We first use a direct current (DC) motor model proposed in [21] to verify the effectiveness of our method. The continuous-time descriptor model of the DC motor is described by (38)where i is the current in the motor winding, is the rotor speed, is the angular position, the model parameters and their values are presented in Table 1. Table 1. DC motor parameters Symbols Physical meanings Values J moment of inertia of the rotor b damping (friction) of the mechanical system back-electromotive force constant back-torque constant armature resistance amplification constant 20 It is assumed that all the state variables are measurable. Moreover, to be more practical, we assume that the current i and angular position are affected by an additive sensor fault f. Finally, by using Euler's forward method, the continuous-time DC motor model (38) can be discretised to yield a discrete-time descriptor system with sample time . Considering input and measurement noise, the discrete-time state-space form is given by (39)where Note that in this example Hence Assumption 2 is satisfied. The method in [23] is also suitable for this example. We will compare the proposed method with the method proposed in [23] in this simulation. To design the fault detection observer, we set , and in Theorem 3. To improve the fault detection effect, here we set the weighted coefficient and solve the optimisation problem (36) by using YALMIP toolbox in MATLAB. Then we obtain , and In the simulation, the disturbance is assumed bounded with . Constant and time-varying sensor faults are injected, respectively Figs. 1 and 2 show the fault detection results of the DC motor model using the proposed method and the method proposed in [23]. The solid lines denote residual evaluation functions and the dashed lines denote thresholds, respectively. In Fig. 1, the residual evaluation function obtained by using the method in [23] seems small and the fault detection effect is not good. The constant fault is detected by using the proposed method in this study. In Fig. 2, the time-varying fault is not detected using the method proposed in [23], while the fault is detected successfully using our proposed method. The proposed method exhibits better fault detection effect than the method proposed in [23] for both constant and time-varying sensor faults. Fig. 1Open in figure viewerPowerPoint Residual evaluation functions and thresholds for constant fault Fig. 2Open in figure viewerPowerPoint Residual evaluation functions and thresholds for time-varying fault 4.2 Example 2 In this part, we study sensor fault detection for the longitudinal control system of a flight vehicle 'ALFLEX' in [19], which is expressed in the descriptor system form as (40)where By using Euler's forward method, the continuous-time longitudinal control system model (40) can be discretised to yield a discrete-time descriptor system with sample time . Considering input and output disturbances, the discrete-time state-space form in (3) is given by (41)where Remark 6.Note that in this example, we find that Hence, Assumption 2 is satisfied. Note that the existing observer-based fault detection methods in [23, 29–31] cannot be applied on this model system directly. To design the fault detection observer, we set in Theorem 3. To improve the fault detection effect, here we set the weighted coefficient and solve the optimisation problem (36) by using YALMIP toolbox in MATLAB. Then we obtain and . Moreover, the parameter matrices in fault detection observer (9) are given by Here the matrix N is not full rank. The condition (10) still holds. In the simulation, the disturbance is assumed bounded with . Constant and time-varying sensor faults are injected, respectively For a constant sensor fault, the residual evaluation function exceeds the calculated threshold in Fig. 3. A similar result of time-varying sensor fault is presented in Fig. 4. By searching for the optimal solutions, the time-varying fault is detected within 30 time steps after the fault occurrence. Both kinds of faults are detected successfully, which shows the effectiveness of the proposed fault detection method for descriptor systems. Fig. 3Open in figure viewerPowerPoint Residual evaluation function and time-varying threshold for constant fault Fig. 4Open in figure viewerPowerPoint Residual evaluation function and time-varying threshold for time-varying fault 5 Conclusion In this study, we have studied an fault detection observer for discrete-time descriptor systems with sensor fault. A new structure of the fault detection observer is proposed by augmenting the output, which broadens the application scope. In a fault detection scheme, a time-varying threshold for residual evaluation is generated based on analysis. The residual generation and threshold calculation are integrated together in our fault detection. Sufficient conditions for the existence of such a fault detection observer are formulated in terms of LMIs which are efficiently solved. 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Publication Year: 2018
Publication Date: 2018-10-01
Language: en
Type: article
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