Abstract
This paper deals with the robust fault detection problem in the finite frequency domain for a class of Lipschitz nonlinear systems with uncertain parameters. The design conditions of fault detection observers are proposed. These conditions make the error estimation system stable, and the residual has disturbance attenuation performance for external disturbances and uncertain parameters. In addition, considering that fault occurs in the finite frequency domain, the residual also has fault sensitivity performance. To ease of solution, the observer design conditions are further transformed into linear matrix inequalities. On this basis, a dynamic threshold design method based on analysis is presented. Finally, simulation results demonstrate the superiority of the proposed method.
1. Introduction
The structure and function of the control system are becoming more complicated, which requires the system has better reliability and testability. Therefore, fault diagnosis techniques are also constantly developed [1–3]. Fault detection based on state observer is one of the most widely used fault detection methods. Generating residuals by comparing the state of the observer and the system and determining whether a fault occurs according to residual exceeds the threshold or not.
Since it is impossible to obtain a complete and accurate mathematical model, therefore, the residual must be robust to external disturbance and uncertain parameters. Patton and Chen proposed a disturbance decoupling method based on characteristic structure configuration [4]. Dong et al. investigated the robust fault detection filter design problem for a class of discrete-time conic-type nonlinear Markov jump systems with jump fault signals [5]. Yin et al. investigated the problem of robust fault diagnosis of stochastic discrete-event systems against model uncertainty [6]. However, our fundamental purpose is to detect the fault accurately and timely, so a balance needs to be struck between robustness and fault sensitivity. Hou and Patton used index to measure the sensitivity of residual to fault and proposed the concept of observer [7]. Estrada and Wei designed fault detection observer for uncertain linear systems, respectively [8, 9]. This method has been applied to uncertain nonlinear systems [10–13].
All the above methods design the observer in full frequency domain. In fact, faults usually occur in finite frequency domain. For example, slow-varying fault usually occurs in low-frequency bandwidth, and the frequency of abrupt fault is zero. In addition, designing an observer in finite frequency domain can reduce the design conservatism [14]. Iwasaki and Hara proposed the generalized Kalman-Yakubovich-Popov lemma and solved the performance analysis problem in finite frequency domain [15]. Subsequently, many works have been done on the design of fault detection observer for linear systems in finite frequency domain [16–19]. Zhai et al. proposed a robust fault detection method for spatial interconnected systems (SISs) with polyhedral uncertainties in the finite frequency domain [20]. Wang considers parametric uncertainties and applies the method to uncertain linear systems [21].
However, as a gain measure of energy-to-energy, the norm has some defects when applied to residual evaluation. The reason is that the disturbance signal is unknown, and it is difficult to calculate the energy of the disturbance signal in the whole time domain. In addition, the periodic signal and the step signal are energy unbounded. While norm describes the peak value of the signal, it is easy to obtain and more suitable for residual calculation. Some achievements have been made on the fault detection observer for linear systems [22–25]. In practical applications, most systems are nonlinear, and the above methods are not applicable. For a class of Lipschitz nonlinear systems, Zhou proposed an observer design method, but this method does not consider the uncertainty of model parameters [26]. Uncertain parameters and the nonlinear term bring great challenges to the index analysis of and in finite frequency domain. Therefore, this paper proposes an fault detection observer design method in finite frequency domain for a class of Lipschitz nonlinear systems with uncertain parameters.
1.1. Notation
This paper uses following mathematical notations: denotes dimensional real matrices. 0 and denote the zero matrix and identity matrix with appropriate dimensions, respectively. For a matrix , and denote the transpose and the complex conjugate transpose of . denotes the sum of matrix and its complex conjugate transpose. For a matrix , or denote is a positive definite matrix or a negative definite matrix. and denote the minimum and maximum eigenvalue of . For a signal , the norm is defined as , the norm is defined as , and denote the Fourier transform of .
2. Problem Description
Considering the following systemwhere , , denote the state, output, and input vectors, respectively. denote the external disturbance and measurement noise. denotes the fault. , , , , , are known matrices with appropriate dimensions, respectively. denotes uncertain parameters. is the nonlinear term and satisfies the Lipschitz conditionwhere is a known scalar representing the Lipschitz constant.
Assumption 1. The time-varying matrixsatisfies, where,are known.is unknown and satisfies.
Assumption 2. satisfies, whereis a known scalar.
Without causing confusion, is denoted as for simplicity, and the other variables are denoted in the same way. Constructing the following fault detection observer aswhere is the state vector of the observer, is the residual vector, and is a matrix that needs to be designed. The state estimate error is denoted asSo we can getwhere . The finite frequency domain can be expressed by [25]where is the frequency of fault signal, and are known scalars. When , , we can get , it means that the fault occurs in the low-frequency bandwidth, where denotes the boundary of low-frequency bandwidth. When , , we can get , and it means that the fault occurs in the middle frequency bandwidth. When , , we can get , and it means that the fault occurs in the high-frequency bandwidth, where denotes the boundary of high-frequency bandwidth. The observer design conditions are proposed as follows.(i)The state estimation error system (5) is stable.(ii)Under zero initial condition, state estimation error system (5) has performance (i)where , and are scalars need to be designed. is a time-varying function related to the initial estimate error value , when , . satisfies where is a known scalar.(i)When a fault occurs in the finite frequency domain (6), the state estimate error satisfiesAnd the state estimation error system (5) has performance
3. Fault Detection Observer Design
3.1. Observer Design Method
The state estimation error system (5) can be rewritten aswhere . When only considering the influence of external disturbance and initial conditions on the residual, (11) can be represented as
When only considering the influence of fault on the residual, (11) can be represented as
3.2. Stability and Disturbance Attenuation Performance
Theorem 1. Given scalars , , , the state estimation error system (5) satisfies (a) and (b) if there exist scalars , , and matrices , such thatwhere .
Proof. We take the following Lyapunov functionThen, we can obtainAccording to (12), is obtained as
Lemma 1. Suppose matrices , have appropriate dimensions, if matrix satisfies , then the following inequality can be established for any scalar and vector [27].According to (19), set , and we can getThe following inequality can be derived asIf satisfieswhere . We derive the following inequality if we integrate (22)According to assumption 2, (23) is equivalent to the following formula:According to assumption 1, we can obtain the following formula:Then,Therefore, when , , satisfies . We obtainIt follows that . When , so that , namely the state estimation error system (5) satisfies (a). According to Lipschitz condition, . If the following inequality is satisfied,Then, the inequality (22) can be derived. Substituting (16) and (21) into (28), it follows thatwhere . According to Schur’s complement lemma, (29) is equivalent to (14). Besides, the following inequality can be obtained from (15)(30) is equivalent to the following formula:Combining (26) with (31), the following formula can be obtained:where , namely the state estimation error system (5) satisfies (b).
3.3. Fault Sensitive Performance
Theorem 2. Given a scalar , the state estimation error system (5) satisfies (c) if there exist scalars , , and matrices , , satisfies such thatwhere,,,,,,.
Proof. We take the following Lyapunov functionThen, we derive the following formula:Considering the fault occurs in finite frequency domain, and the performance index is expressed asUnder zero initial condition, if , it follows thatDefine . According to Parseval’s theorem [28],It can be seen that . If satisfies , then . As a result, we obtainCombining (37) and (39), it follows that , namely the estimation error system (5) satisfies (c). According to Lipschitz condition, . Therefore, the following inequality is a sufficient condition of :Furthermore,Substitute (13) into (41), the following formula can be derived asCombining (35), (40) and (42), it follows thatAnd (43) is equivalent to (33).
As can be seen that the inequalities given in Theorem 1 and Theorem 2 are nonlinear, which are not conducive to the solution, therefore, the following theorem is proposed to convert (14) and (33) into linear matrix inequalities.
Theorem 3. Given scalars , , , , , matrix , the observer (3) is an fault detection observer in finite frequency domain for system (1) if there exist scalars , , , and matrices , , , , , satisfies such thatwhere,,,,.
Proof. (14) can be expressed aswhere , , , . Furthermore,Then, (48) is equivalent to the following inequality:We choose , , then (50) is equivalent to (45). According to Schur’s complement lemma, (47) can be rewritten asAccording to Lemma 1,We obtainwhere , , and . If the left and right sides of are multiplied by and its transpose matrix, respectively, then we can obtain (33). □
3.4. Dynamic Threshold Design Method
It is necessary to design an appropriate residual evaluation function and the detection threshold. In this paper, norm of the residual is selected as the residual evaluation function
The following inequality holds when no fault occurs:
According to (32), it follows that
Therefore, we select the following function as the dynamic detection threshold means that a fault occurs, and means that no fault occurs.
4. Simulation
In this section, we use the state-space model of a single-link robot to demonstrate the effectiveness of the proposed method. The state-space model [14] is given as
Supposing there exists uncertain parameters , , is a time-varying random number between , and is a random vector with values between . It is considered that the fault occurs in low-frequency bandwidth . We choose scalars , , , , , , matrix . Supposing the initial value of the system state is , the initial value of observer state is set as . By solving Theorem 3, the and performance indexes are calculated as , , , and the gain matrix of the fault detection observer is obtained as
Considering the system occurs abrupt fault and slow-varying fault , respectively.
The simulation results are shown in Figure 1 and Figure 2. is the residual evaluation result generated by observer, and is the dynamic threshold. According to the simulation results, we can obtain that remains near 0 when no fault occurs, indicating that is robust to external disturbance and uncertain parameters. When faults occur at 10 s, quickly exceeds the detection threshold, and when faults disappeared at 70 s, also quickly fell below the detection threshold, indicating that the proposed observer has good sensitivity to fault. The simulation proves that the proposed method can be used for fault detection of nonlinear systems with uncertain parameters. To illustrate the superiority of the proposed method, we have made the following comparisons: (a) comparison of the proposed method with observer in finite frequency domain. Supposing the system occurs abrupt fault and slow-varying fault and , respectively. The performance index of fault detection observer is calculated as , , and the gain matrix is obtained as
The comparison results are shown in Figure 3 and Figure 4.
is the residual evaluation result generated by observer, and is the dynamic threshold for observer. As can be seen from Figures 3 and 4 that the observer cannot detect and although has changed when the fault occurs, it is not sensitive enough to fault. The reason is that the robustness and fault sensitivity are contradictory. The disturbance attenuation performance index of fault detection observer is better than the observer we proposed. However, it also inhibits the fault sensitivity. In contrast, we design the index to improve the sensitivity of residual to fault, which illustrates the advantages of the proposed method. (b) Comparison of the proposed method with observer in full frequency domain. Supposing the system occurs abrupt fault and slow-varying fault and , respectively. The and performance indexes of fault detection observer in full frequency domain are calculated as , , , and the gain matrix is obtained as
The comparison results are shown in Figure 5 and 6.
As can be seen from Figures 5 and 6 that the observer in full frequency domain cannot detect fault . For the fault , it can only be detected within , indicating that the observer in full frequency domain is not sensitive enough to the fault. The reason is that it is conservative to design an fault detection observer in full frequency domain without considering the frequency of fault. If the fault amplitude is low, the observer may not be able to detect the fault adequately. In this paper, the frequency domain of fault is considered in the observer design process, so that the observer design limitation is reduced and improved the sensitivity of the residual to fault. (c) Comparison of the proposed method with the method proposed in [14]. Considering the system occurs the following abrupt fault and slow-varying fault , respectively.
Supposing is a random vector with values between . The comparison results are shown in Figure 7 and Figure 8.
As can be seen from Figures 7 and 8, although both the proposed method and the method proposed in [14] can detect the occurrence of faults, the fault detection effect of the proposed method is better: the residual evaluation result obtained by the proposed method completely exceeds the threshold, while the residual evaluation result obtained by the method of [14] does not fully exceed the threshold. The reason is that [14] does not consider the model parameters uncertainty, so it needs to take the model parameters uncertainty as additional noise, which increases the average energy of the noise signal over a time interval. And the fault detection threshold is designed based on the maximum value of the average energy of the noise signal. Therefore, it needs to raise the fault detection threshold. When the noise intensity increases, the fault may not be adequately detected by residual evaluation results. The method proposed in the paper considers the influence of model parameters uncertainty in the design process of the observer and dynamic threshold and can fully detect the occurrence of faults, which shows the superiority of the proposed method.
5. Conclusion
In this paper, we derive and analyze a robust fault detection scheme for uncertain Lipschitz nonlinear systems in finite frequency domain. A design method of fault detection observer is proposed, and the superiority of the proposed method is verified by a numerical example. However, only the matrix has time-varying uncertain parameters, and the other matrices are not considered in the paper. The problem of robust fault detection in finite frequency domain for nonlinear systems with more uncertain parameters is the focus of future work.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.