TY - JOUR
AU - Dong, Jian-Kang
AB - Abstract Sensors are important parts of various control systems and their faults can cause degradation of systems. The problem of fault diagnosis for sensors is considered specially in this article. A time-varying and anti-disturbance fault diagnosis algorithm based on the fault detection observer and the adaptive control theory has been proposed for a class of nonlinear systems that are influenced by disturbances and measurement noise. When external disturbances, measurement noise and faults of sensors exist simultaneously, the designed fault diagnosis algorithm is able to give specific estimated values of state variables and faults, respectively. The asymptotical stability of the fault detection observer in the diagnosis algorithm for sensors is guaranteed by setting a well-designed adaptive adjusting law of the fault vector. Moreover, a theoretically rigorous proof based on Lyapunov stability theory has been given. Two experiments have been carried out to evaluate the performance of the proposed fault diagnosis algorithm. 1. Introduction Sensors are sensitive components in measuring devices, intelligent instruments, automatic control systems and computer information systems. With the fast-growing information technology, sensors have become increasingly important, especially for devices that need huge investment, such as aircrafts and power plants. The qualities of sensors are directly related not only to the operational status of the equipment but also to the vital security issues. Especially for sensors that provide control signals, their working state can affect the state of the whole system. As things stand, the situation is not optimistic. Because of the complex and harsh working environment, sensors that are most prone to fault have become a weak link in automatic control systems, which are shown by Escobar et al. (2014) and Pan et al. (2014). In this case, fault diagnosis for sensors has become more significant. Extensive and comprehensive researches have been dedicated to sensors’ fault diagnosis. Generally, there are mainly two types of research approaches for sensors’ fault diagnosis, one is the data-driven approach and the other is based on analytical models. Nowadays, the solutions of data-driven fault diagnosis have been receiving considerably increasing attention. Data-driven approaches are able to accomplish the fault diagnosis process through analysing and processing operation data without knowing accurate analytical models of systems. Parikh et al. (2008), Cheng et al. (2008), Hosseinabadi et al. (2014) and Yin et al. (2015, 2016) have introduced data-driven approaches such as machine learning methods, multivariate statistical analysis methods, signal processing methods and information fusion methods. Compared with the approaches based on analytical models, data-driven fault diagnosis methods are easier to be applied in practical systems, which are receiving more and more attention, as is illustrated by Yin et al. (2016). However, due to the lack of information about the internal structure and mechanism of systems, it is relatively difficult for data-driven fault diagnosis approaches to give the detailed information of the existing faults, such as the amplitude or the frequency. In general, it can only give a fault warning, which does not meet the requirements of in-depth troubleshooting. Compared with the data-driven solutions, the methods based on analytical models are more sophisticated. By adopting precise mathematical models, the approaches based on analytical models can construct residual signals through the observation of input–output to reflect the discrepancy between the desired systems’ behaviour and the actual operation, and then fault diagnosis can be carried out on the basis of the acquired residual signals, as is shown by Izadi et al. (2008), Haghani et al. (2014), Anibal et al. (2014) and Benallouch & Outbib (2014) that have introduced the approaches such as state estimation methods, parameter estimation approaches and parity space methods. The representative articles are analysed in detail as follows. A fuzzy adaptive observer was designed to estimate the unmeasured state by Li et al. (2014, 2015). A theoretically rigorous stability proof was given and simulation results were provided to show the effectiveness of the approach. However, the specific values of faults were not able to be given. An observer-based fault detection approach with an adaptive threshold was designed for detecting a rotor-position-sensor fault by Choi et al. (2015). To avoid missing or false alarms, the key design issue of the proposed detection method was the adaptive threshold, which was determined by analysing position estimation errors of the observer. Experimental results showed the effectiveness of the designed algorithm. However, throughout the whole diagnostic process, the external disturbances were not fully considered. A sensor fault compensation system, applied to a heat pump’s helical evaporator, was presented by Scelba et al. (2014) and Escobar et al. (2015). The sensors’ fault detection and isolation system was based on a combination of two high-gain observers. The experiments on the evaporator of the absorption heat pump showed the reliability of the method to detect and isolate a sensor fault when a total or partial fault occurred. However, their detection and isolation system did not take into account the impact of the sensors’ measurement noise on the performance of fault diagnosis. In summary, for current solutions of sensors’ fault diagnosis based on analytical models, there are three prominent issues that have not been solved perfectly. First, in the case that the external disturbances and the sensors’ measurement noise exist simultaneously, the current solutions are not very effective. In other words, the disturbances and the noise are different, so they must be handled separately. However, the existing literatures did not notice this problem. The external disturbances can affect the states of the system, whereas the sensors’ measurement noise can always only affect the outputs of the system. Second, the robustness of the designed fault diagnosis algorithms to measurement noise or high frequency disturbances is not very satisfactory. When the frequency of disturbances or noise is high, current diagnostic solutions always cannot provide accurate fault estimation results. Third, the adaptive adjusting law is very difficult to select and inappropriate selection may lead to large errors in fault estimation. In this article, the problem of fault diagnosis for sensors is further pursued for a class of nonlinear systems that are influenced by sensors’ measurement noise and external disturbances simultaneously. A novel adaptive observer-based fault diagnosis algorithm is proposed. The robustness of the state observer in the diagnosis algorithm for sensors is guaranteed by setting a well-designed adaptive adjusting law of the fault vector. The main innovations of this article are reflected in three aspects: First, different from the existing articles, the proposed fault diagnosis algorithm is not based on ideal analytical models, in other words, random measurement noise and external disturbances are taken into account simultaneously during the whole diagnosis process. Second, the designed fault diagnosis algorithm can effectively separate the sensors’ measurement noise, the unknown external disturbances and the internal faults of the control system when they exist at the same time. Moreover, it is able to give specific estimated values of both faults and system’s state variables, respectively, rather than just gives the estimated values of state variables or a fault warning. This is very beneficial for in-depth fault analysis and the following troubleshooting actions. Third, the proposed algorithm is very concise and efficient, which can overcome the defect of ‘explosion of complexity’ and is easy to be applied in actual operations. The structure of this article is organized as follows. In section 2, an analytical model of the nonlinear system influenced by disturbances, measurement noise and faults is proposed. The diagnosis targets and the constraints under which the diagnosis algorithm is developed are identified in Section 3. In addition, a rigorous proof of asymptotical stability in theory has been given. In Section 4, the simulation results are presented. Conclusions and future works are listed in Section 5. 2. Description of the control system Consider the following fault-free nonlinear control system: {x˙(t)=W(x,t)+Bu(t)+Gξ(t)y(t)=Cx(t)+Hη(t),  (2.1) where $$x(t)\in \Re^{n}$$ denotes the state vector and it can be measured directly; $$y(t)$$ denotes an output vector; $$u(t)\in \Re^{r}$$ denotes the control inputs; $$\xi (t)\in \Re^{n}$$ denotes the external time-varying disturbance vector; $$\eta (t)\in \Re^{n}$$ denotes the random measurement noise (white noise) vector; $$W(x,t)$$ is the nonlinear function of state variables $$x(t)$$; $$B$$ and $$C$$ are appropriately dimensional const matrices; $$G$$ and $$H$$ are full rank const matrices. When faults occur in sensors, the fault system can be expressed as: {x˙(t)=W(x,t)+Bu(t)+Gξ(t)y(t)=Cx(t)+Hη(t)+Ff(t),  (2.2) where $$f(t)\in \Re^{r}$$ denotes the unknown time-varying fault vector that occurs in sensors; $$F$$ is an appropriately dimensional const matrix that describes the distribution of faults. Remark 1 From fault system (2.2), it can be found that when there is a sensor fault, the state variables of the control system are not affected (the structure of the control system (2.1) has not been destroyed), but output vector $$y(t)$$ will contain faults’ information. 3. Fault diagnosis algorithm 3.1. Constraint conditions For a start, some constraint conditions need to be satisfied for fault system (2.2) before designing the fault diagnosis algorithm: (1) Nonlinear function $$W(x,t)$$ satisfies the Lipschitz condition about state vector $$x(t)$$, namely there exists a constant $$\mu_{1} >0$$, such that: ∥W(x1)−W(x2)∥⩽μ1∥x1−x2∥. (3.1) Remark 2 To simplify writing, $$x(t)$$ is abbreviated to $$x$$. This kind of writing method is suitable for the subsequent article. (2) External time-varying disturbance vector $$\xi (t)$$ is bounded, i.e.: ∥Gξ(t)∥⩽l0, (3.2) where $$l_{0} \geqslant 0$$ represents the maximum affordable value of disturbance. (3) Measurement noise vector $$\eta (t)$$ is also bounded, i.e.: ∥Hη∥⩽l1, (3.3) where $$l_{1} \geqslant 0$$ represents the maximum amplitude of the measurement noise. (4) Unknown time-varying fault vector $$f$$ is a bounded vector with finite rate of change, i.e.: ∥f∥⩽l2∥f˙∥⩽l3, (3.4) where $$l_{2} \geqslant 0$$ and $$l_{3} \geqslant 0$$ represent the maximum amplitude of the faults and the rate of change, respectively. (5) Just as fault vector $$f$$, state vector $$x$$ is also bounded with finite rate of change, i.e.: ∥x∥⩽l4∥x˙∥⩽l5, (3.5) where $$l_{4} \geqslant 0$$ and $$l_{5} \geqslant 0$$ represent the maximum amplitude of the states and the rate of change, respectively. 3.2. Nonlinear fault detection observer On the basis of fault system (2.2) and constraint conditions (3.1)–(3.5), the following nonlinear fault detection observer has been designed: {x^˙=W(x^)+Bu+K(y−y^)y^=Cx^+Ff^,  (3.6) where $$\hat{x} \in \Re^{n}$$ denotes state estimation; $$\hat{y} \in \Re^{m}$$ denotes output estimation; $$\hat{f} \in \Re^{r}$$ denotes fault estimation; $$K$$ is a finite gain matrix of the designed observer. Here, the deviation signal $$\tilde{x} \in \Re^{n}$$ is defined in the following form: x~=x−x^. (3.7) Remark 3 $$x$$ denotes the real values of states in fault system (2.2), which is measurable and does not contain faults’ information [sensors’ fault $$f$$ does not affect state vector $$x$$, see (2.2)]. However, $$\hat{x}$$ comes from fault detection observer (3.6) and it may contain faults’ information. When faults occur in sensors, namely $$\vert \vert f\| \ne 0$$, $$x$$ is not equal to $$\hat{x}$$, namely $$\| \tilde{x} \| \ne 0$$. In other words, $$\tilde{x}$$ always contains the deviation information caused by sensors’ fault $$f$$. If there are not faults, $$\hat{x}$$ asymptotically approaches $$x$$ and in this case, the fault detection observer (3.6) becomes a standard state observer. So, (3.6) is named as ‘fault detection observer’. By combining fault system (2.2) and fault detection observer (3.6), the following expression can be obtained: x~˙=x˙−x^˙=−KCx~+(W−W^)−KF(f−f^)+Gξ−KHη, (3.8) where $$W(\hat{x} )$$ is written as $$\hat{W}$$. 3.3. Adaptive fault estimation Considering the fault system that has been given and supervised by (2.2), the proposed adaptive fault estimation algorithm for this system is given by: f^˙=−Λ(KF)TMx~, (3.9) where $$\it{\Lambda} = diag\{\it{\Lambda_{11}} ,\it{\Lambda_{22}} ,\ldots,\it{\Lambda _{rr}} \} \forall \it{\Lambda_{ii}} >0,i\in [1,r]$$ denotes the convergence factor matrix, namely the adaptive law, which can influence the convergence speed and accuracy of fault estimation; $$M$$ is a symmetric positive definite matrix and the conditions that need to be satisfied for $$M$$ will be discussed in detail in the following. Here, the error signal of fault estimation $$\tilde{f} \in \Re^{r}$$ is defined in the following expression: f~=f−f^. (3.10) Remark 4 Fault estimation $$\hat{f}$$ and its rate of change $$\dot{\hat{f}}$$ should also meet constraint condition (3.4). In other words, $$\| \hat{f} \| \leqslant l_{2}$$ and $${\| \dot{\hat{f}}} \| \leqslant l_{3}$$ still hold. By constraint condition (3.4) and fault estimation algorithm (3.9), the following form can be obtained: f~˙=f˙−f^˙=f˙+Λ(KF)TMx~. (3.11) Theorem 1 If $$M$$in (3.9) and the symmetric positive definite matrix $$N$$ are selected to satisfy the following condition: (−KC)TM+M(−KC)+μMM+μI=−N, (3.12) where $$\mu >0$$is a positive const that satisfies Lipschitz condition (3.1), $$I$$ is an appropriately dimensional unit matrix, then the fault diagnosis algorithm based on fault detection observer (3.6) and the adaptive fault estimation algorithm (3.9) can ensure that deviation signal $$\tilde{x}$$ and fault estimation error $$\tilde{f}$$ are driven asymptotically to bounded positive constants simultaneously. In the sense that: {limt→∞∥x~(t)∥⩽τ1limt→∞∥f~(t)∥⩽τ2,  (3.13) where $$\tau_{1} >0$$ and $$\tau_{2} >0$$ denote two bounded positive constants. Proof. A Lyapunov function $$V(t)$$ is defined as: V(t)=x~TMx~+f~TΛ−1f~. (3.14) Remark 5 $$V(t)$$ evaluates deviation signal $$\tilde{x}$$ and fault estimation error $$\tilde{f}$$ comprehensively. After taking the time derivative of (3.14) and substituting (3.8) and (3.11) into it, the following expression is obtained: V˙=x~˙TMx~+x~TMx~˙+f~˙TΛ−1f~+f~TΛ−1f~˙. (3.15) In order to facilitate understanding, the four terms on the right of (3.15) are expanded, respectively, into the following expressions: x~˙TMx~ =[x~T(−KC)T+(W−W^)T−f~ T(−KF)T+ξTGT−ηTHTKT]Mx~, (3.16) x~TMx~˙ =x~TM[(−KC)x~+(W−W^)−KFf~+Gξ−KHη], (3.17) f~˙TΛ−1f~ =(f˙−f^˙)TΛ−1f~=f˙TΛ−1f~+x~TMKFf~, (3.18) f~TΛ−1f~˙ =f~TΛ−1f˙+f~T(KF)TMx~. (3.19) After substituting (3.16)–(3.19) into (3.15), $$\dot{V}$$ can be written as: V˙ =x~T[(−KC)TM+M(−KC)]x~+[(W−W^)TMx~+x~TM(W−W^)] +[ξTGTMx~+x~TMGξ]−[ηTHTKTMx~+x~TMKHη]+[2f~TΛ−1f˙]. (3.20) By using (3.20) and Lipschitz condition (3.1), the following inequality is obtained: V˙ ⩽x~T[(−KC)TM+M(−KC)]x~+2∥x~TM∥μ∥x~∥ +2∥Gξ∥λmax(M)∥x~∥+2∥KHη∥λmax(M)∥x~∥+2f~TΛ−1f˙, (3.21) where $$\lambda_{\max } (M)$$ denotes the maximum eigenvalue of matrix $$M$$; $$\mu$$ has been defined in (3.12). Next, the last term in (3.21) is handled as follows: 2f~TΛ−1f˙⩽f~Tf~+f˙TΛ−1Λ−1f˙ ⩽∥f~∥2+∥f˙∥2λmax(Λ−1Λ−1) ⩽∥f~∥2+l32λmax(Λ−1Λ−1), (3.22) where $$l_{3}$$ has been defined in (3.4). Moreover, because measurement noise vector $$\eta (t)$$ is bounded and $$H$$ is a full rank matrix, which means $$\| H\eta \| \leqslant l_{1}$$ [it has already been declared in constraint conditions (3.3)] and because $$K$$is the finite gain matrix of the designed observer, $$KH\eta$$ will also be bounded. So the following inequality can be obtained: ∥KHη∥⩽l6, (3.23) where $$l_{6} >0$$ is a bounded positive constant. Obviously, after substituting (3.22) and (3.23) into (3.21), condition (3.21) can be strengthened to (3.24). V˙ ⩽x~T[(−KC)TM+M(−KC)]x~+μ[∥x~TM∥2+∥x~∥2] +2l0λmax(M)∥x~∥+2l6λmax(M)∥x~∥+∥f~∥2+l32λmax(Λ−1Λ−1) =x~T[(−KC)TM+M(−KC)+μMM+μI]x~ +2(l0+l6)λmax(M)∥x~∥+∥f~∥2+l32λmax(Λ−1Λ−1), (3.24) where $$l_{0}$$ has been declared in (3.2). By using constraint condition (3.4), the following inequality can be obtained: ∥f~∥=∥f−f^∥⩽∥f||+∥f^∥⩽2l2, (3.25) where $$l_{2}$$ has been defined in (3.4). After substituting (3.25) into (3.24), condition (3.24) can be further strengthened to (3.26). V˙ ⩽x~T[(−KC)TM+M(−KC)+μMM+μI]x~ +2Υλmax(M)∥x~∥+4l22+l32λmax(Λ−1Λ−1), (3.26) where $$\Upsilon =\eta_{0} +\eta_{6}$$. To simplify writing, constant $$\Phi$$ is introduced: Φ=4l22+l32λmax(Λ−1Λ−1). (3.27) After substituting (3.12) and (3.27) into (3.26), the condition can be adjusted into the following form: V˙ ⩽−x~TNx~+2Υλmax(M)∥x~∥+Φ ⩽−λmin(N)∥x~∥2+2Υλmax(M)∥x~∥+Φ, (3.28) where $$\lambda_{\min } (N)$$ denotes the minimum eigenvalue of positive definite matrix $$N$$ that satisfies Theorem 1. Next, in order to complete the analysis of asymptotical stability, two cases should be examined. Case 1: $$\| \tilde{x} \| > \delta$$, where $$\delta =\frac{2\Upsilon \lambda_{\max } (M)+\sqrt {4\Upsilon ^{2}\lambda_{\max }^{2} (M)+4 \Phi \lambda_{\min } (N)} }{2\lambda_{\min } (N)}$$. In this case, $$\dot{V} <0$$ is established and (3.6) is a stable fault detection observer of nonlinear fault system (2.2). Case 2: $$\| \tilde{x} \| \leqslant \delta$$. In this case, $$\dot{V} >0$$ is established and fault detection observer (3.6) will be divergent. This will cause the increase of $$\vert \vert \tilde{x} \|$$, and let $$\| \tilde{x} \| >\delta$$, and then, it will satisfy Case 1, which brings $$\vert \vert \tilde{x} \|$$ to reduce, and so on, namely the deviation $$\| \tilde{x} \|$$ will remain at a certain tracking level range. Considering Cases 1 and 2 comprehensively, it is obvious that the fault detection observer is asymptotically stable, and the designed sensors’ fault diagnosis algorithm is able to remain on a satisfactory tracking accuracy for state variable $$x(t)$$ and time-varying fault $$f(t)$$. Remark 6 $$K$$ in Theorem 1 is a finite gain matrix of observer (3.6), which just needs to satisfy the condition that ensures $$(-KC)$$ is a stable matrix. For example, the following choice is reasonable: K=ωMCT, (3.29) where $$\omega >0$$ is an adjustable coefficient. 4. Experiments In order to fully evaluate the performance of the proposed fault diagnosis algorithm, a model of nonlinear system is selected and its parameters are presented as the following form: W(x)=[x2−0.5sin⁡(x1)]B=[1.5001]C=F=G=H=[1001]. (4.1) Obviously, $$W(x)$$ satisfies Lipschitz condition (3.1). The fault detection observer has been constructed, and its parameters are selected according to (3.12) with the following expressions: μ=1.0,M=K=[5.4−4.2−4.26.6],N=[45.8−50.4−50.460.2]. (4.2) 4.1. Experiment 1. The performance test in the absence of faults As mentioned above, if there are not faults that occur in sensors, fault detection observer (3.6) will become a standard state observer. Here, a special experiment has been conducted to evaluate the performance of the state observer. Assuming that there are not faults, namely $$f=[\bar{0}]$$, and the external time-varying disturbances are shown in (4.3). ξ=[0.1cos⁡(62.8⋅t)0.5sin⁡(62.8⋅t)]. (4.3) Sensors’ random measurement noise $$\eta (t)$$ is seen as a band-limited white noise with zero mean and its variance is equal to 1, the height of the power spectral density (PSD) is equal to 0.01, namely {PSD(n)=0.01mean(n)=0.0var(n)=1.0 . (4.4) Control inputs vector $$u$$ is selected as a constant vector with the following expression: u=[0.10.5]. (4.5) Initial state vector $$x$$ and estimation vector $$\hat{x}$$ are selected as follows: x=[10.020.0],x^=[0.00.0]. (4.6) The convergence factor matrix in adaptive fault estimation algorithm (3.9) is selected, respectively, as: Λ1=[0.17000.17]. (4.7) The simulation process lasts for 15 s. The results of state estimation are shown in Figs 1 and 2. The estimated faults are shown in Figs 3 and 4. Fig. 1. View largeDownload slide Estimation of $$x_{1}$$ in Experiment 1: (a) estimation of $$x_{1}$$ (b) enlarged view. Fig. 1. View largeDownload slide Estimation of $$x_{1}$$ in Experiment 1: (a) estimation of $$x_{1}$$ (b) enlarged view. Fig. 2. View largeDownload slide Estimation of $$x_{2}$$ in Experiment 1: (a) estimation of $$x_{2}$$ (b) enlarged view. Fig. 2. View largeDownload slide Estimation of $$x_{2}$$ in Experiment 1: (a) estimation of $$x_{2}$$ (b) enlarged view. Fig. 3. View largeDownload slide Estimation of fault 1 in Experiment 1. Fig. 3. View largeDownload slide Estimation of fault 1 in Experiment 1. Fig. 4. View largeDownload slide Estimation of fault 2 in Experiment 1. Fig. 4. View largeDownload slide Estimation of fault 2 in Experiment 1. 4.2. Experiment 2. The performance test with the presence of disturbances Next, the performance of fault estimation will be discussed in detail. Assume that there are external time-varying disturbances as shown in (4.3), sensors’ random measurement noise $$\eta (t)$$ as shown in (4.4), and internal time-varying faults as shown in (4.8) in the diagnosis process at the same time. The control input vector also uses (4.5); initial state vector $$x$$ and estimation vector $$\hat{x}$$ use (4.6). f=[0.32cos⁡(6.28⋅t)0.5sin⁡(6.28⋅t)]. (4.8) In order to fully compare and analyse the dynamic performance of estimation, the convergence factors in adaptive fault estimation algorithm (3.9) are selected as (4.7) and (4.9), respectively. Λ2=[0.051000.051]. (4.9) The effects of estimation based on $$\it{\Lambda_{1}}$$ are shown in Figs 5–8. The effects of estimation based on $$\it{\Lambda_{2}}$$ are shown in Figs 9–12. Fig. 5. View largeDownload slide Estimation of $$x_{1}$$ with $$\it{\Lambda_{1}}$$ in Experiment 2. Fig. 5. View largeDownload slide Estimation of $$x_{1}$$ with $$\it{\Lambda_{1}}$$ in Experiment 2. Fig. 6. View largeDownload slide Estimation of $$x_{2}$$ with $$\it{\Lambda_{1}}$$ in Experiment 2. Fig. 6. View largeDownload slide Estimation of $$x_{2}$$ with $$\it{\Lambda_{1}}$$ in Experiment 2. Fig. 7. View largeDownload slide Estimation of fault 1 with $$\it{\Lambda_{1}}$$ in Experiment 2: (a) estimation of fault 1 (b) enlarged view. Fig. 7. View largeDownload slide Estimation of fault 1 with $$\it{\Lambda_{1}}$$ in Experiment 2: (a) estimation of fault 1 (b) enlarged view. Fig. 8. View largeDownload slide Estimation of fault 2 with $$\it{\Lambda_{1}}$$ in Experiment 2: (a) estimation of fault 2 (b) enlarged view. Fig. 8. View largeDownload slide Estimation of fault 2 with $$\it{\Lambda_{1}}$$ in Experiment 2: (a) estimation of fault 2 (b) enlarged view. From Experiment 1 to Experiment 2, conclusions can be made as follows: (1) The performance of state estimation is satisfactory when there are not faults that act to the control system. The deviation signals of state estimation in stable state are small enough, namely $$\| \tilde{x} \| \approx 0$$. In addition, the rapidity of state estimation is excellent, both $$\hat{x}$$ and $$\hat{f}$$ are driven asymptotically to their measured or actual values within about 5 s (see Figs 1–4). This shows that the performance of state estimation is satisfactory in the absence of faults. (2) Compared with the actuators’ faults shown by Guo et al. (2015), the external disturbances and measurement noise that act to the sensors have a more obvious influence on the estimation performance. Through Experiment 2, it can be found: with the presence of external time-varying disturbances, sensors’ measurement noise and faults, the values of state estimation have obvious small-amplitude oscillation (see Figs 5, 6, 9 and 10). At this time, the estimated state variables will oscillate around the actual values. In addition, the means of the estimated values just represent the actual state variables. Fig. 9. View largeDownload slide Estimation of $$x_{1}$$ with $$\it{\Lambda_{2}}$$ in Experiment 2. Fig. 9. View largeDownload slide Estimation of $$x_{1}$$ with $$\it{\Lambda_{2}}$$ in Experiment 2. Fig. 10. View largeDownload slide Estimation of $$x_{2}$$ with $$\it{\Lambda_{2}}$$ in Experiment 2. Fig. 10. View largeDownload slide Estimation of $$x_{2}$$ with $$\it{\Lambda_{2}}$$ in Experiment 2. (3) Convergence factors have an obvious impact on the speediness of fault estimation. In Experiment 2, through the comparison of $$\it{\Lambda_{1}}$$ and $$\it{\Lambda_{2}}$$, it can be found: If $$\it{\Lambda_{1}}$$ is selected, the estimated faults come into steady oscillation in about the fifth second (see Figs 7 and 8). However, if $$\it{\Lambda_{2}}$$ are selected, the same effect will be postponed to about the 10th second (see Figs 11 and 12). Therefore, an appropriate increase of convergence factors can increase the rapidity of fault estimation. However, it is also found through experiments that too large values of convergence factors will cause the observer’s divergence or unacceptable overshoot, therefore, careful selection of the convergence factors is always required. Fig. 11. View largeDownload slide Estimation of fault 1 with $$\it{\Lambda_{2}}$$ in Experiment 2: (a) estimation of fault 1 (b) enlarged view. Fig. 11. View largeDownload slide Estimation of fault 1 with $$\it{\Lambda_{2}}$$ in Experiment 2: (a) estimation of fault 1 (b) enlarged view. Fig. 12. View largeDownload slide Estimation of fault 2 with $$\it{\Lambda_{2}}$$ in Experiment 2: (a) estimation of fault 2 (b) enlarged view. Fig. 12. View largeDownload slide Estimation of fault 2 with $$\it{\Lambda_{2}}$$ in Experiment 2: (a) estimation of fault 2 (b) enlarged view. Remark 7 Here, an analytical approach about how to select the optimal values of $$\it{\Lambda}$$ in (3.9) cannot be given. The optimal $$\it{\Lambda}$$need to be adjusted according to the simulation or actual test results. (4) The precision of fault estimation is also satisfactory and acceptable (see Figs 7 and 8). The amplitude and frequency of fault estimation are very close to those of the actual values, but there is a slight constant phase difference [see Figs 7(b) and 8(b)]. According to Theorem 1, the sensors’ fault diagnosis algorithm based on fault detection observer (3.6) and fault estimation algorithm (3.9) can only ensure that deviation signal $$\tilde{x}$$ and fault estimation error $$\tilde{f}$$ are driven asymptotically to bounded positive constants $$\tau_{1}$$ and $$\tau _{2}$$, not to zero. This is why the constant bias is generated, in other words, the proposed fault estimation algorithm is not an ‘unbiased estimator’. However, the bias is not too large and is within the acceptable range. 5. Conclusions A nonlinear fault detection observer and an adaptive fault estimation algorithm have been presented for the diagnosis of a class of nonlinear systems that contain sensors’ faults. Satisfactory experimental results have been achieved. The most important contribution of this article is to propose a novel diagnostic strategy that is able to give specific estimated values of state variables and time-varying faults when external disturbances, random measurement noise and internal faults exist simultaneously. The fault detection observer obtained via a Lyapunov function is proved to be useful in achieving asymptotic stability in the case with the presence of unknown bounded disturbances and measurement noise. Future works will be focused on three aspects. First, the selection of convergence factors $$\it{\Lambda}$$ in (3.9) will be considered. Future works will be dedicated to finding a standard approach of parameters’ selection, which makes the nonlinear system have faster estimation speed and smaller estimation errors. Second, the diagnosis for a class of nonlinear systems that contain both sensors’ faults and actuators’ faults would be further studied. How to deal with the problem about giving accurate estimated values of sensors’ faults and actuators’ faults, respectively, when they exist simultaneously still deserves further research. Third, the engineering implementation of the proposed algorithm needs to be paid much attention. This article only focuses on the theoretical research. In the future, we plan to apply this algorithm to the fault diagnosis of airborne sensors. There are plenty of on-board sensors on the aircraft, which provides a broad application background for the proposed algorithm. Acknowledgements We would like to express our gratitude. 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TI - Fault diagnosis for sensors in a class of nonlinear systems
JF - IMA Journal of Mathematical Control and Information
DO - 10.1093/imamci/dnw053
DA - 2016-10-17
UR - https://www.deepdyve.com/lp/oxford-university-press/fault-diagnosis-for-sensors-in-a-class-of-nonlinear-systems-CMDeTAwxvJ
SP - 1
EP - 391
VL - Advance Article
IS - 2
DP - DeepDyve
ER -