Sensor Fault Diagnosis for a Class of Time Delay Uncertain Nonlinear Systems Using Neural Network

Transcription

1 International Journal of Automation and Computing 05(4), October 2008, DOI: /s Sensor Fault Diagnosis for a Cls of Time Delay Uncertain Nonlinear Systems Using Neural Network Mou Chen Chang-Sheng Jiang Qing-Xian Wu College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing , PRC Abstract: In this paper, a sliding mode observer scheme of sensor fault diagnosis is proposed for a cls of time delay nonlinear systems with input uncertainty bed on neural network. The sensor fault and the system input uncertainty are sumed to be unknown but bounded. The radial bis function (RBF) neural network is used to approximate the sensor fault. Bed on the output of the RBF neural network, the sliding mode observer is presented. Using the Lyapunov method, a criterion for stability is given in terms of matrix inequality. Finally, an example is given for illustrating the availability of the fault diagnosis bed on the proposed sliding mode observer. Uncertain nonlinear system, time delay, radial bis function (RBF) neural network, sliding mode observer, fault diag- Keywords: nosis. 1 Introduction As is well known, there are a large number of research results about sensor fault detection and diagnosis of uncertain nonlinear systems due to their wide applicability in many practical systems [1 3]. Especially, the considerable attention h been paid to the problem of sensor fault detection and diagnosis for the time delay uncertain systems over the pt years [4 6]. Thisisbecausetime-delayisoftenource of instability and performance degradation in many control systems. Recently, sliding mode observer for different systems h been a very active field of research. The observer-bed control problem for a cls of linear delay-differential systems of neutral type w investigated by Park [7]. Wang et al. [8] studied the problem of observer design for a cls of uncertain linear systems with delayed state and parameter uncertainties. Raghavan and Hedrick [9] proposed a sliding mode observer for a cls of nonlinear systems. A new sliding mode observer for a cls of nonlinear uncertain systems w proposed in [10]. In [11], a linear matrix inequality (LMI) bed sliding-mode observer design method w proposed for a cls of multivariable uncertain systems. But the systems with input uncertainty were not considered in these sliding mode observer design methods. The sliding mode observer technology h been used to detect and reconstruct the faults in various fields. Koshkouei and Zinober [12] were concerned with fault diagnosis for a cls of nonlinear systems with time-varying uncertainties bed on robust/sliding mode observers. Two methods for detecting and constructing sensor faults using sliding mode observers were proposed in [13]. In [14], the application of a particular sliding mode observer to the problem of fault detection and isolation w studied. Accordingly, observerbed nonlinear tolerant control scheme w studied for spe- Manuscript received October 9, 2007; revised January 4, 2008 This work w supported by Natural Science Foundation of Jiangsu Province (No. SBK ) and Aeronautical Science Foundation of China (No ). *Corresponding author. address: chenmou@nuaa.edu.cn cific clses of nonlinear systems. However, most of these methods were restricted to faults in certain particular forms which limited the application of these control methods. Recently, the neural network h been successfully used in fault detection and diagnosis of uncertain nonlinear systems and there are many research results about them [15 20]. In [15], some serious challenges were presented to conventional neural-network design procedures in fault detection and diagnosis for some certain real-world applications. The problem of robust model-bed diagnosis of process faults w addressed by means of artificial neural networks in [16]. Bernieri and D Apuzzo [17] investigated the possibilities offered by neural networks for system identification and fault diagnosis problems in dynamic systems. The fault detection and diagnosis with neural network w used in many practical systems such induction motor rotor [18], industrial processes [19], and robotic systems [20]. Combining sliding mode observer technology with a neural network to detect and diagnose sensor faults of uncertain nonlinear systems is receiving more attention due to its advantages. In this paper, a sliding mode observer approach of fault detection and diagnosis with neural network is proposed for a cls of uncertain nonlinear systems with uncertain input. The structure of this paper is follows. Section 2 gives the fault detection and diagnosis problem formulation for a cls of time delay uncertain nonlinear systems. Section 3 describes the design of sliding mode observer bed on neural network. Section 4 studies stability analysis of fault detection and diagnosis bed on sliding mode observer. The simulation results are given in Section 5. Section 6 concludes this paper with some remarks. 2 Problem formulation Consider the time delay uncertain nonlinear systems in the form of ẋ(t) =Ax(t)+Bu(t)+A 1x(t τ)+h(x, u, t)+bd(t) y(t) =Cx(t) (1)

2 402 International Journal of Automation and Computing 05(4), October 2008 with the initial condition function x(t 0 + Θ) = Υ(Θ), Θ [ τ, 0] (2) where x(t) R n is the state vector, u(t) R m is the control input vector, and y(t) R q is the output vector; A R n n, A 1 R n n, B R n m,andc R q n are constant matrices; τ is a known positive constant timedelay; h(x, u, t) is a unknown nonlinear function vector; d(t) is the input uncertainty of the system; and Υ( ) isthegiven continuously differentiable function [ τ,0]. If a sensor fault occurs in system (1), then the model can be written ẋ(t)=ax(t)+bu(t)+a 1x(t τ)+h(x, u, t)+ f(x, u)+bd(t) y(t)=cx(t) (3) where f(x, u) denotes the unknown sensor fault. In this paper, we design a sliding mode observer to detect and diagnose sensor fault f(x, u) bed on radial bis function (RBF) neural network. To proceed with the design of the robust sliding mode observer for the nonlinear fault system, the following sumptions and lemm are required. Assumption 1. The pair (A, C) is an observable pair. Assumption 2. The unknown nonlinear function h(x, u, t) satisfies the Lipschitz condition. That is, with respect to x and Lipschitz constant κ, thereexists h(x 1,u,t) h(x 2,u,t) κ x 1 x 2. (4) Assumption 3. For t 0, there exists a known function ρ( ) R p and an unknown positive parameter vector θ R p such that the disturbance d(t) satisfies the following condition Bd(t) ρ T θ (5) where ρ( ) =[ρ 1( ),ρ 2( ),,ρ p( )] T,andθ =[θ1,θ 2,, θp] T. Assumption 4. Assume that X and Y are vectors or matrices with corresponding dimensions. By choosing a constant α>0, we can obtain the following inequality X T Y + Y T X αx T X + α 1 Y T Y. 3 Sliding mode observer design with RBF neural network The tk of this section is to design the sliding mode observer bed on the RBF neural network. In this paper, the robust sliding mode observer of system (3) can be designed ˆx(t) =Aˆx(t)+Bu(t)+A 1ˆx(t τ)+ν + L(y C ˆx(t))+ h(ˆx, u, t)+f(ˆx, u) (6) where ˆx(t) R n is the state observer vector, ν is an external discontinuous feedforward compensation signal, and L is an observer gain matrix which will be designed. In this paper, the fault f(ˆx, u) is unknown which is approximated by the RBF neural networks. Therefore, the approximation of f(ˆx, u) can be expressed ˆf(ˆx, u) =Ŵ T φ(ˆx) (7) where φ(ˆx) is the be function of the corresponding RBF neural networks. Ŵ =[Ŵ1, Ŵ2,, Ŵn] Rn n, φ(ˆx) = [ϕ 1(ˆx),ϕ 2(ˆx),,ϕ n(ˆx)] T R n 1, ϕ i(ˆx)=exp( ˆx c i 2 / δi 2 ), i =1, 2,,n,andc i and δ i are the center and width of the neural cell of the i-thhiddenlayer. The optimal weight value of RBF neural network is defined W = arg min [sup ˆf(ˆx/Ŵ ) f(ˆx, u) ] (8) Ŵ Ω f ˆx Sˆx where Ω f = {Ŵ : Ŵ M f } is a valid field of the parameter Ŵ, M f is a designed parameter, and Sˆx R n is a variable space of the observable state vector. Under the optimization weight value, f(ˆx, u) canbeexpressed f(ˆx, u) =W T φ(ˆx)+ε (9) where ε =[ε 1,ε 2,,ε n] T R n 1 is the smallest approximation error of RBF neural networks which is unknown. Suppose that ε ε (10) where ε > 0 is the unknown upper bound of the approximation error of f(ˆx, u) using the RBF neural network. The estimation error of the designed sliding mode observer is defined e = x ˆx. (11) Taking the time derivative of e yields ė(t) =A 0e(t)+A 1e(t τ) ν + h(x, u, t) h(ˆx, u, t)+ f(x, u) f(ˆx, u)+bd(t) (12) where A 0 = A LC. Use the RBF neural network to approximate the unknown sensor faults f(x, u) andf(ˆx, u). Then, according to (7) and (9), we can obtain f(x, u) f(ˆx, u) =W T φ(x, u)+ε W rmt φ(ˆx, u)+ W T φ(ˆx, u) Ŵ T φ(ˆx, u). (13) To determine W = W Ŵ, (13) can be written f(x, u) f(ˆx, u) = W T φ(ˆx, u)+w T φ(x, u) W T φ(ˆx, u)+ε. (14) Here, let us define Λ=W T φ(x, u) W T φ(ˆx, u). (15) Considering the expression of φ( ) and the definition of optimal weight value, we can obtain Λ η (16) where η>0 but is unknown. Substituting (14) and (15) into (12) yields ė(t) =(A LC)e(t)+A 1e(t τ) ν + h(x, u, t) (17) h(ˆx, u, t)+ W T φ(ˆx, u)+λ+ε + Bd(t) where L = P 1 C T,andP will be defined in the following section.

3 M. Chen et al. / Sensor Fault Diagnosis for a Cls of Time Delay Uncertain Nonlinear Systems Using Neural Network 403 The sliding mode can be designed P 1 C T Ce e T P (ρ T ˆθ +ˆη + ˆθ) if Ce 0 ν = Ce 2 0 otherwise (18) where ˆθ is the estimation value of θ which satisfies the following updated law: ˆθ =2 e T P Γρ σγˆθ (19) where σ>0, and Γ is a positive-definite constant matrix with corresponding dimension which is a design parameter of the parameter updated law. ˆη is the estimation value of η, which satisfies the following updated law: ˆη =2Γ 1 e T P σ 1Γ 1 ˆη (20) where Γ 1 and σ 1 are positive scalars. Similarly, ˆε is the estimation value of ε, which satisfies the following updated law: ˆε =2Γ 2 e T P σ 2Γ 2ˆε (21) where Γ 2 and σ 2 are positive scalars. The RBF neural weight value of updated law can be chosen where Ŵ i = e ip iϕ i(ˆx) Ŵi M f, e ip i Ŵ T i ϕ i(ˆx) > 0 P ri Ŵi = M f, e ip i Ŵ T i ϕ i(ˆx) 0 (22) e ip i Ŵ i Ŵi T ϕ i(ˆx) P ri [ ] = e ip iϕ i(ˆx)+ (23) Ŵi 2 where > 0 is the learning rate, and P i is the i-th column of P, respectively. 4 Stability analysis of fault detection and diagnosis bed on sliding mode observer Now, we consider the stability of the system which uses the sliding mode observer to detect and diagnose sensor fault when the sensor fault appears. The properties of sensor fault diagnosis using the sliding model observer bed on above design procedure can be summarized in the following theorem. Theorem 1. If there exist a positive-definite matrix P and the positive scalar parameters α, κ, andβ satisfying the following matrix inequality A T P + PA+ βi +2α 1 κ 2 I 2C T C PA 1 A T 1 P βi 2αP 0 2αP 0 αi < 0 (24) then the observer error is bounded. Namely, the sliding mode observer bed on RBF neural network can well detect and diagnose sensor fault when the sensor fault appears. Proof. The Lyapunov function is defined by V = e T Pe+ 1 2 θ T Γ 1 1 n θ + W i + W i T T β e T (s)e(s)ds + 1 t τ 2 Γ 1 1 η Γ 1 2 ε 2 (25) where θ = ˆθ θ, η =ˆη η, and ε =ˆε ε. Considering (12) and (17), the time derivative of V satisfies V =e T (PA+ A T P 2C T C)e +2e T PA 1e(t τ)+ 2e T P [h(x, u, t) h(ˆx, u, t)+bd(t) ν]+ 2e T P [f(x, u) f(ˆx, u)+ θγ 1 θ n 1 +2 T W i Wi+ βe T (t)e(t) 2 ε ε e T (PA+ A T P 2C T C)e +2e T PA 1e(t τ)+ 2e T P W T φ(ˆx)+2 e T P Bd(t) 2e T Pν+ 2e T P [h(x, u, t) h(ˆx, u, t)] + 2e T P (Λ + ε)+ θγ 1 θ n 1 +2 T W Wi + βe T (t)e(t) i 2 ε ε. (26) Considering Assumption 3, we have 2 e T P Bd 2 e T P ρ T θ. (27) Substituting (27) into (26), we can obtain V e T (PA+ A T P 2C T C)e +2e T PA 1e(t τ)+ 2e T P W T φ(ˆx)+2 e T P ρ T θ 2e T Pν+ 2e T P [h(x, u, t) h(ˆx, u, t)] + 2e T P (Λ + ε)+ θγ 1 θ n 1 +2 W i T W i + βe T (t)e(t) 2 ε ε. (28) Substituting (18), (19), and (22) into (28) yields V e T (PA+ A T P 2C T C)e +2e T PA 1e(t τ)+ 2 e T P ρ T θ 2 e T P ρ T ˆθ 2 e T P ˆη 2 e T P ˆε +2 e T P ρ T θ+ 2e T P [h(x, u, t) h(ˆx, u, t)] + 2e T P (Λ + ε)+βe T (t)e(t) 2 ε ε σ θ T ˆθ. (29) It is clear that 2 e T P ρ T θ +2 e T P ρ T θ =2 e T P ρ T ˆθ. (30) Then, (29) can be rewritten V e T (PA+ A T P 2C T C)e +2e T PA 1e(t τ) 2 e T P ˆη 2 e T P ˆε σ θ T ˆθ +2e T P (Λ + ε)+ 2e T P [h(x, u, t) h(ˆx, u, t)] + βe T (t)e(t) 2 ε ε. (31)

4 404 International Journal of Automation and Computing 05(4), October 2008 According to Lemma 1, we can obtain e T P Λ+Λ T Pe 2 e T P η (32) e T Pε+ ε T Pe 2 e T P ε (33) e T P (h(x, u, t) h(ˆx, u, t)) + (h(x, u, t) h(ˆx, u, t)) T Pe αe T PPe+ α 1 (h(x, u, t) h(ˆx, u, t)) 2 = αe T PPe+ α 1 κ 2 e 2 (34) where α>0. Considering E =[e, e(t τ)] T and (32) (34), (31) can be written V 2 e T P η +2 e T P ε 2 e T P ˆη 2 e T P ˆε + E T ME +Γ 1 2 ε ε σ θ T ˆθ (35) where M is [ M = A T P + PA+ βi +2αP P 2C T C PA 1 A T 1 P βi ] ] T, [ 0.3x 2sin(x 1) d = 0.5e 0.1t 0.2e ], 0.3t h = [ ] T. 0.1sint 0.1cost h is treated an unknown nonlinear function, d is treated input uncertainty of the system, and f is treated system sensor fault which uses RBF neural network for detection and diagnosis. According to (6), the sliding mode observer is designed. The external discontinuous feedforward compensation signal ν, the updated law of ˆθ, the updated law of η, the updated law of ˆε, andtheupdatedlawofŵ can be designed according to (18), (19), (20), (21), and (22). Choosing α = 10, β =2.0, κ =0.1, σ = σ 1 = σ 2 =0.1, =0.1, and u =[sin(10t), cos(10t)] T, the positive-definite matrix P can be obtained bed on (24). The simulation results are showninfigs.1and2. Considering (20) and (21), we have V E T ME σ θ T ˆθ σ1 η T ˆη σ 2 ε Tˆε. (36) There are the following facts σ θ T ˆθ σ θt θ + θ θ σ 2 θ 2 + σ 2 θ 2 (37) σ 1 η T ˆη σ 1 η T η + η η σ1 2 η 2 + σ1 2 η 2 (38) σ 2 ε Tˆε σ 2 ε T ε + ε ε σ2 2 ε 2 + σ2 2 ε 2 (39) From (24), we can obtain M < 0. When E T ME > σ 2 θ 2 + σ 1 2 η 2 + σ 2 2 ε 2 and considering (37) (40), we have V < 0 Fig. 1 Fault diagnosis error The above inequality shows that the sliding mode observer error is convergent. Apparently, choosing appropriate design parameters κ, β, σ, σ 1, σ 2,andα can improve the dynamic performance of the system. Namely, the sliding mode observer scheme bed on a neural network can perfectly detect and diagnose sensor faults. 5 Simulation example Consider a time delay uncertain nonlinear system in the form of ẋ(t)=ax(t)+bu(t)+a 1x(t τ)+h(x, u, t)+ f(x, u)+bd(t) y(t)=cx(t) [ ] [ ] where A =, A 1 =, B = [ ] [ ] [, C = 1 1, f = 0.1x 1sin(x 2) Fig. 2 Response of sliding mode observer From these simulation results, we can see that the responses of the sliding mode observer are uniformly ultimately bounded. So the proposed fault detection and diagnosis scheme bed on the sliding mode observer and neural network is effective for the uncertain time delay nonlinear system. 6 Conclusions This paper used the RBF neural network to design a sliding mode observer which is used to detect and diagnose sensor faults. This kind of nonlinear systems h uncertain

5 M. Chen et al. / Sensor Fault Diagnosis for a Cls of Time Delay Uncertain Nonlinear Systems Using Neural Network 405 nonlinear terms and input uncertainty with unknown functional bounds. The sliding mode observer is therefore required to handle such problems, and can generate an alarm signal when a sensor fault occurs and produce an estimate of the faulty behavior. The stability analysis is given using the Lyapunov method bed on the LMI method. When the observer errors satisfy some conditions, this can convergence of the designed observer can be guaranteed. Simulation results have shown the feibility and effectiveness of the proposed scheme. The main contribution of this paper is that we have introduced the input uncertainty to time delay nonlinear system, and we combined sliding mode observer technology with a neural network to detect and diagnose sensor faults of uncertain nonlinear systems to improve the results. References [1] S. Simani, C. Fantuzzi, S. Beghelli. Diagnosis Techniques for Sensor Faults of Industrial Processes. IEEE Transactions on Control Systems Technology, vol. 8, no. 5, pp , [2] P. M. Frank. Fault Detection in Dynamic Systems Using Analytical and Knowledge-bed Redundancy A Survey and Some New Results. Automatica, vol. 26, no. 3, pp , [3] J. J. Gertler. Survey of Model-bed Failure Detection and Isolation in Complex Plants. IEEE Control Systems Magazine, vol. 8, no. 6, pp. 3 11, [4] S. S. Ge, H. Fan, T. H. Lee. Adaptive Neural Network Control of Nonlinear Systems with Unknown Time Delays. IEEE Transactions on Automatic Control, vol. 48, no. 11, pp , [5] F. H. Hsiao, J. D. Hwang. Stabilization of Nonlinear Singularly Perturbed Multiple Time-delay Systems by Dither. Journal of Dynamic Systems, Meurement, and Control, vol. 118, no. 1, pp , [6] S. K. Nguang. Robust Stabilization of a Cls of Time-delay Nonlinear Systems. IEEE Transactions on Automatic Control, vol. 45, no. 4, pp , [7] J. H. Park. On the Design of Observer-bed Controller of Linear Neutral Delay-differential Systems. Applied Mathematics and Computation, no. 150, no. 1, pp , [8] Z. Wang, B. Huang, H. Unbehauen. Robust H Observer Design of Linear Time-delay Systems with Parametric Uncertainty. Systems & Control Letters, vol. 42, no. 4, pp , [9] S. Raghavan, J. K. Hedrick. Observer Design for a Cls of Nonlinear Systems. International Journal of Control, vol. 59, no. 2, pp , [10] Y. Xiong, M. Saif. Sliding Mode Observer for Nonlinear Uncertain Systems. IEEE Transactions on Automatic Control, vol. 46, no. 12, pp , [11] H. H. Choi, K. S. Ro. LMI-bed Sliding-mode Observer Design Method. IEE Proceedings of Control Theory and Application, vol. 152, no. 1, pp , [12] B. Jiang, M. Staroswiecki, V. Cocquempot. Fault Estimation in Nonlinear Uncertain Systems Using Robust/Slidingmode Observers. IEE Proceedings of Control Theory and Application, vol. 151, no. 1, pp , [13] P. T. Chee, E. Christopher. Sliding Mode Observers for Detection and Reconstruction of Sensor Faults. Automatica, vol. 38, no. 10, pp , [14] E. Christopher, S. K. Spurgeon, R. J. Patton. Sliding Mode Observers for Fault Detection and Isolation. Automatica, vol. 36, no. 4, pp , [15] C. Rodriguez, S. Rementeria, J. I. Martin, A. Lafuente, J. Muguerza, J. Perez. A Modular Neural Network Approach to Fault Diagnosis. IEEE Transactions on Neural Networks, vol. 7, no. 2, pp , [16] T. Marcu, L. Mirea. Robust Detection and Isolation of Process Faults Using Neural Networks. IEEE Control Systems Magazine, vol. 17, no. 5, pp , [17] A. Bernieri, M. D Apuzzo, L. Sansone, M. Savtano. A Neural Network Approach for Identification and Fault Diagnosis on Dynamic Systems. IEEE Transactions on Instrumentation and Meurement, vol. 43, no. 6, pp , [18] F. Filippetti, G. Franceschini, C. Tsoni. Neural Networks Aided On-line Diagnostics of Induction Motor Rotor Faults. IEEE Transactions on Industry Applications, vol. 31, no. 4, pp , [19] Y. Maki, K. A. Loparo. A Neural-network Approach to Fault Detection and Diagnosis in Industrial Processes. IEEE Transactions on Control Systems Technology, vol.5, no. 6, pp , [20] A. T. Vemuri, M. M. Polycarpou. Neural-network-bed Robust Fault Diagnosis in Robotic Systems. IEEE Transactions on Neural Networks, vol. 8, no. 6, pp , Mou Chen received the B. Sc. degree in material science and engineering at Nanjing Nanjing, PRC, in 1998, the M. Sc. and the Ph. D. degrees in automatical control engineering at Nanjing University of Aeronautics & Astronautics, Nanjing, PRC, in He is currently an sociate professor in Automation College at Nanjing University of Aeronautics & Astronautics, PRC. His research interests include nonlinear control, artificial intelligence, imagine processing and pattern recognition, and flight control. Chang-Sheng Jiang received his B. Sc. and M. Sc. degrees in automatic control engineering at Nanjing University of Aeronautics & Astronautics, Nanjing, PRC, in 1964 and 1968, respectively. He is currently a professor in Automation College at Nanjing China. His research interests include nonlinear control, artificial intelligence, imagine processing and pattern recognition, and flight control. control. Qing-Xian Wu received his B. Sc. and M. Sc. degrees in automatical control engineering at Southet University, PRC, in 1982 and 1985, respectively. He is currently a professor in Automation College at Nanjing PRC. His research interests include nonlinear control, artificial intelligence, imagine processing and pattern recognition, and flight