H fault detection for a class of T-S fuzzy model-based nonlinear networked control systems

Transcription

1 Preprints of the 9th World ongress he International Federation of Automatic ontrol H fault detection for a class of -S fuzzy model-based nonlinear networked control systems Weiguang Ding Zehui Mao Bin Jiang Peng Shi Xiao He ollege of Automation Engineering Nanjing University of Aeronautics and Astronautics Nanjing 26 hina ( dwg sd@nuaa.edu.cn (W. Ding) zehuimao@nuaa.edu.cn (Z. Mao corresponding author) binjiang@nuaa.edu.cn (B. Jiang)) School of Electrical and ollege of Engineering and Science Victoria University Melbourne VI 8 Australia ( peng.shi@adelaide.edu.au) Department of Automation NListsinghua University Beijing 84 P. R. hina ( hexiao@tsinghua.edu.cn) Abstract: In this paper a robust H fault detection filter is designed for a class of discretetime nonlinear networked control systems via -S fuzzy model with multiple bounded state delay and random packet dropout induced by the limited bandwidth of communication networks. Our aim in this paper is to analyze and design a robust H full-order fault detection filter such that the filtering error dynamics is asymptotically mean-square stable with a prescribed H performance level. Sufficient conditions for the existence of the desired filter are presented. Finally an example is given to illustrate the effectiveness and applicability of the proposed new design method. Keywords: Fault detection networked control systems fuzzy systems.. INRODUION Fault detection (FD) has attracted researchers attention over the past decades because of the increasing demand for high safety and high reliability in many industrial processes. Fruitful results can be found in several books hen et al. (999) Ding X. (28) and excellent papers Jiang et al. (26) Zhong et al. (2). Recently with the rapid developments in network technologies the interests on FD of networked control systems (NSs) are increasing. Networked control systems (NSs) are closed-loop feedback control systems sensor-controller and controlleractuator signal link is through a shared network. ompared with conventional systems NSs have great advantages such as low cost reduced weight and increased system agility which leads to the applications of NSs for many fields ranging from D motors advanced aircraft to manufacturing process. However the introduction of communication networks also brings communication constraints to the control systems e.g. network-induced delays and packet dropouts which all might be potential causes to poor performance instability and even faults. herefore many excellent results have been carried out for NSs with time delays and data drops. For exam- Supported by National Natural Science Foundation of hina( ) the Doctoral Program of Higher Education(23282) the Australian Research ouncil (D- P428) the National Key Basic Research Program hina (2B776 22B2522) and the Project (B28). ples Dong et al. (2) develops a robust H filter for NSs with multiple stochastic communication delays and packet dropouts under discrete-time model. Gao et al. (28) presents a new approach to solve the problem of stabilization for networked control systems. Yue et al. (29) concerns the design of stabilization controllers for linear systems with stochastic input delays. he abovementioned works focus on the design of controller observer and filter. However it should be noted that faults in actuators sensors and other components existing in NSs are similar to other control systems. hus it is important to study FD schemes for these systems to avoid the failures. here exist some results to design FD for NSs. Zhang et al. (2) designs a robust FD filter for NSs with delay distribution characterization under discrete-time model. Wang et al. (29) proposes a fault detection approach for NSs with communication constrains. Mao et al. (27) develops the scheme of robust FD for networked control systems with large transfer delays. Unfortunately the most research works on FD for NSs are reported for linear systems and only limited results are concerned on nonlinear systems. In recent years many fuzzy control approaches and fault diagnosis schemes have been developed for nonlinear systems because of its approximate performance (e.g. see Su et al. (22) Zhou et al. (23) Jiang et al. (2)). Zhang et al. (27) designs the fuzzy filter for signal estimation of nonlinear discrete-time systems with multiple time delays and unknown bounded disturbances. ong et al. opyright 24 IFA 647

2 (23) concerns the problem of adaptive fuzzy tracking control for a class of multi-input and multi-output strictfeedback nonlinear systems with both unknown nonsymmetric dead-zone inputs and immeasurable states. Since nonlinear systems can be approximated by -S fuzzy models an applicable design scheme of the fault detectors for nonlinear systems can be transformed into the fault detection problem for -S fuzzy systems. On the other hand preserving some nonlinearities in -S fuzzy systems can make the systems close to the reality. hus in this paper the nonlinear NSs via -S fuzzy model is taken into account. In this paper the fault detection problem for a discretetime nonlinear NSs with time-varying state delays and random data dropouts is considered. Based on Lyapunov functions sufficient conditions for FD filter design are established. he designed FD filter can detect the fault with the limited information successfully. he paper is organized as follows. Section 2 describes the system model and presents some assumptions and definitions. he H FD filter design method based on Lyapunov functions is derived in section 3. A simulation example is included in section 4 followed by some concluding remarks in section PROBLEM SAEMEN In this paper we consider the following class of discretetime fuzzy systems with sector-bounded nonlinearity: Plant Rule i: IF θ (k) is F i and and θ p (k) is F ip HEN x k+ = A i + A 2i x k + A di x k τk + B i w k + E i f k z k = i + 2i x k + di x k τk + D i w k y k = i x k + D i w k () x k = φ k k = τ u τ u +... i =... r F ij is the fuzzy set r denotes the number of IF- HEN rules θ k = θ (k) θ 2 (k)... θ p (k) is the premise variable vector. x k R n is the state vector; z k R n is the signal to be estimated; y k R n2 is the measured output vector; w k R n w is the unknown disturbance belonging to l 2 ) and f k is the fault vector. A i A 2i A di B i i i 2i di D i D i and E i are all constant matrices with appropriate dimensions. τ k is the time-varying state delay with lower and upper bounds τ l τ τ u τ l and τ u are constant positive scalars. φ k is a given real initial sequence on τ u. is the nonlinear function satisfying the following assumption. Assumption : he nonlinear function g( ) in system () satisfies F x k F 2 x k x k R n (2) F R n n and F 2 R n n are known real constant matrices and F = F F 2 is a symmetric positive definite matrix. According to (2) for any scalar ε > it has xk ε Z F Z Z F2 I xk (3) F = (F F 2 + F2 F )/2 F2 = (F + F2 )/2 and Z = I. onsider the measurement signal with random data dropouts as the following form y k = λ k i x k + D i w k (4) the stochastic variable λ k distributes in the known interval α β ( α β ) with its mathematical expectation E{λ k } = ρ and variance ov{λ k } = σ 2. α β ρ and σ are known real scalars. Remark : Here system (4) is used to describe the random data dropouts induced by the limited bandwidth as He et al. (29) which is usually used to describe the characters of net. When α = β = the data is transferred successfully. While α = β = the data is loss at all. he overall fuzzy model can be inferred as follows: x k+ = µ i (θ k )A i + A 2i x k + A di x k τk +B i w k + E i f k z k = µ i (θ k ) i + 2i x k + di x k τk +D i w k y k = µ i (θ k )λ k i x k + D i w k the fuzzy basis functions are given by η i (θ k ) µ i (θ k ) = r η i(θ k ) η i(θ k ) = Π p j= F ij(θ j (k)) F ij (θ j (k)) represents the grade of membership value of θ j (k) in F ij. herefore µ i (θ k ) has the following basic property: r η i (θ k ) i = 2... r η i(θ k ) k r µ i (θ k ) i = 2... r µ i(θ k ) = k. For the purpose of residual generation the following filter can be constructed: ˆx k+ = µ j (θ k )G j ˆx k + K j y k j= (6) ẑ k = µ j (θ k )L j ˆx k j= ˆx k is the observer state vector ẑ k is an estimation for z k. G j K j and L j are the parameters to be determined. By defining x k = x k ˆx k z k = z k ẑ k and d k = fk wk the filter-error dynamic system can be obtained: (5) x k+ = à + (λ k ρ)ā xk + ÃdZ x k τk + B + B d k (7) z k = x k + d Z x k τk + D + D d k à = j= A2i A2 ρk j i G j ρk G 648

3 Ā = Ã d = B = B = = d = D = D = j= j= j= j= j= j= j= j= K j i Adi Ad Ai A K Ei B i E B K j D i KD 2i L j 2 L di d D i D i. onsidering the existence of the stochastic variable λ k we recall the definition of stochastic stability in the meansquare sense for the filter-error system. Definition Mao X. (997): he FD filter-error dynamic system (7) is said to be asymptotically mean-square stable if with d k = for any initial conditions lim E{ x k 2 } = (8) k Due to the existence of the random variable λ k in the rest of the paper our aim is to design a robust full-order FD filter (6) for system (5). For all disturbance d k and the random variable λ k determine the FD filter-gain matrix G j K j and L j such that FD filter-error system satisfies both the following requirements: ) he FD filter-error system (7) is asymptotically meansquare stable. 2) Under the zero-initial conditions the FD filer-error z k satisfies E{ z k 2 } < γ 2 k= k= E{ d k 2 } (9) for all nonzero d k γ > is a prescribed scalar. o detect the system faults the residual evaluation function is selected as J(k) = { k+l e s=k r s r s } /2 J th = sup J(k) () w k l 2 F k = k is the initial evolution time instant; L e is the evolution time steps and J th is the threshold. Based on this the occurrence of faults can be detected by { J(k) < Jth no fault occurs J(k) J th alarm for fault () From () it can be seen that the threshold J th is proportional to the performance index γ. For the no fault case γ determines the influence of the disturbance to FD filter. he smaller γ is the better of the robustness of FD filter to the disturbance will be. 3. FD FILER DESGIN In this section we aim to develop an innovative approach to guarantee the FD filter-error system (7) satisfying the requirements ) and 2). he following theorem provides a sufficient condition for the existence of a full-order FD filter for system (5). heorem. onsider system (5) with a full-order FD filter of the form (6). Given a prescribed H attenuation level γ >. If there exist matrixes P = P > Q = Q > and a constant scalar ε > such that the following inequality holds: Ω Ω 2 Ω 3 Ω 4 Ω Ω = 22 Ω 23 Ω 24 Ω 33 Ω < (2) 34 Ω 44 Ω = Ã P Ã + σ 2 Ā P Ā P εz F Z + + νz QZ Ω 2 = Ã P Ãd + d Ω 3 = Ã P B εz F2 + D Ω 4 = Ã P B + D Ω 22 = Ã d P Ãd Q + d d Ω 23 = Ã d P B + d D Ω 24 = Ã d P B + d D Ω 33 = B P B εi + D D Ω 34 = B P B + D D Ω 44 = B P B γ 2 I + D D ν = τ u τ l + then the FD filter-error system (7) is asymptotically meansquare stable with the prescribed H attenuation level bound γ given in (9). Proof. onsider the following Lyapunov functions: V ( x k ) = V ( x k ) + V 2 ( x k ) + V 3 ( x k ) V ( x k ) = x k P x k V 2 ( x k ) = V 3 ( x k ) = k τ l j=k τ u + i=j k i=k τ k x i Z QZ x i k x i Z QZ x i (3) P = P > and Q = Q >. From the characters of the random variable λ k it has: E{λ k ρ} = E{λ k ρ 2 } = σ 2 (4) For d k = the difference of each Lyapunov functions can be calculated from (7): E{ V } = E{V ( x k+ )} V ( x k ) 649

4 = x k Z x k τk M M 2 M 3 M 4 M 5 M 6 E{ V 2 } = E{V 2 ( x k+ )} V 2 ( x k ) x k Z x k τk E { x k Z QZ x k x k τ k Z QZ x k τk } k τ l + x i Z QZ x i i=k+ τ u E{ V 3 } = E{V 3 ( x k+ )} V 3 ( x k ) = E { (τ u τ l ) x k Z QZ x k } k τ l i=k+ τ u x i Z QZ x i M = à P à + σ 2 Ā P Ā P M 2 = à P Ãd M 3 = à P B M 4 = à d P Ãd M 5 = à d P B and M 6 = B P B. Further we obtain that E{ V } = E{ V } + E{ V 2 } + E{ V 3 } M M 2 M 3 ζk M4 M 5 ζk (5) M 6 ζ k = x k x k τ k Z g (x k ) M = à P à + σ 2 Ā P Ā P + νz QZ M4 = à d P Ãd Q. Substituting (3) into (5) one can obtain E{ V ( x k )} E{ V ( x k )} { } xk Z F Z Z E ε F2 xk I M εz F Z M 2 M 3 εz F2 = ζk M4 M 5 ζ k M 6 εi ζk Ξζ k (6) Using heorem one can verify that d D Ξ < d d d D < (7) D D hus for all d k = we have E{ V ( x k )} < that is system (7) is asymptotically stable in the mean-square sense. Next for any nonzero d k it follows from (7) (2) and (6) that E{ V ( x k )} + E{ z k z k } γ 2 E{d k d k } E { ξk } Ωξ k < (8) ξ k x k x k τ k Z g (x k ) d k. Now summing up this relationship from to with respect to k yields E{ z k z k } < γ 2 E{d k d k } E{V } + E{V }(9) k= k= Since the system (7) is asymptotically mean-square stable it is obvious that (9) holds under the zero initial condition. his concludes the proof. heorem provides a sufficient condition for the design of the FD filter which contains the coupled matrix variables. o overcome the difficulties caused by the coupling in (2) a decoupling techniques as Gao et al. (25) is used by which (2) can be transformed into a new form. orollary. onsider system (5) with a full-order FD filter of the form (6). Given a prescribed scalar γ > if there exist matrixes P = P > Q = Q > F > K Ḡ L and a constant scalar ε > such that the following LMI holds: Π Π 3 Π 5 Π 6 Π 7 Q d Π 26 εi D Π 36 γ 2 I D Π 46 < (2) I Π 66 Π 77 νq P ε F F ε F2 Π = Π F F 3 = Π 5 = 2 L Π 7 = σ K K A Π 6 = 2 P + ρ K A 2 F + ρ K Ḡ Ḡ Π 26 = A P d A d F Π 36 = A P A F E Π 46 = P E F B P + D K B P + D K P F Π 66 = Π 77 =. F F Proof. Suppose condition (2) holds for matrices P = P > Q = Q > à Ā à d B B D D and Z = I. Using Schur complement Boyd et al. (997) one can show that (2) is equivalent to Π εz F2 à P σā P Q d à d P εi D B P γ 2 I D B P < (2) I P P Π = νz QZ P εz F Z. Let the matrix P be P S partitioned as P = S W P > W > and S I is invertible. Define the invertible matrix J = S W. Performing congruence transformations to (2) by diag{j I I I I J J } yields (2) with changes of variables as: F = S W S K = SK Ḡ = SG W S L = L W S. his ends the proof. 65

5 With the aid of the above corollary we are now in a position to present the filter gain design algorithm. heorem 2. Given system (5) with a full-order H FD filter in the form of filter (6) exists if there exist matrices P = P > Q = Q > F > Kj Ḡ j Lj and a constant scalar ε > such that the following LMIs holds for a given scalar γ > : Φ ij + Φ ji < i j (22) Γ Γ 3 Γ 5 Γ 6 Γ 7 Q di Γ 26 εi i Γ 36 Φ ij = γ 2 I Γ 45 Γ 46 I Γ 66 Γ 77 νq P ε F F Γ = Γ F F 3 = Γ 5 = 2i L Γ 7 = σ i K j i j A Γ 6 = 2i P + ρ i K j A 2iF + ρi K j Ḡ j Ḡ j ε F2 K j Γ 26 = A P di A dif Γ 36 = A P i A if Ei P E i F Γ 46 = Bi P + Di K j Bi P + Di K j P F Γ 45 = Γ 66 = Γ 77 = F F D i In this case the FD filter parameters in filter (6) are given by K j = F Kj G j = F Ḡ j L j = L j. (23) Proof. Set (k) = r µ i i denotes matrices A A 2 A d B 2 d D D K Ḡ or L. For simplicity denote the matrix on the left side of inequality (2) by Φ(k). hen from (22) one can obtain that Φ(k) = = µ i µ j Φ ij j= µ 2 i Φ ii + µ i µ j (Φ ij + Φ ji ) < (24) i<j From orollary the H FD filter design problem has been solved and the parameter matrix functions are given by K = S K G = S Ḡ S W L = L S W W > S is invertible and F = S W S. Under the transformation S W xk the FD filter matrix functions can be of the following form: K = S W ( S K) = F K G = S W ( S Ḡ S W ) W S = F Ḡ L = ( L S W ) W S = L. (25) Hence the parameter matrices in filter (6) are given by (23). he proof is completed. Remark 2: Since the mean value and the variance of the random variable λ k are known the parameters of FD filter can be solved through heorem and heorem 2. For the sensors we just consider the same rate of the data drops. It is well-known that each channel has its own drop rate i.e. the λ k for each sensor is different. he obtained results can be extend to the case of the different drop rate. Remark 3: he H fault detection filter design for nonlinear networked control systems can be readily found by solving the following problem: Minimize γ 2 subject to (22) and (23) which can be solved by the LMI tool box in MALAB. 4. ILLUSRAIVE EXAMPLE In this section we give an illustrative example to demonstrate the effectiveness of presented method. onsider a -S fuzzy system with three plant rules (r = 3) whose membership functions are shown in Figure. he param- mf mf2 rule mf3 rule2 rule3.8 Degree of membership X Fig.. Membership functions of the -S fuzzy system eters of the system are given as follows:.3.5 A = A.33 2 = A d = E..2 = B = 2 =.2 2 =.2.5 D =.5 = A 2 = A.3 22 = B = 2 A d2 =..2.3 E 2 = 2 =. 22 =.3.4 D 2 =.5 2 = A 3 = A = B = 2..2 A d3 = E.2 3 = 3 =.5 23 = D 3 =.5 3 = D = D 2 = D 3 = d = d2 = d3 = =.5sin(x k ) Suppose that the probabilistic variable λ k meets uniform distribution in interval.5 from which the expectation and variance can be easily calculated as ρ =.75 σ = 65

6 .44. he unknown disturbance w k is assumed to be white noise with power of.5. he fault signal f k is simulated as that occurs from to 2 steps. Select γ =.98. Using heorem 2 the H filtering parameters can be calculated through the LMI toolbox: K =.44 G.79 = L = K 2 = G.28 2 = L 2 = K 3 = G.35 3 = L 3 = Simulation results with the initial conditions x k = ˆx k = are shown in Figure 2 and Figure 3. From Figure ime (k) 2 3 Fig. 2. Residual estimation based on common Lyapunv function Residual Evolution Faulty Fault Free ime (k) 2 3 Fig. 3. Residual evolution based on common Lyapunov function 3 it can be seen that the residual evolution increases quickly when the fault f k. hoosing appropriate J th the fault f k can be detected as soon as its occurrence. 5. ONLUSION he problem of FD for a class of nonlinear NSs with probabilistic data dropout and time-varying delay has been considered in this paper. Based on Lyapunov function approach sufficient conditions for the FD filter design are established in terms of a set of LMIs which guarantees the mean-square stability of the FD filtering-error systems as well as prescribed H performance requirement. By solving a LMIs we obtain the FD filter parameters. In the future the work about the fault diagnosis and faulttolerant control for more general nonlinear models will be developed. REFERENES S. Boyd L. E. Ghaoui E. Feron and V. Balakrishnan. Linear matrix inequalities in system and control therory. Proceedings of the IEEE 85: J. hen and R. J. Patton. Robust model-based fault diagnosis for dynamic systems. Kluwer Academic Publishers Boston 999. H. Dong Z. Wang and H. Gao. Robust H filtering for a class of nonlinear networked systems with multiple stochastic communication delays and packet dropouts. IEEE ransactions on Signal Processing 58: X. Ding. Model-based fault diagnosis techniques: design schemes algorithms and tools. Springer 28. H. Gao J. Lam P. Shi and. Wang. Parameterdependent filter design with guranteed H performance. IEE Proceedings-ontrol heory and Applications 52: H. Gao X. Meng and. hen. Stabilization of networked control systems with a new delay characterization. IEEE ransactions on Automatic ontrol 53: X. He Z. Wang and D. Zhou. Robust H filtering for time-delay systems with probabilistic sensor faults. IEEE Singal Processing Letters 6: B. Jiang Z. Mao and P. Shi. H -filter design for a class of networked control systems via S fuzzy-model approach. IEEE ransactions on Fuzzy Systems 8: B. Jiang M. Staroswiecki and V. ocquempot. Fault accommodation for a class of nonlinear dynamic systems. IEEE ransactions on Automatic ontrol 5: X. Mao. Stochastic differential equations and applications. Horwood Pubishing: hichester 997. Z. Mao B. Jiang and P. Shi. H fault detection filter design for networked control systems modeled by discrete Markovian jump systems. IE ontrol heory and Applications : X. Su P. Shi L. Wu and Y. Song. A novel control design on discrete-time akagi-sugeno fuzzy systems with time-varying delays. IEEE ransactions on Fuzzy Systems 2: S. ong and Y. Li. Adaptive fuzzy output feedback control of MIMO nonlinear systems with unknown dead-zone inputs. IEEE ransactions on Fuzzy Systems 2: Y. Wang X. Ding H. Ye L. Wei P. Zhang and G. Wang. Fault detection of networked control systems with packet based periodic communication. International Journal of Adaptive ontrol and Signal Processing 23: D. Yue E. ian Z. Wang and J. Lam. Stabilization of systems with probabilistic interval input delays and its applications to networked control systems. IEEE ransactions on Systems Man and ybernetics Part A: Systems and Humans 39: H. Zhang S. Lun and D. Liu. Fuzzy H filter design for a class of nonlinear discrete-time systems with multiple time delays. IEEE ransactions on Fuzzy Systems 5: M. Zhong D. Zhou and X. Ding. On designing H fault detection filter for linear discrete time-varying systems. IEEE ransactions on Automatic ontrol 53: Q. Zhou P. Shi S. Xu and H. Li. Adaptive output feedback control for nonlinear time-delay systems by fuzzy approximation approach. IEEE ransactions on Fuzzy Systems 2: Y. Zhang and H. Fang. Robust fault detection filter design for networked control systems with delay distribution characterisation. International Journal of Systems Science 42: