Robust Fault Detection for Uncertain System with Communication Delay and Measurement Missing

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1 7 nd International Conerence on Mechatronics and Inormation echnology (ICMI 7) Robust ault Detection or Uncertain System with Communication Delay and Measurement Missing Chunpeng Wang* School o Electrical Engineering Binzhou Polytechnic Binzhou 5663 China uwa8@63.com Keywords: robust ault detection ilter communication delays missing measurements H iltering linear matrix inequality. Abstract: he robust ault detection problem is investigated or a class o norm-bounded uncertain systems with communication delays and multiple measurement pacet dropouts. he networed time-delay and missing measurements are modeled as a linear unction o the stochastic variable satisying Bernoulli random binary distribution. A robust ault detection ilter based on H iltering is designed to mae residual error system be exponentially mean-square stable and satisy a prescribed disturbance attenuation level. A suicient condition is derived in terms o linear matrix inequalities. Numerical examples are given to illustrate the eectiveness o the proposed method.. Introduction he robust ault detection method based on H iltering [-7] which is clear in concept and simple in implementation has attracted much attention o researchers and has gained rather rich research results. However this method is built upon precise mathematical model. In practice modeling errors and unnown inputs in complex engineering systems are unavoidable. As results the research o robust ault detection or a class o uncertain systems is receiving extensive attention. he design o robust ault diagnosis ilter or a class o singular time-delay systems subject to parameter uncertainties and disturbances has been studied in [3 4]. he problem o the robust ault detection or a class o linear uncertain systems via LMI has been studied in [5]. he design o ault detection ilter or a class o uncertain systems with one-step delay has been investigated in [6] however the impact o missing measurements on the ault detection perormance has been not considered in this paper. At present with the general application o networ in industrial production the phenomenon o time-delay and missing measurements has been widespread in the process o data transmission. In the case o incomplete measurement inormation the problem o ault detection has become the ocus o current research. In [7 8] the problem o the iltering and ault detection with random time-delays and measurement pacet dropouts has been discussed respectively. he design o ault detection ilter or linear discrete-time systems with random time-delays and measurement pacet dropouts has been studied in [9]. Parity space-based and observer-based respectively haven been used to solve a class o Marov jump systems with measurement pacet dropouts in [ ]. However the impact o system modeling errors on ault detection is not considered in the above literatures. hereore the robust ault detection problem o linear uncertain systems with data loss is still to be studied. In this paper the robust ault detection problem is investigated or a class o norm-bounded uncertain systems with communication delays and multiple measurement pacet dropouts. he networed time-delay and missing measurements are described by a linear unction o the stochastic variable satisying Bernoulli random binary distribution. By using H control theory the robust ault detection ilter (RD) design can be ormulated as an H iltering problem. A suicient condition or the existence o the desired ault detection ilter is established in terms o linear matrix Copyright (7) rancis Academic Press UK 87

2 inequalities (LMIs) and a numerical example is provided to illustrate the eectiveness and applicability o the proposed design method.. Problem ormulation Consider the ollowing class o uncertain discrete-time linear systems: x( + ) ( A+D A) x( ) + ( Bd +D Bd) d( ) + ( B +DB ) ( ) y( ) Cx( ) + Ddd( ) + D ( ) () n q p where x( ) Ris the state vector; y( ) Ris the signal to be estimated; d( ) R is the unnown r disturbance input; ( ) R is the ault to be detected; A B d B C Dd and D are nown real constant matrices with appropriate dimensions. A Bd and B are unnown matrices describing parameter uncertainties which are assumed to be o the orm: A H ( ) G Bd H ( ) G B H ( ) G3 () where H G G and G 3 are nown real constant matrices o appropriate dimensions and ( ) represents an unnown real-valued time-varying matrix satisying ( ) ( ) I. he insertion o networ cable between measurement node and ault detection ilter causes random time-delay and missing phenomena. urthermore the measurement can be described by y ( ) δ( ) y( ) + δ( ) y( ) + δ( ) y ( ) (3) q where ( ) R is the measured output and the stochastic variable δi ( ) R is a Bernoulli distributed white sequence with Prob{ δ ( ) i} E{ δi( ) } δi Prob{ δ ( ) i} E{ δi( ) } δi E{ ( δi( ) δi) } δi( δi) αi i (4) andδ i R is a nown positive scalar. he core module o the ault detection based on H iltering is the generation o residuals. In the case o system () is said to be asymptotically stable consider the ollowing RD or system with communication delays and multiple measurement pacet dropouts: xˆ ( + ) A ˆ x( ) + B ( ) r( ) C ˆ x( ) (5) n r where xˆ ( ) Ris the state estimate r( ) Ris the so-called residual that is compatible with the ault n n vector and A R n p B R p n and C R are ilter parameters to be determined. Deine the augmented state vector x ( ) x ( ) xˆ ( ). Denote ( ) ω d ( ) ( ) re ( ) r( ) ( ) the augmented system ormed by system () (3) and the ilter (5) can be expressed by ξ( + ) Aξ( ) + Aξ( ) + Bω( ) + Bω( ) (6) re ( ) Cξ( ) + Dω( ) where A A + A + ( δ ( ) δ ) A + ( δ ( ) δ ) A3 A A δbc A + δi A BC A3 I A H A ( )[ ] ( ) G H G A A + ( δ ( ) δ ) A + ( δ ( ) δ ) A3 A δbc δa A BC A3 A B B + B + δ δ B ( ( ) ) 88

3 B Bd B dbd d dbd B Bd B H BD d BD B ( )[ G G3 ] H( ) G B B + ( δ ( ) δ ) B B dbd d dbd B BD d BD C [ C ] D [ I]. Ater the process the ault detection problem stated above has been transormed into the robust iltering problem with networed time-delay and missing measurements. More speciically the desigened ilter (5) such that he ollowing requirements are satisied simultaneously: or a given prescribed scalar γ > the system (6) is exponentially mean-square stable and Er < γeω (7) e ( ) ( ) he inial important tas o ault detections is the residual evaluation. Based on the designed ault [4] detection system the residual evaluation unction J( ) and the threshold J th are determined as ollows: + N J r r J E J d( ) l ( ) where denotes the initial evaluation time instant N denotes the length o time window. Based on the above the ault can be detected by using the ollowing logical relationship: J ( ) > Jth abnormal ault J ( ) Jth normal ( ) ( ) ( ) th sup { ( )} (8) (9) 3. Design o robust ault detection ilter heorem : or a given scalar γ > the system (6) is exponentially mean-square stable and achieves the H norm constraint (7) or all nonzero ω ( ) i there exist positive deinite matrices P and Q satisying (). * * * * * * * * Q * * * * * * * * εgg * * * * * * * γ I * * * * * * PA PA PB PB P * * * * * < () α PA αpb α P * * * * αpa αpb αp * * * αpa3 αpa3 α P * * C D I * H P ε I where P+ Q+ εgg γ I+ εgg and ε R is any positive scalar. Remar : he proo o heorem would require to be combined with deinition in [] lemma in [] lemma 5 in[] and lemma in [3]. he speciic proo process would be omitted. On the basis o heeorem the detailed design method o the robusr ault detection ilter would be derived. Corollary : or a given scalar γ > the system (6) is exponentially mean-square stable and achieves the H norm constraint (7) or all nonzero ω ( ) i there exist positive deinite matricesw Y Zand Z 3 matrices Z M N and S satisying (). 89

4 * * * * * * * * * * * * * * 3 33 * * * * * * γ I * * * * * * * * * < * * * * * y8 y8 y88 * where W+ Z + εgg Z εgg εgg W+ Z + εgg Y+ Z3 + εgg Z Z 3 3 εgg 3 εgg 3 γ I εgg + 33 εgg 3 γ I+ εgg WA WA YA + δsc + M + δ( W Y ) YA + δsc WBd WB δsc δm δsc 53 YBd + dsdd YB + dsd 54 dsdd dsd W W Y α W 6 αsc αsc 63 αsdd αsd 66 αw αy αw 7 αsc αsc 74 αsdd αsd 77 αw αy α W 8 α ( W Y) 8 α M 88 αw αy N 9 I HW HY I 99 ε I. Moreover the desired ilter parameters o (5) can be given by A ( ) W Y M B ( ) W Y S C N () Proo : Partitioning matrices P andu P into the ollowing bloc orms: P P U U P P P U P 3 U U 3 n n where P R n n and U R are symmetric positive deinite matrices P andu are two non-singular matrices. Deine the invertible matrices as ollows: U I I P J U J P UZU UZ which implies that PJ J and let J QJ. ZU Z3 Perorming congruence transormation to () by the transormational matrix Ξ then pre- and post-multiplying the resulting LMI o the above transormation by the transormational matrix Ξ and deining the ollowing matrix variables: U W P Y PA U MW CU NW PB S we can arrive at the result () in Corollary where Ξ diag{ J J I I J J J J I} { } Ξ diag U I U I I I I I U I U I U I U I I I he transer unction matrix ( z) o the RD as shown in (5): ( ) ( ) ( ) ( z) C zi A B N zi ( PU U ) M PU U S. () 9

5 rom P* P I it can be obtained that PU + PU I and PU W W Y. Moreover the desired RD parameters can be given by (). his ends the proo. Remar : Let g g. he sub-optimal robust iltering problem or discrete-time uncertain systems with one-step communication delay and measurement missing can be represented as ollows: min g W YZ Z Z3 MNS (3) st.. ( ) he parameters o the sub-optimal ilters can be determined by () and the sub-optimal H attenuation levels are given by g opt g where g are the sub-optimal solutions o the corresponding convex optimization problems. 4. Simulation example In this section a numerical example is proved to illustrate the eectiveness and applicability o the proposed method. Consider the system described by () with parameters as ollows: A..4.5 B d.3 B.45 H C [ ] D d. D.6 G [..] G G3. ( ).5. Letδ. δ.7 δ.. In other words the measurements can be ideally transmitted over the networ with probability. one-step measurement delay can occur with probability.7 and the measurements are missing with probability.. With the prescribed parameters () can be solved by using the Matlab LMI toolbox. As a result the minimum noise attenuation level bound is γ opt.34 and the parameters o RD are given by A B.5 C [.4..3] or... the unnown input d( ) is supposed to be random noise which is uniormly distributed over[.5.5] and the ault signal ( ) is given as: 6 4 ( ) else By using (8) the threshold J th.43can be obtained. he simulation results indicate that the evaluation unction J( 67) > Jth so the appeared ault can be detected ater 7 time steps. ig. shows the generated residual signal r( ) with stepwise ault signal. ig. demonstrates the evolution o residual evaluation unction in which the blue dashed line stands or ault-ree case and the red solid line or the ault case. he simulation results are given in ig. and ig. rom which eectiveness o the method is illustrated. re ( ) In addition we can obtain ( ).98 < γ.34.he simulation shows that the robustness o residual to unnown inputs and model uncertainties can be eective. 5. Conclusions he problem o RD has been considered in this paper or a class o norm-bounded uncertain systems with communication delays and multiple measurement pacet dropouts. he RD based on 9

6 H iltering has been used as residual generator. hen the design problem o RD has been reduced to H iltering problem. A suicient condition or the existence o the RD has been established by means o LMIs and the parameters o the RD are determined directly by solutions o the LMIs. he simulation result has showed the eectiveness and applicability o the obtained approach. Residual signal r() x r with Stepwise ault signal ig. Residual ( ) Residual evaluation unction J().5 x -3.5 aulty case ault-ree case 5 5 ig. Evolution o residual evaluation J( ) Reerences [] Xu S X Chen W Lam J. Robust H iltering or uncertain Marovian jump systems with mode-dependent time delays [J]. IEEE rans.automatic Control 3 48 (5):9-97. [] Wang Z D Daniel W C Liu X H. Robust iltering under randomly varying sensor delay with variance constraints [J]. IEEE rans. On Circuits & Systems 4 5(6):3-36. [3] Chen Li Zhong MAI-Ying. Designing robust H ault detection ilter or singular time-delay systems with uncertainty [J]. Acta Automatica Sinica 8:34(8): [4] Zhong Maiying Ding Steven X Lam James el at. An LMI approach to design robust ault detection ilter or uncertain LI systems [J].Automatic 3 39(3): [5] Wang Hongru Wang Changhong Gao Huijun. Robust ault detection or uncertain systems via LMIs [J]. Electric Machines and Control 5:9(5): [6] Ruan Yu-bin Yang u-wen Wang Wu. Robust ault detection or networed systems with stochastic communication delays [J]. Journal o uzhou University (Natural Science Edition) 8 36(4): [7] Wang WuYang u-wen. H ilter design or discrete-time systems with missing measurements [J]. Acta Automatica Sinica 6 3():7-. [8] Ruan Yu-bin Wang Wu Yang u-wen. ault detection ilter or networed systems with missing measurements [J]. Control heory and Applications 96(3):9-95. [9] He Xiao Wang Zi dong and Zhou Dong hua. Networed ault detection with random communication delays and pacet losses. International journal o systems Science 8 39(): [] Wang Y Q Ye H Ding S X Wang G Z Zhou D H. Residual generation and evaluation o networed control systems subject to random pacet droput.automatica945(): [] He X Wang Z D Zhou D H Robust ault detection or networed systems with communication delay and data missing.automatica945(): [] Wang Zi-dong Yang u-wen Ho D Cet al. Robust H iltering or stochastic time-delay systems with measurements [J].IEEE ransactions on signal processing 6 54(7):

7 [3] Yang u-wen Wang Zi-dong Hung Y S et al. H Control or networed systems with random communication delays [J].IEEE rans.on Automatic Control 6 5(3):