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  Estimación de Fallas Basada en la Relaciones de Paridad para Sistemas No-lineales
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  International Journal of Automation and Computing 04(2), April 2007, 164-168DOI: 10.1007/s11633-007-0164-7 Parity Relation Based Fault Estimation for NonlinearSystems: An LMI Approach Sing Kiong Nguang 1 ∗ Ping Zhang 2 Steven X. Ding 2 1 The Department of Electrical and Computer Engineering, the University of Auckland, Auckland, New Zealand 2 AKS, Faculty of Engineering, University of Duisburg-Essen, Duisburg, Germany Abstract:  This paper proposes a parity relation based fault estimation for a class of nonlinear systems which can be modelled byTakagi-Sugeno (TS) fuzzy models. The design of a parity relation based residual generator is formulated in terms of a family of linearmatrix inequalities (LMIs). A numerical example is provided to illustrate the effectiveness of the proposed design techniques. Keywords:  Fuzzy systems, nonlinear systems, fault identification, fault detection, fault diagnosis. 1 Introduction Many systems are subject to random variations whichmay result from component and interconnection failures,parameters shifting, tracking, sudden environmental distur-bances, abrupt variations of the operating condition,  etc  .Therefore, in order to avoid production deteriorations ordamage to machines and humans, variations have to be de-tected as quickly as possible and decisions that stop thepropagation of their effects have to be made. Over the lastthree decades various fault detection and isolation (FDI)techniques have been developed [1 − 3] . The parity relationapproach or analytical redundancy relation (ARR) is oneof the most commonly used techniques in FDI, see [3-10].This technique is first proposed in [5] to detect sensor faultsin a flight control system. Then extended in [6] to detectboth sensor and actuator faults. This technique is furthergeneralised to handle systems with unknown disturbanceand uncertainty [8 , 11 − 13] . Recently, an attempt has beenmade to extend the parity relation technique to the inputoutput model of a nonlinear system in [14] based on an in-verse model of the system which in general is not unique.Hence, this extension is not always successful.In the past two decades, many researchers have stud-ied a class of nonlinear systems described by a Takagi-Sugeno(TS) fuzzy model, see [15-21]. In this TS fuzzymodel, local dynamics in different state space regions arerepresented by local linear systems. The overall model of the system is obtained by “blending” of these linear modelsthrough nonlinear membership functions. In other words,a TS fuzzy model is essentially a multi-model approachin which simple sub-models are combined to represent theglobal behavior of the system. This fuzzy model has beenshown to be able to approximate a large class of nonlinearsystems. However, to the best of our knowledge, the prob-lem of parity-based fault estimation for TS fuzzy modelshas not been addressed in the literature.Motivated by the capability of the TS fuzzy model, thispaper generalizes the parity space approach for linear sys- Manuscript received January 10, 2006; revised November 18, 2006.This work was supported by the Alexander von Humboldt Founda-tion.*Corresponding author. E-mail address: sk.nguang@auckland.ac.nz tems to nonlinear systems described by TS fuzzy models.The design procedure is given in terms of a family of LMIswhich can be solved effectively by an algorithm given in[22]. The effectiveness of the proposed design techniques isdemonstrated through a numerical example.This paper is organized as follows. The parity based faultestimation for nonlinear systems is developed in Section 2.The validity of this approach is demonstrated in Section 3.Finally, in Section 4, the conclusion is drawn. 2 Parity-based fault estimation fornonlinear systems Consider the following nonlinear system: x ( k  + 1) =  f  ( x ( k )) + g 1 ( x ( k )) u ( k )+ g 2 ( x ( k )) w ( k ) + g 3 ( x ( k )) f  ( k ) y ( k ) =  h ( x ( k )) x ( k ) + d 1 ( x ( k )) u ( k )+ d 2 ( x ( k )) w ( k ) + d 3 ( x ( k )) f  ( k ) , (1)where  x ( k )  ∈  R n is the state vector,  u ( k )  ∈  R m is the in-put,  w ( k )  ∈  R p is the unknown disturbance/uncertainty, y ( k )  ∈  R  is the measurement,  f  ( k )  ∈  R q is the fault. f  ( x ( k )) , g i ( x ( k )) , h ( x ( k )), and  d i ( x ( k )) are function of  x ( k ).A fuzzy dynamic model has been proposed by Takagiand Sugeno in [15] to represent local linear input/outputrelations of nonlinear systems. This fuzzy linear model isdescribed by IF-THEN rules and has been shown to be ableto approximate a large class of nonlinear systems. The  i -thrule of this fuzzy model for the nonlinear system (1) is of the following form [15] :Plant Rule  i: IF  ν  1 ( k ) is  M  i 1  and  ···  and  ν  ϑ ( k ) is  M  iϑ  THEN x ( k  + 1) =  A i x ( k ) + B i u ( k ) + E  i w ( k ) + G i f  ( k ) y ( k ) =  C  i x ( k ) + D i u ( k ) + J  i w ( k ) + H  i f  ( k )(2)where  i  = 1 , ···  ,r ,  r  is the number of IF-THENrules,  M  ij (  j  = 1 , 2 , ···  ,ϑ ) are fuzzy sets, and  ν  ( k ) =[ ν  1 ( k )  ···  ν  ϑ ( k )] are the premise variables.There are two major ways in constructing a TS fuzzymodel. One is the TS fuzzy model identification [15 , 23 , 24] us-  S. K. Nguang et al./ Parity Relation Based Fault Estimation for Nonlinear Systems: An LMI Approach  165ing input-output data, and the other is the TS fuzzy modelconstruction, by the idea of sector nonlinearity [25 − 27] .The defuzzified output of the TS fuzzy system (2) is rep-resented as follows: x ( k  + 1) =  A ( µ ( k )) x ( k ) + B ( µ ( k )) u ( k )+ E  ( µ ( k ) w ( k ) + G ( µ ( k )) f  ( k ) y ( k ) =  C  ( µ ( k )) x ( k ) + D ( µ ( k )) u ( k )+ J  ( µ ( k )) w ( k ) + H  ( µ ( k )) f  ( k )(3)where A ( µ ( k )) = r X i =1 µ i ( k ) A i , B ( µ ( k )) = r X i =1 µ i ( k ) B i E  ( µ ( k )) = r X i =1 µ i ( k ) E  i , G ( µ ( k )) = r X i =1 µ i ( k ) G i C  ( µ ( k )) = r X i =1 µ i ( k ) C  i , D ( µ ( k )) = r X i =1 µ i ( k ) D i J  ( µ ( k )) = r X i =1 µ i ( k ) J  i , H  ( µ ( k )) = r X i =1 µ i ( k ) H  i and  µ i ( k ) =   i ( ν ( k )) P ri =1   i ( ν ( k ))  with   i ( ν  ( k )) = Q ϑj =1 M  ij ( ν  j ( k )) and  M  ij ( ν  j ( k )) is the grade of mem-bership of   ν  j ( k ) in  M  ij . Notice that   i ( ν  ( k ))  ≥  0 for i  = 1 , 2 ,...,r , and  P ri =1  i ( ν  ( k ))  >  0 ,  ∀ k  ≥  0, hence, µ i ( k )  ≥  0 for  i  = 1 , 2 ,...,r , and  P ri =1 µ i ( k ) = 1  ∀ k  ≥  0.Define S  i  ≡ n x ( k ) | µ i ( x ( k ))  ≥  µ j ( x ( k )) , j  ∈  I  s o , i  ∈  I  s where  I  s  = n 1 , 2 , ···  ,r o  is the set of indexes. Then, theglobal model of the fuzzy dynamic system can also be ex-pressed in each subspace as x ( k  + 1) =  A i x ( k  + 1) + B i u ( k ) + E  i w i ( k ) + G i f  ( k ) + ¯ r i X j ¯ µ j ( k )( A j  − A i ) x ( k ) + ¯ r i X j ¯ µ j ( k )[( B j  − B i ) u ( k ) + ( E  j  − E  i ) w ( k )] + ¯ r i X j ¯ µ j ( k )( G j  − G i ) f  ( k ) y ( k ) =  C  i x ( k ) + D i u ( k ) + J  i w ( k ) + H  i f  ( k ) + ¯ r i X j ¯ µ j ( k )( C  j  − C  i ) x ( k ) + ¯ r i X j ¯ µ j ( k )[( D j  − D i ) u ( k ) + ( H  j  − H  i ) f  ( k )]for  x ( t )  ∈ S  i , where ¯ µ 1 ( k ) ,  ¯ µ 2 ( k ) , ···  ,  ¯ µ ¯ r i ( k ) are the mem-bership functions that are not equal to zero when  i -th playsa dominant role. In a more compact form: x ( k  + 1) =  A i x ( k  + 1) + B i u ( k ) + ¯ E  i  ¯ w i ( k ) + G i f  ( k ) y ( k ) =  C  i x ( k ) + D i u ( k ) + ¯ J  i  ¯ w i ( k ) + H  i f  ( k )(4)for  x ( t )  ∈ S  i , where ¯ E  i  = h ¯ E  1 i ¯ E  2 i ¯ E  3 i ¯ E  4 i i , ¯ J  i  = h ¯ E  5 i ¯ E  6 i ¯ E  7 i ¯ E  8 i i , ¯ E   i  = h ¯ E   i 1  ···  ¯ E   i ¯ ri i ,  i  = 1 , ···  , 8,¯ E  1 ij  =  A j  −  A i , ¯ E  2 ij  =  B j  −  B i , ¯ E  3 ij  =  E  j  −  E  i , ¯ E  4 ij  = G j  − G i , ¯ E  5 ij  =  C  j  − C  i , ¯ E  6 ij  =  D j  − D i , ¯ E  7 ij  =  J  j  − J  i ,¯ E  8 ij  =  H  j  −  H  i , ¯ w i ( k ) = h ¯ v T1 i ( t ) ¯ v T2 i ( k ) ¯ v T3 i ( k ) ¯ v T4 i ( k ) i T ,¯ v 1 i ( t ) = 2664 ¯ µ 1 ( k ) x ( k )...¯ µ ¯ r i ( k ) x ( k ) 3775 , ¯ v 2 i ( t ) = 2664 ¯ µ 1 ( k ) u ( k )...¯ µ ¯ r i ( k ) u ( k ) 3775 ,¯ v 3 i ( t ) = 2664 ¯ µ 1 ( k ) w ( k )...¯ µ ¯ r i ( k ) w ( k ) 3775 , and ¯ v 4 i ( t ) = 2664 ¯ µ 1 ( k ) f  ( k )...¯ µ ¯ r i ( k ) f  ( k ) 3775 .Given the system (3), the residual signal is generated bythe temporal redundancy approach [3 − 11] as follows:IF  x ( t )  ∈ S  i  THEN r s ( k ) =  V   s i 0BBBB@266664 y ( k  − s ) y ( k  − s + 1)... y ( k ) 377775 − H  us i 266664 u ( k  − s ) u ( k  − s + 1)... u ( k ) 3777751CCCCA ∈ R q where  µ s ( k ) = n µ ( k − s ) ,µ ( k − s +1) , ···  ,µ ( k ) o and  H  us i  = 2666664 D i  0  ...  0 C  i B i  D i ............ ... 0 C  i A s − 1 i  B i  ... C  i B i  D i 3777775 with  V   s i  ∈  R ( s +1) m × q isthe parity matrix for the subspace  x ( t )  ∈ S  i , and  s >  0 isthe order of the parity relation.Using (4),  r s ( k ) can be re-expressed as r s ( k ) =  V   s i h H  os i x ( k  − s ) + H  ws i  ¯ w i s ( k ) + H  fs i f  s ( k ) i where H  os i  = 266664 C  i C  i A i ... C  i A si 377775 ,  ¯ w i s ( k ) = 266664 ¯ w i ( k  − s )¯ w i ( k  − s + 1)...¯ w i ( k ) 377775 f  s ( k ) = 266664 f  ( k  − s ) f  ( k  − s + 1)... f  ( k ) 377775  166  International Journal of Automation and Computing 04(2), April 2007  H  ws i  = 2666664 ¯ J  i  0  ...  0 C  i  ¯ E  i  ¯ J  i ............ ... 0 C  i A s − 1 i ¯ E  i  ... C  i  ¯ E  i  ¯ J  i 3777775 and H  fs i  = 2666664 H  i  0  ...  0 C  i G i  H  i ............ ... 0 C  i A s − 1 i  G i  ... C  i G i  H  i 3777775 The main objection of the parity-based fault estimation isto select  V   s i  for each subspace such that the residual sig-nal  r s ( k ) =  f  s ( k ). However, in general, this objective isnot achievable. Hence, for each subspace, find  V   s i  suchthat   V   s i H  os i  2 ,  V   s i H  ws i  2 ,   V   s i H  fs i − I   2 are minimised.This minimisation problem can easily be cast in terms of LMIs as follows: LMI Parity-Based Fault Estimation Problem:  For i  = 1 , ···  ,r , find  V   s i  such that  − Q 1  V   s i H  os i V   s H  T os i  − I  #  <  0  − Q 2  V   s i H  ws i V   s i H  T ws i  − I  #  <  0 24 − Q 3 “ V   s i H  fs i  − I  ”“ V   s i H  fs i  − I  ” T − I  35  <  0and traces of   Q i  >  0 are minimised. 3 Illustrative example Consider a nonlinear mass-spring-damper mechanicalsystem with a nonlinear spring:˙ x 1 ( t ) =  x 2 ( t )˙ x 2 ( t ) =  − x 2 ( t ) − 2 x 1 ( t ) − 2 x 31 ( t ) + f  1 ( t )+ d 1 ( t ) + u ( t ) y 1 ( t ) =  x 1 ( t ) y 2 ( t ) =  x 2 ( t ) + 0 . 1 d 2 ( t ) + f  2 ( k )(5)where  x 1 ( k ) is the spring’s displacement,  x 2 ( k ) = ˙ x 1 ( t ), f  1 ( t ) and  f  2 ( k ) are, respectively, actuator and sensor faults. d 1 ( t ) and  d 2 ( k ), respectively, represent unknown distur-bances/uncertainties in the mechanical system and the sen-sor system. The term  − 0 . 67 x 31  is due to the nonlinearity of the spring. The spring is attached to a fixed wall, thereforethe spring’s displacement  x 1 ( t ) is physically constrained bythe length of the spring and the wall. The length of thespring could be any value, in this paper,  x 1 ( t )  ∈  [ − 0 . 5 ,  0 . 5]is assumed. The lower limit is the minimum length thatthe spring can be compressed. The Euler approximationmethod is employed to discretise the mass-spring-dampersystem (5) at 0 . 1 seconds. x 1 ( k  + 1) =  x 1 ( k ) + 0 . 1 x 2 ( k ) x 2 ( k  + 1) =  x 2 ( k ) − 0 . 1 x 2 ( k ) − 0 . 1 x 1 ( k ) − 0 . 2 x 31 ( k ) + 0 . 1 u ( k ) + 0 . 1 f  1 ( k ) + 0 . 1 d ( k ) y 1 ( k ) =  x 1 ( k ) y 2 ( k ) =  x 2 ( k ) + 0 . 1 d ( k ) + f  2 ( k ) .  (6)The concept of sector nonlinearity [26] is employed toconstruct an exact TS fuzzy model for the mass-spring-damper system. Using the fact that  x 1 ( k )  ∈ [ − 0 . 5 ,  0 . 5], this nonlinear term can be expressed as − 0 . 2 x 31 ( k ) =  − h 1 ( x 1 ( k )) ˆ 0 ˜ x 1 ( k )  −  h 2 ( x 1 ( k )) ˆ 0 . 05 ˜ x 1 ( k )where  h 1 ( x 1 ( k )) = 1 −  x i ( k )0 . 25  and  h 2 ( x 1 ( k )) =  x i ( k )0 . 25  . Using h 1 ( x 1 ( k )) and  h 2 ( x 1 ( k )), we obtain the following TS fuzzymodel which exactly represents (5) under the assumptionon bounds of the state variable  x 1 ( k )  ∈  [ − 0 . 5 0 . 5]: x ( k  + 1) =  P 2 i =1 h i ( z  ( k )) h A i x ( k ) + B i u ( k )+ E  i w ( k ) + G i f  ( k ) i y ( k ) =  P 2 i =1 h i ( z  ( k )) h C  i x ( k ) + D i u ( k )+ J  i w ( k ) + H  i f  ( k ) i (7)where  x ( k ) = ˆ x 1 ( k )  x 2 ( k ) ˜ T , A 1  =  1 0 . 1 − 0 . 1 0 . 9 # ,A 2  =  1 0 . 1 − 0 . 15 0 . 9 # B 1  =  B 2  =  00 . 1 # C  1  =  C  2  =  1 00 1 # D 1  =  D 2  =  0 00 0 # ,E  1  =  E  2  =  0 00 0 . 1 # G 1  =  G 2  =  0 00 . 1 0 # ,H  1  =  H  2  =  0 00 1 # and  J  1  =  J  2  =  0 00 . 1 0 # . For the sake of simplicity,the order of the parity relations is selected to be  s  = 2.Applying the procedure given in Section 3, the followingresidual generator is obtainedIF  h 1 ( x 1 ( k ))  ≥  h 2 ( x 1 ( k )) THEN r s ( k ) =  V   s 1 0BBBB@266664 y ( k  − s ) y ( k  − s + 1)... y ( k ) 377775 − H  us 1 266664 u ( k  − s ) u ( k  − s + 1)... u ( k ) 3777751CCCCA ∈ R q  S. K. Nguang et al./ Parity Relation Based Fault Estimation for Nonlinear Systems: An LMI Approach  167Else r s ( k ) =  V   s 2 0BBBB@266664 y ( k  − s ) y ( k  − s + 1)... y ( k ) 377775 − H  us 2 266664 u ( k  − s ) u ( k  − s + 1)... u ( k ) 3777751CCCCA ∈ R q where V   s 1  =  89 . 87 0  − 187 . 65 0 98 . 76 09 . 99 0 . 99  − 9 . 99 0 0 0 # and V   s 2  =  91 . 38 0  − 189 . 76 0 99 . 87 09 . 99 1 . 00  − 9 . 99 0 0 0 # . In order to compare our result with the linear result, thefollowing residual generator is designed based on the linearmodel obtained by linearising the system (5) at the srcin. r s ( k ) =  V   s L 0BBBB@266664 y ( k  − s ) y ( k  − s + 1)... y ( k ) 377775 − H  us L 266664 u ( k  − s ) u ( k  − s + 1)... u ( k ) 3777751CCCCA ∈ R q where V   s L  =  45 . 50 0  − 94 . 99 0 50 . 00 09 . 90 0 . 99  − 9 . 90 0 0 0 # . Fig.1 Plot of the residual signal  r s 1 ( k ) and the  f  1 ( k )Fig.2 Plot of the residual signal  r s 2 ( k ) and the  f  2 ( k ) u ( k ) = sin(0 . 1 k ), and  d 1 ( k ) and  d 2 ( k ) are white noises,respectively, with 0 . 00001 and 0 . 1 as their noise powers.Histories of the residual signals  r s 1 ( k ) and  r s 2 ( k ) are, re-spectively, depicted in Figs.1 and 2. These two figures showthat for the nonlinear case both  r s 1 ( k ) and  r s 2 ( k ) are able tofollow  f  1 ( k ) and  f  2 ( k ), however, for the linear case there ex-ist a large fault estimation error for  r s 1 ( k ). Hence, this ex-ample clearly demonstrates the superiority of the approachproposed in this paper. 4 Conclusion In this paper, the parity space approach for linear sys-tems has been generalized to nonlinear systems describedby the TS fuzzy models. The design procedure has beenprovided in terms of a family of linear matrix inequali-ties (LMIs) which can be solved effectively by an algorithmgiven in [22]. A numerical example has been given to illus-trate the effectiveness of the proposed design technique. References [1] M. Basseville. Detecting Changes in Signals and Systems –A Survey.  Automatica  , vol. 24, no. 3, pp. 309-326, 1988.[2] R. Isermann. Process Fault Detection Based on Modelingand Estimation Methods – A Survey.  Automatica  , vol. 20,no. 4, pp. 387-404, 1984.[3] P. M. Frank. Fault Diagnosis in Dynamic Systems UsingAnalytical and Knowledge Based Redundancy – A Surveyof Some New Results.  Automatica  , vol. 26, no. 3, pp. 459-474, 1990.[4] R. J. Patton, J. Chen. A Review of Parity Space Approachesto Fault Diagnosis. In  Proceedings of IFAC Symposium onSafeprocess’91 , Hull, UK, vol. 1, pp. 239-255, 1991.[5] J. E. Potter, M. C. Suman. Threshold Redundancy Man-agement with Arrays of Skewed Instruments.  Integrity inElectronic Flight Control Systems  , vol. Agardograph 224,no. 15, pp. 11-25, 1977.[6] E. Y. Chow, A. S. Willsky. Analytical Redundancy and theDesign of Robust Failure Detection Systems,  IEEE Trans-actions on Automatic Control  , vol. 29, no. 7, pp. 603-614,1984.[7] J. J. Gertler. Survey of Model-based Failure Detection andIsolation in Complex Plants,  IEEE Control Systems Maga-zine  , vol. 3, no. 6, pp. 3-11, 1988.
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