Fault Isolation Filter with Linear Matrix Inequality Solution to Optimal Decoupling

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1 Proceeding of the 2006 American Control Conference Minneapoli Minneota USA June ThA07.3 Fault Iolation Filter with Linear Matrix Inequality Solution to Optimal Decoupling Imad M. Jaimoukha Zhenhai Li and Emmanuel Mazar Abtract In thi paper we conider a model baed fault detection and iolation problem for linear time invariant dynamic ytem ubject to fault and diturbance. We ue an oberver cheme that cancel the ytem dynamic and define a reidual vector ignal that i enitive only to fault and diturbance. We then deign a table fault iolation filter uch that the H norm of the tranfer matrix function from diturbance to the reidual i minimized (for fault detection) ubject to the contraint that the tranfer matrix function from fault to reidual i equal to a pre aigned diagonal tranfer matrix (for fault iolation). The optimization of diturbance decoupling i accomplihed via the help of linear matrix inequalitie. A numerical example i alo preented to illutrate the algorithm. I. INTRODUCTION Model baed fault detection and iolation (FDI) cheme exploiting analytic redundancy have received increaing attention in the literature and application 6 18 and 11. The cheme involve the deign of an oberver which effectively cancel the (nominal) proce dynamic and provide a reidual ignal that i enitive only to diturbance plant/model mimatch (often recat a diturbance) and fault. The filter deign objective i then to reduce the enitivity to diturbance and/or plant/model mimatch a well a iolating fault. There are two main approache for achieving the detection objective namely exact and almot diturbance decoupling 21. In the former the aim i to decouple the reidual ignal from diturbance exactly while in the latter the tranfer matrix from diturbance to the reidual ignal i required to be mall in either the H 2 or H norm ene. For the purpoe of iolating fault in both cae the tranfer matrix function from fault to reidual i required to be diagonal. Patton and Chen conidered left and right eigenvector aignment 17 and Chen et al. treated the robut FDI problem by uing an unknown input oberver with diturbance exactly decoupled in the tate etimation error 2. Although thee technique can achieve diturbance decoupling their iolation ability i retricted. In mot perfect decoupling and iolation cae olvability condition are generically difficult to be atified. Hence almot decoupling ha been widely invetigated recently through H technique 21. Frank and Ding developed a matrix factorization method to obtain an Thi work wa partially upported by the Minitry of Defence through the Data & Information Fuion Defence Technology Centre I. M. Jaimoukha Z. Li and E. Mazar are with the Control and Power Group Department of Electrical and Electronic Engineering Imperial College London SW7 2BT UK. i.jaimouka@imperial.ac.uk zhenhai.li@imperial.ac.uk and e.mazar@imperial.ac.uk Fax: Telephone: optimal fault detection filter 8 and Sadrnia et al. utilized a Riccati equation iteration method to contruct an extended H filter 22. Uing thee technique the decoupling problem can be tranformed to a enitivity optimization problem which eek to increae the enitivity of the reidual to fault and imultaneouly reduce the enitivity to diturbance and plant/model mimatch. However iolation i employed indirectly in the above method through the ue of bank of oberver. Thi make it hard to deal with multiple fault (where fault might occur imultaneouly). It i deirable to conider decoupling diturbance approximately and iolating multiple fault uing a ingle oberver. See 9 for a dicuion of thi iue. In thi contribution we contruct an FDI oberver uch that the H norm of the tranfer matrix function from diturbance to the reidual i minimized with the contraint that the tranfer matrix function from fault to reidual i equal to a pre aigned diagonal tranfer matrix. Neceary and ufficient condition in the quare cae and ufficient condition in the nonquare cae for the exitence of uch an oberver are given. Variou verion of thi problem have been conidered in the literature. The problem of deigning a table diagonalizing filter ha been conidered in 14 where a partial olution wa given (when the number of output i equal to the number of fault) however the influence of the diturbance wa not conidered. In 4 the problem of limiting the influence of the diturbance wa conidered although the reulting equation were nonlinear and the olution wa uboptimal. Here we deign a table multiple fault iolating filter and minimize the effect of diturbance (uing the H norm a a meaure) on the reidual uing linear matrix inequalitie (LMI). The LMI technique ued in 12 for fault detection could alo be modified for fault iolation under certain aumption. In thi work we remove thee aumption and preent the work in a more general etting. Thi work i organized a follow. We review the reidual ignal generation for almot detection and iolation filter in Section II and III. In Section IV we divide the iolation problem into two cae and give exitence condition for their olution. The diturbance decoupling problem i treated uing LMI technique in Section V. Section VI give a jet engine example to validate the approach. Finally Section VII conclude the paper. The notation we ue i fairly tandard. The et of real (complex) n m matrice i denoted by R n m (C n m ). For A R n m we ue the notation A T to denote the tranpoe. The et of complex number i denoted by C. The open left half of the complex plane i denoted by C and the cloed /06/$ IEEE 2339

2 right half of the complex plane i denoted by C +. The i th eigenvalue of A C n n i denoted by λ i (A). For a ymmetric matrix A R n n A>0 (A <0) denote that A i poitive definite (negative definite) that i λ i (A) > 0 i (λ i (A) < 0 i). The notation A diag (a 1...a n ) denote that A i a diagonal matrix with diagonal entrie a 1...a n. The n n identity matrix i denoted a I n and the n m null matrix i denoted a 0 nm with the ubcript dropped if they can be inferred from context. R() m p denote the et of all m p proper real rational matrix function of. L m p denote the pace of m p matrix function with entrie bounded on the extended imaginary axi. The ubpace H m p L m p denote matrix function analytic in the cloed right half of the complex plane. A prefix R denote a real rational function o that RH m p denote the et of all m p table real rational matrix function of. For G() RH m p we define G up ω R max i λ i (G( jω) T G(jω)) If G() (A B C D) RH m p with m p then G() i called co outer if G() ha no zero 1 in C + that i if the matrix pencil A μi B C D ha full column rank for all μ C + (equivalently if G() ha a left invere in RH p m 5). The matrix A R n n i called table if λ i (A) C i. The pair (A C) i called detectable if there exit a real matrix L uch that A + LC i table. The pair (A C) i detectable if the matrix pencil A T μi C T T ha full column rank for all μ C+. II. PROBLEM FORMULATION A faulty linear time-invariant (LTI) dynamic ytem ubject to diturbance can be modeled a ẋ(t) Ax(t)+B d d(t)+b f f(t)+bu(t) y(t) Cx(t)+D d d(t)+d f f(t)+du(t) where x(t) R n u(t) R n u and y(t) R n y are the tate input and output vector repectively and d(t) R n d and f(t) R n f are the diturbance and fault vector repectively. Here B f R n n f and D f R n y n f are the component and intrument fault ditribution matrice repectively while B d R n n d and D d R n y n d are the correponding diturbance ditribution matrice 7 6. For implicity tructured model uncertaintie are recat a additive diturbance 17. We aume that the ytem ha at leat a many output a potential fault i.e n y n f which i a general aumption in fault diagnoi. Since enor fault can be recat by actuator fault we may alo aume without lo of generality that D f 0. A tandard filter baed FDI approach to generate a reidual ignal r(t) i the ue of ytem duplication to cancel the the input dynamic 6. Define Gd () G f () A Bd B f R() n y (n d +n f ). C D d 0 Then the reidual dynamic are r() F ()G d ()d()+g f ()f() (1) where F () R() n f n y i a free table pot-filter to be deigned. The reidual dynamic can be generated uing a tate oberver framework a ˆx(t) Aˆx(t)+Bu(t) L (y(t) C ˆx(t) Du(t)) r(t) H(y(t) C ˆx(t) Du(t)) where ˆx(t) R n i the oberver tate and r(t) R n f i the reidual ignal. Here L R n n y and H R n f n y are the oberver and reidual gain matrice repectively and are to be determined. Define the tate etimation error ignal a e(t) x(t) ˆx(t). It follow that the reidual dynamic are given by ė(t) (A + LC)e(t)+(B d + LD d )d(t)+b f f(t) r(t) HCe(t)+HD d d(t). Taking Laplace tranform r() T rf ()f()+t rd ()d() where Trd () T rf () A+LC Bd +LD d B f (2) HC HD d 0 are the tranfer matrice from fault and diturbance to reidual repectively. Then a calculation how that r() F ()G d ()d()+g f ()f() where F () A + LC L R() n f n y HC H (3) o that Trd () T rf () F () G d () G f () (4) To guarantee the exitence of at leat one L uch that A+ LC i table we aume that the pair (A C) i detectable. In fact with thi aumption we how next that there i no lo of generality in auming that A i table. If A i untable and (A C) i detectable there exit L 0 uch that A + L 0 C i table. A imple calculation now how that Trd () T rf () F () Ḡ d () Ḡ f () where Ḡd () Ḡ f () A + L0 C B d + L 0 D d B f C D d 0 and where F () A + L0 C +(L L 0 )C L L 0 HC H The reult now follow by identifying A : A+L 0 C B d : B d +L 0 D d and L : L L 0. Note however that fault detection and iolation for untable ytem may be problematic in practice due to modeling uncertaintie

3 We conider the following detection and iolation problem. Problem 2.1: Find a table filter F () uch that the optimum i achieved where γ o inf T rf T o T rd (5) F () H n f ny T o M(I Λ) 1 (6) i a preaigned table diagonal tranfer matrix and Λ diag(λ 1...λ nf ) R n f n f λ i < 0 i M diag(m 1...m nf ) R n f n f m i > 0 i. (7) It i evident that the firt contraint T rf FG f T o i equivalent to F Ĝf I where Ĝ f G f (I Λ)M 1 G f M 1 G f ΛM 1 A ABf M 1 A Bf ΛM C CB f M 1 1 C 0 A ABf M 1 B f ΛM 1 C CB f M 1. (8) Hence Problem 2.1 i equivalent to the following Problem 2.2: Find a table filter F () uch that the optimum γ o inf FG d (9) F Ĝf I F () H n f ny i achieved. Remark 2.1: The equality contraint F Ĝf I follow from the iolation condition which contruct a directional reidual to achieve fault iolation. Correpondingly the minimization of T rd in the ene of H -norm i interpreted a the detection or diturbance decoupling condition. Remark 2.2: Problem 2.2 i known a an almot diturbance decoupling problem in fault diagnoi 21 where diturbance ignature do not need to be perfectly removed from the reidual. It i alo claified a an optimal approximate decoupling problem in 9 where fault repone are pecified a explicit equality contraint intead of inequalitie contraint. Thi etup allow an eay implementation of iolation function. III. STABILITY CONDITION Before the deign of a reidual generator we give the exitence condition of a table iolation filter firt. Lemma 3.1: Suppoe n y n f. Then the following are equivalent: 1) There exit a table fault iolation filter F (). 2) Ĝ f i co outer. 3) CB f ha full column rank and G f () ha no finite zero in C +. Proof. 1 2 The reult follow from the fact that Ĝ f i table (from (8) and the aumption that A table) and the contraint in (9): F Ĝf I. (10) Follow directly from the definition of co outer function. Let ρ(μ) be defined a ( ) A μi (AB f B f Λ)M ρ(μ) : rank 1 (11) C CB f M 1 For any finite μ C + ME : M(μI Λ) 1 ha full rank ince Λ i table and o ( ) A μi (AB f B f Λ)M ρ(μ) rank 1 I B f E C CB f M 1 0 ME ( ) A μi Bf rank. C 0 It follow that Ĝ f and G f () have the ame finite zero in C + and the concluion follow. Note the fact that due to the noningularity of M CB f having full column rank i equivalent to CB f M 1 having full column rank. Remark 3.1: Part (3) in Lemma 3.1 give a tet independent of Λ and could be ued in the beginning of the deign. Remark 3.2: The condition that CB f ha full column rank eem to be retrictive however it i neceary for the form of T o we have choen (e.g. diagonal with firt order diagonal element). Other choice of T o will require different correponding condition. In thi work we have opted for the implet choice of T o () e.g. fixed diagonal and with firt order diagonal element. An iue for further reearch i to optimize the choice of T o (). Remark 3.3: In the cae that n y >n f then generically Ĝ f ha no zero and i co outer. Thu when the number of output i larger than the number of fault we expect the fault iolation problem to be olvable. IV. FAULT ISOLATION FILTER DESIGN Next we derive an optimal fault iolation filter for two cae: the quare cae (n y n f ) and the general cae (n y > n f ). We will how that the reult in the quare cae can be generalized to the general cae via olving a left invere of a tranfer matrix in tate pace. Theorem 4.1: Suppoe n y n f. Then there exit an optimal filter F () which olve Problem 2.2 if and only if Ĝf i an outer function. Furthermore if a feaible F () exit then the correponding oberver gain in the form of (3) are given a L (AB f B f Λ)(CB f ) 1 (12) H M(CB f ) 1. (13) Proof. Firtly we how that the optimization objective in (9) can be achieved if and only if F Ĝ 1 f exit and table. ( ) According to Lemma 3.1 the exitence of a feaible F implie the exitence of a table Ĝ 1 f. ( ) IfĜ 1 f i table then (9) i equivalent to γ o inf F Ĝf Ĝ 1 f G d F Ĝf I inf F () H n f n y Ĝ 1 f G d 2341

4 where the optimum i achieved when F Ĝ 1 f. Secondly the election of L and H can be verified via comparing (3) with F Ĝ 1 f A R F M 1 S F C M(CB f ) 1 C A R F (CB f ) 1 C M(CB f ) 1 C R F M 1 S F M(CB f ) 1 R F (CB f ) 1 M(CB f ) 1 where R F AB f B f Λ and S F (CB f M 1 ) 1. Finally an outer Ĝf implie that Ĝ f ha full rank over the extended imaginary axi and that Ĝf ) 1 i table which in turn imply that CB f i noningular and A (AB f B f Λ)(CB f ) 1 C i table. In the quare cae the almot iolation condition can be eaily verified by calculating the invere of Ĝ f. Note however that the FI filter i unique and no degree of freedom can be ued to reduce the effect of diturbance. It i natural to generalize the reult to the non quare cae (n y >n f ). However the left-invere of Ĝf i not unique and degree of freedom may be exploited in an oberverbaed form. Hence in the ret of thi ection we conider a tate-pace olution of F Ĝf I in which the freedom in the choice of the olution for L and H will be ued to optimize diturbance decoupling. Lemma 4.1: Suppoe that n y > n f. Then the iolation condition (10) i atified if there exit L and H uch that LCB f B f Λ AB f HCB f M. (14) Furthermore with the aumption Ĝf i co-outer the feaible L and H are given by L L 1 + RL 2 (15) H H 1 + SL 2 (16) where L 1 (B f Λ AB f )D # L 2 I DD # H 1 MD # and D # (D T D) 1 D T i the Moore Penroe generalized invere of D CB f. In addition R R n n y and S R n f n y are free matrice to be decided. Proof. By ubtituting (3) and (8) into (10) we can how that F Ĝf A+LC HC L H A+LC LC 0 A A C AB f M 1 B f ΛM 1 CB f M 1 LCB f M 1 AB f M 1 B f ΛM 1 HC HC HCB f M 1 T 1 A+LC LC T T 1 LCB f M 1 0 A AB f M 1 B f ΛM 1 HC HC T HCB f M 1 A+LC 0 0 A LCB f M 1 +AB f M 1 B f ΛM 1 AB f M 1 B f ΛM 1 HC 0 HCB f M 1 A+LC HC LCB f M 1 +AB f M 1 B f ΛM 1 HCB f M 1 I I I where the imilarity tranformation matrix T. 0 I Hence (14) give a ufficient condition to enure that (10) i atified. According to Lemma 3.1 an co-outer Ĝ f implie that CB f ha full column rank. Therefore Moore Penroe generalized invere can be applied to olve (14) which reult in (15) and (16). V. AN LMI SOLUTION TO OPTIMAL DISTURBANCE DECOUPLING Linear matrix inequality technique have proved to be popular in control ytem analyi and deign due to their numerical reliability. The application of LMI in modelbaed FD have been addreed in and 25 where LMI technique are ued to olve a enitivity optimization problem however iolation ability wa not emphaized. In thi ection we ue the freedom provide in Lemma 4.1 to minimize reidual enitivity to diturbance which i achieved uing the Bounded Real Lemma 23. Lemma 5.1: Let n y >n f. There exit L and H uch that A + LC i table and T rd <γif and only if there exit L H and P P T R n n uch that P>0and (A+LC)T P +P (A+LC) P (B d +LD d ) C T H T (B d +LD d ) T P γi D T d HT < 0. (17) HC HD d γi The Bounded Real Lemma give the olvability condition in the form of a Bilinear Matrix Inequality. The next theorem provide a numerical algorithm to compute L and H via olving an LMI. Note that the condition i only ufficient ince it i baed on Lemma 4.1 which give ufficient condition on L and H to atify the iolation condition (10). Theorem 5.1: Suppoe that n y > n f and Ĝf i co outer. There exit L and H uch that the pecification in Problem 2.2 are atified if there exit Z R n n y S R n f n y and P P T R n n uch that P>0 (18) and P (A+L 1C)+ZL 2C+( ) ( ) ( ) (B d +L 1D d ) T P +D T d LT 2 ZT γi ( ) < 0 (19) H 1 C+SL 2 C H 1 D d +SL 2 D d γi where ( ) denote term readily inferred from ymmetry. Furthermore if (18) and (19) are olved we can contruct L and H a L L 1 + P 1 ZL 2 (20) H H 1 + SL 2. (21) Proof. By ubtituting (15) and (16) into (17) and applying Lemma 5.1 we get (18) and (19). Then (9) i achieved if the LMI have a feaible olution. The correponding filter i obtained by extracting R P 1 Z and S from the LMI. Remark 5.1: Here we deign a table fault iolating filter for the general cae and furthermore derive an LMI algorithm for minimizing the effect of diturbance (uing the H norm a a meaure) on the reidual. The reult obtained generalie thoe of 14 and

5 VI. NUMERICAL EXAMPLE To illutrate the application of the iolation filter cheme in a real ytem a jet engine example i conidered in thi ection. The GE-21 jet engine tate-pace model 20 i given a A B C D The ytem ha n 2 n y 3 and n u 3. To conider the general cae (n y >n f ) we uppoe that thi ytem i ubject to two potential actuator fault. Here the etup i given by B f and the diturbance ditribution matrice are given a B d D d It i eay to ee that Ĝf i a co-outer function via checking condition (3) in Lemma 3.1. A imple election of the diagonal matrix i Λ diag{ 1 2} M diag{1 1} which mean that we allocate the ame priority to each fault. Following Lemma 4.1 we have L L H Then ubtituting into (19) reult in the oberver gain L H The correponding optimal γ o Aume that diturbance 1 i a white noie with mean zero and tandard deviation 1 and diturbance 2 i a contant bia of amplitude 1 applied from the 2nd econd. Fault 1 in actuator 1 imulated by an abrupt jump and fault 2 in actuator 2 imulated by an incipient drift with lope 0.5 are connected from the 4th econd and 6th econd repectively. Figure 1 give the reidual repone. The example make clear that the deigned filter atifie the performance requirement of rapid fault detection and iolation which i ufficiently robut againt diturbance. Amplitude Reidual Time(ec) Fig. 1. Time repone of the reidual vector VII. SUMMARY We have preented the olution of a fault detection and iolation problem uing a tate oberver framework. The fault iolation i accomplihed by olving two equality contraint (ee Lemma 4.1). It wa alo hown that the optimal FDI filter deign reduce to the olution of linear matrix inequalitie (ee Theorem 5.1 ). Our cheme can handle multiple fault (where fault might occur at the ame time) a well a provide robutne to diturbance. A jet engine example i given to clarify our algorithm. VIII. ACKNOWLEDGMENTS Thi work ha been partially upported by the Minitry of Defence through the Data & Information Fuion Defence Technology Centre. REFERENCES 1 F. Callier and C. Deoer Linear Sytem Theory. Springer Verlag New York J. Chen P. Patton and H. Zhang Deign of unknown input oberver and robut fault detection filter Int. J. Control vol. 63 no. 1 pp R. K. Dougla and J. L. Speyer Robut fault detection filter deign in Proc. Amer. Control Conf. Seattle Wahington June 1995 pp IEEE Pre New York. 4 R. Dougla and J. Speyer An H bounded fault detection filter in Proc. Amer. Control Conf. Seattle Wahington June 1995 pp IEEE Pre New York. 5 B. Franci A Coure in H Control Theory. New York: Springer- Verlag: Heidelberg P. M. Frank and X. Ding Survey of robut reidual generation and evaluation method in oberver baed fault detection ytem J. Proc. Cont. vol. 7 no. 6 pp r1 r2 2343

6 7 P. Frank Fault diagnoi in dynamic ytem uing analytical and knowledge baed redundancy: A urvey and ome new reult Automatica vol. 26 no. 3 pp P. Frank and X. Ding Frequency domain approach to optimally robut reidual generation and evaluation for model baed fault diagnoi Automatica vol. 30 no. 5 pp J. J. Gertler and M. M. Kunwer Optimal reidual decoupling for robut fault diagnoi Int. J. Control vol. 61 no. 2 pp M. Hou and R. Patton An LMI approach to H /H fault detection oberver in UKACC International Conference on Control. 96 England 1996 pp R. Iermann Model-baed fault-detection and diagnoi - tatu and application Annual Review in Control vol. 29 pp I. Jaimoukha Z. Li and E. F. M. Mazar Linear matrix inequality olution to the H /H fault detection problem in Proceeding of the Seventh IASTED International Conference on Control and Application Cancun Mexico May 2005 pp M. Kinnaert R. Hanu and P. Arte Fault detection and iolation for untable linear ytem IEEE Tran. Automatic Control vol. 40 no. 4 pp B. Liu and J. Si Fault iolation filter deign for linear time-invariant ytem IEEE Tran. Automatic Control vol. 42 no. 5 pp M.-A. Maoumnia A geometric approach to failure detection and identification in linear ytem Ph.D. diertation Ma. Int. Technol. Feb J. Park G. Rizzoni and W. Ribben On the repreentation of enor fault in fault detection filter Automatica vol. 30 no. 11 pp R. Patton and J. Chen On eigentructure aignment for robut fault diagnoi Int. J. Robut & Nonlinear Control vol. 10 no. 14 pp R. Patton P. Frank and R. C. (ed) Iue of Fault Diagnoi for Dynamic Sytem. Springer F. Rambeaux F. Hamelin and D. Sauter Optimal threholding for robut fault detection of uncertain ytem Int. J. Robut & Nonlinear Control vol. 10 pp M. L. Rank and H. Niemann Norm baed deign of fault detector Int. J. Control vol. 72 no. 9 pp A. Saberi A. Stoorvogel P. Sannuti and H. Niemann Fundamental problem in fault detection and identification Int. J. Robut & Nonlinear Control vol. 10 no. 14 pp M. Sadrnia J. Chen and R. Patton Robut fault diagnoi oberver deign uing H optimiation and μ ynthei IEE Colloquium on Modelling and Signal Proceing for Fault Diagnoi pp. 9/ C. Scherer P. Gahinet and M. Chilali Multi-objective outputfeedback control via LMI optimization IEEE Tran. Automatic Control vol. 42 no. 7 pp J. White and J. Speyer Detection filter deign: pectral theory and algorithm IEEE Tran. Automatic Control vol. 32 no. 7 pp. 0 0 July M. Zhong S. Ding J. Lam and H. Wang An LMI approach to deign robut fault detection filter for uncertain LTI ytem Automatica vol. 39 no. 3 pp

A QMI APPROACH TO THE ROBUST FAULT DETECTION AND ISOLATION PROBLEM. Emmanuel Mazars, Zhenhai Li, Imad M Jaimoukha. Imperial College London

A QMI APPROACH TO THE ROBUST FAULT DETECTION AND ISOLATION PROBLEM Emmanuel Mazars Zhenhai Li Imad M Jaimoukha Imperial College London Abstract: This paper investigates the robust fault detection isolation