Design of Integrated Systems for the Control and Detection of Actuator/Sensor Faults Stoustrup, Jakob; Grimble, M.J.; Niemann, H.H.

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1 Aalborg Univeritet Deign of Integrate Sytem for the Control an Detetion of Atuator/Senor Fault Stoutrup, Jakob; Grimble, MJ; Niemann, HH Publihe in: Senor Review Publiation ate: 1997 Doument Verion Tilig verion ogå kalet pre-print Link to publiation from Aalborg Univerity Citation for publihe verion (APA): Stoutrup, J, Grimble, M J, & Niemann, H H (1997) Deign of Integrate Sytem for the Control an Detetion of Atuator/Senor Fault Senor Review, General right Copyright an moral right for the publiation mae aeible in the publi portal are retaine by the author an/or other opyright owner an it i a onition of aeing publiation that uer reognie an abie by the legal requirement aoiate with thee right? Uer may ownloa an print one opy of any publiation from the publi portal for the purpoe of private tuy or reearh? You may not further itribute the material or ue it for any profit-making ativity or ommerial gain? You may freely itribute the URL ientifying the publiation in the publi portal? Take own poliy If you believe that thi oument breahe opyright pleae ontat u at vbn@aubaauk proviing etail, an we will remove ae to the work immeiately an invetigate your laim Downloae from vbnaauk on: augut 13, 218

2 Reearh artile Deign of integrate ytem for the ontrol an etetion of atuator/enor fault Jakob Stoutrup MJ Grimble an Henrik Niemann The author Jakob Stoutrup i a Profeor at the Department of Control Engineering, Aalborg Univerity, Frerik Bajervej 7, DK-922 Aalborg Eat, Denmark Tel: (45) ; Fax: (45) ; jakob@ontrolaukk MJ Grimble i Profeor an Diretor at the Inutrial Control Centre, Univerity of Strathlye, Graham Hill Builing, 5 George Street, Glagow G1 1QE, UK mgrimble@eeetrathauk Henrik Niemann i Aoiate Profeor at the Department of Automation, Tehnial Univerity of Denmark, Builing 326, DK-28 Lyngby, Denmark hhn@iautuk Abtrat Conier ontrol ytem operating uner potentially faulty onition Diue the problem of eigning a ingle unit whih not only hanle the require ontrol ation but alo ientifie fault ourring in atuator an enor In ommon pratie, unit for ontrol an for iagnoi are eigne eparately Attempt to ientify ituation in whih thi i a reaonable approah an ae in whih the eign of eah unit houl take the other into onieration Preent a omplete haraterization for eah ae an give ytemati eign proeure for both the integrate an non-integrate eign of ontrol an iagnoi unit Show how a ombine moule for ontrol an iagnoi an be eigne whih i able to follow referene an rejet iturbane robutly, ontrol the ytem o that unetete fault o not have iatrou effet, reue the number of fale alarm an ientify whih fault have ourre Senor Review Volume 17 Number pp MCB Univerity Pre ISSN Introution In the ontrol of inutrial ytem, it i rare that a ontrol ytem funtion ontinuouly without hutown throughout the heule life yle of the plant an ontroller harware Owing to wear of mehanial an eletrial omponent, both atuator an enor an fail in more or le ritial way For afetyritial proee, it i of paramount importane to etet when fault are likely to happen an then to ientify thee fault a fat a poible one they have ourre To meet uh inutrial nee, a number of heme for fault etetion an iolation (FDI) have been put forwar in the literature on automati ontrol Muh of the reearh ha ealt with the eign of filter whih monitor a proe an generate alarm when fault have ourre In mot ae, the filter are moel-bae evie whih at inepenently of the omputer-implemente igital ontroller In thi paper, however, the avantage of ombining the ontrol algorithm an the FDI filter in a ingle moule will be iue, an a relatively imple methoology to eign uh ombine moule will be eribe It will be hown that a ombine moule will be benefiial in term of implementation an reliability, but it will alo be hown that the quality of ontrol an the quality of etetion will not improve by uing the integrate eign ompare with the iniviual eign of two omponent provie that a goo nominal moel i available Thi reult i hown to be very general A peial ae uing an algebrai Riati equation approah wa preente in Tyler an Morari (1994) On the other han, if the quality of the available moel i poor, the eign of the ontrol ytem an of the iagnoi ytem ha to be unertaken imultaneouly in orer to improve overall funtionality Ueful urvey about early work on FDI an be foun in Frank (199) an Patton et al (1989) Many of thee tehnique are oberver-bae, uh a Magni an Mouyon (1991) Thee metho have ine been refine an extene A more reent work in thi line of reearh i that of Frank an Ding (1994) The original iea of uing the information alreay available in the oberver part of a ontroller for iagnoti purpoe wa given in Nett et al (1988)

3 Sytem for the ontrol an etetion of atuator/enor fault Jakob Stoutrup, MJ Grimble an Henrik Niemann Senor Review Early work on FDI experiene problem owing to moelling unertaintie In ome ae, fale alarm were ommon ue to imperfet moelling Thi alle for iue of robutne to be inorporate into the FDI eign algorithm Speifi robutne olution to FDI problem were iue in Bokor an Kevizky (1994); Mangoubi et al, (1995); Mura et al, (1996); Patton an Chen (1991); Qiu an Gertler (1993); an Wang an Wu (1993) All thee metho ue frequeny omain tehnique in ontrat to Ajbar an Kantor (1993), whih ue l tehnique An intereting appliation of FDI tehnique wa preente in work by Blanke et al (1995), Jørgenen et al (1995), Grainger et al (1995) an Garia et al (1995), whih uggete uing a ieel-engine atuator a an FDI benhmark problem Problem formulation Figure 1 illutrate a ontrol problem in the tanar ytem onfiguration (ee, for example, Zhou et al (1996) for an introution to the tanar onfiguration paraigm) Here, w an be thought of a a olletion of uneire ignal (iturbane) entering the ytem G() or a et-point The ignal y are the meaurement ue by the ontroller K(), generating the ontrol ignal u in orer to make the output to-be-ontrolle z uffiiently mall The ytem in Figure 1 an be eribe in either the tate pae formulation: ẋ A B B x 1 2 z C1 D11 D12 w y C D D u or, alternatively, in tranfer matrix funtion form: z w G y () u G G w 11() 12() G () G () u C I A B D 1( ) + C ( I A) B + D 2 C( I A) B + D w 1 (1) C2 ( I A) B 2 + D 22 u For the tanar problem hown in Figure 1, a ontroller K(), making the tranfer funtion from w to z mall, an be foun by tanar ontrol optimization tool Popular ontrol eign metho that upport tanar problem optimization omprie: LQG (or H 2 ) metho (Zhou et al, 1996), H metho (Zhou et al,1996), L 1 metho (Dahleh an Pearon, 1987) an µ metho (Zhou et al, 1996) Uually, the moel G() will ontain the plant moel itelf, but it an alo ontain moel of iturbane, meaurement noie, time variation, non-linearitie an unmoelle ynami Hene, making the tranfer funtion from w to z mall enure a number of performane an robutne propertie The everyay operation of uh a feebak ytem epen on reliable atuator an enor However, in mot inutrial environment atuator an enor an fail One way to moel thi i epite in Figure 2 Here, the meaurement ue by the ontroller are y y + f rather than y an the ontrol ating on the plant are u + f a rather than u For example y + f or u + f a an repreent ompletely efetive enor or atuator, repetively For afety-ritial proee in partiular, fault mut be ientifie an ation taken immeiately Two main path of ation an be taken: either the ontrol eign algorithm an be moifie to tolerate minor error; or, uing Figure 1 Control ytem in tanar ontrol onfiguration Figure 2 Control ytem with atuator an enor fault 139

4 Sytem for the ontrol an etetion of atuator/enor fault Jakob Stoutrup, MJ Grimble an Henrik Niemann Senor Review an etimator, the faulty ignal an be ientifie an ation taken by the operator or a uperviory ytem In mot appliation, the latter woul be preferable A metho i now eribe whih allow for either, or both, approahe to be inorporate into a ingle eign tep that alo omprie the ontroller eign Thi i ahieve uing a ingle moule whih generate both the ontrol ation an the fault etimate To ientify iniviual fault uefully, it i eential to have reliable fault moel One way to eribe fault moel i to introue frequeny weighting on the fault ignal: f a W a ()w a an f W ()w where w a an w are ignal that are antiipate to have flat power petra (white noie) Thee are imaginary ignal with the ole purpoe of generating the frequeny-oloure ignal f a an f The moule to be eigne houl, in aition to the ontrol ignal u, alo generate a ignal ontaining etimate of potential fault: fa uf ˆ fˆ Thi ituation i epite in Figure 3 The final tep i to efine a fault etimation error z f a: fa zf uf f Uing thee ignal, a new augmente tanar problem an be etablihe a hown in Figure 4 Defining: x w ξ (2) u x w w u z z a, a,,, y y + f uf zf x w the following new tanar problem i obtaine in tate pae form: ξ A ξ + Bw 1 + B2u z C ξ D11w D12u y C ξ + D w + D u or in tranfer funtion matrix form: z w G G w G y u () 11 12() () G () G () u (3) (The expliit formulae are given below) Uing tanar ontrol optimization metho, a generalize ontroller u K()y for the iagram hown in Figure 5 an now be ompute, whih will be able to generate both ontrol ignal an fault etimate In the following, the olution to the tanar problem epite in Figure 5 will be given, an the interpretation of that olution will be iue H optimization i well uite to thi problem, beaue thi metho provie valuable lue to the proper eletion Figure 4 Stanar moel; for integrate ontrol an FDI Figure 5 Stanar problem for ontrol optimization Figure 3 Control ytem with fault an iagnoti 14

5 Sytem for the ontrol an etetion of atuator/enor fault Jakob Stoutrup, MJ Grimble an Henrik Niemann Senor Review of weighting matrie, whih i ruial for the problem oniere However, the main obervation, whih i a type of eparation priniple, will hol for any riteria of the form: z < 1 z f < 1 ubjet to boune et of iturbane an fault ignal The nominal ae Uing the partition (1), the following expreion for the tanar problem that i equation (3) (epite in Figure 4 an 5) an be erive: z z w G G w zf G y u () 11 12() () G () G 21 22() u y w G 11() G12() Wa() G12() w Wa() a Iw W( ) u G () G () Wa() W() G () u f Introuing the ontrol law u K()y, the following loe-loop formula an be obtaine: w z Tzw wa zf () w where Tzw() G () G () () K I G () () K G ( 22 ) 21() G () G () Wa() Wa () W() G () + I K 12 () ( I ( G22() ) K () ) G () G () W () W () The tranfer matrix G() will often be table owing to inner loop whih are inlue in the tanar ontrol moel The following analyi an be arrie out for untable tanar moel too, but for impliity G() will be aume table below In thi ae, the YJBK parameterization (Youla et al, 1971) of all tabilizing ontroller an be obtaine imply by making the ubtitution: Q () K () I G () () K, ( 22 ) ( 22 ) K () Q () I+ G () Q () ( a ) Partitioning the ontrol enitivity funtion Q() a 141 Qq Q () () Q () the following expreion i obtaine: Now, the ruial obervation in thi expreion i that eah of the two row of the blokpartitione matrix epen on only one of the Q i, i {1, 2} Thi ha the following two onequene: (1) Making the loe-loop tranfer funtion aoiate with the ontrol objetive mall an making the loe-loop tranfer funtion aoiate with the FDI objetive mall an be ahieve inepenently (2) Optimizing inepenently eliminate ome of the onervatim often introue in optimization metho that optimize the norm of a tranfer matrix built by taking tranfer matrie orreponing to ifferent riteria Thi poibility of a eparation priniple hall be exploite in the eign proeure below A eparation priniple imilar in pirit to thi i eribe in Stoutrup an Niemann (1997) Sine the upper row partition of T zw () epen only on Q 1 () an the lower row partition epen on Q 2 (), the tranfer funtion T zw an be optimize by iniviually optimizing the ifferent blok term Hene, after eparating the optimization for z an z f, we are fae with the problem of optimizing the following two tranfer matrie inepenently: ( 2 Tzw() G 11() + G12() Q1() G21() G12()( I + Q1() G22()) Wa() Wa () Q2( ) G21( ) Q2 ( ) G22 ( ) Wa ( ) G () Q () W() 12 1 Q ( ) ( ) 2 W W() G11() + G12() Q1() G21(), G12()( I + Q1() G22()) (4) Wa(), G12() Q1() W() ) an W () a Q2( ) G21( ), Q2 ( ) G22 ( ) Wa ( ), (5) () Q ( ) W ( ) 2 W The tanar ontrol problem orreponing to equation (4) an (5) are in a form known a the moel-mathing problem, whih i a impler, peial ae of the o-alle general four-blok ontrolle problem (ee, for example, Zhou et al, 1996)

6 Sytem for the ontrol an etetion of atuator/enor fault Jakob Stoutrup, MJ Grimble an Henrik Niemann Senor Review The tanar problem formulation orreponing to equation (4) i: z y where u Q1 i the output of the Q 1 () partition an y Q1 i the input to the Q 1 () ubytem For equation (5), the aoiate tanar problem i: (6) (7) where u Q2 i the output of the Q 2 () partition an y Q2 i the input to the Q 2 () ubytem Given Q 1 an Q 2, the olution to the tanar problem of equation (3) i: where K 1 () an K 2 () the feebak ontrol part an the FDI part, repetively an be ompute a: K () Q ()( I + G () Q ()) 1 (8) an Q1 z y G11 G12 Wa G12 () () () () G21() G22() Wa() W() f Q2 Wa() W() G () G () W () W () K ( ) K ( ) 1 K ( ) ( ) K2() Q2()( I + G22() Q1()) Q2() I + G22() Q1()( I + G22() Q1()) Q ()( I G () K ()) I w wa w uq2 (9) Remark 1 It i important to note that the expreion that i equation (8) for K 1 oe not epen on Q 2 but only on Q 1 whih i foun by an optimization whih alo oe not epen on Q 2 Thi mean that, in thi formulation of the problem, the ontrol ation oe not iretly epen on the fault etimator ynami Still, the regulating ontroller an be etune ompare with a et-up in whih fault are not allowe for, ine the ontrol eign algorithm regar the fault a iturbane an noie (a an be een from equation (6)) Where thi i not eirable, ome attention mut be pai to the weighting eletion heme to avoi etuning Alternatively, the optimization problem of equation (4) an be ompletely reformulate by virtue of the eparation priniple eribe above Equation (9) for K 2 epen on Q 1 Thi i phyially obviou, ine the fault etetion an iolation filter ha to ue the oberver part of the ontroller to ientify the fault a w wa w uq1 142 Relationhip to the four-parameter ontroller The four-parameter ontroller wa introue by Nett et al (1988) in onnetion with fault etetion The four-parameter ontroller an be oniere a an extenion of the twoparameter ontroller introue above Let the plant till be given by equation (2) The four-egree-of-freeom ontroller alo ha ae to a referene ignal t a well a the meaurement ignal y, an the ontroller return both a ontrol ignal u an a iagnoti ignal a: u K a K The eign et-up for the four-parameter ontroller an alo be formulate uing the tanar ytem eription given in Figure 4 The generalize ytem G nffp () i then given by: G (1) A above, aume that the ytem G nffp () i open-loop table Then we an again ue the following parameterization of all tabilizing ontroller: G 22 Q () K ()( I K ()) G K () Q ()( I+ Q ()) 22 1 Again, let Q() be partitione a: Q Q Q () (11) Q21 Q22 Uing Q(), the loe-loop tranfer funtion T nffp i then given by: T nffp nffp K K y t G G W G a 12 W a I () W G21 G22Wa W G22 I G 11 + G12Q11G21 G12( I + Q12G22) () W a Wa Q21G21 G12Q12W G12Q 12 W Q W Q (12) (13) Again, note that there i a eparation Q 11 an Q 12 appear only in the firt row of T nffp an Q 21 an Q 22 appear only in the eon row Bae on the Q ontroller, we an alulate the K ontroller by uing equation (11)The ontroller K then take the following form:

7 Sytem for the ontrol an etetion of atuator/enor fault Jakob Stoutrup, MJ Grimble an Henrik Niemann Senor Review Q ( I + G Q ) ( I + Q G ) Q K () 12 Q21( I + G22Q11) Q22 Q21( I + G22Q11) G 22 Q 12 Q ( I + G Q ) ( I K G ) Q Q ( I G K ) Q Q K G Q A in the previou etion, there i a eparation between ontrol an fault-etetion objetive Note that in the implementation, the pure ontrol part K 11 epen on Q 11 only, K 12 epen both on Q 11 an Q 12, K 21 epen on Q 11 an Q 21 an K 22 epen on all element of Q Thi four-parameter ontroller et-up ha been analye by Nett et al (1988) in the nominal ae The et-up applie in Nett et al (1988) i lightly ifferent from that ue in thi paper The eign et-up for the fourparameter ontroller ha not been formulate in the tanar et-up a in Figure 4 One onequene i that the eparation in the ontroller eign oe not appear in the parameterization ue in Nett et al (1988) A eign heme ha been arrie out by Nett et al (1988) bae on the o-alle ingle ontroller priniple role An H olution to the nominal problem To obtain expliit eign formulae, the riterion of optimization nee to be more peifi (14) For a number of purpoe, H optimization i a goo hoie, ine it ontitute a flexible loop-haping tool By appropriately eleting the weighting, we an aume without lo of generality that we are fae with normalize H ontraint, in whih ae equation (4) an (15) take the form: 1 I G where Z 1 () i a quare matrix whih i invertible a an element of RH, an J 1 i a ontant matrix of the form I J1 I with a uitable number of 1 an 1 J 1 i alle the ignature matrix of 1 The moel-mathing problem equation (6) ha a olution if an only if the following ontroller i tabilizing: I I Q1 I Z1 1 I Z1 1 ( ) (16) Moreover, in that ae, all olution are given by: Q YX ( ) where X 1 Z Y 1 an A 1 an B 1 are (free) table rational matrie, et A 1 having all it root in the open left half omplex plane, atifying: AA 12 ZJZ G21G21 G22W W G a a W 22 W G11G 21 G12W W G a a 21 G 21G 11 G21W W G a a 12 I G11G 11 G12WaW ag 12 I G ( ) BB A B an ( G11() + G12() Q1() G21(), G12()( I + Q1() G22()) Wa(), G12() Q1() W() ) < 1 W () a Q2( ) G21( ), Q2 ( ) G22 ( ) Wa ( ), W () Q2 ( ) W ( ) < 1 (15) The only remaining tep in eviing an algorithm for the omputation of the ombine ontrol an FDI evie i to olve the two inequalitie of equation (14) an (15), whih have the two tanar formulation of equation (6) an (7) Uing polynomial H theory (ee Kwakernaak, 1993), the following reult are obtaine Lemma 1 Conier the following J-petral fatorization: 143 Lemma 2 Similarly, for equation (7), onier the following J-petral fatorization: G G G22WaW 22 ag WW Π 2 WWG a a 22 WW G22W awa WW I W W Z J Z a a I W W where Z 2 () i a quare matrix whih i invertible a an element of RH an J 2 i the ignature matrix of 2 The moelmathing problem equation (7) ha a olution if an only if the following ontroller i tabilizing:

8 Sytem for the ontrol an etetion of atuator/enor fault Jakob Stoutrup, MJ Grimble an Henrik Niemann Senor Review I I Q2 I Z2 1 ( ) I Z2 1 ( ) Moreover, all olution are given by: Q2 Y2X2 1 ( 19) where: X 2 Z Y 2 an A 2 an B 2 are (free) table rational matrie, et A 2 having all it root in the open left half omplex plane, atifying: A A B B A B ( 18) Employing the eparation priniple eribe above, an ombining lemma 1 an 2, the main reult an be tate Theorem 3 Conier the et-up epite in Figure 3 where K() i a ombine ontroller an FDI moule The following two tatement are equivalent: (1) There exit a tranfer matrix K() making the tranfer funtion from iturbane to ontrolle output maller than 1, an making the tranfer funtion from atuator an enor fault to the fault etimation error maller than 1 (2) The ontroller Q 1 given by equation (16) tabilize the tanar problem given by equation (6) an, likewie, the ontroller Q 2 given by equation (18) tabilize the tanar problem given by equation (7) Moreover, when thee onition are K atifie, a poible hoie of K () i given by (8) an (9) where Q () 1 K2 1 an Q2( ) are given by (17) an (19), repetively In onluion, thi etion ha hown how an algorithm an be ue for eigning a ingle moule whih omprie both feebak ontrol ation an fault iagnoi an iolation The eign metho i very flexible Manipulating weight, the following four objetive an be expliitly eigne for: (1) following referene an rejeting iturbane robutly; (2) ontrolling the ytem o that unetete fault o not have iatrou effet; (3) reuing the number of fale alarm; (4) ientifying whih fault have ourre Thee objetive are iue in more etail below 144 The algorithm wa bae on a type of eparation priniple whih failitate tranpareny in the eign proe with repet to the funamental trae-off relate to iagnoing an ontrolling a ytem Not only have the proee of eigning a filter an a ontroller been eparate, but alo the eign riteria Thi how that the ontroller oe not nee to be etune to implement a oun fault etetion mehanim Moreover, thi tatement hol for optimization with repet to any hoie of (norm-bae) eign riteria, formulate a one riterion for the ontroller an another for the filter Deign of filter for unertain ytem In the previou etion, the interepenene of the ontroller an the filter eign in the nominal ae have been examine It wa iovere that a eparation exit between ontroller eign an filter eign In thi etion, the way in whih the preene of unertainty affet the reult will be oniere Firt, onier the tanar ontrol onfiguration in Figure 1 where the ytem G i a funtion of an unertain blok, ie G G(, ) The blok repreent the unertain or unmoelle part of the ytem The unertain part of the ytem i normally eribe a aitive moel unertainty given by: G () G() + () or a multipliative moel unertainty given by: G () G()( I+ ()) where G () i the nominal ytem (ee Skogeta an Potlethwaite (1996) for more on moel unertaintie) By inluing a moel unertainty in the et-up of Figure 1, the tanar ontrol onfiguration for robut ontrol a hown in Figure 6 i obtaine The ytem G () in Figure 6 an be eribe in either the tate pae formulation or in tranfer funtion form The tate pae eription i given by: x A B B B e C D D D z C D D D y C D D D x w u Let u onier the et-up from Figure 4 an inlue a moel unertainty Before the tanar et-up for integrate ontrol an FDI for ytem with moel unertaintie are given, a

9 Sytem for the ontrol an etetion of atuator/enor fault Jakob Stoutrup, MJ Grimble an Henrik Niemann Senor Review Figure 6 Control ytem in tanar ontrol onfiguration for ytem with moel unertainty Figure 7 Generalize et-up for robut ontrol an fault etetion with a 2 parameter ontroller ompat notation i introue for the fault ignal in orer to implify the equation in the ret of thi paper Let the fault ignal vetor f be given a: fa f f Further, the weighte fault ignal f i given by: Wa Wa f Wf wf () () W () W By introuing thi notation, the et-up from Figure 4 with moel unertainty i illutrate in Figure 7 It i aume that i ale in uh a way that 1 ω Further, an be truture or untruture The tranfer funtion from w to z efine the performane of the loeloop ontrol ytem, an the tranfer funtion from w f to z f efine the performane for the fault etetion filter The generalize ytem G rf () in Figure 7 i given by: e z w Grf () zf wf y u where: Conier Figure 8, where the p an f blok repreent performane peifiation for the loe-loop tranfer funtion an performane for the fault etetion ignal Introution of uh fititiou perturbation blok i a tanar evie in µ ynthei to obtain robut performane (ee, for example, Zhou et al (1996)) It i aume that weighting matrie on the performane peifiation in Figure 8 are inlue in the generalize ytem G rf () Applying the ame tehnique a in the nominal ae by uing a parameterization of the ontroller, the following loe-loop tranfer funtion T rf i obtaine: Figure 8 Generalize et-up for robut ontrol an fault etetion with performane peifiation repreente by fititiou perturbation blok G rf G e Gew G W G ew f f eu Gz G G W G zw zw f f zu () Wf Gy Gyw Gyw Wf G f yu I (2) In omparion to the ytem ue in The nominal ae etion earlier in the paper, the introution of the unertainty blok hange the poible eign onept onierably, a will be emontrate 145

10 Sytem for the ontrol an etetion of atuator/enor fault Jakob Stoutrup, MJ Grimble an Henrik Niemann Senor Review T rf G + G QG 1 G + G QG 1 G + G QG G + G QG 1 1 QG 2 y QG 2 yw e eu y ew eu yw z zu y zw zu yw G W + G QG W ew f f eu 1 yw f f G W + G QG W zw f f zu 1 yw f f W Q G W f 2 yw f f ( 21) Figure 9 Generalize et-up for robut ontrol with performane peifiation At firt glane, it eem there i again a eparation between the two parameter in Q() Q 1 () appear only in the firt two row an Q 2 appear only in the lat row of T rf However, beaue the feebak loop with, p, an f i oniere iretly in the eign proe, there i no eparation in thi ae ue to the moel unertaintie Thi an be een quite eaily by oniering a eparate eign of a robut feebak ontroller Q 1 an a robut fault etetion filter Q 2 (ee below) Firt, let u onier the eign of a robut tabilizing ontroller followe by a eign of a nominal filter The feebak ontroller eign problem with repet to robut tability i repreente by the following loe-loop tranfer funtion: T () G + G QG 1 ( 22) r e eu y The eign problem i a tanar ontrol optimization problem when i untruture Otherwie, it i a truture optimization problem like a µ eign problem The following eign of a nominal filter i repreente by the following loe-loop tranfer funtion: T () Q G W Q G W ( ) ( ) npf 2 yw f 2 yw f f 23 Here, there i again eparation between the two eign beaue Q 1 appear only in equation (22) an Q 2 only in equation (23) The next eign ae onit of a eign for a feebak ontroller with repet to robut performane followe by the eign of a nominal filter The eign problem for the feebak ontroller i repreente by the following loe-loop tranfer funtion: Ge + GeuQGy Gew + G QG 1 eu 1 yw Trp G G QG G G QG (24) z+ zu 1 y zw + zu 1 yw eign of a feebak ontroller i repreente by equation (22), ie eigne with repet to robut tability The eign problem for a filter with repet to moel unertainty i repreente by the following tranfer funtion: T rpf G e + GeuQG 1 y Gew + G QG eu 1 yw + QG 2 y QG 2 yw Gew Wf + GeuQGyw W f 1 f f ( 25) Wf Q Gyw W 2 f f The filter eign problem i alo a µ eign problem owing to the truture in the (partly fititiou) perturbation (ee Figure 1) Figure 1 Generalize et-up for the eign of a robut fault etetion filter Thi eign problem i a µ eign problem owing to the truture in the perturbation (ee Figure 9) A expete, only the feebak parameter Q 1 appear here The eign of the nominal filter i till given by equation (23); there i alo a eparation between the two eign In the lat two eign ae, the filter i eigne with repet to robut performane In the firt ae, the 146

11 Sytem for the ontrol an etetion of atuator/enor fault Jakob Stoutrup, MJ Grimble an Henrik Niemann Senor Review From equation (25) it an be een that the two eign are no longer eparate Both ontroller appear in the eign problem in equation (25) Another way of eeing that the two eign are ouple i to onier the loe-loop tranfer funtion given by: Tzfwf () Fu ( Trpf, ) W Q G W Q G f 2 yw f f 2 y ( I ( Ge + GeuQG 1 y ) ) ( G W + G QG W ) ew f f eu 1 yw f f It i lear from the above equation that in general the eign of Q 1 epen on Q 2 Generially, it will never be poible to eparate the eign of the two ontroller owing to the feebak with the unertainty blok The latter eign ae entail the eign of a feebak ontroller with repet to robut performane repreente by equation (24) followe by the filter eign from equation (25) One again there i no eparation between the two eign Summary of optimal eign tehnique Depening on whether robutne i oniere important in the eign of the ontrol loop an of the filter, ix lae of eign methoology an be ientifie Thee lae are haraterize in Table I Remark 2 The entry for the filter robut performane ubjet to the ontroller nominal eign preume that the ontrol loop will atually be table, ine the filter annot tabilize an untable ontrol loop Moreover, in thi ae, no optimization metho exit that iretly hanle the oupling an give the ontroller an filter in one eign tep The uggete metho i a reaonable ub-optimal approah Remark 3 The H optimization for filter eign lite in Table I i bae on an aumption that f i untruture More realitially, f will have a blok iagonal truture, orreponing to iniviual fault It i traightforwar to inorporate thi into the eign proeure Solving the H problem epite in Figure 5 implie making ix tranfer funtion mall owing to the efinition (2) of w an z Thee ix tranfer funtion are: (1) T zw tranfer funtion from external iturbane to inferre output; 147 Table I The ix poible eign lae Nominal performane Robut performane of filter of filter Nominal (1) Separate eign (1) Couple eign performane Remark 2 of ontroller (2) H optimization ee (2) H (ontroller) an µ Remark 3 (filter) optimization (3) Equation (14) an (15) (3) Equation (14) (H optimization) an equation (25) (fin Q 2 by µ optimization) Robut (1) Separate eign (1) Couple eign tability of (2) H optimization ee (2) µ optimization ontroller (3) Equation (22) an (23) (3) Equation (25) Robut (1) Separate eign (1) Couple eign performane (2) µ (ontroller) an H (2) µ optimization of ontroller (filter) optimization ee Remark 3 (3) Equation (24) (µ (3) Equation (21) optimization an equation (23) (H optimization) Note: For eah la, item (1) eribe whether the eign of the ontroller an filter eparate; item (2) peifie the metho of optimization require; an item (3) peifie the relevant equation to be ue for the optimization proe (2) T zwa tranfer funtion from atuator fault to inferre output; (3) T zw tranfer funtion from enor fault to inferre output; (4) T zf tranfer funtion from external w iturbane to fault etimation error; (5) T zf tranfer funtion from atuator wa fault to fault etimation error; (6) T zf tranfer funtion from enor w fault to fault etimation error The eential intrument for reating a wellfuntioning moule for ontrol ation, fault etetion an iolation i an optimization algorithm whih make thee tranfer funtion mall an trae off the iniviual funtion by areful weighting eletion Making eah of the ix tranfer funtion mall ha it own (important) interpretation: (1) making T zw mall implie goo iturbane rejetion an robutne, ie the original ontrol objetive are ahieve; (2) making T zwa an T zw mall implie that unetete fault o not aue iater; (3) making T zf mall implie that w iturbane are not reaily interprete a

12 Sytem for the ontrol an etetion of atuator/enor fault Jakob Stoutrup, MJ Grimble an Henrik Niemann Senor Review fault, ie the rik of fale alarm i reue; (4) making T zf wa an T zf mall w implie that u f beome a oun etimate of potential atuator an enor fault From the reult in thi paper, not only i it lear that objetive (1) ha to be trae off againt objetive (2), an that objetive (3) ha to be trae-off againt objetive (4), but alo the eign proe oe not involve a trae-off between the two pair of objetive, one the tanar moel in equation (1) ha been peifie In orer not to ompliate the explanation, the ontrol weight relate to ontrol performane an ontrol robutne have not been expliitly inlue, but they are, of oure, preent in term of the original tanar problem formulation epite in Figure 1 It i evient that the hoie of the original ytem internal weighting i very ignifiant in etermining the overall performane of the ombine ontrol an FDI moule In orer for the optimization in Theorem 3 to give a ueful reult, it i of great importane to hooe the weighting aoiate with the original tanar problem, with the atuator fault an with the enor fault, o that all thee weighting are eparate in a frequeny range Chooing large weight for the iturbane moel mean that the eign algorithm allow for iturbane rejetion an ontrol robutne Alternatively, hooing large weight for the atuator- an enor-fault moel mean that the quality of the fault etimate i emphaize in the eign algorithm, whih enure that very few fault are unetete A mentione in Remark 1, the fault are oniere to be iturbane in the ontrol ub-problem (6) an, ually, the iturbane are impliitly repreente in the etetion ubproblem (7) in term of the tanar problem parameter There i no prinipal limitation in the eign metho uggete in thi paper whih forbi two ifferent tanar problem with ifferent internal weighting to be applie to the ontrol eign ub-problem (6) an the FDI eign ub-problem (7); neverthele, thi woul ompliate the eign proe an the weighting woul have to be well motivate by the peifi appliation 148 Remark 4 It houl be note that fixe weighting for external iturbane have been oniere in thi paper (in fat, they have been aume to be aborbe in the original tanar formulation) Thi i a reaonable point of view for ytem in whih the frequeny petrum of external iturbane i quite well known However, it houl be mentione that ome reearher alo like to onier iturbane weighting a eign parameter, uh that the banwith of the ytem i optimize iniretly In thi ae, the eoupling reult for the nominal ae are mileaing, ine there ha to be a trae-off between thee weighting an thoe of the fault Conluion The integration of feebak ontroller an fault-etetion filter eign ha been oniere for ytem with an without moel unertaintie It ha been foun that the eign of the feebak ontroller an the fault etetion filter an be eparate in the nominal ae In the unertain ae, however, an optimal olution annot be obtaine by eparate robut ontroller an robut filter eign In pite of the eparation in the nominal ae, the integrate eign of both the feebak ontroller an the fault etetion filter i till a oun option, beaue it i poible to ue the ame oberver for both the feebak ontroller an the filter Thi ha been arrie out by Kilgaar et al (1996), where a fourth-orer SIMO ytem ha been oniere an an H eign bae on LMI ha been unertaken The reult wa an integrate feebak ontroller an a fault etetion filter of orer one In the unertain ae, it i eential to traeoff ontrol performane, the effet of unetete fault, the rik of fale alarm an quality of fault etetion arefully In mot ae, thi probably require µ ynthei metho Referene Ajbar, H an Kantor, JC (1993), An l approah to robut ontrol an fault etetion, Proeeing of the Amerian Control Conferene, San Franio, CA, pp Blanke, M, Bøgh, SA, Jørgenen, RB an Patton, RJ (1995), Fault etetion for a ieel engine atuator a benhmark for FDI, Control Engineering Pratie, Vol 3, pp

13 Sytem for the ontrol an etetion of atuator/enor fault Jakob Stoutrup, MJ Grimble an Henrik Niemann Senor Review Bokor, EAJ an Kevizky, L (1994), A H filtering approah to robut etetion of failure in ynamial ytem, Proeeing of the 33r Conferene on Deiion an Control, Lake Buena Vita, FL, pp Dahleh, MA an Pearon, JB (1987), L 1 -optimal feebak ontroller for ontinuou-time ytem, IEEE Tranation on Automati Control, Vol 32 No 1, pp Frank, PM (199), Fault iagnoi in ynami ytem uing analyti an knowlege-bae reunany a urvey an ome new reult, Automatia, Vol 26, pp Frank, PM an Ding, X (1994), Frequeny omain approah to optimally robut reiual generation an evaluation for moel-bae fault iagnoi, Automatia, Vol 3, pp Garia, EA, Køppen-Seliger, B an Frank, PM (1995), A frequeny omain approah to reiual generation for the inutrial atuator benhmark, Control Engineering Pratie, Vol 3, pp Grainger, RW, Holt, J, Iakon, AJ an Ninne, BM (1995), A parametri tatitial approah to FDI for the inutrial atuator benhmark, Control Engineering Pratie, Vol 3, pp Jørgenen, RB, Patton, RJ an Chen, J (1995), An eigentruture aignment approah to FDI for the inutrial atuator benhmark tet, Control Engineering Pratie, Vol 3, pp Kilgaar, S, Rank, ML, Niemann, HH an Stoutrup, J (1996), Simultaneou eign of ontroller an fault etetor, Proeeing of the 35th IEEE Conferene on Deiion an Control, Kobe, Japan Kwakernaak, H (1993), Robut ontrol an H -optimization, Automatia, Vol 29 No 2, pp Magni, JF an Mouyon, P (1991), A generalize approah to oberver for fault iagnoi, Proeeing of the 3th Conferene on Deiion an Control, Brighton, pp Mangoubi, RS, Appleby, BD, Verghee, GC an VanerVele, WE (1995), A robut failure etetion an iolation algorithm, Proeeing of the 34th Conferene on Deiion an Control, New Orlean, LA, pp Mura, GA, Potlethwaite, I an Gu, D-W (1996), A robut eign approah to integrate ontrol an iagnoti, Proeeing of the 13th Worl Congre of IFAC, Vol N, San Franio, CA, pp Nett, CN, Jaobon, CA an Miller, AT (1988), An integrate approah to ontrol an iagnoti: the 4-parameter ontroller, Proeeing of the Amerian Control Conferene, pp Patton, RJ an Chen, J (1991), Robut fault etetion uing eigentruture aignment: a tutorial onieration an ome new reult, Proeeing of the 3th Conferene on Deiion an Control, Brighton, pp Patton, R, Frank, P an Clark, R (1989), Fault Diagnoi in Dynami Sytem Theory an Appliation, Prentie-Hall, Englewoo Cliff, NJ Qiu, Z an Gertler, J (1993), Robut FDI ytem an H optimization, Proeeing of the 32n Conferene on Deiion an Control, San Antonio, TX, pp Skogeta, S an Potlethwaite, I (1996), Multivariable Feebak Control Analyi an Deign, Wiley, New York, NY Stoutrup, J an Niemann, HH (1997), Multi objetive ontrol for multivariable ytem with mixe enitivity peifiation, International Journal of Control (forthoming) Tyler, ML an Morari, M (1994), Optimal an robut eign of integrate ontrol an iagnoti moule, Proeeing of the Amerian Control Conferene, Baltimore, MD, pp 26-4 Wang, YY an Wu, NE (1993), An approah to onfiguration of robut ontrol ytem for robut failure etetion, Proeeing of the 32n Conferene on Deiion an Control, San Antonio, TX, pp Youla, DC, Jabr, HA an Bongiorno, JJ (1971), Moern Wiener-Hopf eign of optimal ontroller, part II, IEEE Tranation on Automati Control, Vol 21, pp Zhou, K, Doyle, JC an Glover, K (1996), Robut an Optimal Control, Prentie-Hall, Englewoo Cliff, NJ 149