Accommodation Rule Based on Navigation Accuracy for Double Faults in Redundant Inertial Sensor Systems

Transcription

1 Internatonal Accommodaton Journal o Rule Control, Based Automaton, on Navgaton and Systems, Accuracy vol. or Double 5, no. 3, Faults pp , n Redundant June 007 Inertal Sensor Systems 39 Accommodaton Rule Based on Navgaton Accuracy or Double Faults n Redundant Inertal Sensor Systems Cheol-Kwan Yang and Duk-Sun Shm* Abstract: hs paper consders a ault accommodaton problem or nertal navgaton systems (INS) that have redundant nertal sensors such as gyroscopes and accelerometers. It s wellknown that the more sensors are used, the smaller the navgaton error o INS s, whch means that the error covarance o the poston estmate becomes less. hus, when t s decded that double aults occur n the nertal sensors due to ault detecton and solaton (FDI), t s necessary to decde whether the aulty sensors should be excluded or not. A new accommodaton rule or double aults s proposed based on the error covarance o trad-soluton o redundant nertal sensors, whch s related to the navgaton accuracy o INS. he proposed accommodaton rule provdes decson rules to determne whch sensors should be excluded among aulty sensors. Monte Carlo smulaton s perormed or dodecahedron conguraton, n whch case the proposed accommodaton rule can be drawn n the decson space o the twodmensonal Cartesan coordnate system. Keywords: Accommodaton rule, ault accommodaton, ault detecton and solaton, nertal sensors, party equaton.. INRODUCION oday all sorts o control, navgaton and communcaton systems consst o varous subsystems and thus the hardware and sotware structure o those systems are complcated. hereore the mportance o relablty o the whole system has been ncreased. he relablty o the whole system can be obtaned by the ault detecton and solaton (FDI) method and ault accommodaton ater FDI. FDI methods have been studed rom the 960 s n varous engneerng problem areas. As reported n lterature such as survey papers [,] and books [3,4], varous methods o FDI have been studed and appled n dverse applcatons. FDI algorthms are desgned to use all redundant normaton o the plant and sensors. Redundancy s broadly classed as hardware redundancy [5-] and analytc redundancy [4,3-6]. Wth hardware redundancy, more than the mnmum number o sensors s used. For example, two or more sensors are Manuscrpt receved January, 006; revsed November 6, 006; accepted February 3, 007. Recommended by Edtoral Board member Hyo-Choong Bang under the drecton o Edtor Jae Weon Cho. hs work was supported by Grant (R ) rom the Basc Research Program o the Korea Scence & Engneerng Foundaton. Cheol-Kwan Yang and Duk-Sun Shm are wth the School o Electrcal and Electroncs Engneerng, Chung-Ang Unversty,, HukSuk-Dong, Dongak-Ku, Seoul , Korea (e-mals: ckyang9@empal.com, dshm@cau.ac.kr). * Correspondng author. used or scalar varables, and our or more or vector varables. Inertal navgaton systems (INS) use bascally three accelerometers and gyroscopes to calculate navgaton normaton such as poston, velocty and atttude. o obtan relablty and to enhance navgaton accuracy, INS may use redundant sensors. Numerous studes on FDI or redundant sensors have been perormed so ar. here are many papers or FDI such as look-up table [5], squared error (SE) method [5], generalzed lkelhood test (GL) method [6] and optmal party test (OP) method [7] or hardware redundancy. Wth analytcal redundancy, addtonal normaton s obtaned rom a system s mathematcal model. hs type o redundancy s based on the dea that nherent redundancy exsts n the dynamc relatonshp between nputs and outputs o the system model. Analytcal redundancy has been studed n many applcatons such as aerospace systems, publc transportaton vehcles, and nuclear power plants. Frank has revewed the state-o-the-art FDI n automatc processes by usng analytcal redundancy [5], and Betta et al. revewed several analytcal redundancy-based technques []. One way o detectng a ault s to x a ault threshold, and the ault estmate s greater than the threshold, t s determned that a ault has occurred. One typcal way to obtan a ault threshold s by usng the probablty o alse alarm. Measurement nose s usually assumed as whte Gaussan. I there s no ault, then the probablty densty uncton o the ault

2 330 Cheol-Kwan Yang and Duk-Sun Shm estmate has Gaussan dstrbuton. hus the probablty o alse alarm s gven, the correspondng ault threshold can be determned. [7] determnes a ault threshold usng both alse alarm n the absence o ault and mss detecton n the presence o ault. [8] suggests a new accommodaton threshold based on the error covarance o an estmated varable, whch s related to the navgaton accuracy o INS. he more sensors are used, the better the navgaton perormance o INS s, whch means that the error covarance o the poston estmate becomes less. hus when INS uses redundant nertal sensors t may happen that even though there s a ault, the aulty sensor should be used so as not to lose the navgaton accuracy o INS. he accommodaton threshold gves a decson rule to determne whether a aulty sensor should be excluded or not. I a ault s less than the accommodaton threshold, t s sad to be a tolerable ault and should not be excluded. I the ault s greater than the accommodaton threshold, t s sad to be a non-tolerable ault and should be excluded. hs paper suggests a new accommodaton rule or double aults n redundant nertal sensors, whle [8] suggests an accommodaton threshold or a sngle ault n redundant nertal sensors. So ar there are no results or the accommodaton rule or the double ault case n the lterature. For the double ault case, the accommodaton rule s gven as a regon n the twodmensonal space, whle the accommodaton rule s gven as a threshold or the sngle ault case.. FAUL DEECION, ISOLAION, AND ACCOMMODAION (FDIA) Consder a typcal measurement equaton or redundant nertal sensors. mt () = Hxt () + () t + ε(), t () where n mt () = [ m m mn ] R : nertal sensor measurement, H = [ h h n ] : n 3 measurement matrx wth rank ( H )=3, 3 x() t R : trad-soluton(acceleraton or angular rate), [ ] () t = R : ault vector, n ε()~ t N(0 n,σ I n): a measurement nose vector, normal dstrbuton (whte nose), Nxy (,): Gaussan probablty densty uncton wth mean x and standard devaton y. A party vector pt ( ) s obtaned usng a matrx V as ollows: n pt () = Vmt () = V() t + Vε(), t () Fg.. FDI and Accommodaton or INS wth redundant sensors. where the matrx V satses ( n 3) n = 0( ) and VV = I V = [ v v v n ] VH V R,. (3) Algorthms to obtan the matrx V above can be ound n the lterature [5-7]. ermnology denton s gven as ollows [4]. Fault detecton: the ndcaton that somethng s gong wrong n the system. Fault solaton: the determnaton o the exact locaton o the ault. Fault dentcaton: the determnaton o the sze and type or nature o the ault. Fault accommodaton: the reconguraton o the system usng healthy components. Fg. shows the block dagram o the FDIA (ault detecton, solaton and accommodaton) procedure n nertal navgaton systems. From the sensor measurement, a party equaton s generated, and FDIA s perormed. rad solutons are calculated by the least square method and entered nto the navgaton equatons. he navgaton accuracy depends on the estmaton error o the trad solutons,.e., acceleraton or angular rate. In ths paper only ault accommodaton s consdered. hree assumptons are made as ollows. Assumpton : Any three sensors are not on the same plane. Assumpton : All sensors have the same nose characterstcs,.e., same standard devaton σ o whte Gaussan nose. Assumpton 3: Fault detecton, solaton, and dentcaton are perormed n advance. 3. ACCOMMODAION RULE FOR SINGLE FAUL BASED ON SYSEM PERFORMANCE Consder measurement equaton (4) the same as (). m= Hx+ + ε, ε ~ N(0, σ I ) (4) rad soluton xˆ = [ xˆx xˆy xˆz] n Fg., whch s acceleraton or angular rate, can be obtaned by the least square method as ollows: n n

3 Accommodaton Rule Based on Navgaton Accuracy or Double Faults n Redundant Inertal Sensor Systems 33 xˆ( t) = ( H H) H m(). t (5) Dene estmaton error o x( t ) as et () = xt ˆ() x(). t Navgaton soluton such as poston, velocty, and atttude s calculated rom xˆ( t ). hus the navgaton accuracy o INS depends on the error covarance Ct ( ) = Eetet [ ( ) ( ) ]. Consder two matrces C+ () t and C () t gven below C () t = () t ( H H) hh ( H H) + + σ ( H H), C () t E[( xˆ () t x())( t xˆ () t x()) t ] = σ ( H W H), (6) (7) where matrces C + and C denote the covarance o ˆx ncludng and excludng the -th sensor respectvely and W s a n n dagonal matrx wth (,) component set to 0 and the other components set to. Lemma [8]: Suppose that -th sensor has ault. For (6) and (7), the ollowng two nequaltes are equvalent: σ ) () t v ) C+ () t C () t 0, where σ and ν are standard devaton o sensor nose and -th column o V matrx, whch satses (3). σ And () t = C+ () t = C (). t v Lemma mples that when the magntude o -th ault s less than σ v, the error covarance o estmate ˆx ncludng -th sensor s less than the error covarance o estmate ˆx excludng t, thus the -th aulty sensor should be used despte ts ault to mprove the navgaton accuracy. From Lemma, we have the excluson threshold σ v as an accommodaton rule. 4. ACCOMMODAION RULE FOR DOUBLE FAULS BASED ON SYSEM PERFORMANCE In ths secton we propose a new accommodaton rule or double aults n redundant sensors. 4.. Navgaton perormance analyss For (), suppose that double aults and occur, whch means that ( t) = [0 0 0 ]. o analyze the navgaton perormance, the error covarance o trad soluton xˆ( t) needs to be calculated. he covarance matrces are dened as ollows. Matrx C + + denotes the error covarance o xˆ( t ) ncludng -th and -th sensor outputs, and C the error covarance o xˆ( t ) excludng -th and -th sensors, and so on or C + and. C Covarance matrx C ++ he error or xˆ( t ) can be calculated as ollows xˆ ++ x = ( H H) { h + h + H ε }, (8) where xˆ+ + = [ xˆ++ x xˆ++ y xˆ++ z]. hen the estmaton error o x can be descrbed as the error covarance matrx C ++ n (9) C E x x x x ++ = ( ˆ++ )( ˆ++ ) = σ ( H H) + ( H H) hh h H H h ( ). 4.. Covarance matrx C he error or ˆx can be calculated as ollows (9) xˆ x = ( H W H) H W ε, (0) where xˆ = [ xˆ x xˆ y xˆ z] and W s a dagonal matrx wth dagonal elements o except (,) component and (,) component whch components are 0. hen the estmaton error o ˆx can be descrbed as the error covarance matrx C n (). where C = E ( xˆ x) ( xˆ x) σ = σ ( H H) + ( H H) hh D () v v v h ( H H), v v v h =, = sn D v v v v v v θ and θ s the angle between two vectors v and v, whch are column vectors o matrx V dened n (3).

4 33 Cheol-Kwan Yang and Duk-Sun Shm 4..3 Covarance matrx C + he error or ˆx can be calculated as ollows F xˆ + x= ( H W H) H W ( V + ε), () where xˆ = [ xˆ xˆ xˆ ] and V = [0 + + x + y + z n 00 0] R wth -th component o, whch results n H WV = h. F hen the estmaton error o ˆx can be descrbed as the error covarance matrx C + n (3) + = ( ) ( ) + σ ( H W H) = ( H W H) hh ( H W H) σ + H H + H H hh H H v C H W H h h H W H σ ( ) ( ) ( ). F (3) 4.. Accommodaton rule or double aults In ths secton three theorems can be obtaned rom the results o Secton 4., whch provde accommodaton rules or double aults. heorem : Consder the measurement equaton () and the trad soluton (5), and suppose that -th and -th sensors have aults. For the two estmaton error covarance matrces (9) and (), the ollowng two nequaltes are equvalent: ) tr( C++ ) < tr( C ) where tr( ) denotes the trace o a matrx. ) ( H H) h + ( H H) h + ( H H) h,( H H) h ζ, < > < where <, > denotes an nner product and ζ D (4) ( H H) h v + ( H H) h v γ = σ, γ = <( H H) h,( H H) h > < v, v >. and Proo: Frst we have + + = ˆ++ x x + ˆ++ y y + E[( xˆ ++ z xz) ] tr( C ) E[( x x ) ] E[( x x ) ] = ˆ x x + ˆ y y + E[( xˆ z xz) ]. tr( C ) E[( x x ) ] E[( x x ) ] Dene A and B as ollows and h A= ( H H) ( H H) h h h = ( H H) h < ( H H) h, ( H H) h > < ( H H) h, ( H H) h > ( H H) h σ v σ v v + D D B =. σ v v σ v + D D hen tr( C++ C ) = tr(ab) < 0 gves the ollowng nequalty wth long manpulaton. ( H H) h + ( H H) h + < ( H H) h,( H H) h > ( H H) h v ( H H) h v σ D + γ Remark : heorem means that aults and occur, and the magntudes o the two aults satsy (4) located nsde an ellpse, then the correspondng aulty sensors should not be excluded to obtan less estmaton error by usng them. heorem 3: Consder the measurement equaton () and the trad soluton (5), and suppose that -th and -th sensors have aults. For the two estmaton error covarance matrces () and (3), the ollowng two nequaltes are equvalent: ) tr( C ) < tr( C ) ) + < ζ, (5) tr( A) where ζ = and tr( B) v v v A= σ ( H H) [ hh ] D v v v ( H H), h hh h v B = ( H W H) h h ( H W H).

5 Accommodaton Rule Based on Navgaton Accuracy or Double Faults n Redundant Inertal Sensor Systems 333 Proo: he proo has the same procedure as heorem. Remark : heorem 3 means that even though aults and are located outsde the ellpse n (4) and <, (5) s satsed, then the -th sensor should not be excluded snce less estmaton error can be obtaned by usng -th sensor. heorem 4: Consder the measurement equaton () and the trad soluton (5), and suppose that -th and -th sensors have aults. For the two estmaton error covarance matrces (9) and (3), the ollowng two nequaltes are equvalent: ) tr( C ) < tr( C ) ) + ++ { ( H H) h ( H WH) h } + ( H H) h ( ),( ) < H H h H H h > σ + > ( H H) h v (6) Proo: From (9) and (3), tr( C + ) and tr( C + + ) can be calculated easly as ollows + = + σ ( ) + H H h, v ++ = + + < ( H H) h,( H H) h > tr( C ) ( H W H) h σ tr(( H H) ) tr( C ) ( H H) h ( H H) h + σ tr(( H H) ). By calculatng tr( C + ) tr( C++ ) < 0, nequalty (6) can be obtaned. Remark 3: heorem 4 means that even though aults and satsy (4), located nsde the ellpse, and <, (6) s satsed, then -th sensor should be excluded snce less estmaton error can be obtaned by ts excluson. Accordng to the results o heorem through heorem 4, double aults can be categorzed nto our groups. Category I: When double aults satsy the ollowng three nequaltes. ) <. < > σ ( H H) h v ( H H) h,( H H) h + < he two aulty sensors should not be excluded. Category II: When double aults satsy the ollowng three nequaltes ) ) ( ) + ( ) + < ( H H) h,( H H) h > < ζ { ( H H) h ( H WH) h } + ( H H) h ( ),( ) < H H h H H h > σ + ( H H) h v < H H h H H h ). he -th sensor should be excluded, but not or the -th sensor. Category III: When double aults satsy the ollowng three nequaltes ) ( ) + ( ) H H h H H h + < ( H H) h,( H H) h > ζ ) < ζ ) <. he -th sensor should be excluded, but not or the -th sensor. Category IV: When double aults satsy the ollowng three nequaltes ) ( ) + ( ) H H h H H h + < ( H H) h,( H H) h > ζ ) ζ ) <. he two aulty sensors should be excluded. Remark 4: For the 4 categores above, we consder only hal o the rst quadrant n two dmensonal space..e., 0 θ π. It can be notced that 4 Category II and Category III gve the same result. ) ) ( ) + ( ) H H h H H h + < ( H H) h,( H H) h > < ζ H H h H WH h + ( H H) h { ( ) ( ) } 4.3 Accommodaton rule or double aults wth dodecahedron conguraton In order to show the decson rule or a real conguraton or redundant nertal sensors, we use the symmetrc dodecahedron conguraton as shown n Fg., whch uses 6 sensors. In ths case the

6 334 Cheol-Kwan Yang and Duk-Sun Shm Z m m m 5 X m 4 m 3 Y m 6 Fg.. Dodecahedron conguraton wth 6 dentcal sensors. measurement matrx and party matrx have the ollowng relatons. H H = I3, h =, v = ( =,,,6) he angles θ ˆ between drecton cosne vectors h o 6 sensors are 63.4 and 6.6. So are angles o party vectors. hus, sn sn ˆ θ = θ = 0.8 and cos θ θˆ = cos = 0. always hold. Inequaltes (4)-(6) become (7)-(9) or the dodecahedron conguraton. able. Four categores o double aults wth dodecahedron conguraton (0 θ /4 π Cate gory I II III IV regon only, : use, ⅹ: excluson). Condtons + + < σ, < σ, 5 < + + < σ, σ, 5 < σ, <.58 σ, < σ,.58 σ, < -th -th aulty aulty sensor sensor ⅹ ⅹ ⅹ ⅹ Fg. 3. Decson rule or excluson o aulty sensors or the dodecahedron conguraton (For π /4 θ π / regon, and should be nterchanged n able )..58 σ, < 6σ, (7) (8) σ. (9) 5 We can obtan able or the symmetrc dodecahedron conguraton summarzng the above observaton. able can be plotted n a twodmensonal plane as n Fg. 3. Remark 5: able s consdered only or 0 θ π /4 regon n the rst quadrant. We need to nterchange and or π / 4 θ π / regon. In Fg. 3, the ellpse = 6σ does not contrbute to the decson rule. However, t does n the second and ourth quadrants. 5. SIMULAIONS In ths secton, Monte Carlo smulatons are perormed 0,000 tmes or each ault to conrm the accommodaton rule or sngle ault case and double ault case, respectvely. Sx dentcal sensors are used wth dodecahedron conguraton [9] as n Fg.. he measurement matrces H and V satsyng VH = 0 and VV = I can be obtaned as ollows: H =

7 Accommodaton Rule Based on Navgaton Accuracy or Double Faults n Redundant Inertal Sensor Systems , V = , (0) where v = v = = v6 = /. Consder the smulaton or the sngle ault case. We assume that rst sensor has a ault lke () t = [ () t 0 0 ], and the ault () t s a bas, and the measurement nose s whte Gaussan wth mean 0 and varance σ =. he excluson threshold stated n Lemma s h = σ or the matrx V obtaned above. Fg. 4 s the Monte Carlo smulaton result o the sngle ault case showng trace( C+ ( t)) and trace ( C ( t)). Horzontal axs denotes the ault sze to nose rato (F/N rato), and the vertcal axs denotes the magntudes o trace( C+ ( t)) and trace( C ( t)). When ault sgnal () t s greater than σ, the nequalty trace( C+ ( t)) > trace( C ( t)) holds. hs nequalty, whch s consstent wth Lemma, means that when a ault sze s greater than σ, the aulty sensor should be excluded to provde less error covarance o xˆ( t ). he excluson o the aulty sensor wll result n superor navgaton accuracy. For the double ault case, we assume that the rst Fg. 4. race( C+ ( t)) and trace( C ( t)) wth respect to F/N rato. Fg. 5. Decson rule or excluson o aulty sensors and the relaton o two ault magntudes or smulaton. Fg. 6. trace( C++ ( t)), trace( C + ( t)) and trace ( C ( t)) wth respect to ault magntude. and second sensors have ault lke () t = [ ], and the aults and are constants and satsy the straght lne as Fg. 5, and the measurement nose s whte Gaussan wth mean 0 and varance σ =. Fg. 6 shows the results o the accommodaton rule or double aults accordng to the ault sze n Fg. 5. When ault and belong to the regon o Category I, the trace o C ++ s the mnmum among three traces. When ault and belong to that o Category II and III, the trace o C + s the mnmum, and to the Category IV, the trace o C s the mnmum CONCLUSIONS We consder a ault accommodaton problem or

8 336 Cheol-Kwan Yang and Duk-Sun Shm redundant nertal sensor systems dependng on navgaton accuracy and propose a new accommodaton rule or the double ault case. he accommodaton rule can be drawn n two-dmensonal decson space. Monte-Carlo smulaton has been perormed or dodecahedron conguraton to conrm the mprovement o the navgaton accuracy or the sngle and double ault cases. he accommodaton rule can be appled to any conguratons and any number o sensors. REFERENCES [] P. M. Frank, Fault dagnoss n dynamc systems usng analytcal and knowledge-based redundancy-a survey and some new results, Automatca, vol. 6, no. 3, pp , 990. [] G. Betta and A. Petrosanto, Instrument ault detecton and solaton: State o the art and new research trends, IEEE rans. on Instrumentaton and Measurement, vol. 49, no., pp , 000. [3] J. J. Gertler, Fault Detecton and Dagnoss n Engneerng Systems, Marcel Dekker, 998. [4] E. Y. Chow and A. S. Wllsky, Analytcal redundancy and the desgn o robust alure detecton systems, IEEE rans. on Automatc Control, vol. AC-9, no. 7, pp , July 984. [5] J. P. Glmore and R. A. McKern, A redundant strapdown nertal reerence unte (SIRU), Journal o Spacecrat, vol. 9, no., pp , July 97. [6] K. C. Daly, E. Ga, and J. V. Harrson, Generalzed lkelhood test or FDI n redundant sensor conguratons, Journal o Gudance and Control, vol., no., pp. 9-7, Jan-Feb 979. [7] H. Jn and H. Y. Zhang, Optmal party vector senstve to desgnated sensor ault, IEEE rans. on Aerospace and Electronc Systems, vol. 35, no. 35 pp. -8, October, 999. [8] J. E. Potter and J. C. Deckert, Mnmax alure detecton and dentcaton n redundant gyro and accelerometer systems, Journal o Spacecrat, vol. 0, no. 4, pp , Dec. 97. [9] J. C. Wlcox, Maxmum lkelhood alure detecton or redundant nertal nstruments, AIAA Paper 7-864, Stanord, Cal., 97. [0] J. C. Wlcox, Compettve evaluaton o alure detecton algorthms or strapdown redundant nertal nstruments, Journal o Spacecrat, vol., no. 7, pp , July 974. [] J. V. Harrson and.. Chen, Falure solaton or a mnmally redundant nertal sensor system, IEEE rans. on Aerospace and Electronc Systems, vol., no. 3, pp , May 975. [].. Chen and M. B. Adams, A sequental alure detecton technque and ts applcaton, IEEE rans. on Automatc Control, vol., no. 5, pp , October 976. [3] R. J. Patton, Fault detecton and dagnoss n aerospace systems usng analytcal redundancy, Computng & Control Engneerng Journal, vol., no. 3, pp. 7-36, 99. [4] X. J. Zhang, Auxlary Sgnal Desgn n Fault Detecton and Dagnoss, Sprnger-Verlag, 989. [5] R. J. Patton, P. M. Frank, and R. N. Clark, Issues o Fault Dagnoss or Dynamc System, st edton, Sprnger Verlag, May 5, 000. [6] G. H. Golub and C. F. Van Loan, Matrx Computatons, he Johns Hopkns Unv. Press, pp. 4-45, 996. [7] J. Stostrup, H. Nemann, and A. l. Cour-Harbo, Optmal threshold unctons or ault detecton and solaton, Proc. o the Amercan Control Conerence, Denver, Colorado, pp , June 4-6, 003. [8] C.-K. Yang and D.-S. Shm, FDI usng multple party vectors or redundant nertal sensors, European Journal o Control, vol., no. 4, pp , December 006. [9] J. Harson and E. Ga, Evaluatng sensor orentatons or navgaton perormance and alure detecton, IEEE rans. on Aerospace and Electronc Systems, vol. 3, no. 6, pp , November 977. Cheol-Kwan Yang receved the B.S., M.S., and Ph.D. degrees n Control and Instrumentaton Engneerng rom Chung-Ang Unversty n 996, 998, and 003, respectvely. Hs research nterests nclude nertal navgaton systems, GPS, robust lterng, and ault detecton and solaton. Duk-Sun Shm receved the B.S. and M.S. degrees n Control and Instrumentaton Engneerng rom Seoul Natonal Unversty, Korea, n 984 and 986, respectvely. He receved the Ph.D. n Aerospace Engneerng rom the Unversty o Mchgan, Ann Arbor n 993. He had been wth the Department o Electrcal Engneerng and Computer Scence n the Unversty o Mchgan as a post-doc rom January 994 to January 995. Snce 995, he has been wth the School o Electrcal and Electroncs Engneerng n Chung-Ang Unversty, Seoul, Korea, and s currently a Proessor. Hs research nterests nclude robust control, lterng, nertal navgaton systems and GPS, ault detecton and solaton.