Introdction In the three papers [NS97], [SG96], [SGN97], the combined setp or both eedback and alt detection lter design problem has been considered.

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1 Robst Falt Detection in Open Loop s. losed Loop Henrik Niemann Jakob Stostrp z Version: Robst_FDI4.tex { Printed 5h 47m, Febrar 9, 998 Abstract The robstness aspects o alt detection and isolation (FDI) or ncertain sstems will be considered in this paper. The standard setp or FDI will be considered here. The FDI design problem will be analzed both in the case where the control inpt signal is considered as a known external inpt signal (open loop) and when the inpt signal is generated b a eedback controller. Kewords: Robst FDI, Feedback control, ncertain sstems. This work is spported b the Danish Technical Research oncil nder grant no Department o Atomation, Technical Uniersit o Denmark, Bilding 326, DK2800 Lngb, Denmark. E mail: hhn@ia.dt.dk. z Department o ontrol Engineering, Aalborg Uniersit, Fredrik Bajersej 7, DK9220 Aalborg East, Denmark. jakob@control.ac.dk. W 3 :

2 Introdction In the three papers [NS97], [SG96], [SGN97], the combined setp or both eedback and alt detection lter design problem has been considered. The design setp is shown in Figre, where a standard ormlation is applied [ZDG95]. z c G(s) w d c s?k+ K(s) c +k a Figre : ontrol sstem with actator alt, a, and sensor alt, s. A complete analsis o the combined eedback controller/alt detection lter has been gien in [SGN97] or both nominal sstems as well as or ncertain sstems. The reslts o this analsis is that there is a separation between the design o the eedback controller and the alt detection lter in the nominal case which does not exist in the ncertain case. The reason or this missing separation in the ncertain case is that there is a tradeo between perormance in the eedback loop and perormance or the alt detection lter. B sing the setp shown in Figre, we are looking at both the eedback controller and the alt detection lter at the same time. The other approach in alt detection is to consider onl the sstem withot taken care o how the control signal is callated. This is shown in Figre 2. G(s) F (s) ^ Figre 2: General strctre or alt detection in open loop. is distrbance inpt and is alt inpt signal. ^ is the estimate o. This setp has been considered in seeral papers, see e.g. [Fra90], [PF89], [Fra96] and the reerences 2

3 therein. The main isse in this paper is to gie an analsis o the FDI design problem both in the case when the relation between and is known and when it is not known. Nominal sstems as well as ncertain sstems will be considered. The rest o this paper is organized as ollows. The sstem setp is gien in Section 2. The analsis reslts are gien in section 3 ollowed b an analsis o a special case in Section 4. A conclsion is gien in Section 5. 2 Problem Formlation onsider the setp gien in Figre 3, which is an extension o the setp shown in Figre 2. (s) G(s) F (s) ^ Figre 3: General setp or robst alt detection in open loop. The sstem G in Figre 3 has the ollowing orm: z = G zw G z G z G z G w G G G 0 w A () where z and are the external otpt signal and the measrement otpt signal respectiel. The inpts are external inpt w rom the ncertain block, distrbance inpt, alt inpt signal and the control inpt signal, respectiel. It will rther be assmed that the pertrbation block is scaled sch that kk ; 8 and the scaling nction is inclded in G. There is no assmption abot the strctre o. Frther, it is also assmed that all other releant weight matrices are inclded in G. 3

4 The design problem is to design a FDI lter F (s) sch that the estimation error e dened b e =? ^ =? (F (s)? F (s)g ) (2) is minimized in some sense. In the next section, the estimation error will be analzed in the general case gien b (). A special case o () will be analzed in Section 4. case. 3 General ase The main analsis reslts will be gien here in or dierent cases. 3. Open Loop FDI Analsis or "Nominal" Sstems In this case, the sstem in () takes the ollowing orm: = The estimation error is then gien b: G G G 0 A (3) e open = (I? F G )? F G (4) It trns directl ot rom (4) that it is not possible to make estimation o i the distrbance is in the same reqenc range as the alt signal is and has the same direction at the sstem. There is a tradeo between alt detection and distrbance attenation. 3.2 losed Loop FDI Analsis or "Nominal" Sstems The control inpt signal is gien b = K(s) (5) where K(s) is a stabilizing eedback controller. The closedloop sstem rom (3) with (5) is then gien b: = (I? G K)? G (I? G K)? G (6) = SG SG where S is an otpt sensitiit nction. The alt estimation error e or the nominal closedloop sstem is gien b: e closed =? ^ (7) = (I? F SG )? F SG The onl dierence between the estimation error or the open loop in (4) and or the closed loop in (7), is that the sensitiit nction S is inclded in (7). So, i the lter in (7) is selected as F closed = F open S?, we get exactl the same eqation or the estimation error as in the open loop case. The open loop and the closed loop cases are eqialent in the nominal case. 4

5 3.3 Open Loop FDI Analsis or Uncertain Sstems When the loop rom z to w is closed b the ncertain block, w = z, the sstem in () takes the ollowing orm: = = where S = (I? G zw )?. G w S G + G G w S G + G G w S G + G 0 G G G 0 The estimation error is then gien b: A A (8) e open = (I? F G w S G? F G )?(F G w S G + F G )?F G w S G (9) With the pertrbation block present in the sstem, three additional terms appear in the eqation or the estimation error, (9), compared to the nominal case. 3.4 losed Loop FDI Analsis or Uncertain Sstems As in Section 3.2, the control inpt signal is based on a eedback controller K(s). losing the loop in (8) with a eedback controller K(s), we get the ollowing sstem (with G, G, and G as dened in (8)): = (I? G K)? G (I? G K)? G (0) The alt estimation error in the closed loop or ncertain sstems is then gien b: e closed = (I? F (I? G K)? G )? F (I? G K)? G () The aboe eqation or the alt detection error signal is qite complicated in the general case. Thereore, a simplied case will be considered in next section, which make it possible to gie a clear analsis. 4 A Special ase A special case o the more general case considered in Section 3 will be considered here. Let s consider a sstem gien b: = G(s)(I + )( + + ) (2) where is a mltiplicatie pertrbation at the plant inpt. Using the same ormlation as in Section 3, the sstem in (2) is gien b: z = 0 I I I G G G G 5 0 B BB@ w A (3)

6 Note that both the alt signal, the distrbance signal and the control inpt signal enter the sstem at the same place. It there is not a separation in reqenc between the alt signal and the distrbance inpt signal, it will not be possible to separate the alt signal rom the distrbance signal. In the nominal case, the two estimation error signals or open loop and or closed loop are b: and respectiel, where S = (I? GK)? is the sensitiit nction. e open = (I? F G)? F G (4) e closed = (I? F SG)? F SG (5) As in the general case, there is no principal dierence between alt detection in open loop and closed loop when we onl consider nominal sstems. To get exactl the same estimation error in the closed loop as in the open loop, we jst need to se the lter gien b F closed = F open S?. Note that when S is small at low reqencies (as it is in general), the gain o F closed will increase eqialentl at low reqencies. This is the drawback o alt detection in closed loop compared with open loop. Now, let s consider the ncertain case. The two estimations errors are gien b: or the open loop and e open = (I? F G(I + ))? F? F G(I? ) (6) e closed = (I? F (I? G(I + )K)? G(I + ))? F G(I? (I + )KG)? (I + ) = (I? F G(I? KG? KG)? (I + ))? F G(I? KG? KG)? (I + ) = (I? F G(S? I? KG)? (I + ))? F G(S? I? KG)? (I + ) = (I? F GS I (I? KGS I )? (I + ))? F GS I (I? KGS I )? (I + ) = (I? F SG(I? T I )? (I + ))? F SG(I? T I )? (I + ) (7) or the closed loop, respectiel, where T I = GK(I?GK)? is the complementar sensitiit nction. The pictre gets mch more complex when we are looking at ncertain sstems. In the open loop, an additional term rom the control inpt signal appears in the estimation error de to the model ncertaint. This term has been remoed in (7) when a eedback controller has been applied. Frther, note that the robstness o the eedback loop has a major inence on the estimation error in (7). This is in accordance with the reslts in [NS97], [SGN97]. Here we can see the connection between the perormance in the eedback loop and the perormance in the alt detection lter. To see this connection, assme that the perormance in the eedback loop is maximized be the design o K(s). This mean that I? T I is close to the robst stabilit bond, i.e. jt I j '. When the inerse o I? T I appear in (7), we will get a er poor estimation o the alt signal. To gie a more complete analsis o the FDI problem or ncertain sstems, let s separate the estimation error into: e = e nom + e nc (8) where e nom is related to the nominal case and e nc is related to the model ncertaint. Frther, as mentioned beore, let s assme that the alt signal and the distrbance signal is separated in reqencies. The estimation error or the open loop is then gien b: e open = e nom + e nc = (I? F G)? F G? F G(I + )? F G (9) 6

7 and or the closed loop e closed = e nom + e nc = (I? F SG)? F SG? F SGT I (I? T I )? (I + )?F SG(I + )? F SGT I (I? T I )? (I + ) (20) Withot loss o generalit, we will onl consider the alt inpt signal in the reqenc range where is separated rom. Onl the alt signal will be considered in the ollowing. The estimation error or the open loop is then gien b: e open = e nom + e nc = (I? F G)? F G? F G (2) where the ncertain part is gien b: and or the closed loop e nc;open =?F G? F G (22) e closed = e nom + e nc = (I? F SG)? F SG? F SGT I (I? T I )? (I + ) (23) where the ncertain part is gien b: e nc;closed =?F SG? F SGT I (I? T I )? (I + ) (24) From the ncertain part in (22) or the open loop, it can be seen that the estimation error, e open, will increase i the model ncertaint is too large in the reqenc range where we want to estimate the alt signal. Frther, the control inpt signal will also redce the estimation precision in the open loop ncertain case de to the term F G. The control inpt signal's inence on the estimation error can be remoed b sing the closedloop ormlation. As a conseqence o closing the loop b a eedback controller K(s), the eect rom the control inpt signal has been remoed rom (24), bt another term appear in the eqation or the estimation error. The new term in (24) incldes (I? T I )? which is related to the robst stabilit o the closedloop sstem. I the perormance o the closed loop sstem is optimized, j T I j will be close to, de to the robst stabilit condition or the sstem in (2), see [SP96]. The conseqence is that (I? T I )? can be er large and the estimation error will increase. onsider e.g. the case where the eedback controller is designed with 50% robstness margin, i.e. j T I j' 0:5. Then j T I (I? T I )? j'. The estimation error rom (23) is then gien b: e closed ' (I? 2F SG)? 2F SG (25) or e closed ' e nom? F SG(I + 2) (26) As a conseqence o (26), we can see that een i the ncertaint is small, a qite large estimation error is obtained de to the term F SG. I the nominal estimation error e nom is small, i.e. j e nom j, we will hae that j F SG j' in the reqenc range where we want to make alt detection. Using this approximation, (26) is then gien b: in a certain reqenc range. e closed '?(I + 2) (27) (27) shows that we get more than 00% estimation error in the case where the eedback controller is designed with 50% robstness margin. 7

8 These reslts are in accordance with the reslts rom [NS97]. In [NS97] it is shown that it is not possible to separate control and alt detection in the ncertain case when a compact setp is applied. The reslts here gie an indication o how the estimation error will increase when a separated design o the controller and the alt detection lter is applied. At rst glance it seems a little strange that we get a worse estimation error when we applied the inormation o how the control inpt signal is deried. The reason is that we in practice decrease the inormation aailable or the alt detection. In the open loop case we se the control inpt signal directl, where as the control inpt signal is onl sed indirectl in the closed loop case. As a matter o act, when we se the open loop ormlation, the ncertaint will not be ed back, becase we se the real signal. Hence, we hae more inormation aailable in the open loop case which makes the dierence. The analsis o the closed loop case aboe shows what can happen when this inormation can not be sed. 5 onclsion Falt detection in open loop s. closed loop has been considered in this paper or both nominal sstems as well as or nceertain sstems. In the nominal case, there is in principle no dierence between open loop and closed loop alt detection. This is not the case or ncertain sstems. In this case, there is a trade o between good alt detection and good perormance o the closed loop sstem. Reerences [Fra90] [Fra96] [NS97] P.M. Frank. Falt diagnosis in dnamic sstems sing analtic and knowledgebased redndanc A sre and some new reslts. Atomatica, 26:459{474, 990. P.M. Frank. Analtical and qalitatie modelbased alt diagnosis A sre and some new reslts. Eropean Jornal o ontrol, 2:6{28, 996. H.H. Niemann and J. Stostrp. Integration o control and alt detection: Nominal and robst design. In SAFEPROESS'97, England, 997. Sbmitted or pblication. [PF89] R. Patton, P. Frank, and R. lark. application. Prentice Hall, 989. Falt diagnosis in dnamic sstems Theor and [SG96] J. Stostrp and M.J. Grimble. Integrated control and alt diagnosis design: A polnomial approach. In Modelling and Signal Processing or Falt Diagnosis, Leicester, U.K., September 996. IEE. [SGN97] J. Stostrp, M.J. Grimble, and H.H. Niemann. Design o integrated sstems or the control and detection o actator/sensor alts, 997. Sbmitted or pblication. [SP96] S. Skogestad and I. Postlethwaite. Mltiariable eedback control Analsis and design. John Wile & Sons, 996. [ZDG95] K. Zho, J.. Dole, and K. Gloer. Robst and optimal control. Prentice Hall,