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Generic solvability of the failure detection and identification problem
Automatica 35 (1999) 887}893
Brief Paper
Generic solvability of the failure detection and identi"cation problem Shahin Hashtrudi Zad *, Mohammad-Ali Massoumnia Systems Control Group, Department of Electrical and Computer Engineering, University of Toronto, 10 King+s College Rd., Toronto, Ontario, Canada M5S 3G4 Electrical Engineering Department, Sharif University of Technology, P.O. Box 11365-8639, Tehran, Iran Received 4 March 1997; revised 28 July 1998; received in "nal form 3 November 1998
Abstract Necessary and su$cient conditions for the generic solvability of &the extended fundamental problem in residual generation' (EFPRG) (Massoumnia et al., 1989, IEEE Trans. Automat. Control 34, 316}321) in the case of simultaneous actuator and sensor failures are presented. In the process of deriving these conditions, the generic forms of supremal controllability and in"mal unobservability subspaces of LTI systems having diagonal state-space matrices are obtained. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Failure detection; Geometric approaches; Linear multivariable systems; Modelling; Controllability
1. Introduction Failure Detection and Identi"cation (FDI) is a twostage process: residual generation and decision making. In this paper, we shall concentrate only on residual generators that are obtained as solutions to the extended fundamental problem in residual generation (EFPRG) (Massoumnia et al., 1989) (see Patton et al. (1989), Ruokonen (1994) and references therein for other approaches to FDI). In this work we examine the generic solvability of EFPRG. A formally stated problem whose solvability depends on a set of parameters is said to be &generically solvable' if it is solvable for almost every set of the parameter values. If the problem is generically unsolvable however, then it might at best be solvable only for special or highly restricted parameter sets and, this is usually undesirable. The issue of generic solvability of EFPRG is treated in Proposition 5 of Massoumnia et al. (1989), but the
***** *Corresponding author. Tel.: #1 4169786289; fax: #1 4169780804; e-mail: [email protected]. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor M. Basseville under the direction of Editor T. SoK derstroK m.
result is applicable when only actuator failures are present. The main result of this paper is a theorem stating necessary and su$cient conditions for the generic solvability of EFPRG in the case of simultaneous actuator and sensor failures. Before this, we will present results on the generic forms of supremal controllability and in"mal unobservability subspaces of LTI systems having diagonal state}space matrices, which will subsequently be used in the proof of the main result. Failure representation and EFPRG will be reviewed in Section 2. In Section 3, generic forms of invariant subspaces of LTI systems having diagonal matrices are obtained. Section 4 contains our main result. We follow the notation and terminology used in Wonham (1985) and Massoumnia (1986b). A few items are recalled here. The set of real numbers is R. If k is a positive integer, then k denotes the set of integers +1, 2,2, k,. We use A2 for the transpose of the matrix A and diag(M ,2, M ) to denote a block-diagonal I matrix with the matrices M on the main diagonal. For G integers n and m, I denotes the n;n identity matrix and L 0 ; the n;m zero matrix. The range of ¸ is L. The L K dimension of the vector space X is d(X). By writing (g) in front of an equation, we mean that the equation holds generically. Finally, m(s) is the Laplace transform of the function m(t).
0005-1098/99/$*see front matter 1999 Elsevier Science Ltd. All rights reserved PII: S 0 0 0 5 - 1 0 9 8 ( 9 8 ) 0 0 2 2 1 - 0
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2. Failure representation and problem formulation We start by reviewing EFPRG. Consider the LTI system (Massoumnia, 1986a; Massoumnia et al., 1989): xR (t)"Ax(t)#Bu(t)# ¸ m (t), G G GZk y(t)"Cx(t),
(1) (2)
where x(t)3 X, u(t) 3U and y(t)3 Y are the state, input and output of the system, respectively, with d(X)"n, d(U)"m and d(Y)"l. The input u(t) and the output y(t) are assumed to be known. The ¸ m (t) are used to model G G actuator failures. (Extension of this framework to include sensor failures will be discussed later in this section.) Here m (t) 3 M (d(M )"k ) is an arbitrary and unknown funcG G G G tion of time and is called the &ith actuator failure mode'. When no failure is present, the failure modes are all equal to zero, by de"nition. They become nonzero only when the corresponding failure mode occurs. The maps ¸ : M PX are referred to as &actuator failure signatures'. G G Since the failure modes are arbitrary, without loss of generality, it is assumed from now on that the failure signatures are one-to-one (monic). Problem 1 (The extended fundamental problem in residual generation (EFPRG; Massoumnia et al., 1989)). Given the system (1) and (2), design LTI "lters: wR (t)"F w (t)!E y(t)#G u(t), i3 k, (3) G G G G G r (t)"M w (t)!H y(t)#K u(t), i 3 k (4) G G G G G (i.e., "nd the F , E , G , M , H and K ) such that for each G G G G G G i 3k: (a) the transfer matrices from u(s) and m (s) ( jOi) to H r (s) become zero, (b) the transfer matrix from m (s) to r (s) G G G becomes input observable and, (c) all of the modes observable from r (t) become asymptotically stable. G In conditions (a)}(c), Eqs. (1)}(4) are considered as a system with the state vector [x2(t), w2 (t)]2 and the G output r (t). Note that a transfer matrix is input observG able if its columns are linearly independent over the "eld of real numbers (Massoumnia et al., 1989; Sain et al., 1969). Obviously the ith designed "lter, which takes u(t) and y(t) as input and generates r (t) at the output, detects G and identi"es the failure mode m (t) and does not respond G to other failure modes nor to the input u(t). In fact, this "lter can be thought of as an observer estimating part of the state space of system (1) and (2) in the presence of the unknown inputs m (t) ( jOi) (Massoumnia, 1986a; H Massoumnia et al., 1989). The r (t) are the residuals. G Proposition 2 (Massoumnia et al., 1989). EFPRG is solvable if and only if
inf S L H HOG
5 L "0, G
i 3 k.
(5)
Here for a subspace L, S(L) denotes the set of (C, A)unobservability subspaces containing L. Unobservability subspaces are dual to controllability subspaces (Wonham, 1985; Massoumnia, 1986b). Now let us include sensor failures in model (1) and (2) to get (Massoumnia et al., 1989) xR (t)"Ax(t)#Bu(t)# ¸ m (t), (6) G G k GZ y(t)"Cx(t)# J n (t), (7) G G GZq where n (t)3 N (d(N )"q ) is the ith sensor failure mode G G G G and, J : N PY is the corresponding sensor failure signaG G ture, assumed to be monic. Sensor failures can be represented by pseudoactuator failures (Massoumnia, 1986a; Massoumnia et al., 1989). For this, each of the sensor failure modes n (t) is taken to be the output of some G auxilary invertible, "nite-dimensional LTI system (C , A , ¸ ), driven by an appropriate input m (t): G G G G xR (t)"A x (t)#¸ m (t), (8) G G G G G n (t)"C x (t). (9) G G G Here x (t)3 X (d(X )"n ), A : X PX , ¸ : N PX G G G G G G G G G G and C : X PN . Note that since the auxilary system G G G (C , A , ¸ ) N (i.e., the number of the columns of ¸ is G G G G G equal to the number of the rows of C which is equal to G q ), ¸ must be monic, C must be epic (surjective), the G G G order of the ith auxilary system n must be equal to or G greater than q . Next the dynamics of these invertible G systems are added to the dynamics of system (6), (7) to get the &extended system' xR (t)"Ax(t)#Bu(t)# ¸ m (t)# ¸ m (t), (10) G G G G GZq GZk y(t)"Cx(t), (11) where x(t) " : [x2(t)"x 2(t)]2,
(12)
A " : diag(A, A ),
(13)
A " : diag(A ,2, A ), O B " : [B2"0]2,
(14) (15)
¸ " : [¸2 "0]2, (16) G G ¸ " : [0"0¸ 20]2, (17) G G C " : [C J C J C 2 J C ]. (18) O O The dimensions of the zero blocks in the above equations are determined by Eqs. (6)}(9). Also note that the ¸ and G the ¸ are monic. G We observe that there are no sensor failure signatures in the extended system (10), (11). Therefore by solving EFPRG for the extended system, we can design k#q residual generators such that each of them detects and
S. Hashtrudi Zad, M.-A. Massoumnia/Automatica 35 (1999) 887}893
identi"es only one of the failure modes of the extended system and does not respond to the other failure modes. By Eq. (5) EFPRG is solvable for the extended system if and only if S* 5L"0, i 3 k, G G S * 5L "0, i 3 q G G with S* " : inf S G
(19) (20)
L# L , i 3 k, H H H Z HOG HZq k
L# L , i 3 q. H H HZk H Z q HOG Here for a subspace L, S(L) denotes the set of all (C, A)-unobservability subspaces containing L. In this paper the generic solvability of EFPRG is examined. The concept of genericity is brie#y reviewed here (Wonham, 1985). Let p"(p ,2, p )2 3R,, for , some N. Suppose the u (j ,2, j ) (i 3 k) are polynomials G , with coe$cients in R. Then the set of common zeros of u ,2, u , I : +p : u (p ,2, p )"0, i 3 k, V" G , is called a variety. A variety V is proper if VOR, and nontrivial if VO. A property % is a function % : R,P+0, 1,; if %(p)"1 (resp. 0), then % holds (resp. fails) at p. If a property % fails only for points belonging to a proper variety V, then we say % holds generically and write: %"1(g). In this case if the parameter set p is selected randomly, then %(p)"1 with a probability of 1. As an example, % can be the solvability of EFPRG. Here p contains the entries of the real matrices A, B, C and the ¸ . If for some p, %(p)"1 (resp. 0), then EFPRG is (resp. G is not) solvable for p. If a problem (e.g. EFPRG) is generically solvable, then that means the problem is solvable for &almost any' choice of the parameter set p. However, if it is generically unsolvable, then it is solvable, at best, only for some highly restricted choices of the parameter set; this is usually undesirable. Note that a "nite union of proper varieties is a proper variety; therefore if a "nite set of properties each holds generically, then the whole set simultaneously holds generically. The issue of the generic solvability of EFPRG is treated in Proposition 5 of (Massoumnia et al. 1989), where it is assumed that the entries of the state}space matrices in EFPRG are all arbitrary. However, we have just seen that in the case of simultaneous actuator and sensor failures, after representing sensor failures by pseudoactuator failures, some "xed zero blocks appear in the state}space matrices of the problem (Eqs. (12)}(18)). Therefore Proposition 5 of Massoumnia et al. (1989) is applicable when only actuator failures are present. In Section 4, we shall deal with the generic solvability of : inf S S *" G
889
EFPRG in the case of simultaneous actuator and sensor failures. We note that instead of representing sensor failures by pseudoactuator failures, one can deal with sensor failures directly (Nikoukhah, 1994). For the study of generic solvability, we found the "rst approach, i.e., using pseudoactuators, easier because the solvability conditions (5) remain the same and are given in terms of unobservability subspaces. For the analysis of the generic forms of these subspaces e$cient tools readily exist in the literature. As it was mentioned earlier in this section, the order of the i-th auxilary system can be taken to be greater than q , enlarging the matrices A , ¸ , C and therefore increasG G G G ing the design parameters. Increasing the orders of the auxilary systems increases the degree of the extended system (10), (11). This generally leads to increase in the orders of the residual generators, which is not desirable. Moreover, increasing the number of the parameters may just complicate the design problem. Hence, it is reasonable and convenient to take n "q , and almost G G always we do so. Therefore from now on we assume n "q . The reader is referred to Hashtrudi Zad (1991) G G for further discussion on this issue and on sensor failure modelling. In the next section, we will present our results on the generic forms of supremal controllability and in"mal unobservability subspaces of LTI systems having diagonal state-space matrices. These will later be used in Section 4 in the proof of our main result.
3. Invariant subspaces of diagonal systems: the generic forms Consider the three matrices A, B and C. We denote the set of all (A, B)-invariant subspaces contained in Ker C by V(Ker C), the set of all (A, B)-controllability subspaces contained in Ker C by R(Ker C), the set of all (C, A) -invariant subspaces containing Im B by W(Im B) and the set of all (C, A)-unobservability subspaces containing Im B by S(Im B). The supremal element of V is denoted by V* and that of R by R* and the in"mal element of W by W* and that of S by S*. The generic forms of V*, R*, W* and S* when the entries of A, B, C are arbitrary are given in the literature (Wonham, 1985). Here we assume the matrices have diagonal structure. Theorem 3. ¸et us assume that A (i 3k), D (i3 q), B and G G C (i 3k) are arbitrary matrices with dimensions n ;n , G G G p ;p , (n#p);m and l ;n , respectively, and G G G G n" : n , p" : p , l" : l, G G G k q GZ GZ GZk n 5l i 3 k, k51, q50. G G
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Also let
sequence:
A" : diag(A ,2, A , D ,2, D ), I O C" : [diag(C ,2, C )"0 ; ]. I J N ¹hen generically
R"0,
(26)
RI>"Ker C5(ARI#Im B), k"0,2, n#p,
(a) V*"sup V(Ker C)"Ker C if m5l, Ker C if m'l, (b) R*"sup R(Ker C)" 0 if m4l.
then
Proof. For two integers m and n de"ne
It can be shown that
(27)
R*"RL>N(g).
mRn " : max(m, n), mn " : min(m, n).
d(RI)"r (g) I
(28)
Obviously rank(C )"n l "l (g), i 3k. Therefore, G G G G d(Ker C)"n#p!l(g). (21)
where
Also
r " : 0,
d(Im B)"m(n#p)(g).
k"0, 1,2, n#p.
Since B and the C are arbitrary matrices, Ker C and G Im B generically intersect transversely (Wonham, 1985), i.e., d(Ker C#Im B)"(n#p)(d(Ker C)#d(Im B)) (g); therefore, d(Ker C#Im B)"(n#p)(n#p!l #m(n#p)) (g).
(22)
(a) If m5l, then by Eq. (22): d(Ker C#Im B)" n#p(g); hence Ker C#Im B"RL>N(g). Therefore, A(Ker C)-Ker C#Im B(g) and "nally V*"Ker C(g). (b) Case 1. m4l. First observe that d(Im B)"m(n#p)"m(g),
(23)
d(Ker C#Im B)"n#p!l#m(g) (by Eq. (22)).
(24)
In Lemma 11.1 of Wonham (1985), Eq. (28) is proved when A and C are arbitrary matrices. In our case these matrices are block-diagonal; however, with the following modi"cations the proof can still be used to show Eq. (28): (1) replace n with n#p (in this paper, the dimension of the state space is n#p), (2) replace D< with C and q' with l to adjust the notation, (3) let j"(j ,2, j )2 be a list , representing the N entries of the matrices A , A , D ,2, D , B, C , C , and (4) use the following data for O the numerical example:
A "
Hence d(Ker C5Im B) "d(Ker C)#d(Im B)!d(Ker C#Im B)
therefore Ker C5Im B"0(g) and as a result (25)
If we de"ne R"0,
0 0 ; 0 J J ************ 0 K\J 0 \ 0
RI>"V*5(ARI#Im B),
k"0,2, n#p, then R*"RL>N (Wonham, 1985). From Eq. (25) it follows that R*"RL>N"2"R"R"0(g). Case 2. m'l. In this case, we "rst prove the theorem assuming k"2. In part (a) it was shown that V*"Ker C(g). Therefore if we construct the following
0 ; 0 J J ************** 0 >L\JK\J 0 \ L\JK\J
0
>L\JK\J D" \ 0
0
N>L\JK\J
if n "l ,
if n 'l ,
L\J K\J
A " 0
"0(g) (by Eqs. (21), (23), (24)),
V*5Im B"0(g).
r " : 0R(((n#p)(r #m))!l), I> I
if pO0
if n "l ,
if n 'l ,
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S. Hashtrudi Zad, M.-A. Massoumnia/Automatica 35 (1999) 887}893
B"[B2 B2 B2]2, B "
[I 0 0] if n "l , J I 0 0 J *************** 2 K\J 0 0 $ $ L\J
2
if n 'l ,
L\JK\J
Theorem 4 is the dual of Theorem 3.
B " [0 ; I 0] if n "l , J J J 0 I 0 J ****************** 2 L\J>K\J L\J> 0 0 $ $
L\J
2
L\JK\J
if n 'l ,
B "
0
2 >L\J >L\JK\J $ $ N>L\J
2
r 4n#p!l. Therefore we can deduce that +r , is I I a nondecreasing sequence that converges to n#p!l in at most n#p!l steps. As a result d(RL>N)" n#p!l(g), and therefore by Eq. (21) and the fact that RL>N-Ker C: RL>N"Ker C(g). Finally using Eq. (27) we conclude that R*"Ker C(g). Here the proof for k"2 is complete. The extension of the proof to the general case of arbitrary k(k51) is obvious and therefore omitted. )
if pO0,
N>L\JK\J
C "[I 0], C "[0 ; I 0]. J J L J If q"0 and p"0, then the matrices D and B will be omitted from the above equations. In the proof of Lemma 11.1 of Wonham (1985), the numerical data provide an example for which the subspaces RI generated by Eq. (26) satisfy d(RI)"r for k"0, I 1,2, n#p. In our case, the matrices A and C are blockdiagonal. But the data in Wonham (1985) does not necessarily have this block-diagonal structure; therefore it has to be replaced. The data provided in this paper obviously has the required block-diagonal structure. Furthermore, it can be seen that the RI generated (by Eq. (26)) from these data satisfy d(RI)"r . Having made the above I changes, we can use the proof of Lemma 11.1 of Wonham (1985), word by word, to establish Eq. (28). The proof is not reproduced here for the sake of brevity. Next observe that r "0R[((n#p)(r #m))!l] I> I 5((n#p)(r #m))!l I 5((n#p)(r #1#l))!l (since m'l). I Therefore if r (n#p!l, then r 5r #1. Also I I> I observe that r "r if r "n#p!l. For the numerI> I I ical example: d(RI)4d(Ker C)"n#p!l; hence
Theorem 4. ¸et us assume that A (i3 k), D (i3 q), B (i 3 k) G G G and C are arbitrary matrices with dimensions n ;n , G G p ;p , n ;m and l;(n#p) respectively and G G G G n" : n, p" : p, m" : m, G G G k q GZ GZ GZ k n 5m i 3 k, k51, q50. G G A" : diag(A ,2, A , D ,2, D ), I O diag(B ,2, B ) I . B" : 0; N K ¹hen generically
(a) W*"inf W (Im B)"Im B, if m4l,
(b) S*"inf S (Im B)"
Im B, if m(l, RL>N, if m5l.
With k"1 and q"0, Theorems 3 and 4 reduce to the results available in the literature (Wonham, 1985). Also using the same technique, similar results may be obtained for systems with other types of structure such as triangular.
4. Generic solvability Now we consider the generic solvability of EFPRG for system (10), (11). Theorem 5. ¸et us assume that in the extended system (10), (11), A, C, ¸ (i 3 k), J (i 3q), A (i 3 q), ¸ (i 3 q) and G G G G C (i 3 q) are arbitrary matrices with dimensions n;n, l;n, G n;k , l;q , n ;n , n ;q and q ;n respectively and G G G G G G G G k50, q50, k#q52 and n "q . EFPRG is generically G G solvable for the extended system (10), (11) if and only if k 4n (if k51), G GZ k k # q !min (+k , i 3k,6+q , i3 q,)(l. G G G G GZ k GZ q
(29) (30)
Proof. It is easy to see that the auxilary systems (C , A , ¸ ) are generically invertible (Wonham, 1985). G G G
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(If ) If k51 and Eq. (29) holds, then the maps ¸ will G generically be monic and the subspaces L will generiG cally form a family of independent subspaces. Therefore,
L# L "0(g) i 3 k (if k51). H H H Z k HOG HZq Also obviously,
L 5 G
L 5 G
L# L "0 i 3 q (if q51). (31) H H HZk H Z q HOG If q51, then by Eq. (30) q (l, i 3q; hence the maps G J will be generically monic. G Now, suppose that q51, k50, k#q52. We have to show that the solvability conditions (19) and (20) hold generically. We start with S * 5L "0. For this, we O O need to obtain the generic form of the unobservability subspace S *. Referring to system (10) and (11), we set O J " : [J C ,2, J C ]. (32) O O We know that C"[C J ]. Next, we put the entries of the matrices A, C, ¸ , A , ¸ and J in the vector p 3 R, G G G A where N "n#l;n#n; k k #2 q q#l; GZ G GZ G q q . By Eqs. (29) and (30) k k 4n (if k51) and GZ G GZ G k k # q O q (l. Therefore by Theorem 4, for GZ G O G GZ G p 3 R, we have
inf S L# L " L# L (g). G G G G GZ k G Z q GOO GZk GZ q GOO (33) Here for a subspace L, S(L) denotes the set of (C, A)unobservability subspaces containing L. As a result, there exists a proper variety V "+p 3 R, : u (p )"0 i 3h,, G such that for the points p ,V , Eq. (33) holds (The u are G polynomials in p , the parameter set). Assume that p* ,V , therefore i* 3 h: u * (p*)O0. (34) G Now in the polynomials u (p ), we replace the entries G of the matrix J with those of the matrices J and G C according to Eq. (32) to obtain polynomials u (p ) G G where p contains the entries of the matrices A, C, ¸ , J , G G A , ¸ , C . Therefore, p 3 R, with N "n#l; G G G n#n; k k #l; q q #3 q q. Consider the varGZ G GZ G GZ G iety V "+p 3 R, : u (p )"0 i 3 h,. The de"nitions G of p and p are such that for every p 3 R, there exists a unique p 3 R,. If p ,V , then the corresponding p does not lie on V (p , V ), and hence Eq. (33) holds. Therefore for points not lying on V , Eq. (33) holds. The variety V is proper. To show this, choose p* in this way: let A, C, ¸ , A , ¸ and J have their corresponding G G G values at p*, set C "I and "nally obtain the J from G OG G J and the C using Eq. (32). Considering this choice and G Eq. (34), we have u * (p*)O0. Since the variety V is G
proper, for p 3R, we have S *" L# L (g). (35) O G G k q O GZ G Z G O By the above equation and Eq. (31): S * 5L "0(g). O O Note that in using Theorem 4 to calculate the generic form of S *, the entries of C have to be arbitrary indeO pendent parameters. If p had been chosen as the para meter set, then the blocks J C of C would have been G G functions of the entries of the J and the C and therefore G G Theorem 4 would not have been applicable. That is why we had to bring in the parameter set p to show Eq. (33) and from there obtain Eq. (35). In this way, we can prove that all of the equations (19) and (20) hold generically and hence EFPRG is generically solvable. Note that for the special cases k"0 and q"0, we do not have to change the proof method (Of course for k"0, Eq. (29) will be omitted). (Only if ) If k"1 and Eq. (29) does not hold, then ¸ will not be monic and hence the transfer matrix from m (s) to r (s) (the corresponding residual) will not be input observable, therefore EFPRG will not be solvable. If k52 and Eq. (29) does not hold, then generically the subspaces LC will not be independent, therefore generiG cally Eq. (19) will not hold and EFPRG will be generically unsolvable. As a result, Eq. (29) is necessary for the generic solvability of EFPRG and from now on we shall assume that it holds. If we assume that Eq. (30) does not hold and that min(+k , i3 k,6+q , i 3 q,)"q , (36) G G G then k k # q O q 5l. Using this inequality GZ G G Z G G G and Theorem 4, we can show that for p 3 R,, * C S "X"RL (g), with p de"ned as in the (if ) part and G nC"n# q q . Now, an argument similar to that of GZ G the (if ) part establishes S *"X"RL (g) for p 3 R, G (with p de"ned as in the (if ) part). Therefore generically the condition S * 5L "0 does not hold and hence G G EFPRG is generically unsolvable. If in Eq. (36), some k is the minimum of the set, then it can be shown G similarly that EFPRG will be generically unsolvable. ) Remark 6. Let us assume that system (6) and (7) has m actuators and l sensors. By solving EFPRG, we want to design residual generators for detecting and identifying failures of k actuators and q sensors. We also assume that the actuator and sensor failure modes are scalar (k "1, i 3 k and q "1, i 3 q), which is usually the case. G G Now, EFPRG is generically solvable if and only if (i) k4n and (ii) k#q4l . This means that the problem is generically solvable if and only if (1) the number of the actuator failure modes is less than or equal to the dimension of the state space of the original system and (2) the number of the failure modes (i.e., the number of the actuators and sensors which may fail) is less than or equal to the number of the output signals of the system
S. Hashtrudi Zad, M.-A. Massoumnia/Automatica 35 (1999) 887}893
(i.e., the total number of the sensors, both those which may fail and those here assumed to be reliable). Condition (i) is not stringent and usually holds. Assuming this, in a system with, say, "ve outputs, generically failures of up to "ve instruments (actuators and/or sensors) can be detected and identi"ed. It should be noted that if (ii) does not hold, then EFPRG is always (not just generically) unsolvable because the transfer matrix from the vector of the failure modes to the output of the system will not be leftinvertible (Massoumnia, 1986a; Massoumnia et al., 1989). Also, since for EFPRG to be solvable, it is necessary that the actuator failure signatures be monic and that their ranges form a family of independent subspaces, we can conclude that if (i) does not hold, then EFPRG is always (not just generically) unsolvable.
5. Conclusion The main result of this paper was a theorem stating necessary and su$cient conditions for the generic solvability of the failure detection and identi"cation problem EFPRG, in the case of simultaneous actuator and sensor failures. For this, we obtained the generic forms of supremal controllability and in"mal unobservability subspaces of LTI systems having diagonal state-space matrices. A nice result of the theorem is that when actuator and sensor failure modes are scalar, EFPRG is generically solvable if and only if (1) the number of the actuator failure modes is less than or equal to the dimension of the state space of the system and (2) the total number of the actuator and sensor failure modes is less than or equal to the number of the system outputs. References Hashtrudi Zad, S. (1991). Computer-aided design of failure detection and identi,cation ,lters. Master's thesis, Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran.
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Massoumnia, M. A. (1986a). A geometric approach to failure detection and identi,cation. Ph.D. thesis, Department of Aeronautics and Astronautics, MIT, Cambridge, MA. Massoumnia, M. A. (1986b). A geometric approach to the synthesis of failure detection "lters. IEEE ¹rans. Automat. Control, 31, 839}846. Massoumnia, M. A., Verghese, G. C., & Willsky, A. S. (1989). Failure detection and identi"cation. IEEE ¹rans. Automat. Control, 34, 316}321. Nikoukhah, R. (1994). Innovations generation in the presence of unknown inputs: Applications to robust failure detection. Automatica, 30, 1851}1867. Patton, R., Frank, P., & Clark, R. (Eds.). (1989). Fault diagnosis in dynamic systems: ¹heory and applications. New York: PrenticeHall. Ruokonen, T. (Ed.). (1994). Proc. IFAC Symp. SAFEPROCESS194, Espoo, Finland. Sain, M. K., & Massey, J. L. (1969). Invertibility of linear timeinvariant dynamical systems. IEEE ¹rans. Automat. Control, 14, 141}149. Wonham, W. M. (1985). ¸inear multivariable control: A geometric approach. New York: Springer.
Shahin Hashtrudi Zad received the B.S. and M.S. degrees in electrical engineering from Sharif University of Technology, Tehran, Iran, in 1989 and 1991, respectively. He is currently working towards the Ph.D. degree at the Department of Electrical and Computer Engineering of the University of Toronto, Canada. His research interests lie in the areas of fault diagnosis, discrete-event systems, hybrid systems and real-time control.
Mohammad-Ali Massoumnia was born on December 18, 1961 in Tehran, Iran. He received his Sc.D. degree in Estimation and Control from Massachusetts Institute of Technology in 1986. From 1986 to 1991 he was with Sharif University of Technology, Tehran, Iran. From 1991 to 1993 he was with Integrated Systems Inc., Santa Clara, CA. Since 1994 he is an Associate Professor of Electrical Engineering at Sharif University of Technology. His research area is currently linear system theory, estimation and inertial navigation systems.
Brief Paper
Generic solvability of the failure detection and identi"cation problem Shahin Hashtrudi Zad *, Mohammad-Ali Massoumnia Systems Control Group, Department of Electrical and Computer Engineering, University of Toronto, 10 King+s College Rd., Toronto, Ontario, Canada M5S 3G4 Electrical Engineering Department, Sharif University of Technology, P.O. Box 11365-8639, Tehran, Iran Received 4 March 1997; revised 28 July 1998; received in "nal form 3 November 1998
Abstract Necessary and su$cient conditions for the generic solvability of &the extended fundamental problem in residual generation' (EFPRG) (Massoumnia et al., 1989, IEEE Trans. Automat. Control 34, 316}321) in the case of simultaneous actuator and sensor failures are presented. In the process of deriving these conditions, the generic forms of supremal controllability and in"mal unobservability subspaces of LTI systems having diagonal state-space matrices are obtained. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Failure detection; Geometric approaches; Linear multivariable systems; Modelling; Controllability
1. Introduction Failure Detection and Identi"cation (FDI) is a twostage process: residual generation and decision making. In this paper, we shall concentrate only on residual generators that are obtained as solutions to the extended fundamental problem in residual generation (EFPRG) (Massoumnia et al., 1989) (see Patton et al. (1989), Ruokonen (1994) and references therein for other approaches to FDI). In this work we examine the generic solvability of EFPRG. A formally stated problem whose solvability depends on a set of parameters is said to be &generically solvable' if it is solvable for almost every set of the parameter values. If the problem is generically unsolvable however, then it might at best be solvable only for special or highly restricted parameter sets and, this is usually undesirable. The issue of generic solvability of EFPRG is treated in Proposition 5 of Massoumnia et al. (1989), but the
***** *Corresponding author. Tel.: #1 4169786289; fax: #1 4169780804; e-mail: [email protected]. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor M. Basseville under the direction of Editor T. SoK derstroK m.
result is applicable when only actuator failures are present. The main result of this paper is a theorem stating necessary and su$cient conditions for the generic solvability of EFPRG in the case of simultaneous actuator and sensor failures. Before this, we will present results on the generic forms of supremal controllability and in"mal unobservability subspaces of LTI systems having diagonal state}space matrices, which will subsequently be used in the proof of the main result. Failure representation and EFPRG will be reviewed in Section 2. In Section 3, generic forms of invariant subspaces of LTI systems having diagonal matrices are obtained. Section 4 contains our main result. We follow the notation and terminology used in Wonham (1985) and Massoumnia (1986b). A few items are recalled here. The set of real numbers is R. If k is a positive integer, then k denotes the set of integers +1, 2,2, k,. We use A2 for the transpose of the matrix A and diag(M ,2, M ) to denote a block-diagonal I matrix with the matrices M on the main diagonal. For G integers n and m, I denotes the n;n identity matrix and L 0 ; the n;m zero matrix. The range of ¸ is L. The L K dimension of the vector space X is d(X). By writing (g) in front of an equation, we mean that the equation holds generically. Finally, m(s) is the Laplace transform of the function m(t).
0005-1098/99/$*see front matter 1999 Elsevier Science Ltd. All rights reserved PII: S 0 0 0 5 - 1 0 9 8 ( 9 8 ) 0 0 2 2 1 - 0
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2. Failure representation and problem formulation We start by reviewing EFPRG. Consider the LTI system (Massoumnia, 1986a; Massoumnia et al., 1989): xR (t)"Ax(t)#Bu(t)# ¸ m (t), G G GZk y(t)"Cx(t),
(1) (2)
where x(t)3 X, u(t) 3U and y(t)3 Y are the state, input and output of the system, respectively, with d(X)"n, d(U)"m and d(Y)"l. The input u(t) and the output y(t) are assumed to be known. The ¸ m (t) are used to model G G actuator failures. (Extension of this framework to include sensor failures will be discussed later in this section.) Here m (t) 3 M (d(M )"k ) is an arbitrary and unknown funcG G G G tion of time and is called the &ith actuator failure mode'. When no failure is present, the failure modes are all equal to zero, by de"nition. They become nonzero only when the corresponding failure mode occurs. The maps ¸ : M PX are referred to as &actuator failure signatures'. G G Since the failure modes are arbitrary, without loss of generality, it is assumed from now on that the failure signatures are one-to-one (monic). Problem 1 (The extended fundamental problem in residual generation (EFPRG; Massoumnia et al., 1989)). Given the system (1) and (2), design LTI "lters: wR (t)"F w (t)!E y(t)#G u(t), i3 k, (3) G G G G G r (t)"M w (t)!H y(t)#K u(t), i 3 k (4) G G G G G (i.e., "nd the F , E , G , M , H and K ) such that for each G G G G G G i 3k: (a) the transfer matrices from u(s) and m (s) ( jOi) to H r (s) become zero, (b) the transfer matrix from m (s) to r (s) G G G becomes input observable and, (c) all of the modes observable from r (t) become asymptotically stable. G In conditions (a)}(c), Eqs. (1)}(4) are considered as a system with the state vector [x2(t), w2 (t)]2 and the G output r (t). Note that a transfer matrix is input observG able if its columns are linearly independent over the "eld of real numbers (Massoumnia et al., 1989; Sain et al., 1969). Obviously the ith designed "lter, which takes u(t) and y(t) as input and generates r (t) at the output, detects G and identi"es the failure mode m (t) and does not respond G to other failure modes nor to the input u(t). In fact, this "lter can be thought of as an observer estimating part of the state space of system (1) and (2) in the presence of the unknown inputs m (t) ( jOi) (Massoumnia, 1986a; H Massoumnia et al., 1989). The r (t) are the residuals. G Proposition 2 (Massoumnia et al., 1989). EFPRG is solvable if and only if
inf S L H HOG
5 L "0, G
i 3 k.
(5)
Here for a subspace L, S(L) denotes the set of (C, A)unobservability subspaces containing L. Unobservability subspaces are dual to controllability subspaces (Wonham, 1985; Massoumnia, 1986b). Now let us include sensor failures in model (1) and (2) to get (Massoumnia et al., 1989) xR (t)"Ax(t)#Bu(t)# ¸ m (t), (6) G G k GZ y(t)"Cx(t)# J n (t), (7) G G GZq where n (t)3 N (d(N )"q ) is the ith sensor failure mode G G G G and, J : N PY is the corresponding sensor failure signaG G ture, assumed to be monic. Sensor failures can be represented by pseudoactuator failures (Massoumnia, 1986a; Massoumnia et al., 1989). For this, each of the sensor failure modes n (t) is taken to be the output of some G auxilary invertible, "nite-dimensional LTI system (C , A , ¸ ), driven by an appropriate input m (t): G G G G xR (t)"A x (t)#¸ m (t), (8) G G G G G n (t)"C x (t). (9) G G G Here x (t)3 X (d(X )"n ), A : X PX , ¸ : N PX G G G G G G G G G G and C : X PN . Note that since the auxilary system G G G (C , A , ¸ ) N (i.e., the number of the columns of ¸ is G G G G G equal to the number of the rows of C which is equal to G q ), ¸ must be monic, C must be epic (surjective), the G G G order of the ith auxilary system n must be equal to or G greater than q . Next the dynamics of these invertible G systems are added to the dynamics of system (6), (7) to get the &extended system' xR (t)"Ax(t)#Bu(t)# ¸ m (t)# ¸ m (t), (10) G G G G GZq GZk y(t)"Cx(t), (11) where x(t) " : [x2(t)"x 2(t)]2,
(12)
A " : diag(A, A ),
(13)
A " : diag(A ,2, A ), O B " : [B2"0]2,
(14) (15)
¸ " : [¸2 "0]2, (16) G G ¸ " : [0"0¸ 20]2, (17) G G C " : [C J C J C 2 J C ]. (18) O O The dimensions of the zero blocks in the above equations are determined by Eqs. (6)}(9). Also note that the ¸ and G the ¸ are monic. G We observe that there are no sensor failure signatures in the extended system (10), (11). Therefore by solving EFPRG for the extended system, we can design k#q residual generators such that each of them detects and
S. Hashtrudi Zad, M.-A. Massoumnia/Automatica 35 (1999) 887}893
identi"es only one of the failure modes of the extended system and does not respond to the other failure modes. By Eq. (5) EFPRG is solvable for the extended system if and only if S* 5L"0, i 3 k, G G S * 5L "0, i 3 q G G with S* " : inf S G
(19) (20)
L# L , i 3 k, H H H Z HOG HZq k
L# L , i 3 q. H H HZk H Z q HOG Here for a subspace L, S(L) denotes the set of all (C, A)-unobservability subspaces containing L. In this paper the generic solvability of EFPRG is examined. The concept of genericity is brie#y reviewed here (Wonham, 1985). Let p"(p ,2, p )2 3R,, for , some N. Suppose the u (j ,2, j ) (i 3 k) are polynomials G , with coe$cients in R. Then the set of common zeros of u ,2, u , I : +p : u (p ,2, p )"0, i 3 k, V" G , is called a variety. A variety V is proper if VOR, and nontrivial if VO. A property % is a function % : R,P+0, 1,; if %(p)"1 (resp. 0), then % holds (resp. fails) at p. If a property % fails only for points belonging to a proper variety V, then we say % holds generically and write: %"1(g). In this case if the parameter set p is selected randomly, then %(p)"1 with a probability of 1. As an example, % can be the solvability of EFPRG. Here p contains the entries of the real matrices A, B, C and the ¸ . If for some p, %(p)"1 (resp. 0), then EFPRG is (resp. G is not) solvable for p. If a problem (e.g. EFPRG) is generically solvable, then that means the problem is solvable for &almost any' choice of the parameter set p. However, if it is generically unsolvable, then it is solvable, at best, only for some highly restricted choices of the parameter set; this is usually undesirable. Note that a "nite union of proper varieties is a proper variety; therefore if a "nite set of properties each holds generically, then the whole set simultaneously holds generically. The issue of the generic solvability of EFPRG is treated in Proposition 5 of (Massoumnia et al. 1989), where it is assumed that the entries of the state}space matrices in EFPRG are all arbitrary. However, we have just seen that in the case of simultaneous actuator and sensor failures, after representing sensor failures by pseudoactuator failures, some "xed zero blocks appear in the state}space matrices of the problem (Eqs. (12)}(18)). Therefore Proposition 5 of Massoumnia et al. (1989) is applicable when only actuator failures are present. In Section 4, we shall deal with the generic solvability of : inf S S *" G
889
EFPRG in the case of simultaneous actuator and sensor failures. We note that instead of representing sensor failures by pseudoactuator failures, one can deal with sensor failures directly (Nikoukhah, 1994). For the study of generic solvability, we found the "rst approach, i.e., using pseudoactuators, easier because the solvability conditions (5) remain the same and are given in terms of unobservability subspaces. For the analysis of the generic forms of these subspaces e$cient tools readily exist in the literature. As it was mentioned earlier in this section, the order of the i-th auxilary system can be taken to be greater than q , enlarging the matrices A , ¸ , C and therefore increasG G G G ing the design parameters. Increasing the orders of the auxilary systems increases the degree of the extended system (10), (11). This generally leads to increase in the orders of the residual generators, which is not desirable. Moreover, increasing the number of the parameters may just complicate the design problem. Hence, it is reasonable and convenient to take n "q , and almost G G always we do so. Therefore from now on we assume n "q . The reader is referred to Hashtrudi Zad (1991) G G for further discussion on this issue and on sensor failure modelling. In the next section, we will present our results on the generic forms of supremal controllability and in"mal unobservability subspaces of LTI systems having diagonal state-space matrices. These will later be used in Section 4 in the proof of our main result.
3. Invariant subspaces of diagonal systems: the generic forms Consider the three matrices A, B and C. We denote the set of all (A, B)-invariant subspaces contained in Ker C by V(Ker C), the set of all (A, B)-controllability subspaces contained in Ker C by R(Ker C), the set of all (C, A) -invariant subspaces containing Im B by W(Im B) and the set of all (C, A)-unobservability subspaces containing Im B by S(Im B). The supremal element of V is denoted by V* and that of R by R* and the in"mal element of W by W* and that of S by S*. The generic forms of V*, R*, W* and S* when the entries of A, B, C are arbitrary are given in the literature (Wonham, 1985). Here we assume the matrices have diagonal structure. Theorem 3. ¸et us assume that A (i 3k), D (i3 q), B and G G C (i 3k) are arbitrary matrices with dimensions n ;n , G G G p ;p , (n#p);m and l ;n , respectively, and G G G G n" : n , p" : p , l" : l, G G G k q GZ GZ GZk n 5l i 3 k, k51, q50. G G
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Also let
sequence:
A" : diag(A ,2, A , D ,2, D ), I O C" : [diag(C ,2, C )"0 ; ]. I J N ¹hen generically
R"0,
(26)
RI>"Ker C5(ARI#Im B), k"0,2, n#p,
(a) V*"sup V(Ker C)"Ker C if m5l, Ker C if m'l, (b) R*"sup R(Ker C)" 0 if m4l.
then
Proof. For two integers m and n de"ne
It can be shown that
(27)
R*"RL>N(g).
mRn " : max(m, n), mn " : min(m, n).
d(RI)"r (g) I
(28)
Obviously rank(C )"n l "l (g), i 3k. Therefore, G G G G d(Ker C)"n#p!l(g). (21)
where
Also
r " : 0,
d(Im B)"m(n#p)(g).
k"0, 1,2, n#p.
Since B and the C are arbitrary matrices, Ker C and G Im B generically intersect transversely (Wonham, 1985), i.e., d(Ker C#Im B)"(n#p)(d(Ker C)#d(Im B)) (g); therefore, d(Ker C#Im B)"(n#p)(n#p!l #m(n#p)) (g).
(22)
(a) If m5l, then by Eq. (22): d(Ker C#Im B)" n#p(g); hence Ker C#Im B"RL>N(g). Therefore, A(Ker C)-Ker C#Im B(g) and "nally V*"Ker C(g). (b) Case 1. m4l. First observe that d(Im B)"m(n#p)"m(g),
(23)
d(Ker C#Im B)"n#p!l#m(g) (by Eq. (22)).
(24)
In Lemma 11.1 of Wonham (1985), Eq. (28) is proved when A and C are arbitrary matrices. In our case these matrices are block-diagonal; however, with the following modi"cations the proof can still be used to show Eq. (28): (1) replace n with n#p (in this paper, the dimension of the state space is n#p), (2) replace D< with C and q' with l to adjust the notation, (3) let j"(j ,2, j )2 be a list , representing the N entries of the matrices A , A , D ,2, D , B, C , C , and (4) use the following data for O the numerical example:
A "
Hence d(Ker C5Im B) "d(Ker C)#d(Im B)!d(Ker C#Im B)
therefore Ker C5Im B"0(g) and as a result (25)
If we de"ne R"0,
0 0 ; 0 J J ************ 0 K\J 0 \ 0
RI>"V*5(ARI#Im B),
k"0,2, n#p, then R*"RL>N (Wonham, 1985). From Eq. (25) it follows that R*"RL>N"2"R"R"0(g). Case 2. m'l. In this case, we "rst prove the theorem assuming k"2. In part (a) it was shown that V*"Ker C(g). Therefore if we construct the following
0 ; 0 J J ************** 0 >L\JK\J 0 \ L\JK\J
0
>L\JK\J D" \ 0
0
N>L\JK\J
if n "l ,
if n 'l ,
L\J K\J
A " 0
"0(g) (by Eqs. (21), (23), (24)),
V*5Im B"0(g).
r " : 0R(((n#p)(r #m))!l), I> I
if pO0
if n "l ,
if n 'l ,
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B"[B2 B2 B2]2, B "
[I 0 0] if n "l , J I 0 0 J *************** 2 K\J 0 0 $ $ L\J
2
if n 'l ,
L\JK\J
Theorem 4 is the dual of Theorem 3.
B " [0 ; I 0] if n "l , J J J 0 I 0 J ****************** 2 L\J>K\J L\J> 0 0 $ $
L\J
2
L\JK\J
if n 'l ,
B "
0
2 >L\J >L\JK\J $ $ N>L\J
2
r 4n#p!l. Therefore we can deduce that +r , is I I a nondecreasing sequence that converges to n#p!l in at most n#p!l steps. As a result d(RL>N)" n#p!l(g), and therefore by Eq. (21) and the fact that RL>N-Ker C: RL>N"Ker C(g). Finally using Eq. (27) we conclude that R*"Ker C(g). Here the proof for k"2 is complete. The extension of the proof to the general case of arbitrary k(k51) is obvious and therefore omitted. )
if pO0,
N>L\JK\J
C "[I 0], C "[0 ; I 0]. J J L J If q"0 and p"0, then the matrices D and B will be omitted from the above equations. In the proof of Lemma 11.1 of Wonham (1985), the numerical data provide an example for which the subspaces RI generated by Eq. (26) satisfy d(RI)"r for k"0, I 1,2, n#p. In our case, the matrices A and C are blockdiagonal. But the data in Wonham (1985) does not necessarily have this block-diagonal structure; therefore it has to be replaced. The data provided in this paper obviously has the required block-diagonal structure. Furthermore, it can be seen that the RI generated (by Eq. (26)) from these data satisfy d(RI)"r . Having made the above I changes, we can use the proof of Lemma 11.1 of Wonham (1985), word by word, to establish Eq. (28). The proof is not reproduced here for the sake of brevity. Next observe that r "0R[((n#p)(r #m))!l] I> I 5((n#p)(r #m))!l I 5((n#p)(r #1#l))!l (since m'l). I Therefore if r (n#p!l, then r 5r #1. Also I I> I observe that r "r if r "n#p!l. For the numerI> I I ical example: d(RI)4d(Ker C)"n#p!l; hence
Theorem 4. ¸et us assume that A (i3 k), D (i3 q), B (i 3 k) G G G and C are arbitrary matrices with dimensions n ;n , G G p ;p , n ;m and l;(n#p) respectively and G G G G n" : n, p" : p, m" : m, G G G k q GZ GZ GZ k n 5m i 3 k, k51, q50. G G A" : diag(A ,2, A , D ,2, D ), I O diag(B ,2, B ) I . B" : 0; N K ¹hen generically
(a) W*"inf W (Im B)"Im B, if m4l,
(b) S*"inf S (Im B)"
Im B, if m(l, RL>N, if m5l.
With k"1 and q"0, Theorems 3 and 4 reduce to the results available in the literature (Wonham, 1985). Also using the same technique, similar results may be obtained for systems with other types of structure such as triangular.
4. Generic solvability Now we consider the generic solvability of EFPRG for system (10), (11). Theorem 5. ¸et us assume that in the extended system (10), (11), A, C, ¸ (i 3 k), J (i 3q), A (i 3 q), ¸ (i 3 q) and G G G G C (i 3 q) are arbitrary matrices with dimensions n;n, l;n, G n;k , l;q , n ;n , n ;q and q ;n respectively and G G G G G G G G k50, q50, k#q52 and n "q . EFPRG is generically G G solvable for the extended system (10), (11) if and only if k 4n (if k51), G GZ k k # q !min (+k , i 3k,6+q , i3 q,)(l. G G G G GZ k GZ q
(29) (30)
Proof. It is easy to see that the auxilary systems (C , A , ¸ ) are generically invertible (Wonham, 1985). G G G
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(If ) If k51 and Eq. (29) holds, then the maps ¸ will G generically be monic and the subspaces L will generiG cally form a family of independent subspaces. Therefore,
L# L "0(g) i 3 k (if k51). H H H Z k HOG HZq Also obviously,
L 5 G
L 5 G
L# L "0 i 3 q (if q51). (31) H H HZk H Z q HOG If q51, then by Eq. (30) q (l, i 3q; hence the maps G J will be generically monic. G Now, suppose that q51, k50, k#q52. We have to show that the solvability conditions (19) and (20) hold generically. We start with S * 5L "0. For this, we O O need to obtain the generic form of the unobservability subspace S *. Referring to system (10) and (11), we set O J " : [J C ,2, J C ]. (32) O O We know that C"[C J ]. Next, we put the entries of the matrices A, C, ¸ , A , ¸ and J in the vector p 3 R, G G G A where N "n#l;n#n; k k #2 q q#l; GZ G GZ G q q . By Eqs. (29) and (30) k k 4n (if k51) and GZ G GZ G k k # q O q (l. Therefore by Theorem 4, for GZ G O G GZ G p 3 R, we have
inf S L# L " L# L (g). G G G G GZ k G Z q GOO GZk GZ q GOO (33) Here for a subspace L, S(L) denotes the set of (C, A)unobservability subspaces containing L. As a result, there exists a proper variety V "+p 3 R, : u (p )"0 i 3h,, G such that for the points p ,V , Eq. (33) holds (The u are G polynomials in p , the parameter set). Assume that p* ,V , therefore i* 3 h: u * (p*)O0. (34) G Now in the polynomials u (p ), we replace the entries G of the matrix J with those of the matrices J and G C according to Eq. (32) to obtain polynomials u (p ) G G where p contains the entries of the matrices A, C, ¸ , J , G G A , ¸ , C . Therefore, p 3 R, with N "n#l; G G G n#n; k k #l; q q #3 q q. Consider the varGZ G GZ G GZ G iety V "+p 3 R, : u (p )"0 i 3 h,. The de"nitions G of p and p are such that for every p 3 R, there exists a unique p 3 R,. If p ,V , then the corresponding p does not lie on V (p , V ), and hence Eq. (33) holds. Therefore for points not lying on V , Eq. (33) holds. The variety V is proper. To show this, choose p* in this way: let A, C, ¸ , A , ¸ and J have their corresponding G G G values at p*, set C "I and "nally obtain the J from G OG G J and the C using Eq. (32). Considering this choice and G Eq. (34), we have u * (p*)O0. Since the variety V is G
proper, for p 3R, we have S *" L# L (g). (35) O G G k q O GZ G Z G O By the above equation and Eq. (31): S * 5L "0(g). O O Note that in using Theorem 4 to calculate the generic form of S *, the entries of C have to be arbitrary indeO pendent parameters. If p had been chosen as the para meter set, then the blocks J C of C would have been G G functions of the entries of the J and the C and therefore G G Theorem 4 would not have been applicable. That is why we had to bring in the parameter set p to show Eq. (33) and from there obtain Eq. (35). In this way, we can prove that all of the equations (19) and (20) hold generically and hence EFPRG is generically solvable. Note that for the special cases k"0 and q"0, we do not have to change the proof method (Of course for k"0, Eq. (29) will be omitted). (Only if ) If k"1 and Eq. (29) does not hold, then ¸ will not be monic and hence the transfer matrix from m (s) to r (s) (the corresponding residual) will not be input observable, therefore EFPRG will not be solvable. If k52 and Eq. (29) does not hold, then generically the subspaces LC will not be independent, therefore generiG cally Eq. (19) will not hold and EFPRG will be generically unsolvable. As a result, Eq. (29) is necessary for the generic solvability of EFPRG and from now on we shall assume that it holds. If we assume that Eq. (30) does not hold and that min(+k , i3 k,6+q , i 3 q,)"q , (36) G G G then k k # q O q 5l. Using this inequality GZ G G Z G G G and Theorem 4, we can show that for p 3 R,, * C S "X"RL (g), with p de"ned as in the (if ) part and G nC"n# q q . Now, an argument similar to that of GZ G the (if ) part establishes S *"X"RL (g) for p 3 R, G (with p de"ned as in the (if ) part). Therefore generically the condition S * 5L "0 does not hold and hence G G EFPRG is generically unsolvable. If in Eq. (36), some k is the minimum of the set, then it can be shown G similarly that EFPRG will be generically unsolvable. ) Remark 6. Let us assume that system (6) and (7) has m actuators and l sensors. By solving EFPRG, we want to design residual generators for detecting and identifying failures of k actuators and q sensors. We also assume that the actuator and sensor failure modes are scalar (k "1, i 3 k and q "1, i 3 q), which is usually the case. G G Now, EFPRG is generically solvable if and only if (i) k4n and (ii) k#q4l . This means that the problem is generically solvable if and only if (1) the number of the actuator failure modes is less than or equal to the dimension of the state space of the original system and (2) the number of the failure modes (i.e., the number of the actuators and sensors which may fail) is less than or equal to the number of the output signals of the system
S. Hashtrudi Zad, M.-A. Massoumnia/Automatica 35 (1999) 887}893
(i.e., the total number of the sensors, both those which may fail and those here assumed to be reliable). Condition (i) is not stringent and usually holds. Assuming this, in a system with, say, "ve outputs, generically failures of up to "ve instruments (actuators and/or sensors) can be detected and identi"ed. It should be noted that if (ii) does not hold, then EFPRG is always (not just generically) unsolvable because the transfer matrix from the vector of the failure modes to the output of the system will not be leftinvertible (Massoumnia, 1986a; Massoumnia et al., 1989). Also, since for EFPRG to be solvable, it is necessary that the actuator failure signatures be monic and that their ranges form a family of independent subspaces, we can conclude that if (i) does not hold, then EFPRG is always (not just generically) unsolvable.
5. Conclusion The main result of this paper was a theorem stating necessary and su$cient conditions for the generic solvability of the failure detection and identi"cation problem EFPRG, in the case of simultaneous actuator and sensor failures. For this, we obtained the generic forms of supremal controllability and in"mal unobservability subspaces of LTI systems having diagonal state-space matrices. A nice result of the theorem is that when actuator and sensor failure modes are scalar, EFPRG is generically solvable if and only if (1) the number of the actuator failure modes is less than or equal to the dimension of the state space of the system and (2) the total number of the actuator and sensor failure modes is less than or equal to the number of the system outputs. References Hashtrudi Zad, S. (1991). Computer-aided design of failure detection and identi,cation ,lters. Master's thesis, Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran.
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Massoumnia, M. A. (1986a). A geometric approach to failure detection and identi,cation. Ph.D. thesis, Department of Aeronautics and Astronautics, MIT, Cambridge, MA. Massoumnia, M. A. (1986b). A geometric approach to the synthesis of failure detection "lters. IEEE ¹rans. Automat. Control, 31, 839}846. Massoumnia, M. A., Verghese, G. C., & Willsky, A. S. (1989). Failure detection and identi"cation. IEEE ¹rans. Automat. Control, 34, 316}321. Nikoukhah, R. (1994). Innovations generation in the presence of unknown inputs: Applications to robust failure detection. Automatica, 30, 1851}1867. Patton, R., Frank, P., & Clark, R. (Eds.). (1989). Fault diagnosis in dynamic systems: ¹heory and applications. New York: PrenticeHall. Ruokonen, T. (Ed.). (1994). Proc. IFAC Symp. SAFEPROCESS194, Espoo, Finland. Sain, M. K., & Massey, J. L. (1969). Invertibility of linear timeinvariant dynamical systems. IEEE ¹rans. Automat. Control, 14, 141}149. Wonham, W. M. (1985). ¸inear multivariable control: A geometric approach. New York: Springer.
Shahin Hashtrudi Zad received the B.S. and M.S. degrees in electrical engineering from Sharif University of Technology, Tehran, Iran, in 1989 and 1991, respectively. He is currently working towards the Ph.D. degree at the Department of Electrical and Computer Engineering of the University of Toronto, Canada. His research interests lie in the areas of fault diagnosis, discrete-event systems, hybrid systems and real-time control.
Mohammad-Ali Massoumnia was born on December 18, 1961 in Tehran, Iran. He received his Sc.D. degree in Estimation and Control from Massachusetts Institute of Technology in 1986. From 1986 to 1991 he was with Sharif University of Technology, Tehran, Iran. From 1991 to 1993 he was with Integrated Systems Inc., Santa Clara, CA. Since 1994 he is an Associate Professor of Electrical Engineering at Sharif University of Technology. His research area is currently linear system theory, estimation and inertial navigation systems.