Automatica 35 (1999) 1701}1706
Technical Communique
Fault isolation "lter design for linear stochastic systems夽 J.-Y. Keller* CRAN-IUT de Longwy, Universite& Henri Poincare& , 186, rue de Lorraine, 54400 Cosnes et Romain, France Received 4 December 1997; received in "nal form 29 March 1999
Abstract This paper is concerned with the problem of detecting and isolating multiple faults by a special structure of the full-order Kalman "lter. A new state "ltering strategy is developed to detect and isolate multiple faults appearing simultaneously or sequentially in discrete time stochastic systems. Under a fault isolation condition, the proposed method can isolate q simultaneous faults with at least q output measurements. The fault isolation "lter generates a reduced output residual vector of dimension q so that its ith component is decoupled from all but the ith fault and so that the e!ect of plant and state noises is minimized. Necessary and su$cient conditions for stability and convergence of the proposed "lter are established. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Fault detection and isolation; Full-order Kalman "lter; State feedback decoupling
1. Introduction The model-based approach to fault detection and isolation (FDI) has received considerable attention during the last two decades. The procedures of using model information to generate additional signals, to be compared with the original measured quantities, is known as analytical redundancy. Willsky (1976), Frank (1990, 1991), Gertler (1988, 1991, 1995), and Patton and Chen (1991) have presented surveys of fault detection theory based on analytical redundancy. For enhancing the isolability of faults, the generation of residuals that have directional properties in response to a particular fault is an attractive idea in order to accomplish fault detection and isolation. The fault detection "lter, a special dynamic observer which generates directional residuals, was "rst developed by Beard (1971) and Jones (1973). The problem was later re-visited by Massoumnia (1986) in the geometric framework and by White and Speyer (1987) in the context of eigenstructure assignment. Further improvements were suggested by Park and Rizzoni (1994a) and Liu and Si (1997). Park and Rizzoni (1994b) have extended the fault detection "lters to stochastic linear systems. After an 夽 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by editor Peter Dorato. * Tel.: 0033-3-8225-9149; fax 0033-3-8223-9793. E-mail address:
[email protected] (J.Y. Keller)
eigenstructure assignment, the remaining degrees of freedom in the design of "lter's gain are used to minimize the e!ect of noises on the output residuals. The obtained fault detection "lter is then viewed as special structure of the Kalman "lter with an additional constraint of directionality on the output residuals. However, the treatment of multiple faults was not studied, convergence and stability conditions of the "lter as well. Recently, Liu and Si (1997) have proposed a full-order observer to detect and isolate multiple faults. They have designed a fault isolation "lter such that faults can be asymptotically detected and isolated. The observer's gain matrix is determined so that the ith component of the output residual is decoupled from all but the ith fault. To satisfy this property, the columns of fault detectability matrix are assigned as eigenvectors of the observer's transition matrix with a set of "xed eigenvalues. This paper proposed to extend this approach in discrete-time stochastic linear systems. The columns of the fault detectability matrix are assigned as eigenvectors of the "lter's transition matrix and the remaining design of freedom is used to minimize the e!ect of plant and state noises on the generated residuals. The obtained fault isolation "lter is very similar to the predictor}corrector structure of the standard Kalman "lter allowing the establishment of its convergence and stability conditions. The paper is organized as follows: The problem formulation is presented in Section 2. The fault isolation "lter is derived in Section 3. Stability and convergence analysis is
0005-1098/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 9 9 ) 0 0 0 7 9 - 5
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J.-Y. Keller / Automatica 35 (1999) 1701}1706
developed in Section 4. Section 5 presents an illustrative example and Section 6 summarizes the paper.
Consider the following "lter: x( "Ax( #Bu #K (y !y( ), I> I I I I I
(5a)
2. Problem statement
y( "Cx( , I I
(5b)
Under actuator or component faults, a discrete-time stochastic system is described by
where x( and y( denote the state and output estimaI I tion vectors. From Eqs. (3) and (5), the estimation errors e "(x !x( )3RL and the output residuals I I I q "(y !y( )3RK propagate as I I I
x "Ax #Bu #Fn #w , (1a) I> I I I I y "Cx #v , (1b) I I I where x 3RL is the state vector, y 3RK the obI I servation vector and u 3RN the inputs vector, I n "[n n 2 nG 2 nO]23RO the vector of fault I I I I I magnitudes and F"[f f 2 f 2 f ]3RL O the dis G O tribution matrix of actuator or component faults. We assume rank(C)"m and rank(F)"q. The noises w and I v are zero mean uncorrelated random sequences with I w = 0 I [w2 v2] " E d where =50. (1c) H H v 0 I IH I The initial state x , uncorrelated with the white noise processes w and v , is a gaussian random variable with I I E+x ,"x and E+(x !x )(x !x )2,"PM . Recall de"nitions of fault detectability indexes and matrix given by Liu and Si (1997).
De5nition 2.1. The linear time invariant system (1) is said to have fault detectability indexes o"+o , o , 2, o , if O o "min+v: CAT\f O0, v"1, 2,2,. G G
e "(A!K C)e #FM n !K v #w , I> I I I I I I
(6a)
q "Ce #v . I I I
(6b)
Due to the additive a!ects of faults occurring at time instant r (with k'r#s), the output residuals q can be I expressed from Eq. (6) as q "q #o [n 2 2 n 2 2 n 2 n 2 ]2 I I I P P I\Q I\ I\
(7a)
with o "C[G FM 2 G FM 2 G FM FM ], I P I\ P I\ I\Q\ I\ (7b) G "G G 2G I\ I\H I\ I\ I\H
(7c)
and G "(A!K C), I I
(7d)
where e and q are estimation errors and the output I I residuals if no fault was present on the system described by
De5nition 2.2. Assume that the system has "nite fault detectability indexes. The fault detectability matrix D is de"ned as
e "(A!K C)e !K v #w , I> I I I I I
(8a)
q "Ce #v . I I I
(8b)
D"C(
Theorem 2.1. Under rank(DM )"q, solution of the algebraic constraint
(2a)
with ("[AM\f AM\f 2 AMO\f ]. (2b) O Let s"max+o ,i"1,2,2,q, the maximum value of fault G detectability indexes. With n "[n 2 n 2 2 n Q2]2 and I I I I FM "[FM FM 2 FM ], where n G 3RO represents the part of Q I G faults having detectability index o and distribution G matrix FM 3RL OG, system (1) can be equivalently rewritten G x "Ax #Bu #FM n #w , (3a) I> I I I I y "Cx #v , (3b) I I I where the fault detectability matrix is then given by DM "C(
(4a)
with ("[FM
AFM 2 AQ\F ]. Q
(4b)
(A!K C)("0 I
(9a)
can be parameterized as K "u%#KM & I I
(9b)
with &"a(I !DM %), %"DM > and u"A(, K
(9c)
where KM 3RL K\O is the reduced gain describing the reI maining design of freedom, DM > the generalized inverse or pseudo-inverse of DM and a3RK\O K an arbitrary matrix determined so that matrix & is of full rows rank. Under Eq. (9a), the ewect of fault on the residual q can I then be expressed as q "q #DM [n 2 n 2 2 n Q2 ]2. I I I\ I\ I\Q
(10)
J.-Y. Keller / Automatica 35 (1999) 1701}1706
Proof. Under rank(DM )"q, we have %DM "I and O &DM "0. So, from Eqs. (9b) and (9c), we have (A!K C)("(A!(u%#KM &)C)( I I "A(!A(%DM !KM &DM I "0. Under Eq. (9a) and from Theorem 2.1, the following relations are satis"ed: CFM "C[FM 0 2 0], CG F[FM FM 2 FM ]"C[0 AFM 2 AFM ], I\ Q Q $ CG FM "C[0 0 2 0 AQ\FM ], I\ I\Q\ Q CG FM "0, I\ I\Q $ CF FM "0, I\ P leading to the output residual q expressed as I q "q #CFM n #CAFM n #2#CAQ\FM n Q I I I\ I\ Q I\Q "q #DM [n 2 n 2 2 n Q2 ]2. I I\ I\ I\Q Now, the fault isolation "lter can be designed by computing the free parameter KM so that the trace of PM "E(e e 2) I I I I (or the trace of E(q q 2)) is minimized. 䊐 I I 3. Design of the fault isolation 5lter The problem of minimizing the trace of the estimation error covariance matrix under algebraic constraints has been studied by Kitanidis (1987) for the design of a reduced-order Kalman "lter and by Nagpal, Helmick and Sims (1987) for the design of a Kalman "lter with unknown inputs. In this paper, the minimization will be made with respect to the free parameter KM avoiding the I use of the Lagrangian and Lagrange multipliers.
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where c "&(y !Cx( ), I I I qP "%(y !Cx( ) I I I has the following properties:
(11h) (11i)
E c 3RK\O is decoupled from the faults, I E qP 3RO satisfy the relation I qP "%q #[n 2 n 2 2 n Q2 ]2, (11j) I I I\ I\ I\Q where the fault n G of detectability index i directly awects I\G the reduced output residual qP with a time delay equals to its I detectability index. qP can also be viewed as a stochastic I deadbeat observer of the fault magnitudes. Proof. From Eq. (8a), the covariance PM "E(e e 2 ) propagates as I> I> I> PM "(A!K C)PM (A!K C)2#K K2#= I> I I I I I or with Eq. (9b)
matrix
PM "(A!(u%#KM &)C)PM (A!(u%#KM &)C)2 I> I I I #(u%#KM &)(u%#KM &)2#=. I I Since %&2"0, the above relation gives PM "(AM !KM CM )PM (AM !KM CM )2#= M #KM I I I I I The trace of PM is then minimized with respect to the I> free parameter KM if and only if I KM "AM PM CM 2(CM PM CM 2# I I I I I PM "(AM !KM CM )PM (AM !KM CM )2#KM I I I I I KM "AM PM CM 2(CM PM CM 2#
