- Email: [email protected]
Sliding mode observers for detection and reconstruction of sensor faults
Automatica 38 (2002) 1815 – 1821
www.elsevier.com/locate/automatica
Brief Paper
Sliding mode observers for detection and reconstruction of sensor faults Chee Pin Tan1 , Christopher Edwards ∗ Control and Instrumentation Group, Department of Engineering, University of Leicester, Leicester, LE1 7RH, UK Received 16 February 2001; received in revised form 9 October 2001; accepted 24 May 2002
Abstract This paper proposes two methods for detecting and reconstructing sensor faults using sliding mode observers. In both methods, 2ctitious systems are introduced in which the original sensor fault appears as an actuator fault. The original sensor faults are then reconstructed using a ‘secondary’ sliding mode observer. For both methods, there are certain conditions which must be satis2ed for successful fault detection and reconstruction. The methods are demonstrated using a chemical process example. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Sliding mode; Observer; Fault detection and isolation; Fault reconstruction; Linear matrix inequalities
1. Introduction Fault detection and isolation (FDI) is becoming an ever increasingly important area. A fault is deemed to occur when the system experiences an abnormal condition, such as a malfunction in the actuators=sensors. The fundamental purpose of an FDI scheme is to generate an alarm when a fault occurs and also to identify its location. Overviews of work in this area appear in Patton, Frank, and Clark (1989), Frank (1996) and Chen and Patton (1999). Observer-based methods are commonly used, where the observer signals are used to infer information about the fault. Residual generation approaches, using linear observers, have been widely used, where the di;erence between the system output and observer output is processed to form the so-called residuals. Ideally, these will be zero during fault-free operation but will give a speci2c response when a certain fault occurs. Residual generation techniques have been demonstrated by Zhang, Basseville, and Benveniste (1998) and Magni and Mouyon (1994). Recently, sliding mode observers have been used for This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Rene Boel under the direction of Editor Tamer Basar. ∗ Corresponding author. Tel.: +44-116-223-1303; fax: +44-116252-2619. E-mail addresses: [email protected] (C.P. Tan), [email protected] (C. Edwards). 1 Present address: School of Engineering and Science, Monash University Malaysia, 2 Jalan Kolaj, Bandar Sunway, 46150 Petaling, Malaysia.
FDI. Sreedhar, Fernandez, and Masada (1993) used sliding mode observers for fault detection, assuming that all states are available. Hermans and Zarrop (1996) designed a sliding mode observer such that, in the presence of faults, the sliding motion is destroyed. Edwards, Spurgeon, and Patton (2000) used the same observer to reconstruct the faults using the so-called ‘equivalent output estimation error injection’ concept. This paper builds on the work of Edwards et al. (2000) in two ways. Firstly, by considering a linear matrix inequality (LMI) observer design method proposed by Tan and Edwards (2000), more complicated examples can be considered. Previous examples in Edwards et al. (2000) and Edwards and Spurgeon (2000), although challenging from certain view points, were of low order and allowed for straightforward tuning of the gains in the observer. Secondly, in the work of Edwards et al. (2000), only the steady-state components of sensor faults were replicated. This paper will present methods to reconstruct sensor faults. The concept of reconstructing faults will enable the detection of subtle drifts in the sensor measurements, which are diFcult to detect using other methods. In this paper, two approaches will be considered. The structure of the paper is as follows: the preliminaries and background work for this paper will be presented 2rst, introducing the sliding mode observer and its design method (Tan & Edwards, 2000) followed by the method to reconstruct actuator faults by Edwards et al. (2000). Next, the 2rst method to reconstruct sensor faults will be presented,
0005-1098/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 2 ) 0 0 0 9 8 - 5
1816
C.P. Tan, C. Edwards / Automatica 38 (2002) 1815–1821
utilizing LMIs to ensure that the sliding motion of a ‘secondary observer’ is stable. The second method to reconstruct sensor faults will then be presented. Finally, both methods will be demonstrated with an example, which is a 17th-order model of a chemical plant. 2. Preliminaries This section introduces the preliminaries and background work that is vital to the work in this paper. 2.1. The sliding mode observer Consider the dynamical system x(t) ˙ = Ax(t) + Bu(t) + Ffi (t);
(1)
y(t) = Cx(t);
(2)
Minimize trace(X ) w.r.t. P, X and Y subject to P Y PA + AT P − YC − (YC)T 0 P −W −1 ¡ 0; (6) YT 0 −V −1 −P I ¡ 0; (7) I −X
where X; P ∈ Rn×n are s.p.d. matrix variables and the matrices W ∈ Rn×n and V ∈ Rp×p are user de2ned s.p.d. weights. The matrix variable P has the structure P11 P12 ¿ 0; P12 := [ P121 0 ]; P= (8) T P12 P22 where P11 ∈ R(n−p)×(n−p) and P121 ∈ R(n−p)×(p−q) . Standard LMI software can be used to synthesize P, X and Y . Given the LMI variables P11 ; P12 ; P22 ; Y , de2ne
where x ∈ Rn , u ∈ Rm and y ∈ Rp . Assume that C and F are full rank. The signal fi ∈ Rq represents an actuator fault and is unknown but bounded so that fi (t) 6 (t; y; u)−0 where : R+ ×Rp ×Rm → R+ is known and 0 is a positive scalar. Assuming that p ¿ q, Edwards and Spurgeon (1994) proposed an observer
then the observer matrix P0 in (4) is chosen as
z˙ = Az + Bu − Gl ey + Gn ;
Gl = P −1 Y;
where the discontinuous term is de2ned by −(t; y; u)CF P0 ey if ey = 0; P0 ey = 0 otherwise;
(3)
(4)
where ey := Cz − y and P0 ∈ Rp×p is a symmetric positive de2nite (s.p.d.) matrix. The matrices P0 , Gl and Gn are to be determined. Edwards and Spurgeon (1994) argue that an observer of the form given in (3) which rejects the e;ect of fi on the estimation error, exists i; A1. rank(CF) = q, A2. invariant zeros of (A; F; C) are stable. Edwards and Spurgeon (1994) have proven that if A1 is satis2ed, then there exists a change of coordinates in which the triple (A; F; C) has the following structure: 0 A11 A12 ; F= ; C = [0 T ]; (5) A= A21 A22 F20 where A11 ∈ R(n−p)×(n−p) ; F20 ∈ Rq×q is non-singular and T ∈ Rp×p is orthogonal. De2ne A211 as the top p − q rows of A21 . It can be shown that the pair (A11 ; A211 ) is detectable and its unobservable modes are the invariant zeros of (A; F; C). De2ne F2 as the bottom p rows of F. Tan and Edwards (2000) presented a method for designing this observer using LMIs. A slight modi2cation to this method will be used in this paper. The design method can be summarized as:
−1 L = [P11 P121
0(n−p)×q ]
(9)
T −1 P0 = T (P22 − P12 P11 P12 )T T
(10)
and the gains GnT = [ − TLT
T ]:
(11)
The motivation for this design is given in Tan and Edwards (2000). Satisfying conditions A1 and A2 guarantees the existence of an observer despite the restriction on the structure of P12 . This does, however, introduce a degree of conservatism into the design. The restriction on P12 arises from the requirement of satisfying a structural constraint between P, F and C. For details, see Edwards and Spurgeon (1994). Additional LMIs to (6) and (7) may be incorporated to force '(A−Gl C) to lie in certain convex regions (Tan & Edwards, 2000). Applying the change of coordinates In−p L (12) TL = 0 T to (A; F; C) and Gn yields, respectively, 0 0 A11 A12 T ; F= ; C = ; A= F2 A21 A22 Ip 0 Gn = ; (13) Ip where F2 = TF2 and A11 = A11 + LA21 is stable. 2.2. Reconstruction of faults Edwards et al. (2000) proposed a method to use sliding mode observers to reconstruct the fault fi . Assume that
C.P. Tan, C. Edwards / Automatica 38 (2002) 1815–1821
1817
observer (3) has been designed, and a sliding motion has occurred (ey = e˙ y = 0). De2ne eq as the equivalent output error injection required to maintain a sliding motion (Utkin, 1992). As argued by Edwards et al. (2000) this can be approximated online to any required accuracy. During the sliding motion (governed by A11 ), the estimation error system in the coordinates of (13) is given by e˙ 1 = A11 e1 ;
(14)
0 = A21 e1 + eq − F2 fi :
(15)
Since A11 is stable, the reconstruction signal fˆi := (FT2 F2 )−1 FT2 eq → fi
(16)
and can be computed online, from eq . Now consider the system in (1) subject to sensor faults. In this case, fi = 0 and Eq. (2) is replaced by y = Cx + f0 ;
(17)
where f0 ∈ Rp is the sensor fault vector. Assume a sliding motion has been attained, the error system in the partition of (13) satis2es e˙ 1 = A11 e1 + A12 f0 ;
(18)
f˙0 = A21 e1 + A22 f0 + eq :
(19)
As argued by Edwards et al. (2000) if A is full rank, then det(A22 − A21 A−1 11 A12 ) = 0 and at (pseudo) steady state −1 f0 ≈ −(A22 − A21 A−1 11 A12 ) eq :
(20)
3. Reconstruction of sensor faults I This paper seeks to improve on the estimate for f0 given above. Consider a new state zf ∈ Rp which is de2ned by z˙f = −Af zf + Af eq ;
(21)
where −Af ∈ Rp×p is a stable 2lter matrix. (Typically, Af represents a diagonal positive de2nite matrix where the diagonal elements represent inverse time constants.) Assume the sensor fault represents a slow incipient drift f˙0 ≈ 0 (step sensor failures are relatively easy to detect because they usually break the sliding motion (Edwards & Spurgeon , 2000)). Combining (18), (19) and (21), the following state-space representation can be obtained e˙ 1 A11 e1 A12 0 = + f0 ; −Af A21 −Af z˙f zf −Af A22
A0
e1
zf = [0 Ip ]
zf C0
F0
(22)
Fig. 1. Schematic of the fault detection structure using the secondary observer.
The signal zf is available since eq is computable online. Eqs. (22) and (23) are now in the form of (1) and (2), where the ‘actuator fault’ is f0 and the ‘output’ is zf . A sliding mode observer as described in Section 2.1 can therefore be used to reconstruct the sensor fault f0 . This will be termed the ‘secondary observer’. The triple (A0 ; B0 ; C0 ) is a square system, hence no design freedom exists associated with the sliding motion (Edwards & Spurgeon, 1994). By applying the following change of coordinates: −1 In−p A12 A−1 22 Af T0 = (24) 0 Ip to the triple (A0 ; F0 ; C0 ) to obtain the canonical form (13), it can be shown that the sliding motion of the secondary observer is governed by '(A11 −A12 A−1 22 A21 ). Notice that C0 F0 =−Af A22 and so is full rank i; A22 is full rank. Thus, necessary requirements for the existence of an appropriate secondary observer are that (A11 − A12 A−1 22 A21 ) is stable and A22 is invertible. The equivalent of Eq. (16), fˆ0 := − A−1 22 Af eq; o where eq; o is the output error injection of the secondary observer, represents a reconstruction of f0 . The complete FDI scheme is shown in Fig. 1. Remark 1. From the Schur complement; if det(A) = 0 and A22 is non-singular; then det(A11 − A12 A−1 22 A21 ) = 0 and so no stable sliding motion exists. Remark 2. A key observation is that the sliding motion is independent of Af ; so whilst Af will a;ect the gains of the observer; theoretically it has no e;ect on the reconstruction signal. In practise; the values of the diagonal elements of Af have been chosen in an e;ort to maintain=establish good numerical conditioning in the system matrices in (22) about which the secondary observer is designed. 3.1. Designing the secondary observer using LMIs
:
(23)
From the previous section, the problem now is to make A22 invertible and A11 − A12 A−1 22 A21 stable by choice of
1818
C.P. Tan, C. Edwards / Automatica 38 (2002) 1815–1821
L, whilst retaining the property that A11 is stable. This will be achieved in two stages. Write L = L1 + L2 where L1 and L2 do not necessarily have the same structure as L in (9). Decompose TL from (12) as I L1 I L2 TL = TL; 2 TL; 1 = : (25) 0 T 0 I
Lemma 4. The matrix K is invertible if A is invertible. Proof. If A is invertible; A˜ in (26) is invertible for any L1 . It can be seen from (31) that K is a Schur complement for ˜ By design; L1 is chosen to make A˜ 22 invertible; and hence A. K is invertible.
˜ the elements where A˜ 22 = A22 − A21 L1 . Applying TL; 2 to A, of A in (13) can then be written as
From Lemma 1, making A11 − A12 A−1 22 A21 stable is equivalent to making KJ −1 stable, which is achievable by making JK −1 stable. (This implies that JK −1 will be invertible, implying that J will be invertible, and from Lemma 3, A22 will be invertible.) Selecting L2 so that JK −1 is stable is equivalent to 2nding a s.p.d. P˜ ∈ R(n−p)×(n−p) and L2 satisfying
A11 = A˜ 11 + L2 A˜ 21 ;
(27)
˜ −1 + (PJK ˜ −1 )T ¡ 0: PJK
A12 = ((A˜ 12 + L2 A˜ 22 ) − (A˜ 11 + L2 A˜ 21 )L2 )T T ;
(28)
A21 = T A˜ 21 ;
(29)
A22 = T (A˜ 22 − A˜ 21 L2 )T T :
(30)
−1 −1 De2ne M =(I +L1 A˜ 22 A˜ 21 )K −1 and N =A˜ 22 A21 K −1 . Choose P˜ =P11 from the Lyapunov matrix in (8). Then recalling that L = L1 + L2 , and using the switch of variables P11 L = P12 from Section 2, inequality (33) becomes
Applying TL; 1 to the matrix A from (5) will yield A˜ 11 A˜ 12 −1 ; A˜ = TL; 1 ATL; 1 = A˜ 21 A˜ 22
(26)
Based on these de2nitions the following lemmas hold: Lemma 1. The matrix (A11 − pressed as KJ −1 where −1 K := A˜ 11 − A˜ 12 A˜ 22 A˜ 21 ;
A12 A−1 22 A21 )
can be ex-
−1 J := I − L2 A˜ 22 A˜ 21 :
(31)
Proof. This follows from straightforward algebra. Lemma 2. If det(A) = 0 then there exists an L1 such that det(A22 − A21 L1 ) = 0. Proof. Under the assumption that det(A) = 0 it follows rank[A21 A22 ] = p. Thus; the matrix pencil [sI − A22 A21 ] associated with the PHB controllability test for (A22 ; A21 ) has full rank at s = 0. This implies that s = 0 is a controllable mode of (A22 ; A21 ). Hence; L1 can always be chosen so that A˜ 22 is full rank. Assuming A˜ 22 is invertible the following can be proven: Lemma 3. J is invertible i4 A22 is invertible. Proof. The I Js := A˜ 21
matrix J in (31) is a Schur complement of L2 (32) A˜ 22
since A˜ 22 is non-singular. Hence J is invertible i; Js is invertible. However Js is invertible i; det(A˜ 22 − A˜ 21 L2 ) = 0 (a Schur complement of Js ). From the de2nitions of L; A˜ 21 and A˜ 22 ; det(A˜ 22 − A˜ 21 L2 ) = det(T T A22 T ) = det(A22 ) and hence det(J ) = 0 ⇔ det(A22 ) = 0.
P11 M + (P11 M )T − P12 N − (P12 N )T ¡ 0:
(33)
(34)
Therefore, if inequality (34) and P11 ¿ 0 have a feasible solution, then A11 − A12 A−1 22 A21 is stable by choice of L. Inequality (34) can be added to inequalities (6) and (7) when designing the primary observer. The LMI variables involved here are P11 and P12 , which are a subset of the variables in the optimization problem in Section 2. The design problem may be summarized as: Minimize trace(X ) with respect to the LMI variables P11 ; P12 ; P22 ; X and Y subject to (6), (7) and (34). Remark. Since P11 JK −1 is aFne in P11 and P12 ; pole clustering methods as described in (Chilali & Gahinet; 1996) can be incorporated to force '(JK −1 ) to lie in certain convex regions of the complex plane. For example; if the eigenvalues of JK −1 lie inside a disc centred at (q˜c ; 0) with a radius r˜c where q˜c = qc =(qc2 − rc2 ); r˜c = rc =(qc2 − rc2 ) with qc ¡ 0 and rc ¡ |qc |; then '(A11 − A12 A−1 22 A21 ) will lie in the disc centred at (qc; ; 0) with a radius rc . 3.2. An example Now the method described above for sensor fault reconstruction will be demonstrated with a chemical process plant, which is a 270 state HDA-plant that produces benzene. Hermann, Spurgeon, and Edwards (2000) linearized and model-reduced the plant to a 17th-order system. The system has 2ve outputs, 2ve control inputs and is subject to two disturbance signals. Due to space constraints the parameters of the plant and the observers will not be shown. It can be veri2ed from Hermann et al. (2000) that CF is full rank and (A; F; C) is minimum phase, where F represents the disturbance distribution matrix. For the main observer, the weighting matrices W = I17 and V = I5 and the 2lter
C.P. Tan, C. Edwards / Automatica 38 (2002) 1815–1821
1819
0.2
0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04
0 -0.2 -0.4 -0.6 -0.8 -1 0
10
20 30 Hours
40
50
0
10
20 30 Hours
40
50
Fig. 2. The left sub2gure is a fault on sensor 1 Sash inlet temperature ◦F). The right sub2gure is a fault on sensor 2 (production rate (lb mol=h)).
0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04
0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0
10
20 30 Hours
40
50
0
10
20 30 Hours
40
50
Fig. 3. The left sub2gure is the reconstruction of the fault on sensor 1. The right sub2gure is the reconstruction of the fault on sensor 2.
0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04 0
0.2 0 -0.2 -0.4 -0.6 -0.8 -1 10
20 30 Hours
40
50
0
10
20 30 Hours
40
50
Fig. 4. The left sub2gure is the ‘reconstruction’ of the fault on sensor 1. The right sub2gure is the ‘reconstruction’ of the fault on sensor 2.
Af = 10I5 . For the secondary observer, the weighting matrices W0 = 0:01I17 and V0 = 100I5 . An additional LMI constraint was added to force the eigenvalues of the secondary observer to lie to the right of a vertical line through −100 in the complex plane. Note that for this high-order example it would have been impossible to do the design by hand using pole placement design. In the following simulations the sensors were subject to white noise. Fig. 2 shows the faults acting on the sensors, and Fig. 3 shows that the secondary observer reconstructs the fault faithfully. In comparison, Fig. 4 shows the fault reconstruction using the ‘steady-state’ method by Edwards et al. (2000), approximating the fault by (20), and using only the primary observer. It can be seen that the method in this paper is an improvement: the shape of the fault is preserved with higher 2delity.
4. Reconstruction of sensor faults II In Section 3, the analysis ignored the e;ect of f˙0 . Depending on the system and the size of f˙0 , this may be signi2cant. This section presents a method to fully reconstruct the sensor fault, hence improving on the method in Section 3. Consider new states zg ∈ Rp that represent a 2ltered version of eq from the primary observer. Hence from (19), z˙g = −Ag zg − Ag A21 e1 − Ag A22 f0 + Ag f˙0 ;
(35)
where −Ag ∈ Rp×p is a stable 2lter matrix as in Section 3. Now 2lter zg to obtain a new state zh z˙h = −Ah zh + Ah zg ;
(36)
1820
C.P. Tan, C. Edwards / Automatica 38 (2002) 1815–1821
where −Ah ∈ Rp×p is a stable 2lter matrix, and de2ne a new state w ∈ Rp as w = zg − Ag f0 :
(37)
Di;erentiating (37) and combining with (35), (18) and (36) an augmented system of order n + p can be formed e˙ 1 A11 e1 0 0 w˙ = −Ag A21 −Ag 0 w z˙h 0 Ah −Ah zh
Aa
−A12
Fig. 5. Schematic of the fault detection structure using the secondary observer.
− Ag (I + A22 ) f0 ; A h Ag
(38)
The particular observer gain will be a;ected but not the reconstruction signal calculations. This FDI scheme is schematically described in Fig. 5.
Fa
e1 w : zh = [0 0 Ip ]
Ca zh
4.1. An example
(39)
The method described in this section will be demonstrated with the chemical plant in Section 3.2. For the main observer, the matrices were the same as those in Section 3.2. The 2lter matrices from (35) and (36) Ag = Ah = 10I5 . The design matrices for the secondary observer Wa = 0:01I22 and Va = 100I5 . The system was subjected to the same conditions (noise=faults) as in Section 3.2. Fig. 6 shows that the secondary observer in this case does reconstruct the sensor fault faithfully. It can be seen that this method is an improvement on the one in Section 3.
Eqs. (38) and (39) are now in the form of (1) and (2) and represent a system with an actuator fault f0 . Hence, the observer in Section 2.1 can be used, driven by the signal zh , to reconstruct f0 . Again, since (Aa ; Fa ; Ca ) is square, there is no freedom to select the sliding motion. The canonical form (13) for the secondary observer can be directly obtained by applying the transformation Ta to (Aa ; Fa ; Ca ) where −1 In−p 0 −A12 A−1 g Ah −1 Ta = (40) −A−1 −(Ag + A22 )A−1 g g Ah 0 0 0 Ip
5. Conclusion
and it can be shown that the sliding motion poles are '(A), hence this method is only applicable to open-loop stable systems. It can be seen that Ca Fa is full rank. After transformation (40), the fault f0 can be reconstructed −1 as fˆ0 := A−1 g Ah eq; a where eq; a is the equivalent output error injection of the secondary observer. Again the 2lter matrices have no theoretical e;ect on the sliding motion.
0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04 0
This paper has proposed two methods to reconstruct sensor faults using sliding mode observers. The paper builds on the work of Edwards et al. (2000) and Tan and Edwards (2000). Here the equivalent output estimation error injection is 2ltered to form a 2ctitious system in which the sensor faults appear as actuator faults. The method to obtain a good approximation of actuator faults described by
0.2 0 -0.2 -0.4 -0.6 -0.8 -1 10
20 30 Hours
40
50
0
10
20 30 Hours
40
50
Fig. 6. The left sub2gure is the reconstruction of the fault on the sensor 1. The right sub2gure is the reconstruction of the fault sensor 2.
C.P. Tan, C. Edwards / Automatica 38 (2002) 1815–1821
Edwards et al. (2000) is then used to estimate the original sensor faults. This approach therefore requires the design of a secondary sliding mode observer. This places additional design requirements on the primary sliding mode observer. The paper shows how LMI methods can be used to solve this problem and to synthesize the required observer gains eFciently. References Chen, J., & Patton, R. J. (1999). Robust model-based fault diagnosis for dynamic systems. Dordrecht: Kluwer Academic Publishers. Chilali, M., & Gahinet, P. (1996). H∞ design with pole placement constraints: An LMI approach IEEE Transactions on Automatic Control, 41, 358–367. Edwards, C., & Spurgeon, S. K. (1994). On the development of discontinuous observers. International Journal of Control, 59, 1211–1229. Edwards, C., & Spurgeon, S. K. (2000). A sliding mode observer based FDI scheme for the ship benchmark. European Journal of Control, 6, 341–356. Edwards, C., Spurgeon, S. K., & Patton, R. J. (2000). Sliding mode observers for fault detection and isolation. Automatica, 36, 541–553.
1821
Frank, P. M. (1996). Analytical and qualitative model-based fault diagnosis—a survey and some new results. European Journal of Control, 2, 6–28. Hermann, G., Spurgeon, S. K., & Edwards, C. (2000). Model-based control of the HDA-plant, a non-linear, large scale chemical process, using sliding mode and H∞ approaches. UKACC Control Conference, Cambridge, 2000. Hermans, F. J. J., & Zarrop, M. B. (1996). Sliding mode observers for robust sensor monitoring. 13th IFAC world congress, San Francisco (pp. 211–216). Magni, J. F., & Mouyon, P. (1994). On residual generation by observer and parity space approaches. IEEE Transactions on Automatic Control, 39, 441–447. Patton, R. J., Frank, P. M., & Clark, R. N. (1989). Fault diagnosis in dynamic systems: Theory and application. New York: Prentice Hall. Sreedhar, R., Fernandez, B., & Masada, G. Y. (1993). Robust fault detection in nonlinear systems using sliding mode observers. IEEE conference on control applications, Vancouver (pp. 715 –721). Tan, C. P., & Edwards, C. (2000). An LMI approach for designing sliding mode observers. IEEE conference on decision and control, Sydney (pp. 2587–2592). Utkin, V. I. (1992). Sliding modes in control optimization. Berlin: Springer. Zhang, Q., Basseville, M., & Benveniste, A. (1998). Fault detection and isolation in nonlinear dynamic systems: A combined input-output and local approach Automatica, 34, 1359–1373.
www.elsevier.com/locate/automatica
Brief Paper
Sliding mode observers for detection and reconstruction of sensor faults Chee Pin Tan1 , Christopher Edwards ∗ Control and Instrumentation Group, Department of Engineering, University of Leicester, Leicester, LE1 7RH, UK Received 16 February 2001; received in revised form 9 October 2001; accepted 24 May 2002
Abstract This paper proposes two methods for detecting and reconstructing sensor faults using sliding mode observers. In both methods, 2ctitious systems are introduced in which the original sensor fault appears as an actuator fault. The original sensor faults are then reconstructed using a ‘secondary’ sliding mode observer. For both methods, there are certain conditions which must be satis2ed for successful fault detection and reconstruction. The methods are demonstrated using a chemical process example. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Sliding mode; Observer; Fault detection and isolation; Fault reconstruction; Linear matrix inequalities
1. Introduction Fault detection and isolation (FDI) is becoming an ever increasingly important area. A fault is deemed to occur when the system experiences an abnormal condition, such as a malfunction in the actuators=sensors. The fundamental purpose of an FDI scheme is to generate an alarm when a fault occurs and also to identify its location. Overviews of work in this area appear in Patton, Frank, and Clark (1989), Frank (1996) and Chen and Patton (1999). Observer-based methods are commonly used, where the observer signals are used to infer information about the fault. Residual generation approaches, using linear observers, have been widely used, where the di;erence between the system output and observer output is processed to form the so-called residuals. Ideally, these will be zero during fault-free operation but will give a speci2c response when a certain fault occurs. Residual generation techniques have been demonstrated by Zhang, Basseville, and Benveniste (1998) and Magni and Mouyon (1994). Recently, sliding mode observers have been used for This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Rene Boel under the direction of Editor Tamer Basar. ∗ Corresponding author. Tel.: +44-116-223-1303; fax: +44-116252-2619. E-mail addresses: [email protected] (C.P. Tan), [email protected] (C. Edwards). 1 Present address: School of Engineering and Science, Monash University Malaysia, 2 Jalan Kolaj, Bandar Sunway, 46150 Petaling, Malaysia.
FDI. Sreedhar, Fernandez, and Masada (1993) used sliding mode observers for fault detection, assuming that all states are available. Hermans and Zarrop (1996) designed a sliding mode observer such that, in the presence of faults, the sliding motion is destroyed. Edwards, Spurgeon, and Patton (2000) used the same observer to reconstruct the faults using the so-called ‘equivalent output estimation error injection’ concept. This paper builds on the work of Edwards et al. (2000) in two ways. Firstly, by considering a linear matrix inequality (LMI) observer design method proposed by Tan and Edwards (2000), more complicated examples can be considered. Previous examples in Edwards et al. (2000) and Edwards and Spurgeon (2000), although challenging from certain view points, were of low order and allowed for straightforward tuning of the gains in the observer. Secondly, in the work of Edwards et al. (2000), only the steady-state components of sensor faults were replicated. This paper will present methods to reconstruct sensor faults. The concept of reconstructing faults will enable the detection of subtle drifts in the sensor measurements, which are diFcult to detect using other methods. In this paper, two approaches will be considered. The structure of the paper is as follows: the preliminaries and background work for this paper will be presented 2rst, introducing the sliding mode observer and its design method (Tan & Edwards, 2000) followed by the method to reconstruct actuator faults by Edwards et al. (2000). Next, the 2rst method to reconstruct sensor faults will be presented,
0005-1098/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 2 ) 0 0 0 9 8 - 5
1816
C.P. Tan, C. Edwards / Automatica 38 (2002) 1815–1821
utilizing LMIs to ensure that the sliding motion of a ‘secondary observer’ is stable. The second method to reconstruct sensor faults will then be presented. Finally, both methods will be demonstrated with an example, which is a 17th-order model of a chemical plant. 2. Preliminaries This section introduces the preliminaries and background work that is vital to the work in this paper. 2.1. The sliding mode observer Consider the dynamical system x(t) ˙ = Ax(t) + Bu(t) + Ffi (t);
(1)
y(t) = Cx(t);
(2)
Minimize trace(X ) w.r.t. P, X and Y subject to P Y PA + AT P − YC − (YC)T 0 P −W −1 ¡ 0; (6) YT 0 −V −1 −P I ¡ 0; (7) I −X
where X; P ∈ Rn×n are s.p.d. matrix variables and the matrices W ∈ Rn×n and V ∈ Rp×p are user de2ned s.p.d. weights. The matrix variable P has the structure P11 P12 ¿ 0; P12 := [ P121 0 ]; P= (8) T P12 P22 where P11 ∈ R(n−p)×(n−p) and P121 ∈ R(n−p)×(p−q) . Standard LMI software can be used to synthesize P, X and Y . Given the LMI variables P11 ; P12 ; P22 ; Y , de2ne
where x ∈ Rn , u ∈ Rm and y ∈ Rp . Assume that C and F are full rank. The signal fi ∈ Rq represents an actuator fault and is unknown but bounded so that fi (t) 6 (t; y; u)−0 where : R+ ×Rp ×Rm → R+ is known and 0 is a positive scalar. Assuming that p ¿ q, Edwards and Spurgeon (1994) proposed an observer
then the observer matrix P0 in (4) is chosen as
z˙ = Az + Bu − Gl ey + Gn ;
Gl = P −1 Y;
where the discontinuous term is de2ned by −(t; y; u)CF P0 ey if ey = 0; P0 ey = 0 otherwise;
(3)
(4)
where ey := Cz − y and P0 ∈ Rp×p is a symmetric positive de2nite (s.p.d.) matrix. The matrices P0 , Gl and Gn are to be determined. Edwards and Spurgeon (1994) argue that an observer of the form given in (3) which rejects the e;ect of fi on the estimation error, exists i; A1. rank(CF) = q, A2. invariant zeros of (A; F; C) are stable. Edwards and Spurgeon (1994) have proven that if A1 is satis2ed, then there exists a change of coordinates in which the triple (A; F; C) has the following structure: 0 A11 A12 ; F= ; C = [0 T ]; (5) A= A21 A22 F20 where A11 ∈ R(n−p)×(n−p) ; F20 ∈ Rq×q is non-singular and T ∈ Rp×p is orthogonal. De2ne A211 as the top p − q rows of A21 . It can be shown that the pair (A11 ; A211 ) is detectable and its unobservable modes are the invariant zeros of (A; F; C). De2ne F2 as the bottom p rows of F. Tan and Edwards (2000) presented a method for designing this observer using LMIs. A slight modi2cation to this method will be used in this paper. The design method can be summarized as:
−1 L = [P11 P121
0(n−p)×q ]
(9)
T −1 P0 = T (P22 − P12 P11 P12 )T T
(10)
and the gains GnT = [ − TLT
T ]:
(11)
The motivation for this design is given in Tan and Edwards (2000). Satisfying conditions A1 and A2 guarantees the existence of an observer despite the restriction on the structure of P12 . This does, however, introduce a degree of conservatism into the design. The restriction on P12 arises from the requirement of satisfying a structural constraint between P, F and C. For details, see Edwards and Spurgeon (1994). Additional LMIs to (6) and (7) may be incorporated to force '(A−Gl C) to lie in certain convex regions (Tan & Edwards, 2000). Applying the change of coordinates In−p L (12) TL = 0 T to (A; F; C) and Gn yields, respectively, 0 0 A11 A12 T ; F= ; C = ; A= F2 A21 A22 Ip 0 Gn = ; (13) Ip where F2 = TF2 and A11 = A11 + LA21 is stable. 2.2. Reconstruction of faults Edwards et al. (2000) proposed a method to use sliding mode observers to reconstruct the fault fi . Assume that
C.P. Tan, C. Edwards / Automatica 38 (2002) 1815–1821
1817
observer (3) has been designed, and a sliding motion has occurred (ey = e˙ y = 0). De2ne eq as the equivalent output error injection required to maintain a sliding motion (Utkin, 1992). As argued by Edwards et al. (2000) this can be approximated online to any required accuracy. During the sliding motion (governed by A11 ), the estimation error system in the coordinates of (13) is given by e˙ 1 = A11 e1 ;
(14)
0 = A21 e1 + eq − F2 fi :
(15)
Since A11 is stable, the reconstruction signal fˆi := (FT2 F2 )−1 FT2 eq → fi
(16)
and can be computed online, from eq . Now consider the system in (1) subject to sensor faults. In this case, fi = 0 and Eq. (2) is replaced by y = Cx + f0 ;
(17)
where f0 ∈ Rp is the sensor fault vector. Assume a sliding motion has been attained, the error system in the partition of (13) satis2es e˙ 1 = A11 e1 + A12 f0 ;
(18)
f˙0 = A21 e1 + A22 f0 + eq :
(19)
As argued by Edwards et al. (2000) if A is full rank, then det(A22 − A21 A−1 11 A12 ) = 0 and at (pseudo) steady state −1 f0 ≈ −(A22 − A21 A−1 11 A12 ) eq :
(20)
3. Reconstruction of sensor faults I This paper seeks to improve on the estimate for f0 given above. Consider a new state zf ∈ Rp which is de2ned by z˙f = −Af zf + Af eq ;
(21)
where −Af ∈ Rp×p is a stable 2lter matrix. (Typically, Af represents a diagonal positive de2nite matrix where the diagonal elements represent inverse time constants.) Assume the sensor fault represents a slow incipient drift f˙0 ≈ 0 (step sensor failures are relatively easy to detect because they usually break the sliding motion (Edwards & Spurgeon , 2000)). Combining (18), (19) and (21), the following state-space representation can be obtained e˙ 1 A11 e1 A12 0 = + f0 ; −Af A21 −Af z˙f zf −Af A22
A0
e1
zf = [0 Ip ]
zf C0
F0
(22)
Fig. 1. Schematic of the fault detection structure using the secondary observer.
The signal zf is available since eq is computable online. Eqs. (22) and (23) are now in the form of (1) and (2), where the ‘actuator fault’ is f0 and the ‘output’ is zf . A sliding mode observer as described in Section 2.1 can therefore be used to reconstruct the sensor fault f0 . This will be termed the ‘secondary observer’. The triple (A0 ; B0 ; C0 ) is a square system, hence no design freedom exists associated with the sliding motion (Edwards & Spurgeon, 1994). By applying the following change of coordinates: −1 In−p A12 A−1 22 Af T0 = (24) 0 Ip to the triple (A0 ; F0 ; C0 ) to obtain the canonical form (13), it can be shown that the sliding motion of the secondary observer is governed by '(A11 −A12 A−1 22 A21 ). Notice that C0 F0 =−Af A22 and so is full rank i; A22 is full rank. Thus, necessary requirements for the existence of an appropriate secondary observer are that (A11 − A12 A−1 22 A21 ) is stable and A22 is invertible. The equivalent of Eq. (16), fˆ0 := − A−1 22 Af eq; o where eq; o is the output error injection of the secondary observer, represents a reconstruction of f0 . The complete FDI scheme is shown in Fig. 1. Remark 1. From the Schur complement; if det(A) = 0 and A22 is non-singular; then det(A11 − A12 A−1 22 A21 ) = 0 and so no stable sliding motion exists. Remark 2. A key observation is that the sliding motion is independent of Af ; so whilst Af will a;ect the gains of the observer; theoretically it has no e;ect on the reconstruction signal. In practise; the values of the diagonal elements of Af have been chosen in an e;ort to maintain=establish good numerical conditioning in the system matrices in (22) about which the secondary observer is designed. 3.1. Designing the secondary observer using LMIs
:
(23)
From the previous section, the problem now is to make A22 invertible and A11 − A12 A−1 22 A21 stable by choice of
1818
C.P. Tan, C. Edwards / Automatica 38 (2002) 1815–1821
L, whilst retaining the property that A11 is stable. This will be achieved in two stages. Write L = L1 + L2 where L1 and L2 do not necessarily have the same structure as L in (9). Decompose TL from (12) as I L1 I L2 TL = TL; 2 TL; 1 = : (25) 0 T 0 I
Lemma 4. The matrix K is invertible if A is invertible. Proof. If A is invertible; A˜ in (26) is invertible for any L1 . It can be seen from (31) that K is a Schur complement for ˜ By design; L1 is chosen to make A˜ 22 invertible; and hence A. K is invertible.
˜ the elements where A˜ 22 = A22 − A21 L1 . Applying TL; 2 to A, of A in (13) can then be written as
From Lemma 1, making A11 − A12 A−1 22 A21 stable is equivalent to making KJ −1 stable, which is achievable by making JK −1 stable. (This implies that JK −1 will be invertible, implying that J will be invertible, and from Lemma 3, A22 will be invertible.) Selecting L2 so that JK −1 is stable is equivalent to 2nding a s.p.d. P˜ ∈ R(n−p)×(n−p) and L2 satisfying
A11 = A˜ 11 + L2 A˜ 21 ;
(27)
˜ −1 + (PJK ˜ −1 )T ¡ 0: PJK
A12 = ((A˜ 12 + L2 A˜ 22 ) − (A˜ 11 + L2 A˜ 21 )L2 )T T ;
(28)
A21 = T A˜ 21 ;
(29)
A22 = T (A˜ 22 − A˜ 21 L2 )T T :
(30)
−1 −1 De2ne M =(I +L1 A˜ 22 A˜ 21 )K −1 and N =A˜ 22 A21 K −1 . Choose P˜ =P11 from the Lyapunov matrix in (8). Then recalling that L = L1 + L2 , and using the switch of variables P11 L = P12 from Section 2, inequality (33) becomes
Applying TL; 1 to the matrix A from (5) will yield A˜ 11 A˜ 12 −1 ; A˜ = TL; 1 ATL; 1 = A˜ 21 A˜ 22
(26)
Based on these de2nitions the following lemmas hold: Lemma 1. The matrix (A11 − pressed as KJ −1 where −1 K := A˜ 11 − A˜ 12 A˜ 22 A˜ 21 ;
A12 A−1 22 A21 )
can be ex-
−1 J := I − L2 A˜ 22 A˜ 21 :
(31)
Proof. This follows from straightforward algebra. Lemma 2. If det(A) = 0 then there exists an L1 such that det(A22 − A21 L1 ) = 0. Proof. Under the assumption that det(A) = 0 it follows rank[A21 A22 ] = p. Thus; the matrix pencil [sI − A22 A21 ] associated with the PHB controllability test for (A22 ; A21 ) has full rank at s = 0. This implies that s = 0 is a controllable mode of (A22 ; A21 ). Hence; L1 can always be chosen so that A˜ 22 is full rank. Assuming A˜ 22 is invertible the following can be proven: Lemma 3. J is invertible i4 A22 is invertible. Proof. The I Js := A˜ 21
matrix J in (31) is a Schur complement of L2 (32) A˜ 22
since A˜ 22 is non-singular. Hence J is invertible i; Js is invertible. However Js is invertible i; det(A˜ 22 − A˜ 21 L2 ) = 0 (a Schur complement of Js ). From the de2nitions of L; A˜ 21 and A˜ 22 ; det(A˜ 22 − A˜ 21 L2 ) = det(T T A22 T ) = det(A22 ) and hence det(J ) = 0 ⇔ det(A22 ) = 0.
P11 M + (P11 M )T − P12 N − (P12 N )T ¡ 0:
(33)
(34)
Therefore, if inequality (34) and P11 ¿ 0 have a feasible solution, then A11 − A12 A−1 22 A21 is stable by choice of L. Inequality (34) can be added to inequalities (6) and (7) when designing the primary observer. The LMI variables involved here are P11 and P12 , which are a subset of the variables in the optimization problem in Section 2. The design problem may be summarized as: Minimize trace(X ) with respect to the LMI variables P11 ; P12 ; P22 ; X and Y subject to (6), (7) and (34). Remark. Since P11 JK −1 is aFne in P11 and P12 ; pole clustering methods as described in (Chilali & Gahinet; 1996) can be incorporated to force '(JK −1 ) to lie in certain convex regions of the complex plane. For example; if the eigenvalues of JK −1 lie inside a disc centred at (q˜c ; 0) with a radius r˜c where q˜c = qc =(qc2 − rc2 ); r˜c = rc =(qc2 − rc2 ) with qc ¡ 0 and rc ¡ |qc |; then '(A11 − A12 A−1 22 A21 ) will lie in the disc centred at (qc; ; 0) with a radius rc . 3.2. An example Now the method described above for sensor fault reconstruction will be demonstrated with a chemical process plant, which is a 270 state HDA-plant that produces benzene. Hermann, Spurgeon, and Edwards (2000) linearized and model-reduced the plant to a 17th-order system. The system has 2ve outputs, 2ve control inputs and is subject to two disturbance signals. Due to space constraints the parameters of the plant and the observers will not be shown. It can be veri2ed from Hermann et al. (2000) that CF is full rank and (A; F; C) is minimum phase, where F represents the disturbance distribution matrix. For the main observer, the weighting matrices W = I17 and V = I5 and the 2lter
C.P. Tan, C. Edwards / Automatica 38 (2002) 1815–1821
1819
0.2
0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04
0 -0.2 -0.4 -0.6 -0.8 -1 0
10
20 30 Hours
40
50
0
10
20 30 Hours
40
50
Fig. 2. The left sub2gure is a fault on sensor 1 Sash inlet temperature ◦F). The right sub2gure is a fault on sensor 2 (production rate (lb mol=h)).
0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04
0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0
10
20 30 Hours
40
50
0
10
20 30 Hours
40
50
Fig. 3. The left sub2gure is the reconstruction of the fault on sensor 1. The right sub2gure is the reconstruction of the fault on sensor 2.
0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04 0
0.2 0 -0.2 -0.4 -0.6 -0.8 -1 10
20 30 Hours
40
50
0
10
20 30 Hours
40
50
Fig. 4. The left sub2gure is the ‘reconstruction’ of the fault on sensor 1. The right sub2gure is the ‘reconstruction’ of the fault on sensor 2.
Af = 10I5 . For the secondary observer, the weighting matrices W0 = 0:01I17 and V0 = 100I5 . An additional LMI constraint was added to force the eigenvalues of the secondary observer to lie to the right of a vertical line through −100 in the complex plane. Note that for this high-order example it would have been impossible to do the design by hand using pole placement design. In the following simulations the sensors were subject to white noise. Fig. 2 shows the faults acting on the sensors, and Fig. 3 shows that the secondary observer reconstructs the fault faithfully. In comparison, Fig. 4 shows the fault reconstruction using the ‘steady-state’ method by Edwards et al. (2000), approximating the fault by (20), and using only the primary observer. It can be seen that the method in this paper is an improvement: the shape of the fault is preserved with higher 2delity.
4. Reconstruction of sensor faults II In Section 3, the analysis ignored the e;ect of f˙0 . Depending on the system and the size of f˙0 , this may be signi2cant. This section presents a method to fully reconstruct the sensor fault, hence improving on the method in Section 3. Consider new states zg ∈ Rp that represent a 2ltered version of eq from the primary observer. Hence from (19), z˙g = −Ag zg − Ag A21 e1 − Ag A22 f0 + Ag f˙0 ;
(35)
where −Ag ∈ Rp×p is a stable 2lter matrix as in Section 3. Now 2lter zg to obtain a new state zh z˙h = −Ah zh + Ah zg ;
(36)
1820
C.P. Tan, C. Edwards / Automatica 38 (2002) 1815–1821
where −Ah ∈ Rp×p is a stable 2lter matrix, and de2ne a new state w ∈ Rp as w = zg − Ag f0 :
(37)
Di;erentiating (37) and combining with (35), (18) and (36) an augmented system of order n + p can be formed e˙ 1 A11 e1 0 0 w˙ = −Ag A21 −Ag 0 w z˙h 0 Ah −Ah zh
Aa
−A12
Fig. 5. Schematic of the fault detection structure using the secondary observer.
− Ag (I + A22 ) f0 ; A h Ag
(38)
The particular observer gain will be a;ected but not the reconstruction signal calculations. This FDI scheme is schematically described in Fig. 5.
Fa
e1 w : zh = [0 0 Ip ]
Ca zh
4.1. An example
(39)
The method described in this section will be demonstrated with the chemical plant in Section 3.2. For the main observer, the matrices were the same as those in Section 3.2. The 2lter matrices from (35) and (36) Ag = Ah = 10I5 . The design matrices for the secondary observer Wa = 0:01I22 and Va = 100I5 . The system was subjected to the same conditions (noise=faults) as in Section 3.2. Fig. 6 shows that the secondary observer in this case does reconstruct the sensor fault faithfully. It can be seen that this method is an improvement on the one in Section 3.
Eqs. (38) and (39) are now in the form of (1) and (2) and represent a system with an actuator fault f0 . Hence, the observer in Section 2.1 can be used, driven by the signal zh , to reconstruct f0 . Again, since (Aa ; Fa ; Ca ) is square, there is no freedom to select the sliding motion. The canonical form (13) for the secondary observer can be directly obtained by applying the transformation Ta to (Aa ; Fa ; Ca ) where −1 In−p 0 −A12 A−1 g Ah −1 Ta = (40) −A−1 −(Ag + A22 )A−1 g g Ah 0 0 0 Ip
5. Conclusion
and it can be shown that the sliding motion poles are '(A), hence this method is only applicable to open-loop stable systems. It can be seen that Ca Fa is full rank. After transformation (40), the fault f0 can be reconstructed −1 as fˆ0 := A−1 g Ah eq; a where eq; a is the equivalent output error injection of the secondary observer. Again the 2lter matrices have no theoretical e;ect on the sliding motion.
0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04 0
This paper has proposed two methods to reconstruct sensor faults using sliding mode observers. The paper builds on the work of Edwards et al. (2000) and Tan and Edwards (2000). Here the equivalent output estimation error injection is 2ltered to form a 2ctitious system in which the sensor faults appear as actuator faults. The method to obtain a good approximation of actuator faults described by
0.2 0 -0.2 -0.4 -0.6 -0.8 -1 10
20 30 Hours
40
50
0
10
20 30 Hours
40
50
Fig. 6. The left sub2gure is the reconstruction of the fault on the sensor 1. The right sub2gure is the reconstruction of the fault sensor 2.
C.P. Tan, C. Edwards / Automatica 38 (2002) 1815–1821
Edwards et al. (2000) is then used to estimate the original sensor faults. This approach therefore requires the design of a secondary sliding mode observer. This places additional design requirements on the primary sliding mode observer. The paper shows how LMI methods can be used to solve this problem and to synthesize the required observer gains eFciently. References Chen, J., & Patton, R. J. (1999). Robust model-based fault diagnosis for dynamic systems. Dordrecht: Kluwer Academic Publishers. Chilali, M., & Gahinet, P. (1996). H∞ design with pole placement constraints: An LMI approach IEEE Transactions on Automatic Control, 41, 358–367. Edwards, C., & Spurgeon, S. K. (1994). On the development of discontinuous observers. International Journal of Control, 59, 1211–1229. Edwards, C., & Spurgeon, S. K. (2000). A sliding mode observer based FDI scheme for the ship benchmark. European Journal of Control, 6, 341–356. Edwards, C., Spurgeon, S. K., & Patton, R. J. (2000). Sliding mode observers for fault detection and isolation. Automatica, 36, 541–553.
1821
Frank, P. M. (1996). Analytical and qualitative model-based fault diagnosis—a survey and some new results. European Journal of Control, 2, 6–28. Hermann, G., Spurgeon, S. K., & Edwards, C. (2000). Model-based control of the HDA-plant, a non-linear, large scale chemical process, using sliding mode and H∞ approaches. UKACC Control Conference, Cambridge, 2000. Hermans, F. J. J., & Zarrop, M. B. (1996). Sliding mode observers for robust sensor monitoring. 13th IFAC world congress, San Francisco (pp. 211–216). Magni, J. F., & Mouyon, P. (1994). On residual generation by observer and parity space approaches. IEEE Transactions on Automatic Control, 39, 441–447. Patton, R. J., Frank, P. M., & Clark, R. N. (1989). Fault diagnosis in dynamic systems: Theory and application. New York: Prentice Hall. Sreedhar, R., Fernandez, B., & Masada, G. Y. (1993). Robust fault detection in nonlinear systems using sliding mode observers. IEEE conference on control applications, Vancouver (pp. 715 –721). Tan, C. P., & Edwards, C. (2000). An LMI approach for designing sliding mode observers. IEEE conference on decision and control, Sydney (pp. 2587–2592). Utkin, V. I. (1992). Sliding modes in control optimization. Berlin: Springer. Zhang, Q., Basseville, M., & Benveniste, A. (1998). Fault detection and isolation in nonlinear dynamic systems: A combined input-output and local approach Automatica, 34, 1359–1373.