Extended results on robust state estimation and fault detection *
Extended results on robust state estimation and fault detection ⋆ C.P. Tan a , F. Crusca b , M. Aldeen c a b
School of Engineering, Monash University Malaysia, 2 Jalan Kolej, 46150 Petaling Jaya, Malaysia
Department of Electrical and Computer Systems Engineering, Building 72, Monash University, Melbourne, Victoria 3800, Australia c
Department of Electrical and Electronic Engineering, University of Melbourne, Parkville, Victoria, 3010, Australia
Abstract A common requirement implicit in current methods for the design of robust state estimators and robust fault detection filters is that the first Markov matrix must be non-zero, and indeed, full rank. We relax both of these restrictions in this paper to allow the applicability to a wider range of systems. The extended results are then applied to an aircraft fault detection for which the restrictive condition is not satisfied. Key words: fault detection, robust estimation, sliding mode, observers
1
Introduction
Fault detection and isolation (FDI) has been the subject of extensive research for some time now, but especially since the early 1990s [3,11]. The interest in this line of research stems from its practical application to a variety of industries such as aerospace, energy systems, and process control to name a few. The main function of an FDI scheme is to detect a fault when it happens, which may then be acted on by sending alarm signals, taking protection measures, or reconfiguring a running control scheme. The most commonly used schemes are observer based approaches [11], where a residual signal is generated and flagged when an abnormal (faulty) condition takes place in the system. Robust state estimation and fault detection methods [4] were introduced in an attempt to achieve accurate fault detection and state estimation in the presence of uncertainties in system parameters, and/or changes in plant operating conditions. The extension to nonlinear observers based on sliding mode theory [6,7], also offered the ability to account for unknown disturbances and model inaccuracy. With respect to sliding mode observer (SMO) based approaches, Edwards et al [7] reported improvements on that of [6] approach where a nonlinear term is used to reconstruct the fault and therefore provide full information about it. It is interesting to observe that a common thread in tech⋆ This paper was not presented at any IFAC meeting. Corresponding author C.P. Tan. Tel. +6-03-56360600 Fax +6-03-56329314 Email addresses: [email protected] (C.P. Tan), [email protected] (F. Crusca), [email protected] (M. Aldeen).
Preprint submitted to Automatica
niques for robust state estimation [6], unknown input observers [10,8,9,5,2], and fault detection filters [14] is that they require satisfaction of the rather strict condition that the first Markov (see e.g. [1]) parameter is non-zero, and indeed, full rank. This severely limits the applicability of these approaches for a wide range of practical systems. In this paper we remove this restrictive assumption and extend the state estimation/fault detection approach of [6] to a wider practical class of systems. This is achieved by introducing a supplementary observer which together with the observer [6] produces full state estimation and fault detection. Thus the removal of the aforementioned constraint has led to the development of a more general approach, which has immediate application to the two areas of: (i) robust state estimation; and (ii) robust fault detection and isolation for systems where the first Markov matrix is rank degenerate. To illustrate the advantages of this approach we apply the new approach to an aircraft control system model where the first Markov matrix is rank deficient. Simulation studies are carried out on this example and results demonstrate the properties of the new approach. 2
Preliminaries and motivation
Consider the dynamic system below ¯ f¯, y = Cx x˙ = Ax + Bu + M n
(1)
m
where x ∈ R is the state, u ∈ R is the control input, ¯,C y ∈ Rp is the output and f¯ ∈ Rq¯ is a fault. Matrices A, M are known and constant. Edwards et al.[6,7] considered this class of systems and developed a sliding mode observer (SMO) that is able to asymptotically estimate x and f¯ upon satisfaction of the following conditions
18 June 2007
¯ ) = q¯ (full column rank) A1 rank(C M ¯ , C) are stable A2 invariant zeros of (A, M
where x ˆ is the estimate of x, and ey = C x ˆ − y is the output estimation error. The matrices G, H ∈ Rn×p are gain matrices that will be described in greater detail later. The term ν is a nonlinear injection term defined by ey ν = −ρ , ey 6= 0, ρ ∈ R+ (5) key k
Subsequently Tan & Edwards [17] extended the class of systems to include unknown signals encapsulating nonlinearites/uncertainties in the system along the lines proposed in [3]. The extended class of systems is now expressed as ¯ y = Cx ¯ f¯ + Q ¯ ξ, x˙ = Ax + Bu + M (2)
In the following we show by examining the error dynamics that the design of [6,7] cannot be used on its own to provide full state estimation and fault detection for the class of systems described by (3). Without loss of generality, the triple (A, M, C) after some coordinate transformation can be written as follows: " # " # h i A1 A2 M1 A= ,M = ,C = 0 T (6) A3 A4 M2
¯
where ξ¯ ∈ Rh is the unknown input signal. They showed that under conditions A1,A2 above, it is possible to design a SMO that would minimize but not completely eliminate the effect of ξ¯ on the state estimation and fault reconstruction. Remark 1: If there are nonlinearities and/or faults in the output equation of (2), they can be easily moved to the state equation of (2) using the procedure in Section 5 of [17] and the existing methods can be used to reconstruct the fault. ♯
where A1 ∈ R(n−p)×(n−p) , M2 ∈ Rp×q , T ∈ Rp×p is orthogonal. From Condition C2, it is clear that rank(M2 ) = r. The matrices A1 , A3 , M1 , M2 can be partitioned as follows # # " " A3,a A3,b A1,a A1,b (7) , A3 = A1 = A3,c A3,d A1,c A1,d " # " # M11 0 0 0 M1 = , M2 = (8) 0 0 0 M22
In this paper we extend the results of [17] to the case where the effect of the unknown signal is totally nullified. This is achieved by proposing a supplementary observer that would complement that of [6] so that when augmented together, asymptotic estimate state estimation and fault detection are achieved irrespective of the unknown signal. The conditions under which such results can be achieved are now stated. Let us re-model system (2) as follows x˙ = Ax + Bu + M f (3) h iT i h ¯ Q ¯ , f ∈ Rq = f¯T ξ¯T . where M ∈ Rn×q = M Assume n > p > q.
where M11 ∈ R(q−r)×(q−r) and M22 ∈ Rr×r are invertible. Furthermore A1,a ∈ R(q−r)×(q−r) and A3,a ∈ R(p−r)×(q−r) . However, A1 , A3 have no particular structure. The fault signal f is scaled and partitioned as follows iT h where U ∈ Rq×q is orthogonal, f → U f = f1T f2T and f1 ∈ Rq−r . Details are available in the Appendix.
Remark 2: When p < q then more than one observer is needed; the exact number is at least equal to the ratio pq rounded up to the next digit, provided both A1 and A2 are satisfied for each observer. For example at least 2 observers are required when p = 3, q = 5. This may be implemented " # i f h 1 . Then one SMO by partitioning: M f = M1 M2 f2 is designed for x˙ = Ax + Bu + M1 f1 , y = Cx and another for x˙ = Ax + Bu + M2 f2 , y = Cx. ♯
Define e := x ˆ − x as the state estimation error to obtain e˙ = Ao e + Hν − M f (9) where Ao = A − GC. In the coordinates of (6) the gain matrix H has the structure h iT H = −LT I (Po T )−1 (10) where Po ∈ Rp×p is symmetric positive definite (s.p.d.) and iT h i h , L1 ∈ R(q−r)×(p−r) . L = Lo 0 , Lo = LT1 LT2
Note that in [6], their results apply to the case where (3) satisfies the following conditions B1 rank(CM ) = q (full column rank) B2 invariant zeros of (A, M, C) are stable
Problem formulation
Apply the change of coordinates TL to the matrices in (6), " # I L where TL = . Then it is straightforward to see that 0 T " # " # A1 + LA3 ∗ M1 + LM2 −1 TL ATL = , TL M = (11) T M2 T A3 ∗ " # " # h i 0 x1 −1 CTL = 0 Ip , TL H = (12) , TL x = −1 y Po
From [6], the SMO for (3) is x ˆ˙ = Aˆ x + Bu − Gey + Hν
where (*) are matrices that play no more role in the following analysis. In this coordinate system, notice that the bottom p rows of the state vector is the output y, due to the structure
It is well known that condition B1 is quite restrictive and may not apply to a wide range of systems. This is due to the requirement that the first Markov matrix must be both nonzero and full rank. We now relax this condition, and propose a new set of milder conditions C1 C, M are full rank matrices C2 rank(CM ) = r < q 3
(4)
2
of CTL−1 , and x1 ∈ Rn−p are the ‘non-output’ states. Also, iT h the error vector now has the structure TL e = eT1 eTy where e1 ∈ Rn−p is the estimation error of x1 . Theorem 1 [17] Assume that there exists a s.p.d. matrix " # P1 P1 L n×n P ∈R = (13) LT P1 LT P1 L + T T Po T
D2 The invariant zeros of (A, M, C) must be stable In the remainder of this section, we develop a constructive proof of Theorem 2. 4.1
In the following we show that under conditions D1 and D2 it is possible to design a supplementary observer to asymptotically estimate x1 . Since y can be estimated by the design procedure in [6], henceforth called the primary observer, the outcome is full state estimation.
that satisfies P Ao + ATo P < 0. Then define two positive scalars µ0 = −λmax P Ao + ATo P , µ1 = kP M k If ρ > µ−1 0 kf k(2kPo T A3 kµ1 +µ0 kPo T M2 k), then in finite time, a sliding motion takes place on S = {e : Ce = 0} Next suppose an ideal sliding motion has taken place on S, then ν becomes νeq 1 and ey = e˙ y = 0. As a result, the observer (4) gives an accurate estimate of the output state since ey has been forced to zero in finite time. However, an accurate estimate of x1 is not possible, as shown next.
Re-arranging (15) gives h := T T Po−1 νeq = −A3 e1 + M2 f . Then partition h according to (7) - (8) # " " " # # 0 A3,a A3,b h1 e1 + =− h= f2 (17) A3,c A3,d h2 M22 Treat h1 as the ‘output’, and partition as in (7) - (8). Then the following system can be obtained from (14) and (17)
Note that LM2 = 0. Pre-multiplying (9) by TL and partitioning according to (11) - (12) yields the set of equations e˙ 1 = (A1 + LA3 )e1 − M1 f (14) −1 0 = T A3 e1 − T M2 f + Po νeq (15)
e˙ 1 = Ae1 e1 − Me1 f1 , h1 = −Ce1 e1 i iT h h T where Me1 = M11 0 , Ce1 = A3,a A3,b and " # A1,a + L1 A3,a A1,b + L1 A3,b Ae1 = A1,c + L2 A3,a A1,d + L2 A3,b
From (14) - (15), it is clear that the observer (4) can track x1 if and only if M1 = 0 [6] which implies that rank(CM ) = rank(M ) and the first Markov matrix CM = T M2 is full rank. Then e1 → 0 and f can be recovered from (M2T M2 )−1 M2T T T Po−1 νeq . Note that we have used the definition of the Markov matrices [1] for the system (A, M, C) as being {H0 , H1 , ..., Hk , ...} with Hk = CAk−1 M . If however CM is not full rank, as is the case for the class of systems in this paper (see Condition C2), then from (14), e1 will not converge to zero since M1 6= 0. If x ˆ1 is the estimate of x1 , then x ˆ1 will not track x1 . Instead, we have x ˆ1 = x1 + e1 (16) We overcome this by proposing a supplementary observer, as outlined in the following sections. 4
State estimation
(18)
Equation (18) shows a state-space system with q − r inputs, n − p states and p − r outputs. It is now possible to design a supplementary SMO to estimate e1 while being insensitive to f1 . This can be accomplished if and only if (18) satisfies conditions D1, D2 which are equivalent to E1 Ce1 Me1 has full column rank E2 The invariant zeros of (Ae1 , Me1 , Ce1 ) are stable Define eˆ1 to be the estimate of e1 , and it is obvious that eˆ1 → e1 . Define a measurable signal x ˆ1,x := x ˆ1 − eˆ1 . Substitute for x ˆ1 from (16), and it is clear that x ˆ1,x → x1 . The primary and supplementary observers are implemented as shown in Figure 1.
Main results
In this section, we extend the design procedure in [6,7] to the wider class of systems described by (3) to allow for the asymptotic estimation of the entire state and fault signals. The following Theorem states the main result of this paper Theorem 2 The states x and faults f¯ from (3) can be asymp¯ by cascading two totically reconstructed in the presence of ξ, SMOs, if and only if the following conditions are satisfied : " # CAM CM D1 rank = rank(CM ) + rank(M ) CM 0
y
Primary - observer for (3) h
1
To reduce chattering, νeq can be obtained by replacing (5) with ey where δ is a small positive scalar. With this, ey no ν = −ρ key k+δ longer slides ideally on S but on a small boundary layer around it, which does not adverselyaffect the accuracy of the state and fault reconstruction. For further details see [7].
x ˆ1 yˆ
νeq
T T Po−1
Supplementary h1 observer for (18) h2
+ ? - m- xˆ1,x − ? ˆ
eˆ1
f1
Ce2
fˆ
- U −1 -
+ ? fˆ2 - m - M −1 +
22
Fig. 1. A diagram showing primary and supplementary observers.
3
4.1.1 Existence conditions Condition E1: It is straightforward to show that # " A3,a M11 Ce1 Me1 = 0
If we partition A, M, C according to (7) - (8), then because T, M11 , M22 are invertible matrices, Po,1 (s) loses rank if and only if Po,2 loses rank where # " −A1,c sI − A1,d Po,2 (s) = −A3,a −A3,b
(19)
To satisfy E1, it is required that rank(A3,a M11 ) = q − r. Proposition 1 The condition rank(A3,a M11 ) = q − r is satisfied if and only if " # CAM CM rank =q+r (20) CM 0
It is now easy to see that the invariant zeros of (A, M, C) are the invariant zeros of (A1,d , A1,c , A3,b , A3,a ) by performing simple manipulations on Po,2 (s). Hence the invariant zeros of (Ae1 , Me1 , Ce1 ) are the invariant zeros of (A, M, C). Notice that condition D1 is less restrictive than B1. This is because if condition B1 is satisfied, then condition D1 is satisfied, but the converse is not necessarily true. Hence the method proposed in this paper can be applied to a wider class of systems that do not satisfy condition B1.
Proof It can be easily shown that "
CAM CM CM
0
#
=
"
A3,a M11 A4,b M11 0
0
# A M A M 0 M 22 3,c 11 4,d 11 0 0 0 0 0 T
T 0
0
0
M22
Since"T is orthogonal, # then CAM CM rank = rank(A3,a M11 ) + 2r CM 0
4.2
The supplementary observer can also be used to reconstruct the fault signal, namely f . As shown in section 4.1, if conditions D1, D2 are satisfied, then the supplementary observer is able to produce accurate reconstruction of f1 and accurate estimate of e1 . Define ah measurablei signal −1 fˆ2 := M22 (h2 + Ce2 eˆ1 ) where Ce2 = A3,c A3,d . Thus, when eˆ1 → e1 , it is clear from (17) that fˆ2 → f2 .
0
(21)
Combining (20) and (21) gives rank(A3,a M11 ) = q − r Condition E2: The Rosenbrock system matrix [12] of (Ae1 , Me1 , Ce1 ) is # " sI − Ae1 −Me1 Pe,1 (s) = Ce1 0
From the supplementary observer, define fˆ1 as the reconstruction of f1 . Then the whole fault signal can now be reh iT constructed from fˆ = U −1 fˆT fˆT → f. 1
Pre-multiplying and post-multiplying by full rank matrices # # " " 0 −In−p−q+r I L2 , H2 = H1 = Iq−r 0 0 I respectively yields H1 Pe,2 (s)H2 =
sI − A1,d −A1,c A3,b
A3,a
2
¯ then f¯ can be easSince f is formed by augmenting f¯ and ξ, ily recovered. Note that robust state estimation and fault reconstruction would not have been possible under the scheme by Edwards et al.[6,7] which requires CM to be full column rank. If CM is full rank (r = q) the method proposed in this paper can also be used, as it would still satisfy condition D1. Remark 3: In this paper, the fault reconstruction is fully ¯ Conditions for the design decoupled from the disturbance ξ. of an observer that is decoupled from disturbances are presented in [15]. Stoorvogel et al.[13] built on the work of [15] and extended it to fault reconstruction that is fully decoupled from the disturbance, however here the fault reconstruction error may not converge asymptotically to zero. Furthermore, state estimation was not considered in [13]. In our paper, full disturbance decoupling, accurate state estimation and fault reconstruction are all provided. ♯
The invariant zeros of (Ae1 , Me1 , Ce1 ) are the values of s that make Pe,1 (s) lose rank. Since M11 is invertible then Pe,1 (s) loses rank if and only if Pe,2 (s) loses rank where " # −(A1,c + L2 A3,a ) sI − (A1,d + L2 A3,b ) Pe,2 (s) = −A3,a −A3,b
"
Fault reconstruction
#
5
Simulation results
The method proposed in this paper will now be demonstrated with an example, which is a 7th order model of an aircraft [16]. There are seven states, two inputs and four measured outputs. The states are the bank angle (rad), yaw rate (rad/s), roll rate (rad/s), sideslip angle (rad), washout filter state, rudder deflection (rad) and aileron deflection (rad), while the inputs are the rudder command (rad) and aileron command
Hence the invariant zeros of (Ae1 , Me1 , Ce1 ) are the invariant zeros of (A1,d , A1,c , A3,b , A3,a ). The Rosenbrock system matrix of (A, M, C) described by (3) is given by " # sI − A −M Po,1 (s) = C 0
4
0.1
(rad). The outputs are the first, second, fourth and seventh states. In the notation of (1) the matrix A can be obtained ¯,Q ¯ are from [16], whereas the matrices M
Fault 1
0.05
0
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0
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20
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T
¯ = 0 0 0 0 0 20 0 M 0 0 0 0 0 0 25 h iT ¯ = 0 0.9501 0.2311 0.6068 0 0 0 Q
Fault 2
0.05
0
−0.05
Fig. 2. The faults (dotted) and their reconstructions
To design the observers, we need to find a s.p.d. matrix P of the structure (13) and a gain matrix G that satisfy P Ao + ATo P < 0. Then, by choosing a large enough value of ρ, sliding motion on S is guaranteed. However, this does not mean that the error signal e1 necessarily converges to zero. To overcome this problem, we design a supplementary observer as outlined in section 4. The combination of the primary and supplementary observers will result in exact state estimation and fault reconstruction. Tan & Edwards [16] have proposed a method to design the sliding mode observer, and will be used to design both the observers. The method by [16] can be summarized to be: Minimize trace(X) with respect to the variables P, X subject to " # " # P A + AT P − C T V −1 C P P I < 0, > 0(22) P −W −1 I X
−20.0018 0
0
20
, Me1 = 0 Ae1 = 0 0.0097 −4 −0.5269 0 −0.9989 0
Ce1 =
0.0010 0
−1
0.4173 0 0.0004
In designing the supplementary observer (the subscript ‘e1’ indicates that they are parameters of the supplementary observer), the weighting matrices were chosen as We1 = 100I3 , Ve1 = I2 and the following gains were obtained
0.2167 5.6626 0.1104 −0.004 , Po,e1= Ge1=He1= 0 0 −0.004 0.4233 −9.0612 −0.0808
where W and V are s.p.d. user-defined weighting matrices to tune the sizes of the gains. The matrix P has the structure # " h i P11 P12 , P12 = P121 0 P = T P22 P12
Notice that G = H; this is because V = I4 was chosen [16]. In the simulations that follow, the constants associated with ν have been chosen to be ρ = 500, δ = 10−5 , ρe1 = 50, δe1 = 10−3 . Furthermore, the system was assumed to have an initial condition of −0.1, 0.0843, 0.1 to the first, fourth and fifth states respectively with all other states set to 0, and the observers were assumed to have zero initial conditions. Both actuators were assumed to be faulty.
where P121 ∈ R(n−p)×(p−r) . Formulating the design problem as an LMI problem, the LMI solver will return values of P and X. The gains can then be calculated in the coordinates −1 P12 , Po = T (P22 − of (6) as G = P −1 C T V −1 ,L = P11 T −1 T P12 P11 P12 )T and H as in (10). For details, see [16]. For the first observer, the weights were chosen as W = 100I7 , V = I4 and the following gains were obtained
The simulation was carried out. A disturbance ξ = sin 0.5t is applied to the system from t = 0s. Figure 2 show the faults that are applied to the actuators as well as their reconstructions; the fault on the first actuator is a ramp signal applied at t = 10s and settles at t = 20s and the fault on the second actuator starts at t = 15s, settles at t = 20s and switches off at t = 25s. In the first observer, the output estimation error ey converges to a small boundary layer near S in a very short time. However, the error e1 does not converge to zero and hence the first observer does not accurately estimate x1 . However, using the method in Section 4.1, the signal x ˆ1,x (which uses the state estimates from the primary and supplementary observer), reconstructs the state x1 accurately. Figure 3 show the system states x as well as the estimates provided by the scheme in this paper. It shows that the estimates converge very quickly to the actual states.
19.6885 −1.5079 −4.6889 −0.1853 −1.5079 10.3727 1.0527 0.0274 −0.2996 2.6239 0.0797 0.0054 G = H = −4.6889 1.0527 10.4536 0.0907 0.0734 0.1097 −0.1926 −0.0014 −0.0139 −0.0208 0.0364 0.0003 −0.1853 0.0274 0.0907 1.9300 0.0572 0.0058 0.0251 0.0042 0.0058 0.0980 −0.0073 −0.0005 Po = 0.0251 −0.0073 0.1077 −0.0025 0.0042 −0.0005 −0.0025 0.5187
It is clear that the scheme proposed in this paper yields fault and state reconstructions that are identical to the actual faults and states. Note that the existing work [6,7] would not be able to do this because rank(CM ) = q is not satisfied.
Hence, the parameters of the supplementary observer were
5
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State 3 State 4 State 5 State 6 State 7
[7] C. Edwards, S.K. Spurgeon, and R.J. Patton. Sliding mode observers for fault detection and isolation. Automatica, 36:541–553, 2000. [8] F.W. Fairman, S.S. Mahil, and L. Luk. Disturbance decoupled observer design via singular value decomposition. IEEE Trans. Automatic Control, 29:84–86, 1984. [9] M. Hou and P.C. Muller. Design of observers for linear systems with unknown inputs. IEEE Trans. Automatic Control, 37:871–875, 1992. [10] P. Kudva, N. Viswanadham, and A. Ramakrishna. Observers for linear systems with unknown inputs. IEEE Trans. Automatic Control, 25:113–115, 1980. [11] R.J. Patton and J. Chen. A survey of robustness problems in quantitative model-based fault diagnosis. Applied Maths and Computer Science, 3:339–416, 1993. [12] H.H. Rosenbrock. State space and multivariable theory. John-Wiley, New York, 1970. [13] A. Saberi, A.A. Stoorvogel, and P. Sannuti. Exact, almost and optimal input decoupled (delayed) observers. Int. J. Control, 73:552–581, 2000. [14] M. Saif and Y. Guan. A new approach to robust fault detection and identification. IEEE Trans. Aerospace and Electronic Systems, 29:685–695, 1993. [15] A.A. Stoorvogel, H.H. Niemann, A. Saberi, and P. Sannuti. Optimal fault signal estimation. Int. J. Robust and Nonlinear Control, 12:697– 727, 2002. [16] C.P. Tan and C. Edwards. An LMI approach for designing sliding mode observers. Int. J. Control, 74:1559–1568, 2001. [17] C.P. Tan and C. Edwards. Sliding mode observers for robust detection and reconstruction of actuator and sensor faults. Int. J. Robust and Nonlinear Control, 13:443–463, 2003.
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Fig. 3. The system states (dotted) and the observer’s estimates
6
A The change of coordinates
Conclusion
Since C has full row rank, then there exists an invertible iT h change of coordinates Tc = Nc C T where Nc spans the null space of C, such that i h iT h T T Mc = Tc M = Mc,1 , Cc = CTc−1 = 0 Ip Mc,2
In this paper, a supplementary observer is introduced so that when cascaded with an existing observer it robustly and asymptotically reconstructs the fault signals and estimates the full states of a linear multivariable system with unknown disturbances and model uncertainties. The conditions for the existence of such an observer are constructed. It is shown that these conditions are much milder than existing ones and therefore apply to a wider class of systems, specifically where the first Markov matrix is of not full rank. A aircraft control system is used as an example to demonstrate the potent points of the new development through computer simulation of fault conditions. The results have successfully validated the proposed new theory.
where Mc,2 ∈ Rp×q . Since rank(CM ) = r, then rank(Mc,2 ) = r. Hence, using the SVD, the following can be obtained : T T Mc,2 U T = diag {0, M22 }, where M22 ∈ Rr×r is invertible, and T, iU are orthogonal. Parh T tition Mc,1 U = Mc,11 Mc,12 where Mc,12 has r columns. Then, let Y1 be an orthogonal matrix such that h iT T Y1 Mc,11 = M11 where M11 ∈ R(q−r)×(q−r) is in0 i h −1 T T . Then apply vertible. Define Y2 = 0 −Y1 Mc,12 M22 the change of coordinates Ta to Mc , Cc where # " Y1 Y2 Ta = 0 TT
References [1] P.J. Antsaklis and A.N. Michel. Linear systems. McGraw Hill, 1997. [2] S.K. Chang and W.T. You. Design of general structured observers for linear systems with unknown inputs. J. Franklin Institute, 334B:213– 232, 1997. [3] J. Chen and R.J. Patton. Robust model-based fault diagnosis for dynamic systems. Kluwer Academic Publishers, 1999. [4] J. Chen, R.J. Patton, and H.Z. Zhang. Design of unknown input observers and robust fault detection filters. Int. J. Control, 63:85– 105, 1996.
and post-multiply Mc by U T to get M11 0 h i , C = Cc Ta−1 = 0 T M = T a Mc U T = 0 0 0 M22
[5] M. Darouach and M. Zasadzinski. Full order observers for linear systems with unknown inputs. IEEE Trans. Automatic Control, 39:606–609, 1994. [6] C. Edwards and S.K. Spurgeon. On the development of discontinuous observers. Int. J. Control, 59:1211–1229, 1994.
6
School of Engineering, Monash University Malaysia, 2 Jalan Kolej, 46150 Petaling Jaya, Malaysia
Department of Electrical and Computer Systems Engineering, Building 72, Monash University, Melbourne, Victoria 3800, Australia c
Department of Electrical and Electronic Engineering, University of Melbourne, Parkville, Victoria, 3010, Australia
Abstract A common requirement implicit in current methods for the design of robust state estimators and robust fault detection filters is that the first Markov matrix must be non-zero, and indeed, full rank. We relax both of these restrictions in this paper to allow the applicability to a wider range of systems. The extended results are then applied to an aircraft fault detection for which the restrictive condition is not satisfied. Key words: fault detection, robust estimation, sliding mode, observers
1
Introduction
Fault detection and isolation (FDI) has been the subject of extensive research for some time now, but especially since the early 1990s [3,11]. The interest in this line of research stems from its practical application to a variety of industries such as aerospace, energy systems, and process control to name a few. The main function of an FDI scheme is to detect a fault when it happens, which may then be acted on by sending alarm signals, taking protection measures, or reconfiguring a running control scheme. The most commonly used schemes are observer based approaches [11], where a residual signal is generated and flagged when an abnormal (faulty) condition takes place in the system. Robust state estimation and fault detection methods [4] were introduced in an attempt to achieve accurate fault detection and state estimation in the presence of uncertainties in system parameters, and/or changes in plant operating conditions. The extension to nonlinear observers based on sliding mode theory [6,7], also offered the ability to account for unknown disturbances and model inaccuracy. With respect to sliding mode observer (SMO) based approaches, Edwards et al [7] reported improvements on that of [6] approach where a nonlinear term is used to reconstruct the fault and therefore provide full information about it. It is interesting to observe that a common thread in tech⋆ This paper was not presented at any IFAC meeting. Corresponding author C.P. Tan. Tel. +6-03-56360600 Fax +6-03-56329314 Email addresses: [email protected] (C.P. Tan), [email protected] (F. Crusca), [email protected] (M. Aldeen).
Preprint submitted to Automatica
niques for robust state estimation [6], unknown input observers [10,8,9,5,2], and fault detection filters [14] is that they require satisfaction of the rather strict condition that the first Markov (see e.g. [1]) parameter is non-zero, and indeed, full rank. This severely limits the applicability of these approaches for a wide range of practical systems. In this paper we remove this restrictive assumption and extend the state estimation/fault detection approach of [6] to a wider practical class of systems. This is achieved by introducing a supplementary observer which together with the observer [6] produces full state estimation and fault detection. Thus the removal of the aforementioned constraint has led to the development of a more general approach, which has immediate application to the two areas of: (i) robust state estimation; and (ii) robust fault detection and isolation for systems where the first Markov matrix is rank degenerate. To illustrate the advantages of this approach we apply the new approach to an aircraft control system model where the first Markov matrix is rank deficient. Simulation studies are carried out on this example and results demonstrate the properties of the new approach. 2
Preliminaries and motivation
Consider the dynamic system below ¯ f¯, y = Cx x˙ = Ax + Bu + M n
(1)
m
where x ∈ R is the state, u ∈ R is the control input, ¯,C y ∈ Rp is the output and f¯ ∈ Rq¯ is a fault. Matrices A, M are known and constant. Edwards et al.[6,7] considered this class of systems and developed a sliding mode observer (SMO) that is able to asymptotically estimate x and f¯ upon satisfaction of the following conditions
18 June 2007
¯ ) = q¯ (full column rank) A1 rank(C M ¯ , C) are stable A2 invariant zeros of (A, M
where x ˆ is the estimate of x, and ey = C x ˆ − y is the output estimation error. The matrices G, H ∈ Rn×p are gain matrices that will be described in greater detail later. The term ν is a nonlinear injection term defined by ey ν = −ρ , ey 6= 0, ρ ∈ R+ (5) key k
Subsequently Tan & Edwards [17] extended the class of systems to include unknown signals encapsulating nonlinearites/uncertainties in the system along the lines proposed in [3]. The extended class of systems is now expressed as ¯ y = Cx ¯ f¯ + Q ¯ ξ, x˙ = Ax + Bu + M (2)
In the following we show by examining the error dynamics that the design of [6,7] cannot be used on its own to provide full state estimation and fault detection for the class of systems described by (3). Without loss of generality, the triple (A, M, C) after some coordinate transformation can be written as follows: " # " # h i A1 A2 M1 A= ,M = ,C = 0 T (6) A3 A4 M2
¯
where ξ¯ ∈ Rh is the unknown input signal. They showed that under conditions A1,A2 above, it is possible to design a SMO that would minimize but not completely eliminate the effect of ξ¯ on the state estimation and fault reconstruction. Remark 1: If there are nonlinearities and/or faults in the output equation of (2), they can be easily moved to the state equation of (2) using the procedure in Section 5 of [17] and the existing methods can be used to reconstruct the fault. ♯
where A1 ∈ R(n−p)×(n−p) , M2 ∈ Rp×q , T ∈ Rp×p is orthogonal. From Condition C2, it is clear that rank(M2 ) = r. The matrices A1 , A3 , M1 , M2 can be partitioned as follows # # " " A3,a A3,b A1,a A1,b (7) , A3 = A1 = A3,c A3,d A1,c A1,d " # " # M11 0 0 0 M1 = , M2 = (8) 0 0 0 M22
In this paper we extend the results of [17] to the case where the effect of the unknown signal is totally nullified. This is achieved by proposing a supplementary observer that would complement that of [6] so that when augmented together, asymptotic estimate state estimation and fault detection are achieved irrespective of the unknown signal. The conditions under which such results can be achieved are now stated. Let us re-model system (2) as follows x˙ = Ax + Bu + M f (3) h iT i h ¯ Q ¯ , f ∈ Rq = f¯T ξ¯T . where M ∈ Rn×q = M Assume n > p > q.
where M11 ∈ R(q−r)×(q−r) and M22 ∈ Rr×r are invertible. Furthermore A1,a ∈ R(q−r)×(q−r) and A3,a ∈ R(p−r)×(q−r) . However, A1 , A3 have no particular structure. The fault signal f is scaled and partitioned as follows iT h where U ∈ Rq×q is orthogonal, f → U f = f1T f2T and f1 ∈ Rq−r . Details are available in the Appendix.
Remark 2: When p < q then more than one observer is needed; the exact number is at least equal to the ratio pq rounded up to the next digit, provided both A1 and A2 are satisfied for each observer. For example at least 2 observers are required when p = 3, q = 5. This may be implemented " # i f h 1 . Then one SMO by partitioning: M f = M1 M2 f2 is designed for x˙ = Ax + Bu + M1 f1 , y = Cx and another for x˙ = Ax + Bu + M2 f2 , y = Cx. ♯
Define e := x ˆ − x as the state estimation error to obtain e˙ = Ao e + Hν − M f (9) where Ao = A − GC. In the coordinates of (6) the gain matrix H has the structure h iT H = −LT I (Po T )−1 (10) where Po ∈ Rp×p is symmetric positive definite (s.p.d.) and iT h i h , L1 ∈ R(q−r)×(p−r) . L = Lo 0 , Lo = LT1 LT2
Note that in [6], their results apply to the case where (3) satisfies the following conditions B1 rank(CM ) = q (full column rank) B2 invariant zeros of (A, M, C) are stable
Problem formulation
Apply the change of coordinates TL to the matrices in (6), " # I L where TL = . Then it is straightforward to see that 0 T " # " # A1 + LA3 ∗ M1 + LM2 −1 TL ATL = , TL M = (11) T M2 T A3 ∗ " # " # h i 0 x1 −1 CTL = 0 Ip , TL H = (12) , TL x = −1 y Po
From [6], the SMO for (3) is x ˆ˙ = Aˆ x + Bu − Gey + Hν
where (*) are matrices that play no more role in the following analysis. In this coordinate system, notice that the bottom p rows of the state vector is the output y, due to the structure
It is well known that condition B1 is quite restrictive and may not apply to a wide range of systems. This is due to the requirement that the first Markov matrix must be both nonzero and full rank. We now relax this condition, and propose a new set of milder conditions C1 C, M are full rank matrices C2 rank(CM ) = r < q 3
(4)
2
of CTL−1 , and x1 ∈ Rn−p are the ‘non-output’ states. Also, iT h the error vector now has the structure TL e = eT1 eTy where e1 ∈ Rn−p is the estimation error of x1 . Theorem 1 [17] Assume that there exists a s.p.d. matrix " # P1 P1 L n×n P ∈R = (13) LT P1 LT P1 L + T T Po T
D2 The invariant zeros of (A, M, C) must be stable In the remainder of this section, we develop a constructive proof of Theorem 2. 4.1
In the following we show that under conditions D1 and D2 it is possible to design a supplementary observer to asymptotically estimate x1 . Since y can be estimated by the design procedure in [6], henceforth called the primary observer, the outcome is full state estimation.
that satisfies P Ao + ATo P < 0. Then define two positive scalars µ0 = −λmax P Ao + ATo P , µ1 = kP M k If ρ > µ−1 0 kf k(2kPo T A3 kµ1 +µ0 kPo T M2 k), then in finite time, a sliding motion takes place on S = {e : Ce = 0} Next suppose an ideal sliding motion has taken place on S, then ν becomes νeq 1 and ey = e˙ y = 0. As a result, the observer (4) gives an accurate estimate of the output state since ey has been forced to zero in finite time. However, an accurate estimate of x1 is not possible, as shown next.
Re-arranging (15) gives h := T T Po−1 νeq = −A3 e1 + M2 f . Then partition h according to (7) - (8) # " " " # # 0 A3,a A3,b h1 e1 + =− h= f2 (17) A3,c A3,d h2 M22 Treat h1 as the ‘output’, and partition as in (7) - (8). Then the following system can be obtained from (14) and (17)
Note that LM2 = 0. Pre-multiplying (9) by TL and partitioning according to (11) - (12) yields the set of equations e˙ 1 = (A1 + LA3 )e1 − M1 f (14) −1 0 = T A3 e1 − T M2 f + Po νeq (15)
e˙ 1 = Ae1 e1 − Me1 f1 , h1 = −Ce1 e1 i iT h h T where Me1 = M11 0 , Ce1 = A3,a A3,b and " # A1,a + L1 A3,a A1,b + L1 A3,b Ae1 = A1,c + L2 A3,a A1,d + L2 A3,b
From (14) - (15), it is clear that the observer (4) can track x1 if and only if M1 = 0 [6] which implies that rank(CM ) = rank(M ) and the first Markov matrix CM = T M2 is full rank. Then e1 → 0 and f can be recovered from (M2T M2 )−1 M2T T T Po−1 νeq . Note that we have used the definition of the Markov matrices [1] for the system (A, M, C) as being {H0 , H1 , ..., Hk , ...} with Hk = CAk−1 M . If however CM is not full rank, as is the case for the class of systems in this paper (see Condition C2), then from (14), e1 will not converge to zero since M1 6= 0. If x ˆ1 is the estimate of x1 , then x ˆ1 will not track x1 . Instead, we have x ˆ1 = x1 + e1 (16) We overcome this by proposing a supplementary observer, as outlined in the following sections. 4
State estimation
(18)
Equation (18) shows a state-space system with q − r inputs, n − p states and p − r outputs. It is now possible to design a supplementary SMO to estimate e1 while being insensitive to f1 . This can be accomplished if and only if (18) satisfies conditions D1, D2 which are equivalent to E1 Ce1 Me1 has full column rank E2 The invariant zeros of (Ae1 , Me1 , Ce1 ) are stable Define eˆ1 to be the estimate of e1 , and it is obvious that eˆ1 → e1 . Define a measurable signal x ˆ1,x := x ˆ1 − eˆ1 . Substitute for x ˆ1 from (16), and it is clear that x ˆ1,x → x1 . The primary and supplementary observers are implemented as shown in Figure 1.
Main results
In this section, we extend the design procedure in [6,7] to the wider class of systems described by (3) to allow for the asymptotic estimation of the entire state and fault signals. The following Theorem states the main result of this paper Theorem 2 The states x and faults f¯ from (3) can be asymp¯ by cascading two totically reconstructed in the presence of ξ, SMOs, if and only if the following conditions are satisfied : " # CAM CM D1 rank = rank(CM ) + rank(M ) CM 0
y
Primary - observer for (3) h
1
To reduce chattering, νeq can be obtained by replacing (5) with ey where δ is a small positive scalar. With this, ey no ν = −ρ key k+δ longer slides ideally on S but on a small boundary layer around it, which does not adverselyaffect the accuracy of the state and fault reconstruction. For further details see [7].
x ˆ1 yˆ
νeq
T T Po−1
Supplementary h1 observer for (18) h2
+ ? - m- xˆ1,x − ? ˆ
eˆ1
f1
Ce2
fˆ
- U −1 -
+ ? fˆ2 - m - M −1 +
22
Fig. 1. A diagram showing primary and supplementary observers.
3
4.1.1 Existence conditions Condition E1: It is straightforward to show that # " A3,a M11 Ce1 Me1 = 0
If we partition A, M, C according to (7) - (8), then because T, M11 , M22 are invertible matrices, Po,1 (s) loses rank if and only if Po,2 loses rank where # " −A1,c sI − A1,d Po,2 (s) = −A3,a −A3,b
(19)
To satisfy E1, it is required that rank(A3,a M11 ) = q − r. Proposition 1 The condition rank(A3,a M11 ) = q − r is satisfied if and only if " # CAM CM rank =q+r (20) CM 0
It is now easy to see that the invariant zeros of (A, M, C) are the invariant zeros of (A1,d , A1,c , A3,b , A3,a ) by performing simple manipulations on Po,2 (s). Hence the invariant zeros of (Ae1 , Me1 , Ce1 ) are the invariant zeros of (A, M, C). Notice that condition D1 is less restrictive than B1. This is because if condition B1 is satisfied, then condition D1 is satisfied, but the converse is not necessarily true. Hence the method proposed in this paper can be applied to a wider class of systems that do not satisfy condition B1.
Proof It can be easily shown that "
CAM CM CM
0
#
=
"
A3,a M11 A4,b M11 0
0
# A M A M 0 M 22 3,c 11 4,d 11 0 0 0 0 0 T
T 0
0
0
M22
Since"T is orthogonal, # then CAM CM rank = rank(A3,a M11 ) + 2r CM 0
4.2
The supplementary observer can also be used to reconstruct the fault signal, namely f . As shown in section 4.1, if conditions D1, D2 are satisfied, then the supplementary observer is able to produce accurate reconstruction of f1 and accurate estimate of e1 . Define ah measurablei signal −1 fˆ2 := M22 (h2 + Ce2 eˆ1 ) where Ce2 = A3,c A3,d . Thus, when eˆ1 → e1 , it is clear from (17) that fˆ2 → f2 .
0
(21)
Combining (20) and (21) gives rank(A3,a M11 ) = q − r Condition E2: The Rosenbrock system matrix [12] of (Ae1 , Me1 , Ce1 ) is # " sI − Ae1 −Me1 Pe,1 (s) = Ce1 0
From the supplementary observer, define fˆ1 as the reconstruction of f1 . Then the whole fault signal can now be reh iT constructed from fˆ = U −1 fˆT fˆT → f. 1
Pre-multiplying and post-multiplying by full rank matrices # # " " 0 −In−p−q+r I L2 , H2 = H1 = Iq−r 0 0 I respectively yields H1 Pe,2 (s)H2 =
sI − A1,d −A1,c A3,b
A3,a
2
¯ then f¯ can be easSince f is formed by augmenting f¯ and ξ, ily recovered. Note that robust state estimation and fault reconstruction would not have been possible under the scheme by Edwards et al.[6,7] which requires CM to be full column rank. If CM is full rank (r = q) the method proposed in this paper can also be used, as it would still satisfy condition D1. Remark 3: In this paper, the fault reconstruction is fully ¯ Conditions for the design decoupled from the disturbance ξ. of an observer that is decoupled from disturbances are presented in [15]. Stoorvogel et al.[13] built on the work of [15] and extended it to fault reconstruction that is fully decoupled from the disturbance, however here the fault reconstruction error may not converge asymptotically to zero. Furthermore, state estimation was not considered in [13]. In our paper, full disturbance decoupling, accurate state estimation and fault reconstruction are all provided. ♯
The invariant zeros of (Ae1 , Me1 , Ce1 ) are the values of s that make Pe,1 (s) lose rank. Since M11 is invertible then Pe,1 (s) loses rank if and only if Pe,2 (s) loses rank where " # −(A1,c + L2 A3,a ) sI − (A1,d + L2 A3,b ) Pe,2 (s) = −A3,a −A3,b
"
Fault reconstruction
#
5
Simulation results
The method proposed in this paper will now be demonstrated with an example, which is a 7th order model of an aircraft [16]. There are seven states, two inputs and four measured outputs. The states are the bank angle (rad), yaw rate (rad/s), roll rate (rad/s), sideslip angle (rad), washout filter state, rudder deflection (rad) and aileron deflection (rad), while the inputs are the rudder command (rad) and aileron command
Hence the invariant zeros of (Ae1 , Me1 , Ce1 ) are the invariant zeros of (A1,d , A1,c , A3,b , A3,a ). The Rosenbrock system matrix of (A, M, C) described by (3) is given by " # sI − A −M Po,1 (s) = C 0
4
0.1
(rad). The outputs are the first, second, fourth and seventh states. In the notation of (1) the matrix A can be obtained ¯,Q ¯ are from [16], whereas the matrices M
Fault 1
0.05
0
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0
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0
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15 Time (s)
20
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30
0.1
T
¯ = 0 0 0 0 0 20 0 M 0 0 0 0 0 0 25 h iT ¯ = 0 0.9501 0.2311 0.6068 0 0 0 Q
Fault 2
0.05
0
−0.05
Fig. 2. The faults (dotted) and their reconstructions
To design the observers, we need to find a s.p.d. matrix P of the structure (13) and a gain matrix G that satisfy P Ao + ATo P < 0. Then, by choosing a large enough value of ρ, sliding motion on S is guaranteed. However, this does not mean that the error signal e1 necessarily converges to zero. To overcome this problem, we design a supplementary observer as outlined in section 4. The combination of the primary and supplementary observers will result in exact state estimation and fault reconstruction. Tan & Edwards [16] have proposed a method to design the sliding mode observer, and will be used to design both the observers. The method by [16] can be summarized to be: Minimize trace(X) with respect to the variables P, X subject to " # " # P A + AT P − C T V −1 C P P I < 0, > 0(22) P −W −1 I X
−20.0018 0
0
20
, Me1 = 0 Ae1 = 0 0.0097 −4 −0.5269 0 −0.9989 0
Ce1 =
0.0010 0
−1
0.4173 0 0.0004
In designing the supplementary observer (the subscript ‘e1’ indicates that they are parameters of the supplementary observer), the weighting matrices were chosen as We1 = 100I3 , Ve1 = I2 and the following gains were obtained
0.2167 5.6626 0.1104 −0.004 , Po,e1= Ge1=He1= 0 0 −0.004 0.4233 −9.0612 −0.0808
where W and V are s.p.d. user-defined weighting matrices to tune the sizes of the gains. The matrix P has the structure # " h i P11 P12 , P12 = P121 0 P = T P22 P12
Notice that G = H; this is because V = I4 was chosen [16]. In the simulations that follow, the constants associated with ν have been chosen to be ρ = 500, δ = 10−5 , ρe1 = 50, δe1 = 10−3 . Furthermore, the system was assumed to have an initial condition of −0.1, 0.0843, 0.1 to the first, fourth and fifth states respectively with all other states set to 0, and the observers were assumed to have zero initial conditions. Both actuators were assumed to be faulty.
where P121 ∈ R(n−p)×(p−r) . Formulating the design problem as an LMI problem, the LMI solver will return values of P and X. The gains can then be calculated in the coordinates −1 P12 , Po = T (P22 − of (6) as G = P −1 C T V −1 ,L = P11 T −1 T P12 P11 P12 )T and H as in (10). For details, see [16]. For the first observer, the weights were chosen as W = 100I7 , V = I4 and the following gains were obtained
The simulation was carried out. A disturbance ξ = sin 0.5t is applied to the system from t = 0s. Figure 2 show the faults that are applied to the actuators as well as their reconstructions; the fault on the first actuator is a ramp signal applied at t = 10s and settles at t = 20s and the fault on the second actuator starts at t = 15s, settles at t = 20s and switches off at t = 25s. In the first observer, the output estimation error ey converges to a small boundary layer near S in a very short time. However, the error e1 does not converge to zero and hence the first observer does not accurately estimate x1 . However, using the method in Section 4.1, the signal x ˆ1,x (which uses the state estimates from the primary and supplementary observer), reconstructs the state x1 accurately. Figure 3 show the system states x as well as the estimates provided by the scheme in this paper. It shows that the estimates converge very quickly to the actual states.
19.6885 −1.5079 −4.6889 −0.1853 −1.5079 10.3727 1.0527 0.0274 −0.2996 2.6239 0.0797 0.0054 G = H = −4.6889 1.0527 10.4536 0.0907 0.0734 0.1097 −0.1926 −0.0014 −0.0139 −0.0208 0.0364 0.0003 −0.1853 0.0274 0.0907 1.9300 0.0572 0.0058 0.0251 0.0042 0.0058 0.0980 −0.0073 −0.0005 Po = 0.0251 −0.0073 0.1077 −0.0025 0.0042 −0.0005 −0.0025 0.5187
It is clear that the scheme proposed in this paper yields fault and state reconstructions that are identical to the actual faults and states. Note that the existing work [6,7] would not be able to do this because rank(CM ) = q is not satisfied.
Hence, the parameters of the supplementary observer were
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State 3 State 4 State 5 State 6 State 7
[7] C. Edwards, S.K. Spurgeon, and R.J. Patton. Sliding mode observers for fault detection and isolation. Automatica, 36:541–553, 2000. [8] F.W. Fairman, S.S. Mahil, and L. Luk. Disturbance decoupled observer design via singular value decomposition. IEEE Trans. Automatic Control, 29:84–86, 1984. [9] M. Hou and P.C. Muller. Design of observers for linear systems with unknown inputs. IEEE Trans. Automatic Control, 37:871–875, 1992. [10] P. Kudva, N. Viswanadham, and A. Ramakrishna. Observers for linear systems with unknown inputs. IEEE Trans. Automatic Control, 25:113–115, 1980. [11] R.J. Patton and J. Chen. A survey of robustness problems in quantitative model-based fault diagnosis. Applied Maths and Computer Science, 3:339–416, 1993. [12] H.H. Rosenbrock. State space and multivariable theory. John-Wiley, New York, 1970. [13] A. Saberi, A.A. Stoorvogel, and P. Sannuti. Exact, almost and optimal input decoupled (delayed) observers. Int. J. Control, 73:552–581, 2000. [14] M. Saif and Y. Guan. A new approach to robust fault detection and identification. IEEE Trans. Aerospace and Electronic Systems, 29:685–695, 1993. [15] A.A. Stoorvogel, H.H. Niemann, A. Saberi, and P. Sannuti. Optimal fault signal estimation. Int. J. Robust and Nonlinear Control, 12:697– 727, 2002. [16] C.P. Tan and C. Edwards. An LMI approach for designing sliding mode observers. Int. J. Control, 74:1559–1568, 2001. [17] C.P. Tan and C. Edwards. Sliding mode observers for robust detection and reconstruction of actuator and sensor faults. Int. J. Robust and Nonlinear Control, 13:443–463, 2003.
0
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−0.5 0.2 0 −0.2 0.5 0 −0.5 0.05 0
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Fig. 3. The system states (dotted) and the observer’s estimates
6
A The change of coordinates
Conclusion
Since C has full row rank, then there exists an invertible iT h change of coordinates Tc = Nc C T where Nc spans the null space of C, such that i h iT h T T Mc = Tc M = Mc,1 , Cc = CTc−1 = 0 Ip Mc,2
In this paper, a supplementary observer is introduced so that when cascaded with an existing observer it robustly and asymptotically reconstructs the fault signals and estimates the full states of a linear multivariable system with unknown disturbances and model uncertainties. The conditions for the existence of such an observer are constructed. It is shown that these conditions are much milder than existing ones and therefore apply to a wider class of systems, specifically where the first Markov matrix is of not full rank. A aircraft control system is used as an example to demonstrate the potent points of the new development through computer simulation of fault conditions. The results have successfully validated the proposed new theory.
where Mc,2 ∈ Rp×q . Since rank(CM ) = r, then rank(Mc,2 ) = r. Hence, using the SVD, the following can be obtained : T T Mc,2 U T = diag {0, M22 }, where M22 ∈ Rr×r is invertible, and T, iU are orthogonal. Parh T tition Mc,1 U = Mc,11 Mc,12 where Mc,12 has r columns. Then, let Y1 be an orthogonal matrix such that h iT T Y1 Mc,11 = M11 where M11 ∈ R(q−r)×(q−r) is in0 i h −1 T T . Then apply vertible. Define Y2 = 0 −Y1 Mc,12 M22 the change of coordinates Ta to Mc , Cc where # " Y1 Y2 Ta = 0 TT
References [1] P.J. Antsaklis and A.N. Michel. Linear systems. McGraw Hill, 1997. [2] S.K. Chang and W.T. You. Design of general structured observers for linear systems with unknown inputs. J. Franklin Institute, 334B:213– 232, 1997. [3] J. Chen and R.J. Patton. Robust model-based fault diagnosis for dynamic systems. Kluwer Academic Publishers, 1999. [4] J. Chen, R.J. Patton, and H.Z. Zhang. Design of unknown input observers and robust fault detection filters. Int. J. Control, 63:85– 105, 1996.
and post-multiply Mc by U T to get M11 0 h i , C = Cc Ta−1 = 0 T M = T a Mc U T = 0 0 0 M22
[5] M. Darouach and M. Zasadzinski. Full order observers for linear systems with unknown inputs. IEEE Trans. Automatic Control, 39:606–609, 1994. [6] C. Edwards and S.K. Spurgeon. On the development of discontinuous observers. Int. J. Control, 59:1211–1229, 1994.
6
