A QMI APPROACH TO THE ROBUST FAULT DETECTION AND ISOLATION PROBLEM Emmanuel Mazars Zhenhai Li Imad M Jaimoukha Imperial College London Abstract: This paper investigates the robust fault detection isolation (FDI) problem for uncertain linear time-invariant (LTI) systems. An FDI filter minimizes the sensitivity of the residual signal to disturbances modeling errors subject to the constraint that the transfer matrix function from the faults to the residual is closed to a diagonal transfer matrix function (for fault isolation). A solution of the optimization problem is presented via the formulation of a quadratic matrix inequality (QMI). A jet engine example is employed to demonstrate the effectiveness of our results. Keywords: Fault detection isolation quadratic matrix inequalities robust H-infinity control multiobjective optimization. 1. INTRODUCTION Nowadays Model based fault detection isolation (FDI) schemes exploiting analytic redundancy have received increasing attention in the literature applications (Frank Ding 1997) (Isermann 2005) (Patton et al. 2000). Several methods for achieving robust FDI were developed such as Patton Chen who explored approaches using eigenstructure assignments see (Patton Chen 2000). Frank Ding developed a matrix factorization method to obtain an optimal fault detection filter (Frank Ding 1994) Sadrnia et al. utilized a Riccati equation iteration method to construct an extended H filter see (Sadrnia et al. 1996). Re- 1 The authors are with the Control Power Group Department of Electrical Electronic Engineering Imperial College London. Email: e.mazars@imperial.ac.uk i.jaimouka@imperial.ac.uk zhenhai.li@imperial.ac.uk. Corresponding author: Emmanuel Mazars Control Power Group Department of Electrical Electronic Engineering Imperial College London SW7 2BT UK. Fax: +44 207 594 6282. Telephone: +44 207 594 6286. Email: e.mazars@imperial.ac.uk cently developed linear matrix inequality (LMI) approaches offer numerically attractive techniques for formulating robust FDI decoupling problems. Hou Patton gave a realization of fault detection observer design based on the bounded real lemma (Hou Patton 1996). Zhong et al. proposed a new performance index by introducing a reference residual model formulated using LMI techniques see (Zhong et al. 2003). Using these techniques the decoupling problem can be transformed to a sensitivity optimization problem which seeks to increase the sensitivity of the residual to faults simultaneously reduce the sensitivity to disturbances plant/model mismatch. In this paper an FDI filter is constructed such that the H norm of the transfer matrix function from both disturbances plant/model mismatch to the residual is smaller than a given value with the constraint that the transfer matrix function from faults to the residual is close to a diagonal transfer matrix function. Sufficient conditions for the existence of such a filter are obtained in the form of QMIs.
The structure of the work is as follows. After defining the notation we review observer based fault isolation techniques for residual signal generation give the problem formulation in Section 2. Section 3 gives a matrix inequality formulation for the problem. Section 4 presents a solution via solving a QMI. Finally a numerical example is presented in Section 5 Section 6 summarizes our results. The notation we use is fairly stard. The set of real (complex) n m matrices is denoted by R n m (C n m ). For A C n m we use the notation A T A to denote the transpose complex conjugate transpose respectively. A matrix A C n n is called Hermitian if A = A. For a Hermitian matrix A C n n A 0 denotes that A is positive semidefinite (that is all the eigenvalues of A are greater than or equal to zero). For a Hermitian matrix A C n n λ(a) denotes the largest λ(a) the smallest eigenvalue of A respectively. For A C n m σ(a) denotes the largest σ(a) the smallest singular values of A respectively. Note that σ(a) = λ(a A) σ(a) = λ(a A). The n n identity matrix is denoted as I n. R(s) m p denotes the set of all m p proper real rational matrix functions of s. L m p denotes the space of m p matrix functions with entries bounded on the extended imaginary axis. The subspace H m p L m p denotes matrix functions analytic in the closed right half of the complex plane. A prefix R denotes a real rational function so that RH m p denotes the set of all m p stable real rational matrix functions of s. G(s) RL m p is a spectral factor of φ(s) if φ(s) = G H ( s)g(s). For G(s) RH m p we define G = sup σ (G(jω)) G = inf σ (G(jω)). ω R ω R In Section 3 we use the following Lemmas. Lemma 1.1. ((Clements et al. 1997)). Let φ(s) = R + B H ( si A H ) 1 C + C H (si A) 1 B + B H ( si A T ) 1 Q(sI A) 1 B with (AB) sign controllable R H = R Q H = Q. Then finding a spectral factor of φ is equivalent to finding hermitian solutions P H = P of the linear matrix inequality: [ PA + A H P + Q PB + C (PB + C) T 0. R Lemma 1.2. (Scherer et al. 1997). Let G(s) s =(ABCD) A R n n let γ >0. Then A is stable G <γ if only if there exists P =P T R n n such that P >0 A T P + PA PB C T B T P γi D T < 0. C D γi 2. PROBLEM FORMULATION Consider a linear time invariant (LTI) dynamic system subject to disturbances modeling errors process sensor actuator faults modeled as ẋ(t) = (A + A)x(t) + (B + B)u(t) +B d d(t) + B f f(t) y(t) = (C + C)x(t) + (D + D)u(t) +D d d(t) + D f f(t) (1) x(t) R n u(t) R nu y(t) R ny are the process state input output vectors respectively d(t) R n d f(t) R n f are the disturbance fault vectors respectively. Here B f R n n f D f R ny n f are the component instrument fault distribution matrices respectively while B d R n n d D d R ny n d are the corresponding disturbance distribution matrices (Frank Ding 1997). The matrices A B C D represent modeling errors are given by: [ [ A B F1 = [ E C D 1 E 2 (2) F 2 F 1 F 2 E 1 E 2 are known matrices is an unknown matrix such that T I. Note that this is a larger class of uncertainties than that considered in (Zhong et al. 2003). In general a residual signal in an FDI system should represent the inconsistency between the actual system variables the mathematical model respond to faults disturbances modeling errors only. The objective is to design a fault detection isolation observer (or filter) of the form ˆx(t) = Aˆx(t) + (B + LD)u(t) L(y(t) Cˆx(t)) r(t) = H(y(t) Cˆx(t) Du(t)) ˆx(t) R n is the observer state r(t) R n f is the residual signal. Here L R n ny H R n f n y are the observer residual gain matrices respectively are to be determined. Define the state estimation error signal as e(t) = x(t) ˆx(t). It follows that the residual dynamics are given by ė(t) = (A + LC)e(t) + ( A + L C)x(t) +(B d + LD d )d(t) + (B f + LD f )f(t) +( B + L D)u(t) r(t) = HCe(t) + H Cx(t) + H Du(t) +HD d d(t) + HD f f(t). 2
[ e(t) By setting z(t) = w(t) = x(t) get: { ż(t) = Ãz(t) + B f f(t) + B w w(t) r(t) = Hz(t) + HD f f(t) + H w w(t) [ d(t) we u(t) (3) [ A + LC A + L C [ Ã= H = HC H C 0 A + A [ Bf + LD f H w = [ HD d H D Bf = [ Bd + LD Bw = d B + L D B + B B d B f. By taking Laplace transforms it is easy to show that r(s) = T rf (s)f(s) + T rw (s)w(s) (4) T rf (s) T rw (s) [à s Bf = HHD f R(s) n f n f (5) [à s Bw = H H R(s) n f (n d +n u) (6) w are the transfer matrices from faults w to the residual respectively. In robust FDI we would ideally like to solve the following optimization problem: minimize αγ 1 + (1 α)γ 2 T rf T o γ 1 T rw γ 2 T o 1 à is stable T o(s) S (7) α [01 is a parameter S is the set of all transfer matrices with a special structure e.g. diagonal block diagonal or triangular. The first third constraints ensure a minimum level of sensitivity of the residual to fault signals. The second ensures that the filter is robust to disturbances model uncertainties. The fourth ensures that the filter is stable while the fifth ensures fault isolation the structure depends on the nature of the fault. Unfortunately this optimization is intractable mainly due to the difficulty in characterizing the set S. It is clear that minimizing γ 2 will improve fault detection it is obvious that minimizing γ 1 will improve fault isolation. According to the design requirements by choosing a suitable α we can vary the emphasis between fault detection fault isolation since optimizing one will generally affect the performance of the other. Our focus in this work is on solving this optimization when S is the set of all diagonal transfer matrices. We will therefore consider the simplest nontrivial version of (7): Problem 1. Assume that (CA) is detectable let α be given. The optimal robust FDI filter design is to find L H to minimize αγ 1 + (1 α)γ 2 such that: à is asymptotically stable (8) T rf T o γ 1 (9) T rw γ 2 (10) T o 1 (11) T o has the following structure : T o = M(sI Λ) 1 + N (12) Λ = diag(λ 1...λ nf ) R n f n f λ i<0 i M = diag(m 1...m nf ) R n f n f m i >0 i N = diag(n 1...n nf ) R n f n f n i >0 i. (13) These settings will ensure that T o is diagonal stable. Remark 2.1. The assumption (C A) detectable is necessary to have à stable. 3. MATRIX INEQUALITY FORMULATION We consider in this section a matrix inequality formulation approach to hle Problem 1. The main idea is to express the inequalities (9) (10) in a matrix inequality formulation using the bounded real lemma then finding an upper-bound on all the terms containing the unknown matrices ( A B C D) using the fact that T I. Consider first the inequality (10). With the help of the version of the bounded real lemma given in Lemma 3.1 we can derive a matrix inequality formulation as follows. Lemma 3.1. Let T rw define in (6). There exists L H such that à is stable Trw < γ 2 if only if there exist L H P = P T R n n such that P > 0 (Ã)T P+P(Ã) P( B w) HT ( B w) T P γ 2I HT w H Hw γ 2I < 0. (14) The solution of the matrix inequality in (14) is not tractable in its current form since it is nonlinear. 3
In order to get a tractable solution we restrict P to have the form: P = diag(p 1 P 2 ). Using the expression of T rw in (6) we can separate the terms involving modeling uncertainties from the other terms since the inequality (14) can be written as ˆT rw = T rw = ˆT rw + T rw < 0 (15) (A+LC) T P 1+( ) ( ) ( ) ( ) ( ) 0 P 2A+( ) ( ) ( ) ( ) (B d +LD d ) T P 1 B T d P2 γ2i ( ) ( ) 0 B T P 2 0 γ 2I ( ) HC 0 HD d 0 γ 2I 0 ( ) ( ) ( ) ( ) ( A+L C) T P 1 P 2 A+( ) ( ) ( ) ( ) 0 0 0 ( ) ( ) ( B+L D) T P 1 B T P 2 0 0 ( ) 0 H C 0 H D 0. Where ( ) denotes terms readily inferred from symmetry. By using the expressions of the modeling errors in (2) it can be verified that T rw = F w E w + E T w T F T w (16) P 1 (F 1 + LF 2 ) P 2 F 1 F w = 0 0 E w = [ 0 E 1 0 E 2 0. HF 2 The next result uses the fact that T I to bound T rw. Lemma 3.2. Let F E be matrices with appropriate dimensions. If T I then F E + E T T F T FF T + E T E It follows from (15) (16) Lemma 3.2 that ˆT rw + T rw ˆT rw + F w F T w + E T we w. Using a Schur complement type argument it follows that T rw < 0 = [ ˆTrw + E T rw = we T w F w I F T w T rw < γ 2 (17). By using (12) it is easy to show that the inequality (9) is equivalent to à 0 Bf 0 Λ I H M HD f N < γ 1. We can hle this inequality using the same procedure as that for inequality (10). Thus ˆT rf = T rf < 0 = T rf T o < γ 1 (18) [ ˆTrf + Ef T T E rf = f F f I F T f P 3(A+LC)+( ) ( ) ( ) ( ) ( ) 0 P 4A+( ) ( ) ( ) ( ) 0 0 P 5Λ+( ) ( ) ( ) (B f +LD f ) T P 3 B T f P4 P5 γ1i ( ) HC 0 M HD f N γ 1I P 3 (F 1 + LF 2 ) P 4 F 1 F f = 0 0 E f = [ 0 E 1 0 0 0. HF 2 Using the next lemma we can express the inequality (11) as a matrix inequality formulation: [ Λ I Lemma 3.3. Let T o = be a diagonal transfer matrix with ΛMN defined previously. And M N define [ Find P = P T such that (η) PΛ+Λ T P+M T M P+M T N 0 P+N T M We get: T o 1 (η) N T N I Proof. Let φ(s) = T o (s)t o (s) I nf. It is easy to show that φ(s) can be written as follows φ(s) = N T N I nf +( si nf Λ H ) 1 M T N + (M T N) H (si nf Λ) 1 +( si nf Λ T ) 1 Q(sI nf Λ) 1. Then by using the iff condition in lemma 1.1 we get: (η) φ(s) has a spectral factor G φ(s) = G H (s)g(s) φ(s) 0 T o T o I nf 0 T o 1. 4
4. FDI FILTER DESIGN VIA QMI In this section we give a state-space solution by formulating a quadratic matrix inequality (QMI). By using (17) (18) Lemma 3.3 we can now reformulate the problem 1 in a matrix inequality formulation. The next theorem gives a sufficient condition for the existence of a robust FDI filter via solving a quadratic matrix inequality (QMI). Theorem 2. Let T rf T rw be as defined previously. Let γ 1 > 0 γ 2 > 0 be given. Assume that (CA) is detectable. There exist L R n ny H R n f n y such that the specifications (8) (9) (10) (11) in Problem 1 are satisfied if there exist Λ M N (defined in (13)) L H a symmetric matrix P 6 R n f a symmetric matrix P = diag(p 1 P 2 P 3 P 4 P 5 ) R 5n such that P > 0 (19) T rf < 0 (20) T rw < 0 (21) [ P 6Λ+Λ T P 6+M T M P 6+M T N P 6+N T M N T N I 0. (22) These QMIs can be solved by using the algorithms guaranteeing global convergence (Goh et al. 1994) (Yamada et al. 1995) as well as local numerical searching algorithms that converge (without a guarantee) much faster (Geromel et al. 1993) (Beran Grigoriadis 1996). A related discussion of the solving algorithms for QMI s can also be found in Saberi et al (Saberi et al. 1995). Remark 4.1. By using some simple settings it is easy to get an LMI formulation from Theorem 2. Indeed by setting P 1 = P 3 P 5 = P 6 R = P 5 Λ Q = P 1 L in (20) (21) we get two LMIs. By adding the constraint that P 5 R are diagonal we are sure to get a diagonal Λ given by Λ = P 1 5 R. It is straightforward to show that R+R T P 5 M T P 5 (1 γ 3)I 0 M 0 (1 γ 3)I N γ 3 I 0 < 0 (22) Where γ 3 is a positive scalar. Then (20) (21) (22) are satisfied if the LMIs have a feasible solution. The corresponding filter design is given by extracting L = P1 1 Q H from the LMIs. Note that the solution given by the LMIs is suboptimal because of the constraints we impose to some of variables. 5. NUMERICAL EXAMPLE To illustrate the effectiveness of the proposed fault detection isolation filter scheme an example is considered in this section. The GE-21 jet engine state-space model is given as [ [ 3.370 1.636 0.586 1.419 1.252 A= B= 0.325 1.896 0.410 1.118 0.139 1 0 0 0 0 C = 0 1 D= 0 0 0. 0.731 0.786 0.267 0.025 0.146 We suppose that this system is subject to two disturbances three potential faults. Here the setup is given by [ 0.2618 0.1745 0.1379 0.6204 B d = D 0.3145 0.1602 d = 0.1047 0.1461 0.3277 0.1737 [ 1.1679 0.3977 1.0846 B f = 0.2444 0.5619 0.1957 0.3986 0.8364 0.4238 D f = 1.0324 0.0069 1.9389 2.8553 0.6071 0.9040 F 1 = [ 0.1 0.1 T F2 = [ 0.1 0.1 0.1 T E 1 = [ 0.1 0.1 E 2 = [ 0.1 0.1 0.1. We implemented the algorithm to solve the LMIs described in Remark 4.1 in MATLAB to minimize αγ 1 + (1 α)γ 2 compute L H. We have chosen α = 0.5 as a compromise between fault detection fault isolation. Even though the LMI formulation for solving Problem2 gives a suboptimal solution we are able to design a filter that achieve fault detection isolation. We get [ 0.2731 0.8457 0.7186 L = 0.4979 0.4048 0.3129 0.2633 0.1196 0.3618 H = 1.0331 0.3361 0.2579 0.1253 0.5739 0.1860 Λ = 100I 3 M = 0.0723I 3 N = I 3 γ 1 = 0.0530 γ 2 = 0.3772. In order to show that our filter is robust against disturbances model uncertainties we introduce the uncertainties defined by the matrices (F 1 F 2 E 1 E 2 ) as well as two disturbances. Simulated through MATLAB SIMULINK these disturbances are b limited white noise with power 0.01 (zeroth-order hold with sampling time 0.1s) positive jump from the 6 th second. Fault f 1 simulated by a unit positive jump is connected from the 14 th second. Fault f 2 simulated by a unit negative jump fault f 3 simulated by a soft bias (slope = 0.5) are both connected from the 20 th second. The input u is taken as a periodic signal. 5
Figure 1 gives the residual responses each fault can be readily distinguished from the others the disturbances. It is worth noting that in order to verify the effectiveness of the optimal filter similar amplitudes for the disturbances faults are assumed which is quite deming from the practical point of view. Magnitude 2 1.5 1 0.5 0 0.5 1 1.5 2 r 1 r 2 r 3 5 10 15 20 25 30 35 Time(secs) Fig. 1. Time response of the residual This example makes clear that the designed filter satisfies the performance requirement of rapid detection direct fault isolation which is sufficiently robust against disturbances modeling errors. Notice that the time responses clearly show that it is possible to set thresholds that allow us to distinguish between faults disturbances. 6. SUMMARY In this paper we have considered a robust fault detection isolation problem using a matrix inequality formulation approach. We derive a sufficient condition for solvability of the robust FDI problem in the form of a QMI formulation furthermore give a construction of a stable FDI filter that bounds the influence of the disturbances model uncertainties on the residual signal measured in terms of the H norm. Our scheme implies that each element of the residual is only corresponding to a specified potential fault therefore can hle multiple faults ( faults might occur at the same time). A real example is given to validate our algorithm. ACKNOWLEDGMENT This work has been partially supported by the Ministry of Defence through the Data & Information Fusion Defence Technology Centre. REFERENCES Beran E. K. Grigoriadis (1996). A combined alternating projections semidefinite programming algorithm for low-order control design. In: In Proc. Prep. 13th IFAC World Congr.. San Francisco CA. pp. 85 90. Clements D.J. B.D.O. Anderson A.J. Laub J.B. Matson (1997). Spectral factorization with imaginary axis zeros. Lin. Alg. Appl. 250 225 252. Frank P.M. X. Ding (1994). Frequency domain approach to optimally robust residual generation evaluation for model based fault diagnosis. Automatica 30(5) 789 804. Frank P.M. X. Ding (1997). Survey of robust residual generation evaluation methods in observer based fault detection systems. J. Proc. Cont. 7(6) 403 424. Geromel J.C. P.L.D. Peres S.R. de Souza (1993). Output feedback stabilization of uncertain systems through a min/max problem. In: Preprints of Papers 12th World Congress I.F.A.C. July 1993 Volume 6. The Inst. of Engineers Australia. pp. 35 38. Goh K. C. M. G. Safonov G. P. Papavassilopoulos (1994). A global optimization approach for the bmi problem. In: Proc. IEEE Conf. Dec. & Control. pp. 2009 2014. Hou M. R.J. Patton (1996). An LMI approach to H /H fault detection observers. In: UKACC International Conference on Control. 96. Engl. pp. 305 310. Isermann R. (2005). Model-based fault-detection diagnosis - status applications. Annual Reviews in Control 29 71 85. Patton R.J. J. Chen (2000). On eigenstructure assignment for robust fault diagnosis. Int. J. Robust & Nonlinear Control 10(14) 1193 1208. Patton R.J. P.M. Frank R.N. Clark (eds) (2000). Issues of Fault Diagnosis for Dynamic Systems. Springer. Saberi A. P. Sannuti B. M. Chen (1995). h 2 optimal control system control engineering. London UK. Sadrnia M.A. J. Chen R.J. Patton (1996). Robust fault diagnosis observer design using H optimisation µ synthesis. IEE Colloquium on Modelling Signal Processing for Fault Diagnosis pp. 9/1 16. Scherer C. P. Gahinet M. Chilali (1997). Multi-objective output-feedback control via LMI optimization. IEEE Trans. Automatic Control 42(7) 896 911. Yamada Y. S. Hara H. Fujioka (1995). Global optimization for constantly scaled h control problem. In: Proc. American Contr. Conf. pp. 427 430. Zhong M. S.X. Ding J. Lam H. Wang (2003). An LMI approach to design robust fault detection filter for uncertain LTI systems. Automatica 39(3) 543 550. 6