Vol. 35, No. 11 ACTA AUTOMATCA SNCA November, 29 Fault Detection Observer Design in Low Frequency Domain for Linear Time-delay Systems L Xiao-Jian 1, 2 YANG Guang-Hong 1 Abstract This paper deals with the robust fault detection (FD) problem in low frequency domain for linear time-delay systems. The H norm and H index are used to measure the robustness to unknown inputs and the sensitivity to faults, respectively. The main results include derivation of a sufficient condition for the existence of a robust FD observer and its construction based on the linear matrix inequality (LM) solution parameters. Finally, numerical simulations show the effectiveness of the presented methodology. Key words Fault detection (FD), time-delay system, low frequency range, linear matrix inequality (LM) The research and application of robust fault detection (FD) in automated processes have received considerable attention during last decades and a great number of results have been achieved 14. The main challenge in robust FD is to distinguish faults from other disturbances. There have been a number of results using robust control theory to solve this problem, e.g., the H /H approach 4, multi-objective H approach 5, H filter approach 6, robust fault estimation schemes 79, and recently developed H /H approach 114. However, most of those works considered the whole frequency spectrum. n practice, however, faults usually emerge in the low frequency domain, e.g., for an incipient signal, the fault information is contained within a low frequency band as the fault development is slow 1, and the actuator stuck failures that occur in the flight control systems just belong to the low frequency domain 15. This motivated the FD observer design for linear time-invariant (LT) systems in low frequency 1617. On the other hand, time delays are frequently encountered in industry and are often the source of performance degradation of a system. So, this paper focuses on the FD observer design problem in the low frequency domain for linear time-delay systems with unknown inputs. The proposed design methodology of this paper is based on the following idea: by combining the new results in 18 and H /H observer approach, the FD problem is converted into a detection observer design problem in the low frequency domain, and LM-based design conditions are then derived. The following notations are used throughout this paper. For a matrix A, A denotes its complex conjugate transpose. The rmitian part of a square matrix A is denoted by (A) = A A. The symbol H n stands Received April 29, 28; in revised form October 27, 28 Supported by the State Key Program of National Natural Science of China (65341), the National Basic Research Program of China (973 Program) (29CB3264), the National Natural Science foundation of China (667421, 68424, 697443), the Funds for Creative Research Groups of China (682163), the 111 Project (B815), and Research Fund for the Doctoral Program of Higher Education of China (2614519) 1. Key Laboratory of ntegrated Automation for the Process ndustry, Ministry of Education, and College of nformation Science and Engineering, Northeastern University, Shenyang 114, P. R. China 2. College of Sciences, Northeastern University, Shenyang 114, P. R. China DO: 1.3724/SP.J.14.29.1465 for the set of n n rmitian matrices. denotes the identity matrix with an appropriate dimension. For matrices Φ and P, Φ P means the Kronecker product. For matrices G C n m and Π H nm, a function σ : C n m H nm H m is defined by G G σ(g, Π) = Π (1) m m 1 Problem formulation and preliminaries 1.1 Problem formulation Definition 1 17. The H index of a transfer function matrix G(s) is defined as G(s) Ω = inf σg(jω) (2) ω Ω where σ denotes the minimum singular value, Ω is a subset of real numbers as shown in Table in 17. Faults considered in this paper are assumed to be in the low frequency domain, i.e., ω Ω = ϖ, ϖ, where ϖ is a positive scalar. n this paper, we consider the following linear time-delay systems: ẋx(t) = Ax(t) A d x(t τ) B f f(t) B d d(t) y(t) = Cx(t) D f f(t) D d d(t) (3) where x(t) R n is the state space vector, y(t) R ny denotes the measurement output vector, d(t) R n d is the unknown input vector satisfying d(t) L 2, f(t) R n f denotes the fault to be detected. A, A d, B f, B d C, D f, and D d are known matrices with appropriate dimensions and τ is a known constant time-delay. Without loss of generality, we assume (A, C) to be observable and omit the control input. We propose to use the following FD observer: ˆx(t) = Aˆx(t) A dˆx(t τ) H(y ŷy) ŷy(t) = Cˆx(t) r(t) = y ŷy (4) where ˆx(t) R n and ŷy(t) R ny represent the state and output estimation vectors, respectively. r(t) R nr is the so-called residual signal. The design parameter is observer gain matrix H. Remark 1. The disturbances considered in this paper are assumed to be in the same frequency range as that of faults because disturbances that belong to the high frequency domain can be decoupled by designing a low-pass filter after the residual outputs. Denoting e(t) = x(t) ˆx(t) and augmenting the model of system (3) to include the states of FD observer (4), we obtain the following augmented system: ėe(t) = Āe(t) A d e(t τ) B d d(t) B f f(t) r(t) = Ce(t) D d d(t) D f f(t) (5) where Ā = A HC, B d = B d HD d, and B f = B f HD f. Then, the FD observer design problem can now be formulated as follows: 1) System (5) is asymptotically stable; 2) G rf (jω) ϖ,ϖ > β 1; 3) G rd (jω) ϖ,ϖ < β 2, where G rf (s) = C(s Ā eds A d ) 1 Bf D f (6) G rd (s) = C(s Ā eds A d ) 1 Bd D d (7)
1466 ACTA AUTOMATCA SNCA Vol. 35 and β 1, β 2 are two given positive scalars. 1.2 Preliminaries n this subsection, some useful lemmas are given. We introduce the main results of 18. Given a linear time-delay system ẋx(t) = Ax(t) A d x(t τ) Bd(t) y(t) = Cx(t) Dd(t) (8) where x(t) R n is the state space vector, y(t) R ny denotes the measurement output vector, and d(t) R nϖ is the disturbance input vector. A, A d, B, C, and D are known matrices with appropriate dimensions, and τ is a constant time-delay. The transfer function matrix G(λ) from d to y is denoted by G(s) = C(s A e τs A d ) 1 B D (9) Given a rmitian matrix Π, the specification can be described by where σ(g(λ), Π) <, λ Λ(Φ, Ψ) (1) Λ(Φ, Ψ) = {λ C σ(λ, Φ) =, σ(λ, Ψ) } (11) and Λ = Λ if Λ is bounded, Λ = Λ {} if is unbounded. Lemma 1 18. Let matrices A C n n, A d C n n, B C n n ω, C C ny n, D C ny n ω, Π H nyn ω, and Φ, Ψ H 2 be given and define Λ by (11). Suppose Λ represents curves on the complex plane. Then σ(g(λ), Π) < holds for all λ Λ(Φ, Ψ) if there exist P = P, Q = Q >, and X = X > such that A B Ad A B Ad (Φ P Ψ Q) C D C D X Π < X (12) Remark 2. n the rest of this paper we choose 1 1 Φ = and Ψ = 1 ϖ 2, then λ Λ(Φ, Ψ) is equivalent to ω ϖ, ϖ, where λ = jω. Furthermore, for the later development, the following Lemmas are required. Lemma 2 (Finsler s Lemma). Let ξ C n, P C n n and H C n m. Let H be any matrix such that H H =. The following statements are equivalent: 1) ξ Pξ <, H ξ =, ξ ; 2) H PH < ; 3) µ R : P µhh < ; 4) X R m n : P HX X H < Lemma 3 (Elimination Lemma). Let Γ, Λ, and Θ = Θ be given matrices. There exists a matrix F to solve the matrix inequality ΓF Λ (ΓF Λ) Θ < if and only if the following conditions are satisfied Γ ΘΓ <, and Λ ΘΛ < 2 Main result 2.1 Fault sensitivity condition n this section, the fault sensitivity condition is considered. Let d(t) = in (5), we have ėe(t) = Āe(t) A d e(t τ) B f f(t) r(t) = Ce(t) D f f(t) (13) f we choose Π = β1 2 and Φ, Ψ as given in Remark 2, then for system (13), the performance (1) becomes G rf (jω) ϖ,ϖ > β 1, ω ϖ, ϖ where G rf (s) = C(s Ā eds A d ) 1 Bf D f. Theorem 1. For system (13), let a symmetric matrix Π 1 = β1 2 R (nrn f ) (n rn f ) and Φ, Ψ, β 1 > are given, then there is an FD observer satisfying G rf (jω) ϖ,ϖ > β 1, if there exist P 1 = P1, Q 1 = Q 1 >, and X 1 = X1 >, W, V f1, V f2, and K such that the following inequality T Π 1 X 1 T < X 1 W R 1 V f1 V f2 A W R 1 C KR 1 V f1 C V f2 B f W R 1 D f KR 1 D f V f2 A dw R 1 (14) holds with the assumption that R 1 C n (4nn f n r) satisfies Y T Π 1 X 1 T Y X 1 Y = µ 1Y R 1R 1Y < A C H C B f D f H D f A d (15) (16) where µ 1 > is a real scalar and T is the permutation matrix such that M 1, M 2, M 3, M 4, M 5, M 6T = M 1, M 2, M 3, M 5, M 4, M 6 (17) for arbitrary matrices M 1, M 2, M 3, M 4, M 5, and M 6 with column dimensions n, n, n r, n, n f and n, respectively. The observer gain matrix is given by K = H W (18) Proof. By Lemma 1, the performance G rf (jω) ϖ,ϖ > β 1 is satisfied if the following
No. 11 L Xiao-Jian and YANG Guang-Hong: Fault Detection Observer Design in Low Frequency Domain For 1467 inequality = Ξ T T Ξ < (19) Ξ = A C H C B f D f H D f A d Π 1 X 1 X 1 (2) holds, where T is defined by (17). We let P = T T, H = Ξ, and H =. Then, condition (19) is equivalent to 2) of Lemma Ξ 2 and the following inequality T Π 1 X 1 T < X 1 ( Ξ X ) (21) is equivalent to 4) of Lemma 2, where X is a multiplier. So, by Lemma 2, condition (19) is equivalent to condition (21). However, in Lemma 2, we should notice the equivalence between 4) and other items needs that the structure of X in 4) has no constraint; once we add an additional constraint to X, 4) will be a sufficient condition for other items. To make the problem tractable, similar to that of 19, we restrict the class of multiplier X to be X = W R 1 V f (22) where W C n n, det(w ), V f C (nnr) (4nn f n r) and R 1 C n (4nn f n r) is a multiplier to be chosen. Then, (21) will be held if T Π 1 X 1 X 1 A C H C Bf Df H A d Df T < W R1 V f (23) holds. Defining K = H Vf1 W and V f =, with some V f2 matrix manipulations, we have that (23) is equivalent to (14), so we can conclude that (14) provides a sufficient condition for performance index G rf (jω) ϖ,ϖ > β 1. Remark 3. As pointed out in 18 19, we can choose R 1 to satisfy (15) with Y defined by (16). f R 1 is given, condition (14) is an LM in P 1, Q 1, X 1, W, V f1, V f2, and K. Remark 4. By the condition (17), we can get T = 1 2 3 5 4 6 where n(n = 1, 2,, 6) denote identity matrices with appropriate dimensions. 2.2 Robustness condition re, we study the robustness requirement of system (5). Let f(t) = in (5), we have ėe(t) = Āe(t) A d e(t τ) B d d(t) r(t) = Ce(t) D d d(t) (24) To attenuate the disturbance influence, we give the following theorem. Theorem 2. For system (24), let a symmetric matrix Π 2 = β2 2 R (nrn d) (n rn d ) and Φ, Ψ, β 2 > are given, then there is an FD observer satisfying G rd (jω) ϖ,ϖ < β 2, if there exist P 2 = P2, Q 2 = Q 2 >, and X 2 = X2 >, W, V d1, V d2, and K such that the following inequality T Φ P 2 Ψ Q 2 Π 2 X 2 X 2 W R 2 V d1 V d2 A W R 2 C KR 2 V d1 C V d2 B d W R 2 D dkr 2 D dv d2 A dw R 2 T < holds with the assumption that R 2 C n (4nn d nr) satisfies Φ P 2 Ψ Q 2 Y T Π 2 X 2 T Y X 2 Y = µ 2Y R 2R 2Y < A C H C B d D dh D d A d (26) (25) (27) where µ 2 > is a real scalar and T is defined by (17). And the observer gain matrix is given by K = H W (28) Proof. The proof is similar to Theorem 1, so it is omitted.
1468 ACTA AUTOMATCA SNCA Vol. 35 2.3 Stability condition Conditions (14) and (25) do not ensure a stable observer, so we wish to add an additional constraint to guarantee the stability of system (5). Lemma 4. System (5) is asymptotically stable if there exist matrices W, K, P 3 = P3 that (Φ P 3) W A W C K A dw >, and X 3 = X 3 > such X 3 X 3 q p < (29) where r = p q C 2 is an arbitrary fixed vector satisfying rφr <. Proof. From the Lyaponov stability conditions for timedelay systems, system (5) is stable if there exist symmetric matrices P 3 > and X 3 > such that Ā A d P3 Ā A d P 3 X 3 < (3) X 3 Notice that Ā A d is the null space of Ā. According to Lemma 3, (3) will be held if the following inequality P3 X 3 X 3 < Ā A d A d W R (31) holds, where W is chosen as that in Theorem 1 and Theorem 2, and R = q p, where r = p q C 2 is an arbitrary fixed vector satisfying rφr <. f K = H W, then Lemma 4 is completed. and Ψ =, then we get the LM conditions for fault de- Remark 5. f we choose Φ = 1 tection in the full frequency domain. 1 1 2.4 Detection observer design By combining Theorems 1 and 2, and Lemma 4, conditions 1) 3) given in Section 1 will be satisfied if LMs (14), (25), and (29) hold simultaneously. Theorem 3. System (5) is asymptotically stable and conditions G rf (jω) ϖ,ϖ > β 1, G rd (jω) ϖ,ϖ < β 2 are satisfied if there exist symmetric matrices P 1, P 2, P 3 >, Q 1 >, Q 2 >, X 1 >, X 2 >, X 3 >, and matrices W, V f1, V f2, V d1, V d2, K such that (14), (25) and (29) hold, where K = H W. Given β 2 >, the observer gain matrix H can be determined through the following optimization: 3 Numerical simulations max β 1 s.t. (14), (25), (29) (32) Consider a linear time-delay system of the form in (3) with the following parameters.9231.5422 A =.9442.6764.6264.7227 A d =.117.161.4141, B f =.3287, B d =.293.1224 C =.5432.4595, D f =.7525, D d =.834 The frequency range is restricted in (.1,.1). We let q = 1, p = 1, β 2 =.2 R 1 = 2 2 R 13 2 R 15 4 1 R 13 =, R 15 = 1 R 2 = R 21 R 22 2 2 R 21 = R 22 = 2 and T be given in Remark 4. Solving the optimization problem (32), we obtain the.6557 observer gain matrix H low = and β.2478 1 optlow =.5297. The actual achieved value of β 1 in the low frequency domain is.6328. n the full frequency domain, the observer gain matrix 1.3882 H full = and β.2243 1 optfull =.3578. The actual achieved value of β 1 in the full frequency domain is.3744. To show the effectiveness of our method more clearly, some simulations are also given. The system is assumed to be affected by stuck faults such that f(t) = 5, t 6 s, and f(t) = elsewhere. As clearly seen from Fig. 1, the residual is more sensitive to faults in the low frequency domain than in the full range. Fig. 1 The residual outputs with the disturbance d(t) = sin(.5t) After designing the FD observer, the important task is the evaluation of the generated residual. A threshold J th and residual evaluation function J r can be determined by 1 t J r = r t T (τ)r(τ)dτ J th = sup J r f=, d L 2, ω (.1,.1) Based on this, the occurrence of faults can be detected by
No. 11 L Xiao-Jian and YANG Guang-Hong: Fault Detection Observer Design in Low Frequency Domain For 1469 the following logic rules: J r > J th with faults alarm J r J th no faults Using Matlab, we can obtain J th =.1469. The residual evaluation function J r and threshold J th are reported in Figs. 2 and 3. From Fig. 2, we can conclude that faults can be effectively detected by using finite frequency FD observer and Fig. 3 illustrates that the finite frequency FD observer can receive better results than the full frequency FD observer. Fig. 2 The residual evaluation and the threshold Fig. 3 4 Conclusion The residual evaluation n this paper, we have investigated the problem of FD for linear time-delay systems in the low frequency domain. The H norm and H index have been used to measure the robustness to unknown inputs and the sensitivity to fault, respectively. A design method has been presented in terms of solutions to a set of LMs and a numerical example has been given to illustrate the effectiveness of the proposed method. References 1 Chen J, Patton P R. Robust Model-Based Fault Diagnosis For Dynamic Systems. London: Kluwer Academic Publishers, 1999 2 Yoshimura M, Frank P M, Ding X. Survey of robust residual generation and evaluation methods in observer-based fault detection systems. Journal of Process Control, 1997, 7(6): 43424 3 Zhong M Y, Ding X S, Lam J, Wang H B. An LM approach to design robust fault detection filter for uncertain LT systems. Automatica, 23, 39(3): 54355 4 Ding X S, Zhong M Y, Tang B Y, Zhang P. An LM approach to the design of fault detection filter for time-delay LT systems with unknown inputs. n: Proceedings of the American Control Conference. Arlington, USA: EEE, 21. 21372142 5 Kim Y M, Watkins J M. A new approach for robust and reduced order fault detection filter design. n: Proceedings of the American Control Conference. New York, USA: EEE, 27. 11371142 6 Nobrega E G, Abdalla M O, Grigoriadis K M. LM-based filter design for fault detection and isolation. n: Proceedings of the 39th EEE Conference on Decision and Control. Sydney, Australia: EEE, 2. 43294334 7 Gao Z, Ding X S, Ma Y. Robust fault estimation approach and its application in vehicle lateral dynamic systems. Optimal Control Applications and Methods, 26, 28(3): 143156 8 Peter A M, Marquez H J, Zhao Q. LM-based sensor fault diagnosis for nonlinear Lipschitz systems. Automatica, 27, 43(8): 14641469 9 Yan X G, Edwards C. Nonlinear robust fault reconstruction and estimation using a sliding mode observer. Automatica, 27, 43(9): 1651614 1 Liu J, Wang J L, Yang G H. An LM approach to minimum sensitivity analysis with application to fault detection. Automatica, 25, 41(11): 199524 11 Hou M, Patton P J. An LM approach to H /H fault detection observers. n: Proceedings of UKACC nternational Conference on Control. London, UK: EEE, 1996. 3531 12 Wang J L, Yang G H, Liu J. An LM approach to H index and mixed H /H fault detection observer design. Automatica, 27, 43(9): 16561665 13 Wang H B, Wang J L, Liu J, Lam J. terative LM approach for robust fault detection observer design. n: Proceedings of the 42nd EEE Conference on Decision and Control. Hawaii, USA: EEE, 23. 19741979 14 nry D, Zolghadri A. Design and analysis of robust residual generators for systems under feedback control. Automatica, 25, 41(2): 251264 15 Yang G H, Wang J L, Sco Y C. Reliable H controller design for linear systems. Automatica, 21, 37(5): 717725 16 Wang H B, Wang J L, Lam J. An optimization approach for worst-case fault detection observer design. n: Proceedings of the American Control Conference. Boston, USA: EEE, 24. 2475248 17 Wang H, Yang G H. Fault detection observer design in low frequency domain. n: Proceedings of nternational Conference on Control Applications. Singapore, Singapore: EEE, 27. 976981 18 Zhang X N, Yang G H. Dynamic output feedback control with finite frequency specifications for time-delay systems. n: Proceedings of the 47th EEE Conference on Decision and Control. Mexico, USA: EEE, 28, 35593564 19 wasaki T, Hara S. Robust control synthesis with general frequency domain specifications: static gain feedback case. n: Proceedings of the American Control Conference. Bonston, USA: EEE, 24. 46134618 L Xiao-Jian Ph. D. candidate at the College of nformation Science and Engineering, Northeastern University. is also an assistant at the College of Sciences, Northeastern University. His research interest covers fault diagnosis and robust control. Corresponding author of this paper. E-mail: lixiaojian8@yahoo.com.cn YANG Guang-Hong Professor at Northeastern University. His research interest covers fault-tolerant control, fault detection and isolation, and robust control. E-mail: yangguanghong@ise.neu.edu.cn