Robust fault detection in linear systems based on ... - Semantic Scholar

Journal of Control Theory and Applications 2007 5 (4) 325–330

DOI 10.1007/s11768-006-6073-4

Robust fault detection in linear systems based on full-order state observers Aiguo WU, Guangren DUAN (1. Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin Heilongjiang 150001, China; 2. Institute for Information and Control, Harbin Institute of Technology Shenzhen Graduate School, Shenzhen Guangdong 518055, China ) Abstract: A parametric approach to robust fault detection in linear systems with unknown disturbances is presented. The residual is generated using full-order state observers (FSO). Based on an analytical solution to a type of Sylvester matrix equations, the parameterization of the observer gain matrix is given. In terms of the design degrees of freedom provided by the parametric observer design and a group of introduced parameter vectors, a sufficient and necessary condition for fullorder state observer design with disturbance decoupling is then established. By properly constraining the design parameters according to this proposed condition, the effect of the disturbance on the residual signal is also decoupled, and a simple algorithm is developed. The presented approach offers all the degrees of design freedom. Finally, a numerical example illustrates the effect of the proposed approach. Keywords: Robust fault detection; Full-order state observers; Linear systems; Parametric approach

1

Introduction

Fault detection and isolation (FDI) is considered in practical processes for the purpose of safety and has attracted the attention of a considerable number of investigators [1∼4]. Among the techniques for realizing FDI, the class of observer-based approaches has been extensively investigated. The basic idea underlying the observer-based approaches is that observers are employed for the role of residual generation. Generally, residual signals, which act as the indicators of the faults, are constructed by a properly weighted output estimation error [5∼7]. When the considered system is subject to an unknown disturbance, which can be used to describe some kinds of modeling uncertainties, such as linearization and model reduction errors, etc. (see, Patton and Chen [4], and the references therein), the residual may give a false alarm. To avoid such a case, one can formulate a robust fault detection scheme whose residual is sensitive to faults but insensitive to disturbances. The main aim of such a scheme is to reduce the effect of disturbances on residual signals and to enhance the influence of faults to residuals [8]. The problems concerning observer-based robust fault detection includes minimizing the influence of the disturbance and maximizing the influence of faults. Such a robust fault detection problem can be converted into some optimization problems with performance indices often expressed by H2

or H∞ norms (see, Patton and Liu [10], Zhong and Ding and their co-authors [11, 12], Frisk and Nielsen [13]). To solve the optimization, some investigators employed genetic algorithms [3], some researchers applied the well-established H2 and H∞ control theories [13], while other investigators utilized LMI approaches to the minimization of the H∞ and H2 norms [11, 12]. Another type of problem for observer-based robust fault detection is to decouple the residuals from the disturbances or unknown inputs. Such problems have attracted a lot of attention in the development of the theories of FDI [4, 6, 9]. Concerning observer-based approaches to disturbance decoupling, many researchers employed Luenberger function observers. This type of residual generators were widely studied [1, 6] because such residual generators have lower dynamical orders than the system dimension and offer more degrees of design freedom by the five coefficient matrices to be designed. Another type of observer-based residual generators with the function of disturbance decoupling is based on full-order observers [5,8,9,15] because such an approach has only one coefficient matrix to be designed and hence is simpler. References [4, 8, 9, 15] employed the technique of eigenstructure assignment. In [8, 9], some left eigenvectors of the observer system are assigned to be orthogonal to the disturbance distribution direction, while in [4] some of the observer right eigenvectors are assigned to the disturbance distribution direction. Unfortunately, in [5,8,9], only a suffi-

Received 6 May 2006; revised 30 September 2006. This work was supported by the National Natural Science Foundation of China (No. 60374024) and the Program for Changjiang Scholars and Innovative Research Team in University.

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A. WU et al. / Journal of Control Theory and Applications 2007 5 (4) 325–330

cient condition for full-order state observer design with disturbance decoupling is proposed. In this paper, we also deal with the problem of robust fault detection based on full-order state observers. The same as [5], the design purpose is also to design a residual generator so that the effect of the disturbances is decoupled from the residual signal. Different from [5], this paper gives a sufficient and necessary condition for full-order state observer design with disturbance decoupling. On the basis of the solution to a type of generalized Sylvester matrix equations, parametric expressions of the observer gain matrix and the left eigenvector matrix of the observer system are established in terms of a group of free parameters si , gi , i = 1, 2, · · · , n. Based on these parametric expressions and a group of introduced vectors pi , i = 1, 2, · · · , n, the condition for disturbance decoupling is converted into some constraints on these design parameters si , gi , pi , i = 1, 2, · · · , n. In this case, the parametric expression of the weighting matrix is given. The whole design is finally converted into a problem of selecting a set of design parameters which satisfy a group of constraints. By properly choosing the parameters satisfying the established constraints, robust fault detection is realized. The proposed approach can offer all the degrees of design freedom which can be further utilized to achieve some additional performance specifications.

2

Problem formulation Consider the following linear system: ( x˙ = Ax + Bu + Dd + F f, y = Cx,

(1)

where x ∈ Rn , y ∈ Rm and u ∈ Rr are the state vector, the output vector, and the input vector, respectively; d ∈ Rq is the unknown disturbance vector and f ∈ Rτ is the fault vector; A, B, C, D and F are known real matrices with appropriate dimensions and satisfy the following assumptions. Assumption 1 B, D and F are all of full column rank, and C is of full row rank; Assumption 2 Matrix pair (A, C) is observable. Remark 1 The term Dd can be used to describe additive disturbance as well as a number of different kinds of modelling uncertainties [4]. In order to realize fault detection for system (1), a fullorder sate observer in the following form is used: ·

x ˆ = Aˆ x + Bu + L(y − C x ˆ), n

where x ˆ ∈ R is the state estimation vector, and L ∈ R is the observer gain matrix.

(2) n×m

Definition 1 System (2) is said to be a full-order state observer for system (1) with d = 0 and f = 0, if for arbitrary initial conditions x(0) and x ˆ(0) and any input u(t),

the following relation holds: lim (ˆ x(t) − x(t)) = 0.

t→∞

Let e(t) = x ˆ(t) − x(t), then from (1) and (2) the following relation is obtained: e˙ = Ao e − Dd − F f, Ao = A − LC.

(3)

When d = 0 and f = 0, system (3) becomes e˙ = Ao e.

(4)

From linear system theory, it follows that the system (2) forms a full state observer for system (1) with d = 0 and f = 0 if and only if system (4) is stable, i.e., all the eigenvalues of matrix Ao have negative real parts. Define the residual vector, generated from the difference between the measured and estimated outputs, as r = W (y − C x ˆ),

(5)

σ×m

where 0 6= W ∈ R is a proper weighting matrix to be determined. When the disturbance vector d and the fault vector f are not present and the system (4) is stable, we have r(t) → 0. However, when the disturbance vector d and the fault vector f are present, this residual vector is no longer zero. Thus, using (3) and (5), we can express the contribution of the disturbance d and the fault f to the residual r, in the frequency domain, as r(s) = W C(sI − Ao )−1 Dd(s) +W C(sI − Ao )−1 F f (s).

(6)

The residual signal is used as an indicator of a fault in fault detection. For robustness consideration, the effect of the disturbance d on the residual signal r needs to be eliminated in order to avoid a false alarm, and this requires W C(sI − Ao )−1 D = 0.

(7)

In addition, non-defective matrices whose Jordan form is diagonal have lower eigenvalue sensitivities than defective ones, so we restrict the matrix Ao to being non-defective. When condition (7) holds, the residual signal r does not respond to the disturbance d. However, since the residual signal acts as the indicator of faults, it must be ensured that the residual signal r really responds to the fault f. To guarantee this, the following relation must hold W C(sI − Ao )−1 F 6= 0.

(8)

From the above discussion, we now can state the problem of robust fault detection based on full-order state observers as follows: Problem FSRFD Given system (1) satisfying Assumptions 1 and 2, determine the observer gain matrix L and a proper weighting matrix W such that the following conditions are met: 1) The matrix Ao in (3) is non-defective and stable; 2) Robust fault detection conditions (7) and (8) are satisfied.

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3

Preliminaries

the following equation through (12), (13) and (14).

Problem FSRFD (Full-order State Robust Fault Detection) is concerned with the solution to an observer gain matrix L and a weighting gain matrix W satisfying two requirements. As a preliminary, this section copes with the first requirement in Problem FSRFD based on a result of eigenstructure assignment by state feedback, i.e., the following Problem FSO. Problem FSO Given matrices A ∈ Rn×n , C ∈ Rm×n satisfying Assumptions 1 and 2, determine the observer gain matrix L such that the matrix Ao in (3) is stable and nondefective. Since matrix Ao is required to be non-defective, it has a diagonal Jordan form Λ = diag(s1 , s2 , · · · , sn ),

(9)

where si , i = 1, 2, · · · , n, are not necessarily distinct, but satisfy the following constraint to ensure that Ao is real and stable: Constraint 1 {si , i = 1, 2, · · · , n} is self-conjugate and Re(si ) < 0, i = 1, 2, · · · , n. Denote the left eigenvector of matrix Ao associated with eigenvalue si by ti , then, the following relations hold by definition tTi Ao

=

si tT i ,

i = 1, 2, · · · , n.

(10)

T = [ t1 t2 · · · tn ]. (11)

Under the conditions of Assumptions 1 and 2, there exist a pair of right coprime polynomial matrices N (s) ∈ Rn×m [s] and D(s) ∈ Rr×r [s] satisfying the following right coprime factorization: (12)

Based on the above statement and the results in [16], we obtain the following theorem to solve Problem FSO. Theorem 1 Let ( T = [ t1 t2 · · · tn ], (

ti = N (si )gi , i = 1, 2, · · · , n, Z = [ z1 z2 · · · zn ], zi = D(si )gi , i = 1, 2, · · · , n.

Moreover, the matrix T parameterized by (13) is the left eigenvector matrix of matrix Ao .

4 Solution to Problem FSRFD Based on the preliminary work in Section 3, in this section we give a parametric approach to Problem FSRFD. This problem is concerned with the solution of an observer gain matrix L and a weighting gain matrix W satisfying two requirements. In Theorem 1, the parameterization of the observer gain matrix satisfying the first requirement in Problem FSRFD has been presented; in order to solve Problem FSRFD, it suffices only to consider the second requirement, i.e., equations (7) and (8). It follows from (11) and (9) that W C(sI − Ao )−1 D = W C(T T )−1 [sI − T T Ao (T T )−1 ]−1 T T D = W C(T T )−1 (sI − Λ)−1 T T D.

(16)

Let W C(T T )−1 = P = [ p1 p2 · · · pn ], pi ∈ Cσ , i = 1, 2, · · · , n,

(17)

then using (13), (16) and (17), we obtain = [ p1 p2 · · · pn ]diag(

Then, it follows from (10) and (9) that

(sI − AT )−1 C T = N (s)D−1 (s).

(15)

W C(sI − Ao )−1 D

Construct the left eigenvector of matrix Ao by

T T Ao = ΛT T .

L = −[T T ]−1 Z T .

(13)

1 1 , ··· , )T T D s − s1 s − sn

n p tT P i i )D i=1 s − s1 n p g T N T (s )D P i i i = . s − s1 i=1 In view of the arbitrariness of the variable s, the robust fault detection condition (7) is equivalent to the following constraint with respect to parameters pi , gi , si , i = 1, 2, · · · , n. Constraint 4 pi giT N T (si )D = 0, i = 1, 2, · · · , n. In this case, the weighting matrix W can be obtained by (17). In view of (13), (17) is equivalent to n P WC = pi giT N T (si ). (18)

=(

i=1

(14)

Then Problem FSO has a solution if and only if there exist a group of parameter vectors gi ∈ Cm , i = 1, 2, · · · , n and a group of parameter scalars si , i = 1, 2, · · · , n satisfying Constraint 1 and the following constraints: Constraint 2 detT 6= 0; Constraint 3 gi = g¯l if si = s¯l . In this case, all the solutions to Problem FSO are given by

It is obvious that the above equation has a real solution with respect to W if and only if there holds the two constraints given below:    Constraint 5 rankC =rank n+p P i=1

C pi giT N T (si )

 ;

Constraint 6 pi = p¯l if si = s¯l . It is easy to see that when the following relation holds: W CF 6= 0,

(19)

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the relation (8) is met. By using (18), condition (19) can be converted into the following constraint with respect to parameters pi , gi , si , i µ = 1, 2, · · · , n. ¶ n P Constraint 7 pi giT N T (si )F 6= 0. i=1

When the above constraint is met, in view that matrix C is of full-row rank, it follows from (18) that the weighting matrix W is given by n P W = ( pi giT N T (si ))C T (CC T )−1 . (20) i=1

From the above reasoning, we can give the following theorem to solve the problem FSRFD in Section 2. Theorem 2 Let Assumptions 1 and 2 hold. Then, Problem FSRFD has a solution if there exist parameters si ∈ C, pi ∈ Cσ , gi ∈ Cm , i = 1, 2, · · · , n, satisfying Constraints 1∼7. In this case, the observer gain matrices are given by (15) through (12), (13) and (14), and the weighting matrix W is obtained by (20). On the basis of Theorem 2, we can give the following algorithm to solve the problem of robust fault detection based on full-order state observers. Algorithm FSRFD: Step 1 Solve a pair of right coprime polynomial matrices N (s) and D(s) satisfying the right coprime factorization (12). Step 2 Express Constraints 1∼7 in forms represented by the parameters si ∈ C, pi ∈ Cσ , gi ∈ Cm , i = 1, 2, · · · , n. Step 3 Find a group of parameters si , pi , gi , i = 1, 2, · · · , n, satisfying Constraints 1∼7. If such parameters do not exist, this approach is invalid. Step 4 Based on the parameters si , pi , gi , i = 1, 2, · · · , n, obtained in Step 3, calculate the observer gain matrix L according to (13), (14) and (15), and the weighting matrix W according to (20). For the above algorithm, we give the following remarks. Remark 2 Regarding solutions to the right coprime factorization equation (12), one can refer to [16]. Remark 3 Constraint 3 is clearly seen to be ‘almost always’ valid because the parameters satisfying Constraint 3 form a Zarisky open set. Therefore, in practical applications, Constraint 3 may often be neglected when only one specific solution is of interest. Remark 4 The proposed approach can offer all the degrees of design freedom which can be further utilized to meet some additional performance specifications beyond the performance of robust fault detection.

5 A numerical example Consider a system in the form of (1) with the following parameters     " # 0 3 4 1 0     0 1 0     A = 1 2 3 , B = 0 0 , C = , 0 0 1 0 2 5 0 1 (21)     1 1 −3        D =  2  , F =  −0.5 1  . −1 0.5 0 It is obvious that Assumptions 1 and 2 are met. In the following, we solve, using the Algorithm FSRFD proposed in Section 4, the robust fault detection problem based on a fullorder state observer in this example system. The weighting matrix W is restricted to a row vector, i.e., σ = 1. Step 1 Using the algorithm proposed in [16], yield the right coprime polynomial matrix pair satisfying the right coprime factorization (12)   " # 1 0   s2 − 2s − 3 −2   . (22) N (s) =  s 0  , D(s) = −3s − 4 s − 5 0 1 Step 2 For simplicity, we restrict the closed-loop poles s1 , s2 , s3 to real ones, then Constraint 1 is si < 0, i = 1, 2, 3.

(23)

In this case, the vectors gi , i = 1, 2, 3 can also be restricted to real ones, then Constraint 3 holds automatically. Denote " # gi1 gi = , i = 1, 2, 3. gi2 Thus, Constraint 2 and Constraint 4 are respectively derived as   g11 g21 g31    det  (24)  s1 g11 s2 g21 s3 g31  6= 0, g12 g22 g32 and pi (gi1 − gi2 + 2si gi1 ) = 0, i = 1, 2, 3,

(25)

where pi , i = 1, 2, 3, are real, thus Constraint 6 is automatically met. In this case, Constraint 5 is   0 1 0     0 0 1 rank   = 2, 3 3 3 P  P P pi gi1 si pi gi1 pi gi2 i=1

i=1

i=1

which is equivalent to the following relation: p1 g11 + p2 g21 + p3 g31 + p4 g41 = 0,

(26)


andConstraint 7 is  3 3 3 P P P  i=1 pi gi1 − 0.5 i=1 si pi gi1 + 0.5 i=1 pi gi2    6= 0. (27) 3 3   P P −3 pi gi1 + si pi gi1 i=1

i=1

Step 3 Specially, we choose the following parameters satisfying (23)∼(27) s1 = −1, s2 = −2, s3 = −1, p1 = 1, p2 = 2, p3 = 0, " # " # " # −1 1 0 g1 = , g2 = , g3 = . 1 −3 1 Step 4 Based on the parameters given in the above step and using (13) and (14), we obtain   " # −1 1 0   −2 11 −2  T =  1 −2 0  , Z = −5 23 −6 . 1 −3 1 In this case, from (15) and (20) the observer gain matrix and the weighting matrix are respectively obtained as   5 7    L=  5 6  , W = [ −1 −2 ]. 2 6

6

Conclusion

Robust fault detection based on full-order state observers in linear systems with unknown disturbance is considered. On the basis of the solution to a type of generalized Sylvester matrix equations, a parametric approach is established. In terms of the free parameters proposed by the parametric observer design and a group of introduced parameters, the condition for disturbance decoupling is converted into some constraints with respect to the design parameters. The proposed approach can offer all the degrees of design freedom. An illustrative example shows the effect of the proposed approach. References [1] H. Hou, P. C. Muller. Fault detection and isolation observers[J]. International Journal of Control, 1994, 60(5): 827 – 846. [2] H. Yang, M. Saif. State observation, failure detection and isolation (FDI) in bilinear systems[J]. International Journal of Control, 1997, 67(6): 901 – 920. [3] R. J. Patton, J. Chen, G. Liu. Robust fault detection of dynamic systems via genetic algorithm[C]// Genetic Algorithms in Engineering Systems: Innovation and Applications. Stevenage, England: IEE, 1995, 414: 511 – 516 [4] R. J. Patton, J. Chen. On eigenstructure assignment for robust fault diagnosis[J]. International Journal of Robust and Nonlinear Control, 2000, 10(14): 1193 – 1208. [5] G. Duan, R. J. Patton, J. Chen, Z. Chen. A parametric approach for fault detection in linear systems with unknown disturbances[C]// Proceedings of the IFAC Symposium on Fault Detection, Supervision

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and Safety for Technical Processes. Hull, England: Kingston Upon Hull, 1997: 318 – 322 [6] G. Duan, R. J. Patton. Robust fault detection using Luenberger-type unknown input observers – a parametric approach[J]. International Journal of Systems Science, 2001, 32(4): 533 – 540. [7] Y. Xiong, M. Saif. A novel design for robust fault diagnostic observer[C]// Proceedings of the 37th IEEE Conference on Decision and Control. Piscataway: Institute of Electrical and Electronics Engineers Inc., 1998: 592 – 597. [8] R. J. Patton, J. Chen. A robust disturbance decoupling approach to fault detection in process systems[C]// Proceedings of the 30th Conference on Decision and Control. Piscataway: Institute of Electrical and Electronics Engineers Inc., 1991: 1543 – 1548. [9] R. J. Patton, J. Chen. Robust fault detection using eigenstructure assignment: a tutorial consideration and some new results[C]// Proceedings of the 30th Conference on Decision and Control. Piscataway: Institute of Electrical and Electronics Engineers Inc., 1991: 2242 – 2247. [10] R. J. Patton, G. Liu. Robust fault detection of dynamic systems via genetic algorithms[C]// Genetic Algorithms in Engeering Systems: Innovations and Applications. Sheffield, England: IEE, 1995, 414: 511 – 516 [11] M. Zhong, S. Ding, J. Lam, H. Wang. An LMI approach to design robust fault detection filter for uncertain LTI systems[J]. Automatica, 2003, 39(3): 543 – 540. [12] M. Zhong, S. Ding, B. Tang. An LMI approach to robust fault detection filter design for discrete-time systems with model uncertainty[C]// Proceedings of the 40th Conference on Decision and Control. Piscataway: Institute of Electrical and Electronics Engineers Inc., 2001: 3613 – 3618. [13] E. Frisk, L. Nielesen. Robust residual generation for diagnosis including a reference model for residual behavior[J]. Automatica, 2006, 42(3): 437-445. [14] T. Dalton, R. J. Patton, J. Chen. An application of eigenstructure assignment to robust residual design for FDI[C]// International Conference on Control. Hull England: IEE, 1996, 427: 78 – 83 [15] Y. Xiong, M. Saif. Robust fault detection and isolation via a diagnostic observer[J]. International Journal of Robust and Nonlinear Control, 2000, 10(14): 1175 – 1192. [16] G. Duan. Solutions to matrix equation AV + BW = V F and their application to eigenstructure assignment in linear systems[J]. IEEE Transactions On Automatic Control, 1993, 38(2): 276 – 280.

Aiguo WU was born in Gong’an County, Hubei Province on September 20, 1980. He received his B. Eng. degree in Automation in 2002 and M. Eng. degree in Navigation, Guidance and Control in 2004 from Harbin Institute of Technology. Currently, he is pursuing his Ph. D degree in the Center for Control Theory and Guidance Technology at Harbin Institute of Technology. He has ever served as a General Secretary of Program Committee of the 25th Chinese Control Conference, which was held in Harbin during August 7 – 11, 2006. His research interests include robust control and estimation, observer design and descriptor linear systems. E-mail: [email protected]; [email protected].

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Guangren DUAN was born in Heilongjiang Province on April 5, 1962. He received his B. S. degree in Applied Mathematics, and both his M. Sc. and Ph. D. degrees in Control Systems Theory. From 1989 to 1991, he was a post-doctoral researcher at Harbin Institute of Technology, where he became a professor of control systems theory in 1991. Prof. Duan visited the University of Hull, UK, and the University of Sheffield, UK from December 1996 to October 1998, and

worked at the Queen’s University of Belfast, UK from October 1998 to October 2002. Since August 2000, he has been elected Specially Employed Professor at Harbin Institute of Technology sponsored by the Cheung Kong Scholars Program of the Chinese government. He is currently the Director of the Center for Control Theory and Guidance Technology at Harbin Institute of Technology. He is the author and co-author of over 400 publications. His main research interests include robust control, eigenstructure assignment, descriptor systems, missile autopilot control and magnetic bearing control. Dr. Duan is a Charted Engineer in the UK, a Senior Member of IEEE and a Fellow of IEE. E-mail: [email protected].