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Actuator fault diagnosis for a class of bilinear systems with uncertainty
Journal of the Franklin Institute 339 (2002) 361–374
Actuator fault diagnosis for a class of bilinear systems with uncertainty Bin Jianga,*, Jian Liang Wangb a
LAIL-CNRS, Bat. P2, UPRESA 8021, Universit!e des Sciences et Technologies de Lille, 59655 Villeneuve d’Ascq, Cedex, France b School of EEE, Nanyang Technological University, Nanyang Avenue, Block S2, Singapore 639798, Singapore
Received 10 May 2000; received in revised form 30 May 2001; accepted 27 November 2001
Abstract In this paper, the actuator fault diagnosis problem for a class of bilinear systems with uncertainty is discussed. The system is transformed into two different subsystems. One is not affected by actuator fault, so an adaptive observer can be designed such that, under certain conditions, the observer error dynamics is stable. The other whose states can be measured is affected by the faults. The observation scheme is then used for model-based fault diagnosis. Finally, an example of a semiactive suspension system is used to illustrate the applicability of the proposed method. r 2002 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved. Keywords: Fault detection and isolation (FDI); Bilinear systems; Adaptive observer; Uncertainty
1. Introduction The process of detection and isolation of system faults has been of considerable interest during the last two decades, as can be seen from the survey paper [1–3] and the books [4–6]. Research is still under way into the development of more effective solutions for fault detection and isolation (FDI) in automatic control systems. Until now, most research work on FDI tends to focus on linear systems, though some
*Corresponding author. Tel./fax: +33-3-20-43-4743. E-mail address: [email protected] (B. Jiang). 0016-0032/02/$22.00 r 2002 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 6 - 0 0 3 2 ( 0 1 ) 0 0 0 5 1 - 5
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initial results for nonlinear systems are available, for example, see [7,8] based on nonlinear observers and [9] based on parity space approach. Bilinear systems are a special class of nonlinear systems in which the control appears in both additive and multiplicative terms. Bilinear systems arise in a variety of physical situations, and many physical processes are of this type, for example nuclear reactors, field controlled d.c. motors, heat exchangers and many biomedical processes such as the immune system are known to be described by bilinear systems [10,11]. In [12,13] the unknown input fault detection observer approach is extended to bilinear systems. Recently, Yang and Saif [14] discussed FDI for a class of bilinear systems based on reduced order observer, and the estimation error of the observer discussed there is dependent on inputs. However, no system uncertainties have been considered in that paper. In practice, since an accurate mathematical model of a physical process is not available, there is a mismatch between the actual process and its mathematical model even if there is no fault in the process. Such mismatch causes fundamental methodological difficulties in FDI applications and is a source of false alarms which can corrupt the performance of the FDI system. Furthermore, in some cases such as unstructured uncertainties in the system, perfect decoupling is not possible, see for example, [15]. In this paper, we consider the robust FDI problem for a class of bilinear systems with uncertainties. At first, the system is transformed into two subsystems. The 1st subsystem is decoupled from the actuator fault and the other is affected by the fault, but its states can be measured directly. As a generalization of the observer design approach in [16], a stable adaptive observer is proposed for the 1st subsystem. By using estimated states and unknown parameters, we can approximate the fault from of the 2nd subsystem. When the fault signal is greater than certain threshold, the fault can be detected or even isolated. This paper is organized as follows. Section 2 gives conditions for the existence of a transformation that transforms the original uncertain system into a bilinear canonical form. Then an adaptive observer is given based on the transformed system. In Section 3, actuator FDI are discussed. An application example is included in Section 4, followed by some concluding remarks in Section 5.
2. Adaptive observer design Consider the following system with uncertainty: xðtÞ ’ ¼ AxðtÞ þ
m X
Ai ui ðtÞxðtÞ þ BuðtÞ þ Efa ðtÞ þ DxðtÞ;
ð1Þ
i¼1
yðtÞ ¼ CxðtÞ ¼ ½O
IxðtÞ;
ð2Þ
where the state is xARn ; the input is u ¼ ½u1 ; y; um T ARm ; and the output is yARr : The fault is modelled as an additional disturbance input fa ARq ; with qor; E is of full
B. Jiang, J.L. Wang / Journal of the Franklin Institute 339 (2002) 361–374
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column rank, and xðtÞ denotes unknown parameter vector of the system which is assumed to be uniformly bounded. Note that in Eq. (2), the special form for the matrix C is not a restriction. Since as long as C is of full row rank, there exists a similarity transformation that can bring the output equation in that desired form.
Assumption 1. rankðCEÞ ¼ q: Remark 1. Assumption 1 is required for fault identification, as in [14,15]. Partition x as x ¼ ½x1
x3 T
x2
with x1 ARnr ; x2 ARrq ; and x3 ARq : Then the system described by Eqs. (1) and (2) can be written as 2 3 2 32 3 2 i 3 2 3 x’ 1 A11 A12 A13 x1 A11 Ai12 Ai13 x1 m 6 7 6 76 7 X6 i 7 i i 7 6 4 x’ 2 5 ¼ 4 A21 A22 A23 54 x2 5 þ 4 A21 A22 A23 5ui 4 x2 5 i¼1 x’ 3 A31 A32 A33 x3 Ai31 Ai32 Ai33 x3 2 3 2 3 2 3 E1 B1 D1 6 7 6 7 6 7 ð3Þ þ 4 B2 5u þ 4 E2 5fa þ 4 D2 5x; B3 " y ¼ Cx ¼
E3
D3
0 IðrqÞðrqÞ
0
0
Iq
0
# ð4Þ
x:
As a generalization of the definition of bilinear canonical form (BCF) for fault-free bilinear systems in [16], the following definition is made. Definition. The generalized bilinear canonical form (GBCF) of the bilinear system in Eqs. (3) and (4) is described as 2 2 3 2 32 3 3 i i 3 2 0 A% 12 A% 13 z’1 A% 11 A% 12 A% 13 z1 z1 m 7 6 7 6 % 76 7 X6 7 i i 7 6 6 A% i 4 z’2 5 ¼ 4 A21 A% 22 A% 23 54 z2 5 þ 4 21 A% 22 A% 23 5ui 4 z2 5 i¼1 i i i z’3 z3 z3 A% 31 A% 32 A% 33 A% 31 A% 32 A% 33 2 3 2 3 2 3 B% 1 D% 1 0 6 % 7 6 % 7 6 % 7 ð5Þ þ 4 B2 5u þ 4 E2 5fa þ 4 D2 5xðtÞ; B% 3
" y¼
0 IðrqÞðrqÞ 0 0
E% 3 #
0 W % z ¼ Cz: Iq
D% 3
ð6Þ
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Define " T¼
T1
# ;
T2
E1 ; T1 ¼ ½A111 A211 ? Am 11 "" #" # " #" ## A121 A221 Am E2 21 T2 ¼ ? m : 1 2 A31 A31 A31 E3
Assumption 2. rankðCT Þ ¼ rankðT2 Þ ¼ rankðTÞ: Remark 2. Assumption 2 characterizes the existence condition under which a bilinear system can be transformed into GBCF described by Eqs. (5) and (6). Some practical systems indeed satisfy this assumption, see for example, Section 4. Lemma 1. Suppose that Assumption 2 holds. Then there exists F ARðnrÞr such that the bilinear system described by Eqs. (3) and (4) can be transformed into GBCF by using the similarity transformation z ¼ Nx with " N¼
IðnrÞðnrÞ
F
0
Irr
# ;
FT2 ¼ T1 :
Proof. The proof of Lemma 1 is similar to that in [16] and thus is omitted.
ð7Þ
&
Remark 3. Lemma 1 has extended the obtained results in [16] for fault-free system (i.e., E ¼ 0 in Eq. (1)) to that in the faulty case. % it is % ¼ q: From the structure of matrix C; According to Assumption 1, rankðC% EÞ easy to show that "
E% 2 rank E% 3
# ¼ q:
ð8Þ
% From Eq. (8), EE% 23 is of full column rank. Without loss of generality, it can be assumed that E% 3 is nonsingular. Let 2
I 6 S ¼ 40 0
3 0 0 7 I E2 E31 5: 0
I
ð9Þ
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For i ¼ 1; y; m; j ¼ 1; 2; 3; define W 1 Hj ¼ A% 2j E% 2 E% 3 A% 3j ; W 1 R ¼ B% 2 E% 2 E% 3 B% 3 ;
W i 1 i Gij ¼ A% 2j E% 2 E% 3 A% 3j ; W M ¼ D% 2 E2 E31 D% 3 :
By pre-multiplying Eq. (9) into Eq. (5), we have that 2 3 2 3 2 32 3 i i z’1 0 A% 12 A% 13 A% 11 A% 12 A% 13 z1 7 6 7 6 76 7 Xm 6 6 Gi1 Gi2 Gi3 7 4 y’ 1 E% 2 E% 1 ’ 2 5 ¼ 4 H 1 H 2 H 3 5 4 y1 5 þ 5 3 y i¼1 4 i i i A% 31 A% 32 A% 33 y2 y’ 2 A% 31 A% 32 A% 33 2 3 2 3 2 3 2 3 B% 1 0 z1 D% 1 6 7 6 7 6 7 6 7 ui 4 y1 5 þ 4 R 5u þ 4 0 5fa þ 4 M 5xðtÞ: ð10Þ E% 3 B% 3 D% 3 y2 It is clear that in Eq. (10), the actuator faults enter only through the third block row, while the other two block rows are not affected by any faults. The first and second block rows of Eq. (10) can be rewritten as Xm z’1 ðtÞ ¼ A% 11 z1 ðtÞ þ sðtÞ þ D% 1 xðtÞ; vðtÞ ¼ H1 z1 þ Gi1 ui z1 þ MxðtÞ; ð11Þ i¼1
where sðtÞ ¼ A% 12 y1 þ A% 13 y2 þ
Xm i¼1
i i ðA% 12 ui y1 þ A% 13 ui y2 Þ þ B% 1 u;
vðtÞ ¼ y’ 1 E% 2 E% 3 y’ 2 H2 y1 H3 y2 1
Xm i¼1
ðGi2 ui y1 þ G13 ui y2 Þ Ru:
ð12Þ ð13Þ
Since the dynamical system represented in Eq. (11) is driven by known input sðtÞ and uncertain term D% 1 xðtÞ; its state can be estimated by using a adaptive observer # þ K½vðtÞ v#ðtÞ; z’#1 ðtÞ ¼ A% 11 z#1 ðtÞ þ sðtÞ þ D% 1 xðtÞ Xm # Gi1 ui z#1 ðtÞ M xðtÞ; ð14Þ v#ðtÞ ¼ H1 z#1 ðtÞ i¼1 # is the estimation of xðtÞ which will be designed where K is the observer’s gain, xðtÞ later. Assumption(3. The set O; defined by ) K2 Gi1 ¼ 0; i ¼ 1; 2; y; m; and O ¼ K2 : the pair ðA% 11 ; K2 H1 Þ is completely observable
ð15Þ
is not empty. Remark 4. It should be noted that in practice, matrices Gi1 ði ¼ 1; y; mÞ may be sparse and dimension of A% 11 is not high such that K2 may be found. As a result, the above assumption is not expected to be too restrictive. Under Assumption 3, the observer gain in Eq. (14) can be written as K ¼ K1 K2 ; where K2 AO and K1 is selected so that all the eigenvalues of ðA% 11 K1 K2 H1 Þ are
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placed in the left-hand side of the complex plane. Substituting Eqs. (12), (13) and (15) into Eq. (14) yields
1 z’#1 ¼ ðA% 11 KH1 Þ#z1 þ Kðy’ 1 E% 2 E% 3 y’ 2 Þ þ ðB% 1 KRÞu þ ðA% 12 KH2 Þ i Xm % i12 ui KGi2 ui Þ y1 þ ð A i¼1 h i Xm % i13 ui KGi3 ui Þ y2 þ ðA% 13 KH3 Þ þ ð A i¼1 # þ ðD% 1 KMÞxðtÞ:
ð16Þ
Note that the above observer uses the derivative of the outputs which is not available for direct measurement. In order to eliminate the need for differentiating the output, a new variable W is defined as follows: W ¼ z#1 Kðy1 E% 2 E% 3 y2 Þ: 1
ð17Þ
Then Eq. (16) can be expressed in the following form: ’ ¼ ðA% 11 KH1 ÞW þ ðB% 1 KRÞu W h i Xm % i12 ui KGi2 ui Þ þ KðA% 11 KH1 Þ y1 ð A þ ðA% 12 KH2 Þ þ i¼1 h i Xm i 1 þ ðA% 13 KH3 Þ þ ðA% 13 ui KGi3 ui Þ KðA% 11 KH1 ÞE% 2 E% 3 y2 i¼1
# þ ðD% 1 KMÞxðtÞ:
ð18Þ
From Eqs. (11) and (14), the error dynamics of z*1 ¼ z1 z#1 is given by * z’*1 ¼ ðA% 11 KH1 Þ*z1 þ ðD% 1 KMÞxðtÞ;
ð19Þ
# is the estimation error of the unknown parameter vector. * ¼ xðtÞ xðtÞ where xðtÞ Define v*ðtÞ ¼ vðtÞ v#ðtÞ; considering Eqs. (11), (14) and (19), we can obtain the following input–output expression of the error equation: * v*ðtÞ ¼ QðsÞxðtÞ;
ð20Þ
where % ðA% 11 KH1 Þ1 ðD% 1 KMÞ; QðsÞ ¼ M þ H½sI Xm H% ¼ H1 þ Gi1 ui : i¼1
ð21Þ
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Using the augmented error technique in Narendra (1989), it can be shown that # xðtÞQðsÞ1ðtÞ # v*ðtÞ ¼ x* ðtÞZðtÞ þ QðsÞxðtÞ þ dðtÞ; T
ð22Þ
where ZðtÞ ¼ QðsÞ1ðtÞ; dðtÞ decays exponentially and can be neglected. We can define # xðtÞQðsÞ1ðtÞ # eðtÞ ¼ v*ðtÞ ½QðsÞxðtÞ
ð23Þ
as the augmented error signal. Therefore, it can be obtained that eðtÞ ¼ x* ðtÞZðtÞ: T
ð24Þ
Eq. (24) is in the standard form of the error equations widely used in adaptive control theory (Narendra, 1989). The adaptive updating laws used in adaptive * control can be directly applied to tune xðtÞ: This can be summarized in the following theorem. Theorem 1. Consider the bilinear system ð1Þ–ð2Þ: If Assumptions 1–3 hold, then the adaptive observer given by Eq. ð14Þ and the following parameter update law: ’* ¼ g xðtÞ
eðtÞZðtÞ ; 1 þ ZT ðtÞZðtÞ
ð25Þ
* * ¼ 0 and realizes a bounded xðtÞ and z*1 ðtÞAL2 : Furthermore, limt-N xðtÞ limt-N z*1 ðtÞ ¼ 0 under persistent excitation, where g > 0 is a weighting matrix. Proof. Using the same method as in Narendra ð1989Þ; we can prove that the * adaptive observer (14) and parameter update law (25) can result in a bounded xðtÞ 2 and z*1 ðtÞAL : The persistent excitation means that there exist two positive constants s and tT such that for all t; the following inequality holds: Z tþtT T % H% ðuðsÞÞHðuðsÞÞ dsXsIðnrÞ ; ð26Þ t
where H% is given by Eq. (21). Under the persistent excitation, an accurate estimation of xðtÞ can be obtained, which is regarded as the parameter in standard format of adaptive control, i.e., * ¼ 0: limt-N xðtÞ
ð27Þ
From Eqs. (19) and (27) and the fact that ðA% 11 KH1 Þ is stable, one can conclude that limt-N z*1 ðtÞ ¼ 0: This completes the proof. &
3. Actuator FDI In this section, a simple approach for detecting and isolating the actuator fault is proposed to estimate the magnitude of the actuator fault fa :
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The third row of Eq. (10) is written as Xm i i i ðA% 31 ui z1 þ A% 32 ui y1 þ A% 33 ui y2 Þ y’ 2 ¼ A% 31 z1 þ A% 32 y1 þ A% 33 y2 þ i¼1 þ B% 3 u þ E% 3 fa þ D% 3 xðtÞ:
ð28Þ
Assuming that no fault occurs during the initial transient of the observer, using the estimation z#1 and x# for z1 and x; we can estimate the actuator fault as 1
f#a ðtÞ ¼ E% 3 y’ 2 ðtÞ A% 32 y1 ðtÞ A% 33 y2 ðtÞ A% 31 z#1 ðtÞ i Xm i i i ðA% 31 ui ðtÞ#z1 ðtÞ þ A% 32 ui ðtÞy1 ðtÞ þ A% 33 ui ðtÞy2 ðtÞÞ B% 3 uðtÞ D% 3 x# : i¼1
ð29Þ The estimate error of the actuator fault is given by Xm 1 % i31 ui z*1 E% 1 % * A f*a ðtÞ ¼ fa ðtÞ f#a ðtÞ ¼ E% 3 A% 31 þ 3 D3 x: i¼1
ð30Þ
Note that faults are not involved in estimation of z1 and x; according to Theorem 1, for given e1 > 0; e2 > 0; there exists t0 such that for t > t0 jj*z1 ðtÞjjpe1 ;
* jjxðtÞjjpe 2:
ð31Þ
Therefore Xm 1 % i31 ui jje1 þ jjE% 1 % A jjf*a ðkÞjjpjjE% 3 A% 31 þ 3 D3 jje2 : i¼1 The threshold value is defined as Xm 1 % i31 ui e1 þ jjE% 1 % A Tr ¼ E% 3 A% 31 þ 3 D3 jje2 : i¼1
ð32Þ
ð33Þ
Then the fault detection can be carried out as follows: The alarm is on; The alarm isoff;
jjf#a jj > Tr ; jjf# jjpTr : a
ð34Þ
Remark 5. Compared with existing results on FDI for bilinear systems, our proposed method has two advantages. One is that it can detect not only occurrence but also amplitudes of faults. The other is that it can be used for systems with uncertainties that cannot be decoupled from faults or residuals. Remark 6. In Eq. (29), calculation of derivative of output is required to estimate the fault. Because of the presence of noise in practice, it is not easy to compute the signal derivative. Evaluating derivative of output from noisy signals can be done using
B. Jiang, J.L. Wang / Journal of the Franklin Institute 339 (2002) 361–374
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specific algorithms such as polynomial approximation-based scheme, which has been extensively investigated, for example in [17].
4. An illustrative example To illustrate the proposed method, the example of a two degree of freedom (DOF) quarter car model of vehicle suspension given in [11] is examined. Such a bilinear system is described as follows: xðtÞ ’ ¼ AxðtÞ þ A1 uðtÞxðtÞ þ Efa ðtÞ þ DxðtÞ;
ð35Þ
yðtÞ ¼ CxðtÞ;
ð36Þ
where x ¼ ½x1 x2 x3 x4 T is a state in which x1 is the suspension deflection, x2 is the quarter car body velocity, x3 is the wheel displacement from the road, x4 is the wheel velocity, and u the variable damping. The measurement outputs are y ¼ ½y1 y2 y3 T ; where y1 is the suspension defection, y2 is the wheel velocity, and y3 is the body displacement with respect to the road. fa denotes the actuator fault and x stands for disturbance which affects the vehicle body velocity. The system matrices are given by 2
6 3:6 6 A¼6 4 0
0
1
0:5 0
0 0
0:5 7 7 7; 1 5
5
3600
360 2
0
3
6 0:1 7 6 7 E¼6 7; 4 0 5
B ¼ 0;
3
1
0
1
5
2
0
60 6 A1 ¼ 6 40 0
2 3 0 617 6 7 D ¼ 6 7; 405 0
2
1
6 C ¼ 40 1
0
0
0:004 0 0 0 0:04
0
0
3
0 0
0 0
0
3
0:004 0
7 7 7; 5
0:04
7 0 1 5: 1 0
Since the matrix C in the above system description is not in the desired form, a transformation is needed to get the system in the proper form. The similarity transformation matrices are chosen as follows: 2
3
0
1
0 0
61 6 N% ¼ 6 40
0 0
0 07 7 7: 0 15
1
0
1 0
B. Jiang, J.L. Wang / Journal of the Franklin Institute 339 (2002) 361–374
370
As a result, we get the following system: 2 2 3 0:004 0:5 36 0:5 0 6 6 1 7 0 1 0 7 6 0 6 x’ ¼ 6 7x þ 6 4 0:04 4 5 5 3960 5 3600 1 2
0 2 3 0 1 6 0 7 607 6 7 6 7 þ6 7fa þ 6 7x; 4 0 5 405 3
0
1 2
0
0
0:004
0 0
0 07 7 7ux 0:04 0 5
0
0
0
3
0
0
0
0 6 y ¼ 40
1 0
0 1
3 0 7 0 5x:
0
0
0
1
Using the following similarity transformation: 2 3 1 0 0:1 0 60 1 0 07 6 7 N¼6 7; 40 0 1 05 0 0
0
1
the GBCF is obtained as 2
0 61 6 z’ ¼ 6 45 1
2
360 0 3960 0
0 1:1 5:5 0:1
0 6 y ¼ 40
1 0
0 1
3 0 7 0 5z:
0
0
0
1
2 3 0 360 6 0 0 7 6 7 7z þ 6 4 0:04 3600 5 0 0
0 0 0 0 0 0:004 0 0
3 2 3 2 3 0 0 1 6 0 7 607 07 7 6 7 6 7 7fa þ 6 7x; 7uz þ 6 4 1 5 405 05 0
0
0
From the above model, one can easily get " # " # 1 0 A% 11 ¼ 0; H1 ¼ ; G11 ¼ : 1 0 The observer gain K can be chosen as K ¼ K1 K2 ;
K2 ¼ ½1
0;
K1 > 0:
Therefore, Assumptions 1–3 and the persistent excitation condition (26) in this paper all hold.
B. Jiang, J.L. Wang / Journal of the Franklin Institute 339 (2002) 361–374 0.4
371
0.06 0.04 output y2
y1
0.2 0 −0.2 −0.4
0.02 0 −0.02 −0.04
0
2
4
−0.06
6
0
2
0.6
6
4
6
0.2
0.4 0.2 0 −0.2 −0.4
4 time (sec)
estimation error of z1
z1 (solid) and estimates(dashed)
time (sec)
0
2
4
6
0 −0.2 −0.4 −0.6
0
2
time (sec)
time (sec)
Fig. 1. State estimation under fault occurrence.
0.25
estimation of actuator fault
0.2
0.15
0.1
0.05
0
−0.05 0
0.5
1
1.5
2
2.5 time (sec)
3
3.5
Fig. 2. Fault identification and the estimation.
4
4.5
5
B. Jiang, J.L. Wang / Journal of the Franklin Institute 339 (2002) 361–374
372
In the simulation, x is a uniformly distributed random number on the interval ð0; 0:5Þ; and the gain K1 is selected to be 25, the g in Theorem 1 is 10. The actuator fault is considered, respectively, as follows: ( 0 for 0ptp1; fa ¼ ð37Þ 0:2 for 1otp5; ( fa ¼
0
for 0ptp1;
1 þ 0:1 rand
for 1otp5:
ð38Þ
Fig. 1 shows the response of the system along with that of the observer when there is a fault described by Eq. (37) in the system. Fig. 2 illustrates the simulation results of actuator fault described by Eq. (37). Figs. 3 and 4 depict corresponding results of actuator fault described by Eq. (38). It can be seen that good estimation of fault can be achieved despite the disturbance (noise) in the system.
5. Conclusion The fault diagnostic approach in this paper uses an adaptive observer to detect and identify actuator faults with satisfactory accuracy for a class of bilinear systems with uncertainty. A semiactive suspension example is given to illustrate the proposed
2 0.1
1
output y2
output y1
1.5
0.5 0 −0.5 −1
0
−0.1
0
2
4
6
0
2
2
0
0
6
4
6
0.2
1
−1
4 time (sec)
estimation error of z1
z1 (solid) and estimates(dashed)
time (sec)
2
4 time (sec)
6
0 −0.2 −0.4 −0.6
0
2 time (sec)
Fig. 3. State estimation under fault occurrence.
B. Jiang, J.L. Wang / Journal of the Franklin Institute 339 (2002) 361–374
373
1.2
1
estimation of actuator fault
0.8
0.6
0.4
0.2
0
−0.2
0
0.5
1
1.5
2
2.5 time (sec)
3
3.5
4
4.5
5
Fig. 4. Fault identification and the estimation.
scheme. Using robust or adaptive observer, FDI for more general nonlinear systems with uncertainties will be investigated in future.
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Actuator fault diagnosis for a class of bilinear systems with uncertainty Bin Jianga,*, Jian Liang Wangb a
LAIL-CNRS, Bat. P2, UPRESA 8021, Universit!e des Sciences et Technologies de Lille, 59655 Villeneuve d’Ascq, Cedex, France b School of EEE, Nanyang Technological University, Nanyang Avenue, Block S2, Singapore 639798, Singapore
Received 10 May 2000; received in revised form 30 May 2001; accepted 27 November 2001
Abstract In this paper, the actuator fault diagnosis problem for a class of bilinear systems with uncertainty is discussed. The system is transformed into two different subsystems. One is not affected by actuator fault, so an adaptive observer can be designed such that, under certain conditions, the observer error dynamics is stable. The other whose states can be measured is affected by the faults. The observation scheme is then used for model-based fault diagnosis. Finally, an example of a semiactive suspension system is used to illustrate the applicability of the proposed method. r 2002 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved. Keywords: Fault detection and isolation (FDI); Bilinear systems; Adaptive observer; Uncertainty
1. Introduction The process of detection and isolation of system faults has been of considerable interest during the last two decades, as can be seen from the survey paper [1–3] and the books [4–6]. Research is still under way into the development of more effective solutions for fault detection and isolation (FDI) in automatic control systems. Until now, most research work on FDI tends to focus on linear systems, though some
*Corresponding author. Tel./fax: +33-3-20-43-4743. E-mail address: [email protected] (B. Jiang). 0016-0032/02/$22.00 r 2002 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 6 - 0 0 3 2 ( 0 1 ) 0 0 0 5 1 - 5
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initial results for nonlinear systems are available, for example, see [7,8] based on nonlinear observers and [9] based on parity space approach. Bilinear systems are a special class of nonlinear systems in which the control appears in both additive and multiplicative terms. Bilinear systems arise in a variety of physical situations, and many physical processes are of this type, for example nuclear reactors, field controlled d.c. motors, heat exchangers and many biomedical processes such as the immune system are known to be described by bilinear systems [10,11]. In [12,13] the unknown input fault detection observer approach is extended to bilinear systems. Recently, Yang and Saif [14] discussed FDI for a class of bilinear systems based on reduced order observer, and the estimation error of the observer discussed there is dependent on inputs. However, no system uncertainties have been considered in that paper. In practice, since an accurate mathematical model of a physical process is not available, there is a mismatch between the actual process and its mathematical model even if there is no fault in the process. Such mismatch causes fundamental methodological difficulties in FDI applications and is a source of false alarms which can corrupt the performance of the FDI system. Furthermore, in some cases such as unstructured uncertainties in the system, perfect decoupling is not possible, see for example, [15]. In this paper, we consider the robust FDI problem for a class of bilinear systems with uncertainties. At first, the system is transformed into two subsystems. The 1st subsystem is decoupled from the actuator fault and the other is affected by the fault, but its states can be measured directly. As a generalization of the observer design approach in [16], a stable adaptive observer is proposed for the 1st subsystem. By using estimated states and unknown parameters, we can approximate the fault from of the 2nd subsystem. When the fault signal is greater than certain threshold, the fault can be detected or even isolated. This paper is organized as follows. Section 2 gives conditions for the existence of a transformation that transforms the original uncertain system into a bilinear canonical form. Then an adaptive observer is given based on the transformed system. In Section 3, actuator FDI are discussed. An application example is included in Section 4, followed by some concluding remarks in Section 5.
2. Adaptive observer design Consider the following system with uncertainty: xðtÞ ’ ¼ AxðtÞ þ
m X
Ai ui ðtÞxðtÞ þ BuðtÞ þ Efa ðtÞ þ DxðtÞ;
ð1Þ
i¼1
yðtÞ ¼ CxðtÞ ¼ ½O
IxðtÞ;
ð2Þ
where the state is xARn ; the input is u ¼ ½u1 ; y; um T ARm ; and the output is yARr : The fault is modelled as an additional disturbance input fa ARq ; with qor; E is of full
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column rank, and xðtÞ denotes unknown parameter vector of the system which is assumed to be uniformly bounded. Note that in Eq. (2), the special form for the matrix C is not a restriction. Since as long as C is of full row rank, there exists a similarity transformation that can bring the output equation in that desired form.
Assumption 1. rankðCEÞ ¼ q: Remark 1. Assumption 1 is required for fault identification, as in [14,15]. Partition x as x ¼ ½x1
x3 T
x2
with x1 ARnr ; x2 ARrq ; and x3 ARq : Then the system described by Eqs. (1) and (2) can be written as 2 3 2 32 3 2 i 3 2 3 x’ 1 A11 A12 A13 x1 A11 Ai12 Ai13 x1 m 6 7 6 76 7 X6 i 7 i i 7 6 4 x’ 2 5 ¼ 4 A21 A22 A23 54 x2 5 þ 4 A21 A22 A23 5ui 4 x2 5 i¼1 x’ 3 A31 A32 A33 x3 Ai31 Ai32 Ai33 x3 2 3 2 3 2 3 E1 B1 D1 6 7 6 7 6 7 ð3Þ þ 4 B2 5u þ 4 E2 5fa þ 4 D2 5x; B3 " y ¼ Cx ¼
E3
D3
0 IðrqÞðrqÞ
0
0
Iq
0
# ð4Þ
x:
As a generalization of the definition of bilinear canonical form (BCF) for fault-free bilinear systems in [16], the following definition is made. Definition. The generalized bilinear canonical form (GBCF) of the bilinear system in Eqs. (3) and (4) is described as 2 2 3 2 32 3 3 i i 3 2 0 A% 12 A% 13 z’1 A% 11 A% 12 A% 13 z1 z1 m 7 6 7 6 % 76 7 X6 7 i i 7 6 6 A% i 4 z’2 5 ¼ 4 A21 A% 22 A% 23 54 z2 5 þ 4 21 A% 22 A% 23 5ui 4 z2 5 i¼1 i i i z’3 z3 z3 A% 31 A% 32 A% 33 A% 31 A% 32 A% 33 2 3 2 3 2 3 B% 1 D% 1 0 6 % 7 6 % 7 6 % 7 ð5Þ þ 4 B2 5u þ 4 E2 5fa þ 4 D2 5xðtÞ; B% 3
" y¼
0 IðrqÞðrqÞ 0 0
E% 3 #
0 W % z ¼ Cz: Iq
D% 3
ð6Þ
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Define " T¼
T1
# ;
T2
E1 ; T1 ¼ ½A111 A211 ? Am 11 "" #" # " #" ## A121 A221 Am E2 21 T2 ¼ ? m : 1 2 A31 A31 A31 E3
Assumption 2. rankðCT Þ ¼ rankðT2 Þ ¼ rankðTÞ: Remark 2. Assumption 2 characterizes the existence condition under which a bilinear system can be transformed into GBCF described by Eqs. (5) and (6). Some practical systems indeed satisfy this assumption, see for example, Section 4. Lemma 1. Suppose that Assumption 2 holds. Then there exists F ARðnrÞr such that the bilinear system described by Eqs. (3) and (4) can be transformed into GBCF by using the similarity transformation z ¼ Nx with " N¼
IðnrÞðnrÞ
F
0
Irr
# ;
FT2 ¼ T1 :
Proof. The proof of Lemma 1 is similar to that in [16] and thus is omitted.
ð7Þ
&
Remark 3. Lemma 1 has extended the obtained results in [16] for fault-free system (i.e., E ¼ 0 in Eq. (1)) to that in the faulty case. % it is % ¼ q: From the structure of matrix C; According to Assumption 1, rankðC% EÞ easy to show that "
E% 2 rank E% 3
# ¼ q:
ð8Þ
% From Eq. (8), EE% 23 is of full column rank. Without loss of generality, it can be assumed that E% 3 is nonsingular. Let 2
I 6 S ¼ 40 0
3 0 0 7 I E2 E31 5: 0
I
ð9Þ
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For i ¼ 1; y; m; j ¼ 1; 2; 3; define W 1 Hj ¼ A% 2j E% 2 E% 3 A% 3j ; W 1 R ¼ B% 2 E% 2 E% 3 B% 3 ;
W i 1 i Gij ¼ A% 2j E% 2 E% 3 A% 3j ; W M ¼ D% 2 E2 E31 D% 3 :
By pre-multiplying Eq. (9) into Eq. (5), we have that 2 3 2 3 2 32 3 i i z’1 0 A% 12 A% 13 A% 11 A% 12 A% 13 z1 7 6 7 6 76 7 Xm 6 6 Gi1 Gi2 Gi3 7 4 y’ 1 E% 2 E% 1 ’ 2 5 ¼ 4 H 1 H 2 H 3 5 4 y1 5 þ 5 3 y i¼1 4 i i i A% 31 A% 32 A% 33 y2 y’ 2 A% 31 A% 32 A% 33 2 3 2 3 2 3 2 3 B% 1 0 z1 D% 1 6 7 6 7 6 7 6 7 ui 4 y1 5 þ 4 R 5u þ 4 0 5fa þ 4 M 5xðtÞ: ð10Þ E% 3 B% 3 D% 3 y2 It is clear that in Eq. (10), the actuator faults enter only through the third block row, while the other two block rows are not affected by any faults. The first and second block rows of Eq. (10) can be rewritten as Xm z’1 ðtÞ ¼ A% 11 z1 ðtÞ þ sðtÞ þ D% 1 xðtÞ; vðtÞ ¼ H1 z1 þ Gi1 ui z1 þ MxðtÞ; ð11Þ i¼1
where sðtÞ ¼ A% 12 y1 þ A% 13 y2 þ
Xm i¼1
i i ðA% 12 ui y1 þ A% 13 ui y2 Þ þ B% 1 u;
vðtÞ ¼ y’ 1 E% 2 E% 3 y’ 2 H2 y1 H3 y2 1
Xm i¼1
ðGi2 ui y1 þ G13 ui y2 Þ Ru:
ð12Þ ð13Þ
Since the dynamical system represented in Eq. (11) is driven by known input sðtÞ and uncertain term D% 1 xðtÞ; its state can be estimated by using a adaptive observer # þ K½vðtÞ v#ðtÞ; z’#1 ðtÞ ¼ A% 11 z#1 ðtÞ þ sðtÞ þ D% 1 xðtÞ Xm # Gi1 ui z#1 ðtÞ M xðtÞ; ð14Þ v#ðtÞ ¼ H1 z#1 ðtÞ i¼1 # is the estimation of xðtÞ which will be designed where K is the observer’s gain, xðtÞ later. Assumption(3. The set O; defined by ) K2 Gi1 ¼ 0; i ¼ 1; 2; y; m; and O ¼ K2 : the pair ðA% 11 ; K2 H1 Þ is completely observable
ð15Þ
is not empty. Remark 4. It should be noted that in practice, matrices Gi1 ði ¼ 1; y; mÞ may be sparse and dimension of A% 11 is not high such that K2 may be found. As a result, the above assumption is not expected to be too restrictive. Under Assumption 3, the observer gain in Eq. (14) can be written as K ¼ K1 K2 ; where K2 AO and K1 is selected so that all the eigenvalues of ðA% 11 K1 K2 H1 Þ are
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placed in the left-hand side of the complex plane. Substituting Eqs. (12), (13) and (15) into Eq. (14) yields
1 z’#1 ¼ ðA% 11 KH1 Þ#z1 þ Kðy’ 1 E% 2 E% 3 y’ 2 Þ þ ðB% 1 KRÞu þ ðA% 12 KH2 Þ i Xm % i12 ui KGi2 ui Þ y1 þ ð A i¼1 h i Xm % i13 ui KGi3 ui Þ y2 þ ðA% 13 KH3 Þ þ ð A i¼1 # þ ðD% 1 KMÞxðtÞ:
ð16Þ
Note that the above observer uses the derivative of the outputs which is not available for direct measurement. In order to eliminate the need for differentiating the output, a new variable W is defined as follows: W ¼ z#1 Kðy1 E% 2 E% 3 y2 Þ: 1
ð17Þ
Then Eq. (16) can be expressed in the following form: ’ ¼ ðA% 11 KH1 ÞW þ ðB% 1 KRÞu W h i Xm % i12 ui KGi2 ui Þ þ KðA% 11 KH1 Þ y1 ð A þ ðA% 12 KH2 Þ þ i¼1 h i Xm i 1 þ ðA% 13 KH3 Þ þ ðA% 13 ui KGi3 ui Þ KðA% 11 KH1 ÞE% 2 E% 3 y2 i¼1
# þ ðD% 1 KMÞxðtÞ:
ð18Þ
From Eqs. (11) and (14), the error dynamics of z*1 ¼ z1 z#1 is given by * z’*1 ¼ ðA% 11 KH1 Þ*z1 þ ðD% 1 KMÞxðtÞ;
ð19Þ
# is the estimation error of the unknown parameter vector. * ¼ xðtÞ xðtÞ where xðtÞ Define v*ðtÞ ¼ vðtÞ v#ðtÞ; considering Eqs. (11), (14) and (19), we can obtain the following input–output expression of the error equation: * v*ðtÞ ¼ QðsÞxðtÞ;
ð20Þ
where % ðA% 11 KH1 Þ1 ðD% 1 KMÞ; QðsÞ ¼ M þ H½sI Xm H% ¼ H1 þ Gi1 ui : i¼1
ð21Þ
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Using the augmented error technique in Narendra (1989), it can be shown that # xðtÞQðsÞ1ðtÞ # v*ðtÞ ¼ x* ðtÞZðtÞ þ QðsÞxðtÞ þ dðtÞ; T
ð22Þ
where ZðtÞ ¼ QðsÞ1ðtÞ; dðtÞ decays exponentially and can be neglected. We can define # xðtÞQðsÞ1ðtÞ # eðtÞ ¼ v*ðtÞ ½QðsÞxðtÞ
ð23Þ
as the augmented error signal. Therefore, it can be obtained that eðtÞ ¼ x* ðtÞZðtÞ: T
ð24Þ
Eq. (24) is in the standard form of the error equations widely used in adaptive control theory (Narendra, 1989). The adaptive updating laws used in adaptive * control can be directly applied to tune xðtÞ: This can be summarized in the following theorem. Theorem 1. Consider the bilinear system ð1Þ–ð2Þ: If Assumptions 1–3 hold, then the adaptive observer given by Eq. ð14Þ and the following parameter update law: ’* ¼ g xðtÞ
eðtÞZðtÞ ; 1 þ ZT ðtÞZðtÞ
ð25Þ
* * ¼ 0 and realizes a bounded xðtÞ and z*1 ðtÞAL2 : Furthermore, limt-N xðtÞ limt-N z*1 ðtÞ ¼ 0 under persistent excitation, where g > 0 is a weighting matrix. Proof. Using the same method as in Narendra ð1989Þ; we can prove that the * adaptive observer (14) and parameter update law (25) can result in a bounded xðtÞ 2 and z*1 ðtÞAL : The persistent excitation means that there exist two positive constants s and tT such that for all t; the following inequality holds: Z tþtT T % H% ðuðsÞÞHðuðsÞÞ dsXsIðnrÞ ; ð26Þ t
where H% is given by Eq. (21). Under the persistent excitation, an accurate estimation of xðtÞ can be obtained, which is regarded as the parameter in standard format of adaptive control, i.e., * ¼ 0: limt-N xðtÞ
ð27Þ
From Eqs. (19) and (27) and the fact that ðA% 11 KH1 Þ is stable, one can conclude that limt-N z*1 ðtÞ ¼ 0: This completes the proof. &
3. Actuator FDI In this section, a simple approach for detecting and isolating the actuator fault is proposed to estimate the magnitude of the actuator fault fa :
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The third row of Eq. (10) is written as Xm i i i ðA% 31 ui z1 þ A% 32 ui y1 þ A% 33 ui y2 Þ y’ 2 ¼ A% 31 z1 þ A% 32 y1 þ A% 33 y2 þ i¼1 þ B% 3 u þ E% 3 fa þ D% 3 xðtÞ:
ð28Þ
Assuming that no fault occurs during the initial transient of the observer, using the estimation z#1 and x# for z1 and x; we can estimate the actuator fault as 1
f#a ðtÞ ¼ E% 3 y’ 2 ðtÞ A% 32 y1 ðtÞ A% 33 y2 ðtÞ A% 31 z#1 ðtÞ i Xm i i i ðA% 31 ui ðtÞ#z1 ðtÞ þ A% 32 ui ðtÞy1 ðtÞ þ A% 33 ui ðtÞy2 ðtÞÞ B% 3 uðtÞ D% 3 x# : i¼1
ð29Þ The estimate error of the actuator fault is given by Xm 1 % i31 ui z*1 E% 1 % * A f*a ðtÞ ¼ fa ðtÞ f#a ðtÞ ¼ E% 3 A% 31 þ 3 D3 x: i¼1
ð30Þ
Note that faults are not involved in estimation of z1 and x; according to Theorem 1, for given e1 > 0; e2 > 0; there exists t0 such that for t > t0 jj*z1 ðtÞjjpe1 ;
* jjxðtÞjjpe 2:
ð31Þ
Therefore Xm 1 % i31 ui jje1 þ jjE% 1 % A jjf*a ðkÞjjpjjE% 3 A% 31 þ 3 D3 jje2 : i¼1 The threshold value is defined as Xm 1 % i31 ui e1 þ jjE% 1 % A Tr ¼ E% 3 A% 31 þ 3 D3 jje2 : i¼1
ð32Þ
ð33Þ
Then the fault detection can be carried out as follows: The alarm is on; The alarm isoff;
jjf#a jj > Tr ; jjf# jjpTr : a
ð34Þ
Remark 5. Compared with existing results on FDI for bilinear systems, our proposed method has two advantages. One is that it can detect not only occurrence but also amplitudes of faults. The other is that it can be used for systems with uncertainties that cannot be decoupled from faults or residuals. Remark 6. In Eq. (29), calculation of derivative of output is required to estimate the fault. Because of the presence of noise in practice, it is not easy to compute the signal derivative. Evaluating derivative of output from noisy signals can be done using
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specific algorithms such as polynomial approximation-based scheme, which has been extensively investigated, for example in [17].
4. An illustrative example To illustrate the proposed method, the example of a two degree of freedom (DOF) quarter car model of vehicle suspension given in [11] is examined. Such a bilinear system is described as follows: xðtÞ ’ ¼ AxðtÞ þ A1 uðtÞxðtÞ þ Efa ðtÞ þ DxðtÞ;
ð35Þ
yðtÞ ¼ CxðtÞ;
ð36Þ
where x ¼ ½x1 x2 x3 x4 T is a state in which x1 is the suspension deflection, x2 is the quarter car body velocity, x3 is the wheel displacement from the road, x4 is the wheel velocity, and u the variable damping. The measurement outputs are y ¼ ½y1 y2 y3 T ; where y1 is the suspension defection, y2 is the wheel velocity, and y3 is the body displacement with respect to the road. fa denotes the actuator fault and x stands for disturbance which affects the vehicle body velocity. The system matrices are given by 2
6 3:6 6 A¼6 4 0
0
1
0:5 0
0 0
0:5 7 7 7; 1 5
5
3600
360 2
0
3
6 0:1 7 6 7 E¼6 7; 4 0 5
B ¼ 0;
3
1
0
1
5
2
0
60 6 A1 ¼ 6 40 0
2 3 0 617 6 7 D ¼ 6 7; 405 0
2
1
6 C ¼ 40 1
0
0
0:004 0 0 0 0:04
0
0
3
0 0
0 0
0
3
0:004 0
7 7 7; 5
0:04
7 0 1 5: 1 0
Since the matrix C in the above system description is not in the desired form, a transformation is needed to get the system in the proper form. The similarity transformation matrices are chosen as follows: 2
3
0
1
0 0
61 6 N% ¼ 6 40
0 0
0 07 7 7: 0 15
1
0
1 0
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370
As a result, we get the following system: 2 2 3 0:004 0:5 36 0:5 0 6 6 1 7 0 1 0 7 6 0 6 x’ ¼ 6 7x þ 6 4 0:04 4 5 5 3960 5 3600 1 2
0 2 3 0 1 6 0 7 607 6 7 6 7 þ6 7fa þ 6 7x; 4 0 5 405 3
0
1 2
0
0
0:004
0 0
0 07 7 7ux 0:04 0 5
0
0
0
3
0
0
0
0 6 y ¼ 40
1 0
0 1
3 0 7 0 5x:
0
0
0
1
Using the following similarity transformation: 2 3 1 0 0:1 0 60 1 0 07 6 7 N¼6 7; 40 0 1 05 0 0
0
1
the GBCF is obtained as 2
0 61 6 z’ ¼ 6 45 1
2
360 0 3960 0
0 1:1 5:5 0:1
0 6 y ¼ 40
1 0
0 1
3 0 7 0 5z:
0
0
0
1
2 3 0 360 6 0 0 7 6 7 7z þ 6 4 0:04 3600 5 0 0
0 0 0 0 0 0:004 0 0
3 2 3 2 3 0 0 1 6 0 7 607 07 7 6 7 6 7 7fa þ 6 7x; 7uz þ 6 4 1 5 405 05 0
0
0
From the above model, one can easily get " # " # 1 0 A% 11 ¼ 0; H1 ¼ ; G11 ¼ : 1 0 The observer gain K can be chosen as K ¼ K1 K2 ;
K2 ¼ ½1
0;
K1 > 0:
Therefore, Assumptions 1–3 and the persistent excitation condition (26) in this paper all hold.
B. Jiang, J.L. Wang / Journal of the Franklin Institute 339 (2002) 361–374 0.4
371
0.06 0.04 output y2
y1
0.2 0 −0.2 −0.4
0.02 0 −0.02 −0.04
0
2
4
−0.06
6
0
2
0.6
6
4
6
0.2
0.4 0.2 0 −0.2 −0.4
4 time (sec)
estimation error of z1
z1 (solid) and estimates(dashed)
time (sec)
0
2
4
6
0 −0.2 −0.4 −0.6
0
2
time (sec)
time (sec)
Fig. 1. State estimation under fault occurrence.
0.25
estimation of actuator fault
0.2
0.15
0.1
0.05
0
−0.05 0
0.5
1
1.5
2
2.5 time (sec)
3
3.5
Fig. 2. Fault identification and the estimation.
4
4.5
5
B. Jiang, J.L. Wang / Journal of the Franklin Institute 339 (2002) 361–374
372
In the simulation, x is a uniformly distributed random number on the interval ð0; 0:5Þ; and the gain K1 is selected to be 25, the g in Theorem 1 is 10. The actuator fault is considered, respectively, as follows: ( 0 for 0ptp1; fa ¼ ð37Þ 0:2 for 1otp5; ( fa ¼
0
for 0ptp1;
1 þ 0:1 rand
for 1otp5:
ð38Þ
Fig. 1 shows the response of the system along with that of the observer when there is a fault described by Eq. (37) in the system. Fig. 2 illustrates the simulation results of actuator fault described by Eq. (37). Figs. 3 and 4 depict corresponding results of actuator fault described by Eq. (38). It can be seen that good estimation of fault can be achieved despite the disturbance (noise) in the system.
5. Conclusion The fault diagnostic approach in this paper uses an adaptive observer to detect and identify actuator faults with satisfactory accuracy for a class of bilinear systems with uncertainty. A semiactive suspension example is given to illustrate the proposed
2 0.1
1
output y2
output y1
1.5
0.5 0 −0.5 −1
0
−0.1
0
2
4
6
0
2
2
0
0
6
4
6
0.2
1
−1
4 time (sec)
estimation error of z1
z1 (solid) and estimates(dashed)
time (sec)
2
4 time (sec)
6
0 −0.2 −0.4 −0.6
0
2 time (sec)
Fig. 3. State estimation under fault occurrence.
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373
1.2
1
estimation of actuator fault
0.8
0.6
0.4
0.2
0
−0.2
0
0.5
1
1.5
2
2.5 time (sec)
3
3.5
4
4.5
5
Fig. 4. Fault identification and the estimation.
scheme. Using robust or adaptive observer, FDI for more general nonlinear systems with uncertainties will be investigated in future.
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