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Parameter fault detection and estimation of a class of nonlinear systems using observers
ARTICLE IN PRESS
Journal of the Franklin Institute 342 (2005) 725–736 www.elsevier.com/locate/jfranklin
Parameter fault detection and estimation of a class of nonlinear systems using observers$ Bin Jianga,, Fahmida N. Chowdhuryb a
College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, 29 YuDao Street, Nanjing, 210016, PR China b Department of ECE, P.O. Box 43890, University of Louisiana at Lafayette, Lafayette, LA 70504-3890, USA
Abstract The focus of this paper is on the detection and estimation of parameter faults in nonlinear systems with nonlinear fault distribution functions. The novelty of this contribution is that it handles the nonlinear fault distribution function; since such a fault distribution function depends not only on the inputs and outputs of the system but also on unmeasured states, it causes additional complexity in fault estimation. The proposed detection and estimation tool is based on the adaptive observer technique. Under the Lipschitz condition, a fault detection observer and adaptive diagnosis observer are proposed. Then, relaxation of the Lipschitz requirement is proposed and the necessary modification to the diagnostic tool is presented. Finally, the example of a one-wheel model with lumped friction is presented to illustrate the applicability of the proposed diagnosis method. r 2005 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Fault detection; Fault estimation; Adaptive observer; Nonlinear systems; Lipschitz condition
$ Partially supported by NASA under award NCC5-573, contract NASA/LEQSF(2001-04)-1, Scientific Research fund of NUAA (S0456-031), ‘‘Liu Da Ren Cai Gao Feng’’ scheme of Jiangsu Province (China) and key project of Natural Science Foundation of China (60234010). Corresponding author. E-mail addresses: [email protected] (B. Jiang), [email protected] (F.N. Chowdhury).
0016-0032/$30.00 r 2005 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2005.04.007
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1. Introduction Fault detection and isolation (FDI) algorithms and their applications to a wide range of industrial processes have been the subject of intensive research over the past two decades. Fruitful results can be found in several survey papers [1–3] and books [4–6]. In general, there are several kinds of schemes for model-based FDI: observer based, [7–9], parity space based, [10,11], eigenstructure assignment based [12] and parameter-identification based [13,14]. In some cases, fault accommodation strategies are needed, i.e. the control algorithm must be adapted based on FDI to go on controlling the faulty system. Aircraft flight control systems [15] are good examples for applications of fault accommodation/active reliable control. In such cases, it is important to carry out fault estimation in addition to detection. Some recent results based on adaptive or robust observers can be found in [16–20]; see [21,22] for methods using learning approach. In this paper, detection and estimation of parameter faults for nonlinear systems are investigated using adaptive observers. Compared with [21,23] and our previous work presented in [19], the novelty of this contribution is that we consider difficult cases where the nonlinear fault distribution function depends not only on the inputs and outputs but also on unmeasured states. Furthermore, in order to expand the applicability of the proposed method, we propose a relaxation of the Lipschitz condition which is typically required (e.g., [24,25,19]) in FDI literature. This paper is organized as follows. Section 2 describes a class of nonlinear systems with parameter fault and introduces some preliminaries. Under Lipschitz condition, fault detection observer is proposed in Section 3. An adaptive diagnostic rule is designed in Section 4. Discussion on relaxation of Lipschitz requirement is presented in Section 5. Results of applying our technique to a one-wheel model with lumped friction are presented in Section 6, followed by some concluding remarks in Section 7.
2. Nonlinear dynamic model Consider the following nonlinear system _ ¼ AxðtÞ þ gðuðtÞ; yðtÞÞ þ ByðtÞf ðuðtÞ; yðtÞ; xðtÞÞ, xðtÞ
(1)
yðtÞ ¼ CxðtÞ,
(2)
where the state is x 2 Rn , the input is u 2 Rm , and the output is y 2 Rr . The pair ðA; CÞ is observable. The nonlinear term gðuðtÞ; yðtÞÞ depends on uðtÞ and yðtÞ which are directly available. The f ðuðtÞ; yðtÞ; xðtÞÞ 2 Rr is a nonlinear vector function of uðtÞ; yðtÞ and xðtÞ. The yðtÞ 2 R is a parameter which changes unexpectedly when a fault occurs. It is assumed that yðtÞ is bounded, i.e., jyðtÞjpy0 . In the fault-free case, we have yðtÞ ¼ yH , where yH is a known scalar (subscript H stands for healthy case). Remark 1. It should be noted that there are quite a few practical systems which can be described by this particular form of nonlinear model. For example, see one-wheel
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model with lumped friction in [26], flexible joint robotic model in [27], and series DC motor in [28].
3. Detection observer design Prior to fault detection observer design, the following assumptions are made: Assumption 1. There exists a known positive constant L0 such that for any norm bounded x1 ðtÞ; x2 ðtÞ 2 Rn , the following inequality holds: kf ðuðtÞ; yðtÞ; x1 ðtÞÞ f ðuðtÞ; yðtÞ; x2 ðtÞÞkpL0 kx1 ðtÞ x2 ðtÞk.
(3)
Remark 2. Assumption 1 is the known Lipschitz condition, which is typically required in the literature on FDI for nonlinear systems, e.g., [24,25,19]. The removal of the Lipschitz condition for detection observer design will be discussed later in the paper. Assumption 2. C½sI ðA KCÞ 1 B is strictly positive real (SPR), where K 2 Rn r is chosen such that A KC is stable. Remark 3. The SPR requirement in the above assumption is equivalent to the following: For a given positive definite matrix Q40 2 Rn n , there exists matrices P ¼ PT 40 2 Rn n and a scalar R such that ðA KCÞT P þ PðA KCÞ ¼ Q,
(4)
PB ¼ C T R.
(5)
To detect the fault, the following observer is constructed: _^ ¼ AxðtÞ ^ þ gðuðtÞ; yðtÞÞ þ ByH f ðuðtÞ; yðtÞ; xðtÞÞ ^ ^ xðtÞ þ KðyðtÞ yðtÞÞ,
(6)
^ ¼ C xðtÞ, ^ yðtÞ
(7)
n
^ 2 R is the state estimate. Since it has been assumed that the pair ðA; CÞ is where xðtÞ observable, the observer gain matrix K can be selected such that ðA KCÞ is a stable matrix. We define ^ ex ðtÞ ¼ xðtÞ xðtÞ;
^ ey ðtÞ ¼ yðtÞ yðtÞ.
(8)
Then the observation error and output error equations are given by ^ e_x ðtÞ ¼ ðA KCÞex ðtÞ þ B½yðtÞf ðuðtÞ; yðtÞ; xðtÞÞ yH f ðuðtÞ; yðtÞ; xðtÞÞ , ey ðtÞ ¼ Cex ðtÞ.
(9) (10)
The convergence of the above observer is guaranteed by the following theorem: Theorem 1. Consider the system described by (1), (2) and its observer described by (6), (7). Under Assumptions 1 and 2, the observer is asymptotically convergent when no faults occurs ðyðtÞ ¼ yH Þ, i.e., limt!1 ex ðtÞ ¼ 0 and thus limt!1 ey ðtÞ ¼ 0.
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Proof. Consider the following Lyapunov function V ðtÞ ¼ eTx ðtÞPex ðtÞ,
(11)
where P is given by (4) and (5), Q is chosen such that r1 9lmin ðQÞ 2kCk jRjyH L0 40. Along the trajectory of the fault-free system (9), its derivative with respect to time is V_ ðtÞ ¼ eTx ðtÞ½PðA KCÞ þ ðA KCÞT P ex ðtÞ ^ þ 2eTx ðtÞPByH ½f ðuðtÞ; yðtÞ; xðtÞÞ f ðuðtÞ; yðtÞ; xðtÞÞ .
ð12Þ
From (3), (4), and (5), one can further obtain that V_ ðtÞp eTx ðtÞQex ðtÞ þ 2key k jRjyH L0 kex k p r1 kex k2 p0.
ð13Þ
Thus, limt!1 ex ðtÞ ¼ 0 and limt!1 ey ðtÞ ¼ 0. This completes the proof.
&
According to Theorem 1, after the convergence of the observer, the fault detection can be carried out as follows: ey ðtÞ ¼ 0
no fault occurred,
ey ðtÞa0 fault has occurred.
ð14Þ
The observer given by (6) and (7) is referred to as a detection observer for the system described by (1) and (2) under Assumptions 1 and 2.
4. Fault estimation using an adaptive observer To estimate the fault after the alarm (14) has been generated, the following observer is constructed: _^ ¼ AxðtÞ ^ ^ þ gðuðtÞ; yðtÞÞ þ ByðtÞf ^ ^ xðtÞ ðuðtÞ; yðtÞ; xðtÞÞ þ KðyðtÞ yðtÞÞ,
(15)
^ ¼ C xðtÞ, ^ yðtÞ
(16)
^ is an estimate of yðtÞ. The value ^ 2 R is the observer state vector and yðtÞ where xðtÞ ^ is set to yH until a fault is detected. It is assumed that after a fault occurs, of yðtÞ yðtÞ ¼ yf ¼ constantayH ; jyf jpy0 . We denote n
^ ex ðtÞ ¼ xðtÞ xðtÞ;
^ ey ðtÞ ¼ yðtÞ yðtÞ;
^ ey ðtÞ ¼ yf yðtÞ,
(17)
then it can be obtained that ^ ^ e_x ðtÞ ¼ ðA KCÞex ðtÞ þ B½yf f ðuðtÞ; yðtÞxðtÞÞ yðtÞf ðuðtÞ; yðtÞ; xðtÞÞ ,
(18)
ey ðtÞ ¼ Cex ðtÞ.
(19)
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^ As a result, the purpose of fault diagnosis is to find a diagnostic algorithm for yðtÞ such that lim ex ðtÞ ¼ 0;
t!1
lim ey ðtÞ ¼ 0.
t!1
(20)
The following theorem produces a convergent adaptive diagnostic algorithm for estimating the faulty parameter yf . Theorem 2. Under the Assumptions 1 and 2, the observer described by (15) and (16) and following diagnostic algorithm ^ dyðtÞ ^ ¼ Gf T ðuðtÞ; yðtÞ; xðtÞÞRe (21) y ðtÞ dt can realize limt!1 ex ðtÞ ¼ 0 and a bounded ey ðtÞ 2 L2 . Furthermore, limt!1 ey ðtÞ ¼ 0 under a persistent excitation, where R is given by (5), G40 is a weighting scalar. Proof. Consider the following Lyapunov function V ðtÞ ¼ eTx ðtÞPex ðtÞ þ G1 e2y ðtÞ.
(22)
From (18) and (21), its derivative with respect to time is ^ V_ ðtÞ ¼ eTx ðtÞ½PðA KCÞ þ ðA KCÞT P ex ðtÞ 2ey ðtÞf T ðuðtÞ; yðtÞ; xðtÞÞRe y ðtÞ T ^ ^ þ 2e ðtÞPB½yf f ðuðtÞ; yðtÞ; xðtÞÞ yðtÞf ðuðtÞ; yðtÞ; xðtÞÞ . ð23Þ x
According to Assumptions 1 and 2, one can further obtain that ^ V_ ðtÞp eTx ðtÞQex ðtÞ 2ey ðtÞf T ðuðtÞ; yðtÞ; xðtÞÞRe y ðtÞ ^ þ yf ½f ðu; y; xÞ f ðu; y; xÞ g ^ þ 2eTx C T Rfey f ðu; y; xÞ p r2 kex k2 p0,
ð24Þ
where r2 9lmin ðQÞ 2kCk jRjy0 L0 , jyf jpy0 , Q40 is chosen such that r2 40. Inequality (24) implies the stability of the origin ex ¼ 0, ey ¼ 0 and the uniform boundedness of ex and ey with ex 2 L2 . On the other hand, from (18), e_x is uniformly bounded as well. According to Barbalat’s Lemma (see [29]), one can get lim ex ðtÞ ¼ 0.
t!1
(25)
The persistent excitation condition means that there exist two positive constants s and t0 such that for all t the following inequality holds: Z tþt0 f T ðyðsÞ; uðsÞ; xðsÞÞBT Bf T ðyðsÞ; uðsÞ; xðsÞÞ dsXsI. (26) t
Then from (18), (21), (25) and (26), one can conclude that limt!1 ey ðtÞ ¼ 0. This completes the proof. & Remark 4. The choice/design of K; Q; P and R is summarized as follows: (i) Since ðA; CÞ is observable, the observer gain matrix K is selected such that ðA KCÞ is stable and C½sI ðA KCÞ 1 B is strictly positive real (SPR).
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(ii) In Eq. (4), Q40 is chosen such that the above defined r2 40. (iii) The existence of solutions P and R to Eqs. (4) and (5) can be guaranteed by Remark 2.
5. Relaxing the Lipschitz requirement In the above sections, fault detection and estimation observers have been designed under the Lipschitz condition which does not always hold for practical systems. In this section, we replace the Lipschitz requirement by the following, weaker condition: Assumption 10 . There exists a positive function qðÞ and a positive scalar q0 such that the following inequality holds: kf ðuðtÞ; yðtÞ; xðtÞÞkpqðkxkÞpq0 .
(27)
The focus of this section is to design an adaptive estimation observer under Assumption 10 instead of Assumption 1. The detection observer can be designed similarly and is omitted. To diagnose the fault after detection, the following observer is constructed: _^ ¼ AxðtÞ ^ ^ þ gðuðtÞ; yðtÞÞ þ ByðtÞf ^ ^ xðtÞ ðuðtÞ; yðtÞ; xðtÞÞ þ KðyðtÞ yðtÞÞ ^ þ BsgnðRÞ½y0 ðq0 þ qðkxkÞÞsgnðyðtÞ yðtÞÞ ,
ð28Þ
^ ¼ C xðtÞ, ^ yðtÞ
(29) ^ ^ 2 R is the observer state vector and yðtÞ is an estimate of yðtÞ, which is where xðtÞ obtained by the diagnostic algorithm (21). n
Remark 5. From (28), it can be seen that the Lipschitz condition can be relaxed at the cost of an additional term in the estimation observer design. Using the same notations of ex ðtÞ; ey ðtÞ and ey ðtÞ as in Section 4, then the observation error and output error equations can be described by ^ ^ ðuðtÞ; yðtÞ; xðtÞÞ
e_x ðtÞ ¼ ðA KCÞex ðtÞ þ B½yf f ðuðtÞ; yðtÞ; xðtÞÞ yðtÞf ^ BsgnðRÞ½y0 ðq0 þ qðkxkÞÞsgnðyðtÞ yðtÞÞ ,
ð30Þ
ey ðtÞ ¼ Cex ðtÞ.
(31) 0
Theorem 3. Under Assumptions 1 and 2, the observer described by (28) and (29) and the estimation algorithm described by (21) can realize limt!1 ex ðtÞ ¼ 0 and a bounded ey ðtÞ 2 L2 . Furthermore, limt!1 ey ðtÞ ¼ 0 under persistent excitation conditions. Proof. Considering the same Lyapunov function as (22), from (21), (30) and (31), its derivative with respect to time is ^ V_ ðtÞ ¼ eTx ðtÞ½PðA KCÞ þ ðA KCÞT P ex ðtÞ 2ey ðtÞf T ðuðtÞ; yðtÞ; xðtÞÞRe y ðtÞ ^ ^ þ 2eT ðtÞPB½yf f ðuðtÞ; yðtÞ; xðtÞÞ yðtÞf ðuðtÞ; yðtÞ; xðtÞÞ
x 2eTx PBsgnðRÞ½y0 ðq0
^ þ qðkxkÞÞsgnðyðtÞ yðtÞÞ .
ð32Þ
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According to Assumptions 10 and 2, one can further obtain that ^ V_ ðtÞp eTx ðtÞQex ðtÞ þ 2eTx C T Ryf ½f ðu; y; xÞ f ðu; y; xÞ
2eTx C T RsgnðRÞ½y0 ðq0 þ qðkxkÞÞsgnðey ðtÞÞ
p eTx ðtÞQex ðtÞ.
ð33Þ
As in the proof of Theorem 2, one can conclude that limt!1 ex ðtÞ ¼ 0 and limt!1 ey ðtÞ ¼ 0 under persistent excitation. This completes the proof. & Remark 6. This work is an extension of that in [17] to Lipschitz and bounded nonlinear fault distribution functions. Remark 7. It is worth pointing out that a similar issue has been investigated by some researchers, for example, [8] based on high-gain observer, [9] using differential geometric theory, [22] via learning approach. The contribution of the results obtained in this paper is that it not only enables parameter fault detection, but also provides the estimation of the fault, which is required for fault accommodation such as application to aircraft systems [15,30]. Remark 8. Using a similar method presented in [19], the obtained results can be extended to detection and diagnosis for time-varying parameter faults in nonlinear systems, where convergence of fault estimation error to a given residual set can be achieved.
6. Application example: a one-wheel model with lumped friction To illustrate the effectiveness of the proposed technique in this paper, let us consider a one-wheel model with the lumped tire/road friction model as described in [26]: m_v ¼ F n ðs0 z þ s1 z_Þ þ F n s2 vr ,
(34)
_ ¼ rF n ðs0 z þ s1 z_Þ so o þ ur , Jo
(35)
z_ ¼ vr y
s0 jvr j z, gðvr Þ
(36) 1=2
where gðvr Þ ¼ mc þ ðms mc Þejvr =vs j ; s0 is the normalized rubber longitudinal lumped stiffness, s1 the normalized rubber longitudinal lumped damping, s2 the normalized viscous relative damping, so the viscous rotational friction, mc the normalized coulomb friction, ms the normalized static friction, mc pms 2 ½0; 1 , vs the Stribeck relative velocity, vr ¼ ðro vÞ the relative velocity, v the linear velocity, o the angular velocity, F n the normal force, z the internal friction state. y denotes the parameter related to the unexpected changes in the road conditions, which can be interpreted as being the coefficient of road adhesion. In normal case, y ¼ 1. It is assumed that only o is measurable.
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By introducing the following transformation of coordinates into the system described by (34)–(36): x ¼ rmv þ Jo,
(37)
Z ¼ Jo þ rF n s1 z.
(38)
One can obtain
_x ¼ F n s2 x þ JF n s2 þ r2 F n s2 so o þ ur , m m s0 s0 Z_ ¼ Z þ J so o þ ur , s1 s1 z_ ¼ ðro vÞ ys0
(39)
(40)
jro vj z, gðvr Þ
(41)
1 ðZ rF n s1 zÞ ¼ o. (42) J hxi By defining x ¼ zZ , u ¼ ur , y ¼ o, one can rewrite the above system as state-space form 3 2 F n s2 0 0 2 3 7 6 m 0 7 6 7 6 s0 6 7 s0 jro vj 6 0 07 0 7 x_ ¼ 6 7x þ 6 4 5y gðvr Þ z s 1 7 6 7 6 1 5 4 1 0 0 rm 3 2 JF n s2 2 2 3 þ r F n s2 so 1 7 6 m 7 6 6 7 7 6 7 ð43Þ þ6 7y þ 6 s0 4 1 5u, 7 6 J s o 5 4 s1 0 r y¼
2 6 y¼4
0
3T
7 1=J 5 x. rF n s1 =J
(44)
As in [26], the following values of parameters for the wheel are taken: s0 ¼ 40 ð1=mÞ; mc ¼ 0:5;
s1 ¼ 4:9487 ðs=mÞ;
ms ¼ 0:9;
r ¼ 0:25 ðmÞ;
s2 ¼ 0:0018 ðs=mÞ;
vs ¼ 12:5 ðm=sÞ;
m ¼ 5 ðKgÞ;
J ¼ 0:2344 ðKgm2 Þ;
F n ¼ 14 ðKgm2 =s2 Þ.
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Using the above numerical values of parameters, the system matrices in Eqs. (1) and (2) are: 2 3T 2 3 2 3 0 0:0050 0 0 0 6 7 6 7 6 7 0 8:0923 0 5; B ¼ 4 0 5; C ¼ 4 4:2662 5 . A¼4 73:8927 0:80 0 0 1 It is easy to see that rank½C CA CA2 ¼ 3, thus ðA; CÞ is an observable pair. s0 jro vj z gðvr Þ s0 p jro vj jzj mc s0 p ðjryj þ jvjÞ jzj mc
s0 J jxj p rþ ymax þ jzj rm rm mc
f ðu; y; xÞ ¼
9qðkxkÞ and that s0 ^ f ðu; y; xÞp mc
"
# ^ J jxj rþ þ y j^zj rm max rm
^ 9qðkxkÞ. That is, Assumption 10 holds for this example. However, it can be checked that f ðu; y; xÞ in this example does not satisfy Lipschitz condition (i.e., a Lipschitz constant does not exist), thus those fault detection methods subjected to such a condition cannot be used for this practical example. The persistent excitation condition implies that the relative velocity vr ¼ ðro vÞ should not tend to zero in order for the estimated parameter to converge. This in turn implies that the internal friction state zðtÞ will not asymptotically converge to zero. According to Remark 4, we choose the following K and Q such that C½sI ðA KCÞ 1 B is strictly positive real and defined r2 is positive: 2 3 2 3 0:0548 0:2738 9:0206 0:1250 6 7 6 7 29:4727 5. K ¼ 4 0:2308 5; Q ¼ 4 0:2738 38:1191 0:0661
9:0206
29:4727
Solving Eqs. (4) and (5), it follows that 2 3 5:4797 0:0305 0 6 7 2:51 4:2662 5; P ¼ 4 0:0305 0 4:2662 73:8927
576:3141
R ¼ 1.
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Assume that the parameter fault is created as follows: 8 > < 1; yðtÞ ¼ 0:2; > : 2:4;
0pto2 ðsecÞ; 2pto6 ðsecÞ; 6ptp12 ðsecÞ: Fauit Detection
2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5
0
2
4
6 (sec)
8
10
12
10
12
Fig. 1. Detection signal: key ðtÞk.
Fault diagnosis
5 4 3 2 1 0 -1 0
2
4
6 (sec)
8
Fig. 2. The parameter fault y (dotted) and its estimation y^ (solid).
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Thus it is verified that all the assumptions are satisfied in this practical example. By taking the parameter G ¼ 200 and the sampling period T ¼ 0:01, the simulation results are shown in Figs. 1 and 2. It can be seen that the piece-wise constant fault in the nonlinear system is detected and estimated quite well, using the proposed technique.
7. Conclusions In this paper, adaptive observer-based detection and estimation of parameter faults in a class of nonlinear systems with nonlinear fault distribution is proposed. An adaptive observer design for fault detection and estimation is presented, and it is shown that the Lipschitz condition can be relaxed at the cost of an additional term in the observer design. A one-wheel model with lumped friction is used as the application example to illustrate the effectiveness of the approach.
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[17] H. Wang, Z.J. Huang, S. Daley, On the use of adaptive updating rules for actuator and sensor fault diagnosis, Automatica 33 (1997) 217–225. [18] A.P. Xu, Q.H. Zhang, Fault detection and isolation based on adaptive observers for linear time varying systems, Proceedings of IFAC World Congress, Barcelona, 2002, CD-ROM (562.pdf). [19] B. Jiang, J.L. Wang, Y.C. Soh, An adaptive technique for robust diagnosis of faults with independent effect on system outputs, Int. J. Control 75 (2002) 792–802. [20] B. Jiang, M. Staroswiecki, V. Cocquempot, Fault estimation in nonlinear uncertain systems using robust/sliding-mode observers, IEE Proc. Control Theor. Appl. 151 (2004) 29–37. [21] A.T. Vemuri, M.M. Polycarpou, Robust nonlinear fault diagnosis in input–output systems, Int. J. Control 68 (1997) 343–360. [22] X. Zhang, M. Polycarpou, T. Parisini, A robust detection and isolation scheme for abrupt and incipient faults in nonlinear systems, IEEE Trans. Automat. Control 47 (2002) 576–593. [23] A. Trunov, M. Polycarpou, Robust nonlinear fault diagnosis: Application to robotic systems, Proceedings of the IEEE CCA, Hawai’i, USA, 1999, 1424–1429. [24] V. Krishnaswami, G. Rizzoni, A survey of observer based residual generation for FDI, Proceedings of the IFAC Symposium on Fault Detection, Supervision and Safety for Technical Process: SAFEPROCESS, 1994, 34–39. [25] G. Schreier, J. Ragot, R.J. Patton, P.M. Frank, 1997, Observer design for a class of nonlinear systems, in: Proceedings of the IFAC Symposium on Fault Detection, Supervision and Safety for Technical Process: SAFEPROCESS, UK, 1997, 483–488. [26] Canudas de Wit, C.H. Olsson, K.J. Astrm, P. Lischinsky, A new model for control of systems with friction, IEEE Trans. Autom. Control 40 (1995) 419–425. [27] M. Spong, Modeling and control of elastic joint robots, ASME J. of Dynamic Systems, Measurement and Control 109 (1987) 310–319. [28] J.N. Chiasson, M. Bodson, Nonlinear control of a shunt DC motor, IEEE Trans. Automat. Control 38 (1993) 1662–1666. [29] K.S. Narendra, A.M. Annaswamy, Stable Adaptive Systems, Prentice-Hall, Englewood Cliffs, NJ, 1989. [30] B. Jiang, F.N. Chowdhury, Fault estimation and accommodation for linear MIMO discrete-time systems, IEEE Trans. Control Syst. Technol. 13 (2005) 493–499.
Journal of the Franklin Institute 342 (2005) 725–736 www.elsevier.com/locate/jfranklin
Parameter fault detection and estimation of a class of nonlinear systems using observers$ Bin Jianga,, Fahmida N. Chowdhuryb a
College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, 29 YuDao Street, Nanjing, 210016, PR China b Department of ECE, P.O. Box 43890, University of Louisiana at Lafayette, Lafayette, LA 70504-3890, USA
Abstract The focus of this paper is on the detection and estimation of parameter faults in nonlinear systems with nonlinear fault distribution functions. The novelty of this contribution is that it handles the nonlinear fault distribution function; since such a fault distribution function depends not only on the inputs and outputs of the system but also on unmeasured states, it causes additional complexity in fault estimation. The proposed detection and estimation tool is based on the adaptive observer technique. Under the Lipschitz condition, a fault detection observer and adaptive diagnosis observer are proposed. Then, relaxation of the Lipschitz requirement is proposed and the necessary modification to the diagnostic tool is presented. Finally, the example of a one-wheel model with lumped friction is presented to illustrate the applicability of the proposed diagnosis method. r 2005 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Fault detection; Fault estimation; Adaptive observer; Nonlinear systems; Lipschitz condition
$ Partially supported by NASA under award NCC5-573, contract NASA/LEQSF(2001-04)-1, Scientific Research fund of NUAA (S0456-031), ‘‘Liu Da Ren Cai Gao Feng’’ scheme of Jiangsu Province (China) and key project of Natural Science Foundation of China (60234010). Corresponding author. E-mail addresses: [email protected] (B. Jiang), [email protected] (F.N. Chowdhury).
0016-0032/$30.00 r 2005 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2005.04.007
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1. Introduction Fault detection and isolation (FDI) algorithms and their applications to a wide range of industrial processes have been the subject of intensive research over the past two decades. Fruitful results can be found in several survey papers [1–3] and books [4–6]. In general, there are several kinds of schemes for model-based FDI: observer based, [7–9], parity space based, [10,11], eigenstructure assignment based [12] and parameter-identification based [13,14]. In some cases, fault accommodation strategies are needed, i.e. the control algorithm must be adapted based on FDI to go on controlling the faulty system. Aircraft flight control systems [15] are good examples for applications of fault accommodation/active reliable control. In such cases, it is important to carry out fault estimation in addition to detection. Some recent results based on adaptive or robust observers can be found in [16–20]; see [21,22] for methods using learning approach. In this paper, detection and estimation of parameter faults for nonlinear systems are investigated using adaptive observers. Compared with [21,23] and our previous work presented in [19], the novelty of this contribution is that we consider difficult cases where the nonlinear fault distribution function depends not only on the inputs and outputs but also on unmeasured states. Furthermore, in order to expand the applicability of the proposed method, we propose a relaxation of the Lipschitz condition which is typically required (e.g., [24,25,19]) in FDI literature. This paper is organized as follows. Section 2 describes a class of nonlinear systems with parameter fault and introduces some preliminaries. Under Lipschitz condition, fault detection observer is proposed in Section 3. An adaptive diagnostic rule is designed in Section 4. Discussion on relaxation of Lipschitz requirement is presented in Section 5. Results of applying our technique to a one-wheel model with lumped friction are presented in Section 6, followed by some concluding remarks in Section 7.
2. Nonlinear dynamic model Consider the following nonlinear system _ ¼ AxðtÞ þ gðuðtÞ; yðtÞÞ þ ByðtÞf ðuðtÞ; yðtÞ; xðtÞÞ, xðtÞ
(1)
yðtÞ ¼ CxðtÞ,
(2)
where the state is x 2 Rn , the input is u 2 Rm , and the output is y 2 Rr . The pair ðA; CÞ is observable. The nonlinear term gðuðtÞ; yðtÞÞ depends on uðtÞ and yðtÞ which are directly available. The f ðuðtÞ; yðtÞ; xðtÞÞ 2 Rr is a nonlinear vector function of uðtÞ; yðtÞ and xðtÞ. The yðtÞ 2 R is a parameter which changes unexpectedly when a fault occurs. It is assumed that yðtÞ is bounded, i.e., jyðtÞjpy0 . In the fault-free case, we have yðtÞ ¼ yH , where yH is a known scalar (subscript H stands for healthy case). Remark 1. It should be noted that there are quite a few practical systems which can be described by this particular form of nonlinear model. For example, see one-wheel
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model with lumped friction in [26], flexible joint robotic model in [27], and series DC motor in [28].
3. Detection observer design Prior to fault detection observer design, the following assumptions are made: Assumption 1. There exists a known positive constant L0 such that for any norm bounded x1 ðtÞ; x2 ðtÞ 2 Rn , the following inequality holds: kf ðuðtÞ; yðtÞ; x1 ðtÞÞ f ðuðtÞ; yðtÞ; x2 ðtÞÞkpL0 kx1 ðtÞ x2 ðtÞk.
(3)
Remark 2. Assumption 1 is the known Lipschitz condition, which is typically required in the literature on FDI for nonlinear systems, e.g., [24,25,19]. The removal of the Lipschitz condition for detection observer design will be discussed later in the paper. Assumption 2. C½sI ðA KCÞ 1 B is strictly positive real (SPR), where K 2 Rn r is chosen such that A KC is stable. Remark 3. The SPR requirement in the above assumption is equivalent to the following: For a given positive definite matrix Q40 2 Rn n , there exists matrices P ¼ PT 40 2 Rn n and a scalar R such that ðA KCÞT P þ PðA KCÞ ¼ Q,
(4)
PB ¼ C T R.
(5)
To detect the fault, the following observer is constructed: _^ ¼ AxðtÞ ^ þ gðuðtÞ; yðtÞÞ þ ByH f ðuðtÞ; yðtÞ; xðtÞÞ ^ ^ xðtÞ þ KðyðtÞ yðtÞÞ,
(6)
^ ¼ C xðtÞ, ^ yðtÞ
(7)
n
^ 2 R is the state estimate. Since it has been assumed that the pair ðA; CÞ is where xðtÞ observable, the observer gain matrix K can be selected such that ðA KCÞ is a stable matrix. We define ^ ex ðtÞ ¼ xðtÞ xðtÞ;
^ ey ðtÞ ¼ yðtÞ yðtÞ.
(8)
Then the observation error and output error equations are given by ^ e_x ðtÞ ¼ ðA KCÞex ðtÞ þ B½yðtÞf ðuðtÞ; yðtÞ; xðtÞÞ yH f ðuðtÞ; yðtÞ; xðtÞÞ , ey ðtÞ ¼ Cex ðtÞ.
(9) (10)
The convergence of the above observer is guaranteed by the following theorem: Theorem 1. Consider the system described by (1), (2) and its observer described by (6), (7). Under Assumptions 1 and 2, the observer is asymptotically convergent when no faults occurs ðyðtÞ ¼ yH Þ, i.e., limt!1 ex ðtÞ ¼ 0 and thus limt!1 ey ðtÞ ¼ 0.
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Proof. Consider the following Lyapunov function V ðtÞ ¼ eTx ðtÞPex ðtÞ,
(11)
where P is given by (4) and (5), Q is chosen such that r1 9lmin ðQÞ 2kCk jRjyH L0 40. Along the trajectory of the fault-free system (9), its derivative with respect to time is V_ ðtÞ ¼ eTx ðtÞ½PðA KCÞ þ ðA KCÞT P ex ðtÞ ^ þ 2eTx ðtÞPByH ½f ðuðtÞ; yðtÞ; xðtÞÞ f ðuðtÞ; yðtÞ; xðtÞÞ .
ð12Þ
From (3), (4), and (5), one can further obtain that V_ ðtÞp eTx ðtÞQex ðtÞ þ 2key k jRjyH L0 kex k p r1 kex k2 p0.
ð13Þ
Thus, limt!1 ex ðtÞ ¼ 0 and limt!1 ey ðtÞ ¼ 0. This completes the proof.
&
According to Theorem 1, after the convergence of the observer, the fault detection can be carried out as follows: ey ðtÞ ¼ 0
no fault occurred,
ey ðtÞa0 fault has occurred.
ð14Þ
The observer given by (6) and (7) is referred to as a detection observer for the system described by (1) and (2) under Assumptions 1 and 2.
4. Fault estimation using an adaptive observer To estimate the fault after the alarm (14) has been generated, the following observer is constructed: _^ ¼ AxðtÞ ^ ^ þ gðuðtÞ; yðtÞÞ þ ByðtÞf ^ ^ xðtÞ ðuðtÞ; yðtÞ; xðtÞÞ þ KðyðtÞ yðtÞÞ,
(15)
^ ¼ C xðtÞ, ^ yðtÞ
(16)
^ is an estimate of yðtÞ. The value ^ 2 R is the observer state vector and yðtÞ where xðtÞ ^ is set to yH until a fault is detected. It is assumed that after a fault occurs, of yðtÞ yðtÞ ¼ yf ¼ constantayH ; jyf jpy0 . We denote n
^ ex ðtÞ ¼ xðtÞ xðtÞ;
^ ey ðtÞ ¼ yðtÞ yðtÞ;
^ ey ðtÞ ¼ yf yðtÞ,
(17)
then it can be obtained that ^ ^ e_x ðtÞ ¼ ðA KCÞex ðtÞ þ B½yf f ðuðtÞ; yðtÞxðtÞÞ yðtÞf ðuðtÞ; yðtÞ; xðtÞÞ ,
(18)
ey ðtÞ ¼ Cex ðtÞ.
(19)
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^ As a result, the purpose of fault diagnosis is to find a diagnostic algorithm for yðtÞ such that lim ex ðtÞ ¼ 0;
t!1
lim ey ðtÞ ¼ 0.
t!1
(20)
The following theorem produces a convergent adaptive diagnostic algorithm for estimating the faulty parameter yf . Theorem 2. Under the Assumptions 1 and 2, the observer described by (15) and (16) and following diagnostic algorithm ^ dyðtÞ ^ ¼ Gf T ðuðtÞ; yðtÞ; xðtÞÞRe (21) y ðtÞ dt can realize limt!1 ex ðtÞ ¼ 0 and a bounded ey ðtÞ 2 L2 . Furthermore, limt!1 ey ðtÞ ¼ 0 under a persistent excitation, where R is given by (5), G40 is a weighting scalar. Proof. Consider the following Lyapunov function V ðtÞ ¼ eTx ðtÞPex ðtÞ þ G1 e2y ðtÞ.
(22)
From (18) and (21), its derivative with respect to time is ^ V_ ðtÞ ¼ eTx ðtÞ½PðA KCÞ þ ðA KCÞT P ex ðtÞ 2ey ðtÞf T ðuðtÞ; yðtÞ; xðtÞÞRe y ðtÞ T ^ ^ þ 2e ðtÞPB½yf f ðuðtÞ; yðtÞ; xðtÞÞ yðtÞf ðuðtÞ; yðtÞ; xðtÞÞ . ð23Þ x
According to Assumptions 1 and 2, one can further obtain that ^ V_ ðtÞp eTx ðtÞQex ðtÞ 2ey ðtÞf T ðuðtÞ; yðtÞ; xðtÞÞRe y ðtÞ ^ þ yf ½f ðu; y; xÞ f ðu; y; xÞ g ^ þ 2eTx C T Rfey f ðu; y; xÞ p r2 kex k2 p0,
ð24Þ
where r2 9lmin ðQÞ 2kCk jRjy0 L0 , jyf jpy0 , Q40 is chosen such that r2 40. Inequality (24) implies the stability of the origin ex ¼ 0, ey ¼ 0 and the uniform boundedness of ex and ey with ex 2 L2 . On the other hand, from (18), e_x is uniformly bounded as well. According to Barbalat’s Lemma (see [29]), one can get lim ex ðtÞ ¼ 0.
t!1
(25)
The persistent excitation condition means that there exist two positive constants s and t0 such that for all t the following inequality holds: Z tþt0 f T ðyðsÞ; uðsÞ; xðsÞÞBT Bf T ðyðsÞ; uðsÞ; xðsÞÞ dsXsI. (26) t
Then from (18), (21), (25) and (26), one can conclude that limt!1 ey ðtÞ ¼ 0. This completes the proof. & Remark 4. The choice/design of K; Q; P and R is summarized as follows: (i) Since ðA; CÞ is observable, the observer gain matrix K is selected such that ðA KCÞ is stable and C½sI ðA KCÞ 1 B is strictly positive real (SPR).
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(ii) In Eq. (4), Q40 is chosen such that the above defined r2 40. (iii) The existence of solutions P and R to Eqs. (4) and (5) can be guaranteed by Remark 2.
5. Relaxing the Lipschitz requirement In the above sections, fault detection and estimation observers have been designed under the Lipschitz condition which does not always hold for practical systems. In this section, we replace the Lipschitz requirement by the following, weaker condition: Assumption 10 . There exists a positive function qðÞ and a positive scalar q0 such that the following inequality holds: kf ðuðtÞ; yðtÞ; xðtÞÞkpqðkxkÞpq0 .
(27)
The focus of this section is to design an adaptive estimation observer under Assumption 10 instead of Assumption 1. The detection observer can be designed similarly and is omitted. To diagnose the fault after detection, the following observer is constructed: _^ ¼ AxðtÞ ^ ^ þ gðuðtÞ; yðtÞÞ þ ByðtÞf ^ ^ xðtÞ ðuðtÞ; yðtÞ; xðtÞÞ þ KðyðtÞ yðtÞÞ ^ þ BsgnðRÞ½y0 ðq0 þ qðkxkÞÞsgnðyðtÞ yðtÞÞ ,
ð28Þ
^ ¼ C xðtÞ, ^ yðtÞ
(29) ^ ^ 2 R is the observer state vector and yðtÞ is an estimate of yðtÞ, which is where xðtÞ obtained by the diagnostic algorithm (21). n
Remark 5. From (28), it can be seen that the Lipschitz condition can be relaxed at the cost of an additional term in the estimation observer design. Using the same notations of ex ðtÞ; ey ðtÞ and ey ðtÞ as in Section 4, then the observation error and output error equations can be described by ^ ^ ðuðtÞ; yðtÞ; xðtÞÞ
e_x ðtÞ ¼ ðA KCÞex ðtÞ þ B½yf f ðuðtÞ; yðtÞ; xðtÞÞ yðtÞf ^ BsgnðRÞ½y0 ðq0 þ qðkxkÞÞsgnðyðtÞ yðtÞÞ ,
ð30Þ
ey ðtÞ ¼ Cex ðtÞ.
(31) 0
Theorem 3. Under Assumptions 1 and 2, the observer described by (28) and (29) and the estimation algorithm described by (21) can realize limt!1 ex ðtÞ ¼ 0 and a bounded ey ðtÞ 2 L2 . Furthermore, limt!1 ey ðtÞ ¼ 0 under persistent excitation conditions. Proof. Considering the same Lyapunov function as (22), from (21), (30) and (31), its derivative with respect to time is ^ V_ ðtÞ ¼ eTx ðtÞ½PðA KCÞ þ ðA KCÞT P ex ðtÞ 2ey ðtÞf T ðuðtÞ; yðtÞ; xðtÞÞRe y ðtÞ ^ ^ þ 2eT ðtÞPB½yf f ðuðtÞ; yðtÞ; xðtÞÞ yðtÞf ðuðtÞ; yðtÞ; xðtÞÞ
x 2eTx PBsgnðRÞ½y0 ðq0
^ þ qðkxkÞÞsgnðyðtÞ yðtÞÞ .
ð32Þ
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According to Assumptions 10 and 2, one can further obtain that ^ V_ ðtÞp eTx ðtÞQex ðtÞ þ 2eTx C T Ryf ½f ðu; y; xÞ f ðu; y; xÞ
2eTx C T RsgnðRÞ½y0 ðq0 þ qðkxkÞÞsgnðey ðtÞÞ
p eTx ðtÞQex ðtÞ.
ð33Þ
As in the proof of Theorem 2, one can conclude that limt!1 ex ðtÞ ¼ 0 and limt!1 ey ðtÞ ¼ 0 under persistent excitation. This completes the proof. & Remark 6. This work is an extension of that in [17] to Lipschitz and bounded nonlinear fault distribution functions. Remark 7. It is worth pointing out that a similar issue has been investigated by some researchers, for example, [8] based on high-gain observer, [9] using differential geometric theory, [22] via learning approach. The contribution of the results obtained in this paper is that it not only enables parameter fault detection, but also provides the estimation of the fault, which is required for fault accommodation such as application to aircraft systems [15,30]. Remark 8. Using a similar method presented in [19], the obtained results can be extended to detection and diagnosis for time-varying parameter faults in nonlinear systems, where convergence of fault estimation error to a given residual set can be achieved.
6. Application example: a one-wheel model with lumped friction To illustrate the effectiveness of the proposed technique in this paper, let us consider a one-wheel model with the lumped tire/road friction model as described in [26]: m_v ¼ F n ðs0 z þ s1 z_Þ þ F n s2 vr ,
(34)
_ ¼ rF n ðs0 z þ s1 z_Þ so o þ ur , Jo
(35)
z_ ¼ vr y
s0 jvr j z, gðvr Þ
(36) 1=2
where gðvr Þ ¼ mc þ ðms mc Þejvr =vs j ; s0 is the normalized rubber longitudinal lumped stiffness, s1 the normalized rubber longitudinal lumped damping, s2 the normalized viscous relative damping, so the viscous rotational friction, mc the normalized coulomb friction, ms the normalized static friction, mc pms 2 ½0; 1 , vs the Stribeck relative velocity, vr ¼ ðro vÞ the relative velocity, v the linear velocity, o the angular velocity, F n the normal force, z the internal friction state. y denotes the parameter related to the unexpected changes in the road conditions, which can be interpreted as being the coefficient of road adhesion. In normal case, y ¼ 1. It is assumed that only o is measurable.
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By introducing the following transformation of coordinates into the system described by (34)–(36): x ¼ rmv þ Jo,
(37)
Z ¼ Jo þ rF n s1 z.
(38)
One can obtain
_x ¼ F n s2 x þ JF n s2 þ r2 F n s2 so o þ ur , m m s0 s0 Z_ ¼ Z þ J so o þ ur , s1 s1 z_ ¼ ðro vÞ ys0
(39)
(40)
jro vj z, gðvr Þ
(41)
1 ðZ rF n s1 zÞ ¼ o. (42) J hxi By defining x ¼ zZ , u ¼ ur , y ¼ o, one can rewrite the above system as state-space form 3 2 F n s2 0 0 2 3 7 6 m 0 7 6 7 6 s0 6 7 s0 jro vj 6 0 07 0 7 x_ ¼ 6 7x þ 6 4 5y gðvr Þ z s 1 7 6 7 6 1 5 4 1 0 0 rm 3 2 JF n s2 2 2 3 þ r F n s2 so 1 7 6 m 7 6 6 7 7 6 7 ð43Þ þ6 7y þ 6 s0 4 1 5u, 7 6 J s o 5 4 s1 0 r y¼
2 6 y¼4
0
3T
7 1=J 5 x. rF n s1 =J
(44)
As in [26], the following values of parameters for the wheel are taken: s0 ¼ 40 ð1=mÞ; mc ¼ 0:5;
s1 ¼ 4:9487 ðs=mÞ;
ms ¼ 0:9;
r ¼ 0:25 ðmÞ;
s2 ¼ 0:0018 ðs=mÞ;
vs ¼ 12:5 ðm=sÞ;
m ¼ 5 ðKgÞ;
J ¼ 0:2344 ðKgm2 Þ;
F n ¼ 14 ðKgm2 =s2 Þ.
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Using the above numerical values of parameters, the system matrices in Eqs. (1) and (2) are: 2 3T 2 3 2 3 0 0:0050 0 0 0 6 7 6 7 6 7 0 8:0923 0 5; B ¼ 4 0 5; C ¼ 4 4:2662 5 . A¼4 73:8927 0:80 0 0 1 It is easy to see that rank½C CA CA2 ¼ 3, thus ðA; CÞ is an observable pair. s0 jro vj z gðvr Þ s0 p jro vj jzj mc s0 p ðjryj þ jvjÞ jzj mc
s0 J jxj p rþ ymax þ jzj rm rm mc
f ðu; y; xÞ ¼
9qðkxkÞ and that s0 ^ f ðu; y; xÞp mc
"
# ^ J jxj rþ þ y j^zj rm max rm
^ 9qðkxkÞ. That is, Assumption 10 holds for this example. However, it can be checked that f ðu; y; xÞ in this example does not satisfy Lipschitz condition (i.e., a Lipschitz constant does not exist), thus those fault detection methods subjected to such a condition cannot be used for this practical example. The persistent excitation condition implies that the relative velocity vr ¼ ðro vÞ should not tend to zero in order for the estimated parameter to converge. This in turn implies that the internal friction state zðtÞ will not asymptotically converge to zero. According to Remark 4, we choose the following K and Q such that C½sI ðA KCÞ 1 B is strictly positive real and defined r2 is positive: 2 3 2 3 0:0548 0:2738 9:0206 0:1250 6 7 6 7 29:4727 5. K ¼ 4 0:2308 5; Q ¼ 4 0:2738 38:1191 0:0661
9:0206
29:4727
Solving Eqs. (4) and (5), it follows that 2 3 5:4797 0:0305 0 6 7 2:51 4:2662 5; P ¼ 4 0:0305 0 4:2662 73:8927
576:3141
R ¼ 1.
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Assume that the parameter fault is created as follows: 8 > < 1; yðtÞ ¼ 0:2; > : 2:4;
0pto2 ðsecÞ; 2pto6 ðsecÞ; 6ptp12 ðsecÞ: Fauit Detection
2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5
0
2
4
6 (sec)
8
10
12
10
12
Fig. 1. Detection signal: key ðtÞk.
Fault diagnosis
5 4 3 2 1 0 -1 0
2
4
6 (sec)
8
Fig. 2. The parameter fault y (dotted) and its estimation y^ (solid).
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Thus it is verified that all the assumptions are satisfied in this practical example. By taking the parameter G ¼ 200 and the sampling period T ¼ 0:01, the simulation results are shown in Figs. 1 and 2. It can be seen that the piece-wise constant fault in the nonlinear system is detected and estimated quite well, using the proposed technique.
7. Conclusions In this paper, adaptive observer-based detection and estimation of parameter faults in a class of nonlinear systems with nonlinear fault distribution is proposed. An adaptive observer design for fault detection and estimation is presented, and it is shown that the Lipschitz condition can be relaxed at the cost of an additional term in the observer design. A one-wheel model with lumped friction is used as the application example to illustrate the effectiveness of the approach.
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