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Robust fault reconstruction for a class of nonlinear systems
Automatica xxx (xxxx) xxx
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Robust fault reconstruction for a class of nonlinear systems✩ ∗
Wen-Shyan Chua a,b , Joseph Chang Lun Chan a,c , Chee Pin Tan a , , Edwin Kah Pin Chong d , Sajeeb Saha e a
School of Engineering, Monash University Malaysia, Selangor 47500, Malaysia Selangor Human Resource Development Centre, Selangor 40100, Malaysia Division of Electronic Engineering, Jeonbuk National University, 567 Baekje-daero, Jeonju 54896, Republic of Korea d Department of Electrical and Computer Engineering, Colorado State University, CO 80523, USA e School of Engineering, Deakin University, Geelong, VIC 3220, Australia
b c
article
info
Article history: Received 1 May 2019 Received in revised form 15 October 2019 Accepted 29 October 2019 Available online xxxx Keywords: Nonlinearity Observers Fault detection Fault identification Robust estimation
a b s t r a c t This paper proposes two novel observer schemes for reconstructing faults in systems where the fault enters the state and output equations via nonlinear functions, which has not been considered in the literature. Two design methods are presented: one for the case where the fault dynamics are known and can be expressed as a polynomial function of time, and another for the case where the fault dynamics are unknown. The gains of the observer are designed using linear matrix inequalities (LMIs) such that the root-mean-square (RMS) gain from the uncertainties (or disturbances) to the fault reconstruction error is bounded. Necessary conditions for the feasibility of the LMIs are presented. Finally, a simulation example is shown to demonstrate the efficacy of the proposed scheme. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Modern engineering systems are prone to faults, which could have costly consequences if not properly addressed. Hence it is important that faults be detected, and even better reconstructed in real-time so that timely and appropriate corrective action can be taken. A popular such method is the observer, which consists of a model of the system being monitored, and processes the system’s inputs and outputs to reconstruct the fault (Chen, Edwards, & Alwi, 2018; Ifqir, Ichalal, Ait Oufroukh, & Mammar, 2018). Practical engineering systems are often nonlinear (An, Liu, Wang, & Wu, 2016; Laghrouche, Liu, Ahmed, Harmouche, & Wack, 2015; Liu, Luo, Yang, & Wu, 2016). Most observer schemes for nonlinear systems only consider faults entering the system linearly (Delshad, Johansson, Darouach, & Gustafsson, 2016; Gao, Liu, Sun, Liu, & Wu, 2019; Wu, Liu, Xiong, & Wu, 2018; Yan & ✩ This work was supported by the Ministry of Higher Education Malaysia under the Fundamental Research Grant Scheme (FRGS) Grant No. FRGS/1/2017/STG06/MUSM/02/1. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Maria Letizia Corradini under the direction of Editor André L. Tits. ∗ Corresponding author. E-mail addresses: [email protected] (W.-S. Chua), [email protected] (J.C.L. Chan), [email protected] (C.P. Tan), [email protected] (E.K.P. Chong), [email protected] (S. Saha).
Edwards, 2007; Zhao, Wang, Yan, & Shen, 2019). Comparatively, schemes to reconstruct faults entering the system via nonlinear functions are less developed. Many of them only considered faults in the state equation (Jiang & Chowdhury, 2005; Laghrouche et al., 2015; Liu et al., 2016; Xiong & Saif, 2001). Few works consider the case where faults enter through nonlinear functions in both the state and output equations, such as Saha, Aldeen, and Tan (2011, 2013) who linearised the system about an operating point, which may not represent the actual system if the states evolve far from the operating point, and Chua, Tan, Aldeen, and Saha (2017) who assumed the fault to be constant. This paper proposes observer schemes to reconstruct timevarying faults entering the system through nonlinear functions in both the state and output equations. Two methods are presented: the first considers polynomial faults (which can represent most time-varying faults (Gao, Ding, & Ma, 2007)), and the second considers even more general non-polynomial faults. The proposed methods improve on Chua et al. (2017) and Saha et al. (2011, 2013) where the faults can be time-varying and there is no need to linearise the system. The paper is structured as follows: Sections 2 and 3 present observer schemes to robustly reconstruct polynomial and nonpolynomial faults, respectively. A simulation verifies the efficacy of the proposed schemes in Section 4, and Section 5 draws conclusions.
https://doi.org/10.1016/j.automatica.2019.108718 0005-1098/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: W.-S. Chua, J.C.L. Chan, C.P. Tan et al., Robust fault reconstruction for a class of nonlinear systems. Automatica (2019) 108718, https://doi.org/10.1016/j.automatica.2019.108718.
2
W.-S. Chua, J.C.L. Chan, C.P. Tan et al. / Automatica xxx (xxxx) xxx
˜ x, 2 where M
2. Robust reconstruction of polynomial faults
[
= C
Consider the following faulty nonlinear system: x˙ = Ax + Bu + Mx (x, f ) + Qx ξ ,
where A ∈ R , B ∈ R , C ∈ R , Qx ∈ R , and Qy ∈ R are known and constant, whilst x ∈ Rn , y ∈ Rp , u ∈ Rm , f ∈ Rq , and ξ ∈ Rh are functions of time representing states, outputs, control inputs, faults, and disturbances, respectively. The aim is to reconstruct f using only u and y. Without loss of generality, Mx (x, f ) and My (x, f ) can be decomposed as follows: n×n
n×m
p×n
n×h
Mx (x, f ) = Fx f + Mx,2 (x, f ),
where Fx ∈ Rn×q and Fy ∈ Rp×q are known and constant. Denote the root-mean-square (RMS) of a signal b as R (b) = √ limT →∞
∫T
1 T
∥b∥2 dt.
0
Assumption 1. The following are assumed in this paper:
• The RMS of ξ is finite, i.e., R (ξ ) < ∞. • The nonlinear functions Mx,2 and My,2 are Lipschitz with Lipschitz constants γ1 , γ2 ∈ R+ , respectively. ♯ Assumption 2. In this section, it is assumed that f is a polynomial function of time as follows: f (t) = m0 + m1 t + m2 t 2 + · · · + mk−1 t k−1 ,
(3)
where mi (i = 0, 1, . . . , k − 1) are unknown constants, and k is known. ♯ Define f (i) as the ith derivative of f . From (3) it can be seen that f (k) = 0, and therefore (3) can be re-expressed as:
⎡
⎡
0
0
⎢f (k−1) ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ (k−2) ⎥ ⎢ ⎢f ⎥ ⎢ ⎢ . ⎥=⎢ ⎢ . ⎥ ⎢ ⎢ . ⎥ ⎢ ⎢ (2) ⎥ ⎢ ⎣ f ⎦ ⎣
Iq
0
0
Iq
f (k)
⎤
f (1)
⎡ ⎤ z˙ ⎣ ⎦ f˙
.. .
.. .
0
0
0
0
··· ··· ··· .. .
⎤⎡
f (k−1)
⎤
0
0
0
0
0
0
0
0 ⎥ ⎢f (k−2) ⎥ ⎥⎢ ⎥ 0 ⎥ ⎢f (k−3) ⎥
.. .
⎥⎢
⎥
⎢ .. ⎥ ⎥⎢ . . ⎥ ⎢ .. ⎥⎢ 0 ⎦ ⎣ f (1)
.. .
· · · Iq 0 · · · 0 Iq ⎡ ⎤ ˜1 0 A ⎣ ⎦ A˜ 2 0
0
⎥. ⎥ ⎥ ⎥ ⎦
(4)
f
⎡ ⎤
z f
x˙ = Ax + Bu + Fx f + Mx,2 (x, f ) + Qx ξ , z˙ = A˜ 1 z ,
(5)
y = Cx + Du + Fy f + My,2 (x, f ) + Qy ξ , f˙ = A˜ 2 z .
˙ z˙ˆ = A˜ 1 zˆ + L2 ey , fˆ = A˜ 2 zˆ + L3 ey ,
(6)
)
ey = y − C xˆ − Du − Fy fˆ − My,2 xˆ , fˆ . fT
]T
, xˆ¯ =
[
xˆ T
zˆ T
fˆ T
]T
, e = x¯ − xˆ¯ =
˜ x,2 = Mx,2 (x, f ) − Mx,2 (xˆ , fˆ ). By using (5) eTx eTz ef , and M and (6), the following error system is obtained:
[
⎡
A e˙ = ⎣0 0
0 A˜ 1 A˜ 2
A¯ 1
⎤
L
Q¯ x
Ao
˜x M
Lo
[
]
( ) ˜x M + Q¯ x − LQy ξ . ˜ My
(8)
Qo
˜o M
(
)
˜ o = Mo (x, f ) − Mo xˆ , fˆ . Observer (6) estimates f via where M fˆ , but from (8), ξ corrupts fˆ , and hence observer (6) needs to be designed so that fˆ is robust against ξ . Definition 1. Consider two signals wu : [0, ∞) → R and wy : [0, ∞) → R, and let Ξ be the set of all admissible signals wu . The RMS gain from wu to wy is given by
{ γ wu , wy = sup
(
)
R wy
(
}
)
R (wu )
: R (wu ) > 0, wu ∈ Ξ .
(9)
Proposition 3. Let P ∈ R(n+kq)×(n+kq) = P T > 0, Z = PL, and A = P A¯ 1 − Z A¯ 3 , and ϵ and µ be two positive scalars. Suppose there exists a set of values for P , Z , ϵ, γ , and µ that satisfies the following linear matrix inequality (LMI):
⎡ ¯ A ⎢ T ⎢P ⎢ ⎢−Z T ⎢ ⎢ ⎣ Cf
P
−Z
CfT
−ϵ In+kq
0
0
0
−ϵ Ip
0
0
0
−µIn+kq
0
0
T
0
Q
⎤
Q
⎥ ⎥ 0 ⎥ ⎥ ≤0, ⎥ 0 ⎦ 0 ⎥
(10)
−µIh
¯ = A + A + ϵγ 2 In+kq , and Q = P Q¯ x − ZQy . where Cf = 0 Iq , A ) ( Then the RMS gain from ξ to ef is bounded by µ, i.e., γ ξ , ef ≤ µ. ♯
]
[
T
Proof. Define a Lyapunov function V = eT Pe (≥ 0 and differ) entiate it with respect to time to get V˙ = eT PAo + ATo P e + ˜ o + 2eT PQo ξ . Recall the inequality X T Y + Y T X ≤ 1 X T X + 2eT PLo M ϵ T ˜ o , and ϵ Y Y (Yan & Edwards, 2007); let X T = eT PLo , Y = M [ ]T Mo = MxT (x, f ) 01×kq MyT (x, f ) . From Assumption 1, Mo is Lipschitz and its Lipschitz constant γ is bounded by the triangle ˜ o = Mo (x, f ) − inequality ( ) (i.e., γ ≤ γ1 + γ2 ). Then from (8), M
V˙ ≤eT
(
PAo + ATo P +
1
e PLo LTo Pe + ϵγ 2 ∥e∥2 .
1 T
ϵ
ϵ
PLo LTo P + γ 2 ϵ In+kq
)
e + 2eT PQo ξ .
(11)
Θ
PAo + ATo P + ϵγ 2 In+kq − P˜
(7)
[
− 1ϵ In+kq+p
0
− µ1 In+kq+h
0
]
P˜ T ≤ 0, (12)
where P˜ = PLo CfT PQo . Then substitute Θ from (11) into (12), expand to get Θ + µ1 CfT Cf + µ1 PQo QoT P ≤ 0, and applying
[
]
the Schur complement yields Ω :=
⎤
⎡
[ ] [ ] Fx ˜ x, 2 L1 Qx M 0 ⎦ e − L2 ey + 0 ξ + ⎣ 0 ⎦ , L3 0 0 0
] −L
Next, substitute Ao , Qo from (8) into (10) and apply the Schur complement to obtain
x˙ˆ = Axˆ + Bu + Fx fˆ + Mx,2 (xˆ , fˆ ) + L1 ey ,
zT
Consider the following observer for system (5):
[ T x ] T T
[
Substitute this into V˙ to get
System (1) can therefore be re-expressed using (2)–(4) as
Let x¯ =
)
(
˜o ≤ Mo xˆ , fˆ , and therefore 2eT PLo M
⎣ ⎦
(
0p×(k−1)q
e˙ = A¯ 1 − LA¯ 3 e + In+kq
p×h
(2)
My (x, f ) = Fy f + My,2 (x, f ),
]
stituting for y from (5) and ey from (6) into (7) yields (1)
y = Cx + Du + My (x, f ) + Qy ξ ,
¯ Mx,2 (x, f ) − Mx,2 (xˆ , fˆ ). Next, ( define ) A3 ˜ y = My,2 (x, f ) − My,2 xˆ , fˆ . SubFy and M
=
[
implying eT
[ ]
] e ξT Ω ξ
[
Θ + µ1 CfT Cf QoT P
PQo −µIh
]
≤ 0,
≤ 0. Then substitute for e, Cf , and
V˙ from (7), (10), and (11), respectively, and apply the Schur complement to get µ1 eTf ef ≤ µξ T ξ − V˙ . Integrate both sides from
Please cite this article as: W.-S. Chua, J.C.L. Chan, C.P. Tan et al., Robust fault reconstruction for a class of nonlinear systems. Automatica (2019) 108718, https://doi.org/10.1016/j.automatica.2019.108718.
W.-S. Chua, J.C.L. Chan, C.P. Tan et al. / Automatica xxx (xxxx) xxx
0 to T , divide by T , and use V (T ) ≥ 0 to get − µ
∫T
1 µT
2
∫T
V (0) T
≤
V (T )−V (0) T
≤
2
∥ξ ∥ dt − ∥ef ∥ dt. Let T → ∞ and take square roots 0 T 0 ( ) to get R ef ≤ µR (ξ ). Divide both ( sides ) by R (ξ ), take sup over all ξ ∈ Ξ , and use (9) to get γ ξ , ef ≤ µ, thus completing the proof. □
Proof. For (10) to hold, A + A(T + ϵγ) 2 In+kq ≤ 0 must hold. This implies A is stable, and that A¯ 1 , A¯ 3 is detectable (i.e., their unobservable modes must be stable). From 1 in Chua ( Corollary ) ¯ 1 , A¯ 3 are the values et al. (2017), the unobservable modes of A [ ] sIn+kq − A¯ 1 of s that make R(s) = lose rank (i.e., the values of s A¯ 3 that cause rank (R(s)) < n + kq). Substituting for A¯ 1 from (7) and A¯ 3 from (8) yields
⎡
sIn − A
0
0
sI(k−1)q − A˜ 1
0
A˜ 2
sIq
0
Fy
⎢
C
(
−Fx
⎤
0 ⎥ ⎥
⎦.
(13)
)
Substituting for A˜ 1 , A˜ 2 from (4), it can be seen that if s ̸ = 0 is an unobservable mode )mode of (A, C ), or if s = 0 is an( unobservable ) ( of A, Fx , C , Fy , then R(s) loses rank. Hence A¯ 1 , A¯ 3 is detectable if and ( only if (A), C ) is detectable, and Fx and Fy are chosen such that A, Fx , C , Fy does not have zeros at the origin, thus proving the necessary condition for (10) to hold. □ Remark 5. This section focused on estimating faults modelled as polynomials. The results can be extended to the case of piecewise polynomial functions (where m0 , . . . , mk−1 are piecewise constant) provided that, for the case where k ≥ 2, f and its derivatives up to order k − 2 are continuous; this ensures that f (k) = 0 (almost everywhere). Such faults include constant and ramp incipient faults, or a combination of both. The continuity condition above is not practically restrictive, as the main challenge in fault estimation is dealing with slowly varying faults that are not easy to detect (Alwi, Edwards, & Tan, 2009). Finally, note that the bound in Proposition 3 holds regardless of initial conditions. ♯
˜ o . Since Remark 6. If ξ = 0, (8) becomes e˙ = Ao e + Lo M P > 0 and Θ < 0 from (10), in the absence of external disturbances, observer (6) is able to asymptotically reconstruct x¯ (and therefore f ). ♯ 3. Robust reconstruction of non-polynomial faults The scheme in Section 2 relies on the fault being (piecewise) polynomial and that its kth time derivative equal to zero for some finite k ∈ N+ as in (3). This section proposes an alternative method to relax this assumption. Remark 7. It can be seen that non-polynomial faults cover a larger class of faults than in (3), as it also includes periodic signals with non-zero kth time-derivatives for all k ∈ N+ , such as sinusoidal signals as in Chan, Tan, Trinh, and Kamal (2019), Chan, Tan, Trinh, Kamal, and Chiew (2019) and Delshad et al. (2016). ♯ Let the observer for system (1) and (2) have the structure:
[ e˙ =
A
Fx
0
0
]
)
(
)
ey = y − C xˆ − Du − Fy fˆ − My,2 xˆ , fˆ .
[
[ ] e−
L1 L2
A¯ 1
][ ] [ ] ˜ x,2 ξ M , + 0 Iq f˙ 0
[ ey +
L
[
Qx
0
Q¯ x
ξ˜
] T T
ef
.
(15)
˜x M
Define A¯ 3 = C Fy and Q¯ y = Qy 0p×q . Substituting for y ˜ y from (8), and ey from from (1), Mx (x, f ) and My (x, f ) from (2), M (14) into (15) results in
[
]
[
]
[ ] ˜x ) [ ] M ( ) ¯ ¯ e˙ = A1 − LA3 e + In+q −L + Q¯ x − LQ¯ y ξ˜ . ˜ My Ao Lo Qo (
(16)
˜o M
Proposition 8. Let P ∈ R(n+q)×(n+q) = P T > 0, Z = PL, and A = P A¯ 1 − Z A¯ 3 , and ϵ and µ be two positive scalars. Suppose there exists a set of values for P , Z , ϵ, γ , and µ that satisfies the following LMI:
⎡ ¯ A ⎢ PT ⎢ ⎢ T ⎢−Z ⎢ ⎣ Cf QT
P
−Z
CfT
−ϵ In+q
0
0
0
−ϵ Ip
0
0
0
−µIn+q
0
0
0
Q
⎤
0 ⎥ ⎥
0 ⎥ ≤ 0,
⎥ ⎥ 0 ⎦ −µIh
(17)
¯ = A + AT + ϵγ 2 In+q , and Q = P Q¯ x − Z Q¯ y . where Cf = 0 Iq , A ) ( Then the RMS gain from ξ˜ to ef is bounded by µ, i.e., γ ξ˜ , ef ≤ µ. ♯
[
]
Proof. Notice that (8) and (16) have identical structures. Therefore by using Assumption 1, the same arguments in Proposition 3 can be used on (16) to derive LMI ( (17),) which results in observer (14) being designed such that γ ξ˜ , ef ≤ µ, thus completing the proof. □ Proposition 9. For LMI (17) to be feasible, it is necessary that (A, Fx , C , Fy ) be minimum phase. ♯ T 2 Proof. For LMI (17) to be feasible,( A + A ) + ϵγ In+q ≤ 0 must hold. This implies A is stable and A¯ 1 , A¯ 3 is detectable. The de[ ] ( ) ¯ sI n+q − A1 ¯ ¯ tectability of A1 , A3 can be analysed using R(s) = . A¯ 3 ¯ ¯ By substituting for A1 from (15) and A3 from (16), it can be seen that R(s) loses rank if and only if s is a zero of (A, Fx , C , Fy ), thus completing the proof. □
Remark 10. This paper has shown how to estimate dynamic faults appearing in nonlinear functions; this has not been considered in the fault-estimation literature. Moreover, by exploiting weak assumptions about the fault (i.e., being expressed as a (piecewise) polynomial function of time as in (3) in Section 2), the necessary condition in Proposition 9 can be relaxed to the one in Proposition 4, i.e., that (A, Fx , C , Fy ) not have zeros at the origin and that (A, C ) be detectable. ♯ 4. Simulation example To demonstrate the efficacy of the proposed scheme, consider the following three-phase current motor model (Xiong & Saif, 2001): x˙ 1 = x2 , x˙ 3 = −0.3222x3 + 1.9 cos(x1 ) + u2 ,
˙ x˙ˆ = Axˆ + Bu + Fx fˆ + Mx,2 xˆ , fˆ + L1 ey , fˆ = L2 ey ,
(
3
]T
Let x¯ = xT f , xˆ¯ = xˆ T fˆ T , and e = x¯ − xˆ¯ = eTx Using (1), (2), (7), and (14), the following is obtained:
Proposition 4. For LMI (10) to hold, it is necessary that)(A, C ) be ( detectable, and that Fx , Fy be chosen such that A, Fx , C , Fy does not have zeros at the origin. ♯
R(s) = ⎢ ⎣
] T T
[
(14)
x˙ 2 = − 0.2703x2 − 12.01x3 sin(x1 ) + 48.04 sin(2x1 )
(18)
+ u1 + 12.01fx3 sin(x1 ) + ξ ,
Please cite this article as: W.-S. Chua, J.C.L. Chan, C.P. Tan et al., Robust fault reconstruction for a class of nonlinear systems. Automatica (2019) 108718, https://doi.org/10.1016/j.automatica.2019.108718.
4
W.-S. Chua, J.C.L. Chan, C.P. Tan et al. / Automatica xxx (xxxx) xxx
where x1 and f represent the rotor angle and fault, respectively. The inputs are given as u1 = 36.19, u2 = 1.9333, while the outputs are the rotor angle x1 and rotor acceleration x˙ 2 . The scheme by An et al. (2016) did not consider fault reconstruction, and is therefore not applicable. System (18) is clearly nonlinear, and thus the linear schemes (Chen et al., 2018; Gao et al., 2007; Ifqir et al., 2018) are inapplicable. The schemes in Saha et al. (2011, 2013) cannot be utilised without linearising (18). Furthermore, f is multiplicative and affects the output, and therefore the schemes in Delshad et al. (2016), Gao et al. (2019), Jiang and Chowdhury (2005), Laghrouche et al. (2015), Liu et al. (2016), Wu et al. (2018), Xiong and Saif (2001), Yan and Edwards (2007) and Zhao et al. (2019) are also not applicable. [The initial]T condition of the system was set as x(0) = 0.1 0 1 , while the observer was set to have zero initial conditions. Two fault scenarios were considered: (i) a polynomial fault f (t) = 0.1(t − 3) ∗ r(t − 3) − 0.1(t − 8) ∗ r(t − 8) − 0.05(t − 10) ∗ r(t − 10) + 0.05(t − 13) ∗ r(t − 13), and a non-polynomial fault f (t) = 0.2 sin(t) + 0.1(t − 3) ∗ r(t − 3) − 0.1(t − 10) ∗ r(t − 10), where r(t) is the Heaviside unit step function. Since f is not constant, the scheme in Chua et al. (2017) is also not applicable. Both fault scenarios were simulated with two cases: ξ = 0 for all t, and ξ (t) = 0.05 sin (0.2t + π /6) + 0.2. In scenario (i), f is a piecewise polynomial function of time, and therefore the method presented in Section 2 is applicable. The following design variables were chosen: k = 2, γ = 0.01, Fx = [ ]T [ ]T 0 1 0 , and Fy = 0 1 - it can be verified that the conditions in Proposition 4 are satisfied. LMI (10) was solved, yielding µ = 1.285, ϵ = 1359, and
⎡
732.8
⎢−425.6 ⎢
P =⎢ ⎢
0
−425.6 512.1
0 0
−1.240 −0.3894
0
584.2
0
0.0067
⎤
−0.5597⎥ ⎥ ⎥, 0 ⎥ −21.95 ⎦
⎣−1.240 −0.3984 0 242.6 0.0067 −0.5597 0 −21.95 4.005 ⎤ ⎡ ⎤ ⎡ −2.313 3.606 0.0003 ⎥ ⎢ ⎢ −1.293 ⎥ ⎢ 2.993 0.9993⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ L=⎢ 0 ⎥ ⎥ , λ (A0 ) = ⎢−0.1092⎥ . ⎢ 0 ⎥ ⎢ ⎣0.1161 0.0576⎦ ⎣−0.5275⎦ 1.085 0.6364 −0.3222
Fig. 2. Left: Non-polynomial fault f (solid) and its reconstructions - fˆn (dashed) for when ξ = 0, and fˆr (dash-dotted) for when ξ ̸ = 0. Right: The empirical RMS gain.
(19)
]T
0 1 - it can be verified that the condition in Proposition 9 is satisfied. LMI (17) was solved, yielding µ = 2.273, ϵ = 497.1, and
⎡
274.8
⎢−158.6
P =⎢ ⎣
0 0.1309
⎡
3.510
⎢ 2.962
L=⎢ ⎣
0
0.6213
−158.6 188.3
0
0
211.7
0
0.1309
⎤
−0.5885⎥ ⎥, ⎦ 0
−0.5885 0 2.003 ⎡ ⎤ ⎤ −2.096 −0.0009 ⎢ ⎥ ⎢ ⎥ 0.9948 ⎥ ⎥ , λ (A0 ) = ⎢ −1.416 ⎥ . ⎢ ⎥ ⎦ 0 ⎣ −1.128 ⎦ 1.129 −0.3222
sub-figure of Fig. 1 shows the empirical ( ) RMS gain ( ) (at time instant T ) from ξ to ef , defined as: γT ξ , ef = RT ef /RT (ξ ), where
√ ∫ T
Then in scenario (ii), f is clearly non-polynomial, and therefore the method in Section 3 is required.[ The following design ]T variables were chosen: γ = 0.01, Fx = 0 1 0 , and Fy =
[
Fig. 1. Left: Polynomial fault f (solid) and its reconstructions - fˆn (dashed) for when ξ = 0, and fˆr (dash-dotted) for when ξ ̸ = 0. Right: The empirical RMS gain.
(20)
The left sub-figure of Fig. 1 shows the polynomial fault f and its reconstructions. After the initial transients arising from the mismatch in the initial conditions, f is faithfully reconstructed when ξ = 0, demonstrating the efficacy of the scheme as per Remark 6. This is also observed in the case of ξ ̸ = 0. The right
RT (w) = T1 0 ∥w∥2 dt. We can see that γT ξ , ef is ultimately bounded by µ as calculated in (19), showing the robustness of the polynomial fault reconstruction method in Section 2. The same observations can also be drawn from Fig. 2 for the nonpolynomial fault reconstruction method in Section 3, thus also showing its efficacy.
(
)
5. Conclusion This paper presented two observer schemes to robustly reconstruct faults that enter the state and output equations of the system via nonlinear functions. The first scheme considers the case where the fault can be expressed as a polynomial function of time, while the second considers non-polynomial faults with unknown dynamics. LMI techniques were used to design the observer such that the RMS gain from the disturbances to the fault reconstruction error is bounded. Necessary conditions for the feasibility of the LMI were presented. Lastly, two simulation examples were performed and their results verify the efficacy of the proposed system. References Alwi, H., Edwards, C., & Tan, C. P. (2009). Sliding mode estimation schemes for incipient sensor faults. Automatica, 45(7), 1679–1685. An, H., Liu, J., Wang, C., & Wu, L. (2016). Disturbance observer-based antiwindup control for air-breathing hypersonic vehicles. IEEE Transactions on Industrial Electronics, 63(5), 3038–3049.
Please cite this article as: W.-S. Chua, J.C.L. Chan, C.P. Tan et al., Robust fault reconstruction for a class of nonlinear systems. Automatica (2019) 108718, https://doi.org/10.1016/j.automatica.2019.108718.
W.-S. Chua, J.C.L. Chan, C.P. Tan et al. / Automatica xxx (xxxx) xxx Chan, J. C. L., Tan, C. P., Trinh, H., & Kamal, M. A. S. (2019). State and fault estimation for a class of non-infinitely observable descriptor systems using two sliding mode observers in cascade. Journal of the Franklin Institute, 356(5), 3010–3029. Chan, J. C. L., Tan, C. P., Trinh, H., Kamal, M. A. S., & Chiew, Y. S. (2019). Robust fault reconstruction for a class of non-infinitely observable descriptor systems using two sliding mode observers in cascade. Applied Mathematics and Computation, 350, 78–92. Chen, L., Edwards, C., & Alwi, H. (2018). Sensor fault estimation using LPV sliding mode observers with erroneous scheduling parameters. Automatica, 101, 66–77. Chua, W. S., Tan, C. P., Aldeen, M., & Saha, S. (2017). A robust fault estimation scheme for a class of nonlinear systems. Asian Journal of Control, 19(2), 799–804. Delshad, S. S., Johansson, A., Darouach, M., & Gustafsson, T. (2016). Robust state estimation and unknown inputs reconstruction for a class of nonlinear systems: Multiobjective approach. Automatica, 64, 1–7. Gao, Z., Ding, S. X., & Ma, Y. (2007). Robust fault estimation approach and its application in vehicle lateral dynamic systems. Optimal Control Applications & Methods, 28(3), 143–156. Gao, Y., Liu, J., Sun, G., Liu, M., & Wu, L. (2019). Fault deviation estimation and integral sliding mode control design for Lipschitz nonlinear systems. Systems & Control Letters, 123, 8–15. Ifqir, S., Ichalal, D., Ait Oufroukh, N., & Mammar, S. (2018). Robust interval observer for switched systems with unknown inputs: Application to vehicle dynamics estimation. European Journal of Control, 44, 3–14.
5
Jiang, B., & Chowdhury, F. N. (2005). Parameter fault detection and estimation of a class of nonlinear systems using observers. Journal of the Franklin Institute, 342(7), 725–736. Laghrouche, S., Liu, J., Ahmed, F. S., Harmouche, M., & Wack, M. (2015). Adaptive second-order sliding mode observer-based fault reconstruction for PEM fuel cell air-feed system. IEEE Transactions on Control Systems Technology, 23(3), 1098–1109. Liu, J., Luo, W., Yang, X., & Wu, L. (2016). Robust model-based fault diagnosis for PEM fuel cell air-feed system. IEEE Transactions on Industrial Electronics, 63(5), 3261–3270. Saha, S., Aldeen, M., & Tan, C. P. (2011). Fault detection in transmission networks of power systems. International Journal of Electrical Power & Energy Systems, 33(4), 887–900. Saha, S., Aldeen, M., & Tan, C. P. (2013). Unsymmetrical fault diagnosis in transmission/distribution networks. International Journal of Electrical Power & Energy Systems, 45(1), 252–263. Wu, C., Liu, J., Xiong, Y., & Wu, L. (2018). Observer-based adaptive fault-tolerant tracking control of nonlinear nonstrict-feedback systems. IEEE Transactions on Neural Networks and Learning Systems, 29(7), 3022–3033. Xiong, Y., & Saif, M. (2001). Sliding mode observer for nonlinear uncertain systems. IEEE Transactions on Automatic Control, 46(12), 2012–2017. Yan, X.-G., & Edwards, C. (2007). Nonlinear robust fault reconstruction and estimation using a sliding mode observer. Automatica, 43(9), 1605–1614. Zhao, Y., Wang, J., Yan, F., & Shen, Y. (2019). Adaptive sliding mode fault-tolerant control for type-2 fuzzy systems with distributed delays. Information Sciences, 473, 227–238.
Please cite this article as: W.-S. Chua, J.C.L. Chan, C.P. Tan et al., Robust fault reconstruction for a class of nonlinear systems. Automatica (2019) 108718, https://doi.org/10.1016/j.automatica.2019.108718.
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Technical communique
Robust fault reconstruction for a class of nonlinear systems✩ ∗
Wen-Shyan Chua a,b , Joseph Chang Lun Chan a,c , Chee Pin Tan a , , Edwin Kah Pin Chong d , Sajeeb Saha e a
School of Engineering, Monash University Malaysia, Selangor 47500, Malaysia Selangor Human Resource Development Centre, Selangor 40100, Malaysia Division of Electronic Engineering, Jeonbuk National University, 567 Baekje-daero, Jeonju 54896, Republic of Korea d Department of Electrical and Computer Engineering, Colorado State University, CO 80523, USA e School of Engineering, Deakin University, Geelong, VIC 3220, Australia
b c
article
info
Article history: Received 1 May 2019 Received in revised form 15 October 2019 Accepted 29 October 2019 Available online xxxx Keywords: Nonlinearity Observers Fault detection Fault identification Robust estimation
a b s t r a c t This paper proposes two novel observer schemes for reconstructing faults in systems where the fault enters the state and output equations via nonlinear functions, which has not been considered in the literature. Two design methods are presented: one for the case where the fault dynamics are known and can be expressed as a polynomial function of time, and another for the case where the fault dynamics are unknown. The gains of the observer are designed using linear matrix inequalities (LMIs) such that the root-mean-square (RMS) gain from the uncertainties (or disturbances) to the fault reconstruction error is bounded. Necessary conditions for the feasibility of the LMIs are presented. Finally, a simulation example is shown to demonstrate the efficacy of the proposed scheme. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Modern engineering systems are prone to faults, which could have costly consequences if not properly addressed. Hence it is important that faults be detected, and even better reconstructed in real-time so that timely and appropriate corrective action can be taken. A popular such method is the observer, which consists of a model of the system being monitored, and processes the system’s inputs and outputs to reconstruct the fault (Chen, Edwards, & Alwi, 2018; Ifqir, Ichalal, Ait Oufroukh, & Mammar, 2018). Practical engineering systems are often nonlinear (An, Liu, Wang, & Wu, 2016; Laghrouche, Liu, Ahmed, Harmouche, & Wack, 2015; Liu, Luo, Yang, & Wu, 2016). Most observer schemes for nonlinear systems only consider faults entering the system linearly (Delshad, Johansson, Darouach, & Gustafsson, 2016; Gao, Liu, Sun, Liu, & Wu, 2019; Wu, Liu, Xiong, & Wu, 2018; Yan & ✩ This work was supported by the Ministry of Higher Education Malaysia under the Fundamental Research Grant Scheme (FRGS) Grant No. FRGS/1/2017/STG06/MUSM/02/1. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Maria Letizia Corradini under the direction of Editor André L. Tits. ∗ Corresponding author. E-mail addresses: [email protected] (W.-S. Chua), [email protected] (J.C.L. Chan), [email protected] (C.P. Tan), [email protected] (E.K.P. Chong), [email protected] (S. Saha).
Edwards, 2007; Zhao, Wang, Yan, & Shen, 2019). Comparatively, schemes to reconstruct faults entering the system via nonlinear functions are less developed. Many of them only considered faults in the state equation (Jiang & Chowdhury, 2005; Laghrouche et al., 2015; Liu et al., 2016; Xiong & Saif, 2001). Few works consider the case where faults enter through nonlinear functions in both the state and output equations, such as Saha, Aldeen, and Tan (2011, 2013) who linearised the system about an operating point, which may not represent the actual system if the states evolve far from the operating point, and Chua, Tan, Aldeen, and Saha (2017) who assumed the fault to be constant. This paper proposes observer schemes to reconstruct timevarying faults entering the system through nonlinear functions in both the state and output equations. Two methods are presented: the first considers polynomial faults (which can represent most time-varying faults (Gao, Ding, & Ma, 2007)), and the second considers even more general non-polynomial faults. The proposed methods improve on Chua et al. (2017) and Saha et al. (2011, 2013) where the faults can be time-varying and there is no need to linearise the system. The paper is structured as follows: Sections 2 and 3 present observer schemes to robustly reconstruct polynomial and nonpolynomial faults, respectively. A simulation verifies the efficacy of the proposed schemes in Section 4, and Section 5 draws conclusions.
https://doi.org/10.1016/j.automatica.2019.108718 0005-1098/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: W.-S. Chua, J.C.L. Chan, C.P. Tan et al., Robust fault reconstruction for a class of nonlinear systems. Automatica (2019) 108718, https://doi.org/10.1016/j.automatica.2019.108718.
2
W.-S. Chua, J.C.L. Chan, C.P. Tan et al. / Automatica xxx (xxxx) xxx
˜ x, 2 where M
2. Robust reconstruction of polynomial faults
[
= C
Consider the following faulty nonlinear system: x˙ = Ax + Bu + Mx (x, f ) + Qx ξ ,
where A ∈ R , B ∈ R , C ∈ R , Qx ∈ R , and Qy ∈ R are known and constant, whilst x ∈ Rn , y ∈ Rp , u ∈ Rm , f ∈ Rq , and ξ ∈ Rh are functions of time representing states, outputs, control inputs, faults, and disturbances, respectively. The aim is to reconstruct f using only u and y. Without loss of generality, Mx (x, f ) and My (x, f ) can be decomposed as follows: n×n
n×m
p×n
n×h
Mx (x, f ) = Fx f + Mx,2 (x, f ),
where Fx ∈ Rn×q and Fy ∈ Rp×q are known and constant. Denote the root-mean-square (RMS) of a signal b as R (b) = √ limT →∞
∫T
1 T
∥b∥2 dt.
0
Assumption 1. The following are assumed in this paper:
• The RMS of ξ is finite, i.e., R (ξ ) < ∞. • The nonlinear functions Mx,2 and My,2 are Lipschitz with Lipschitz constants γ1 , γ2 ∈ R+ , respectively. ♯ Assumption 2. In this section, it is assumed that f is a polynomial function of time as follows: f (t) = m0 + m1 t + m2 t 2 + · · · + mk−1 t k−1 ,
(3)
where mi (i = 0, 1, . . . , k − 1) are unknown constants, and k is known. ♯ Define f (i) as the ith derivative of f . From (3) it can be seen that f (k) = 0, and therefore (3) can be re-expressed as:
⎡
⎡
0
0
⎢f (k−1) ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ (k−2) ⎥ ⎢ ⎢f ⎥ ⎢ ⎢ . ⎥=⎢ ⎢ . ⎥ ⎢ ⎢ . ⎥ ⎢ ⎢ (2) ⎥ ⎢ ⎣ f ⎦ ⎣
Iq
0
0
Iq
f (k)
⎤
f (1)
⎡ ⎤ z˙ ⎣ ⎦ f˙
.. .
.. .
0
0
0
0
··· ··· ··· .. .
⎤⎡
f (k−1)
⎤
0
0
0
0
0
0
0
0 ⎥ ⎢f (k−2) ⎥ ⎥⎢ ⎥ 0 ⎥ ⎢f (k−3) ⎥
.. .
⎥⎢
⎥
⎢ .. ⎥ ⎥⎢ . . ⎥ ⎢ .. ⎥⎢ 0 ⎦ ⎣ f (1)
.. .
· · · Iq 0 · · · 0 Iq ⎡ ⎤ ˜1 0 A ⎣ ⎦ A˜ 2 0
0
⎥. ⎥ ⎥ ⎥ ⎦
(4)
f
⎡ ⎤
z f
x˙ = Ax + Bu + Fx f + Mx,2 (x, f ) + Qx ξ , z˙ = A˜ 1 z ,
(5)
y = Cx + Du + Fy f + My,2 (x, f ) + Qy ξ , f˙ = A˜ 2 z .
˙ z˙ˆ = A˜ 1 zˆ + L2 ey , fˆ = A˜ 2 zˆ + L3 ey ,
(6)
)
ey = y − C xˆ − Du − Fy fˆ − My,2 xˆ , fˆ . fT
]T
, xˆ¯ =
[
xˆ T
zˆ T
fˆ T
]T
, e = x¯ − xˆ¯ =
˜ x,2 = Mx,2 (x, f ) − Mx,2 (xˆ , fˆ ). By using (5) eTx eTz ef , and M and (6), the following error system is obtained:
[
⎡
A e˙ = ⎣0 0
0 A˜ 1 A˜ 2
A¯ 1
⎤
L
Q¯ x
Ao
˜x M
Lo
[
]
( ) ˜x M + Q¯ x − LQy ξ . ˜ My
(8)
Qo
˜o M
(
)
˜ o = Mo (x, f ) − Mo xˆ , fˆ . Observer (6) estimates f via where M fˆ , but from (8), ξ corrupts fˆ , and hence observer (6) needs to be designed so that fˆ is robust against ξ . Definition 1. Consider two signals wu : [0, ∞) → R and wy : [0, ∞) → R, and let Ξ be the set of all admissible signals wu . The RMS gain from wu to wy is given by
{ γ wu , wy = sup
(
)
R wy
(
}
)
R (wu )
: R (wu ) > 0, wu ∈ Ξ .
(9)
Proposition 3. Let P ∈ R(n+kq)×(n+kq) = P T > 0, Z = PL, and A = P A¯ 1 − Z A¯ 3 , and ϵ and µ be two positive scalars. Suppose there exists a set of values for P , Z , ϵ, γ , and µ that satisfies the following linear matrix inequality (LMI):
⎡ ¯ A ⎢ T ⎢P ⎢ ⎢−Z T ⎢ ⎢ ⎣ Cf
P
−Z
CfT
−ϵ In+kq
0
0
0
−ϵ Ip
0
0
0
−µIn+kq
0
0
T
0
Q
⎤
Q
⎥ ⎥ 0 ⎥ ⎥ ≤0, ⎥ 0 ⎦ 0 ⎥
(10)
−µIh
¯ = A + A + ϵγ 2 In+kq , and Q = P Q¯ x − ZQy . where Cf = 0 Iq , A ) ( Then the RMS gain from ξ to ef is bounded by µ, i.e., γ ξ , ef ≤ µ. ♯
]
[
T
Proof. Define a Lyapunov function V = eT Pe (≥ 0 and differ) entiate it with respect to time to get V˙ = eT PAo + ATo P e + ˜ o + 2eT PQo ξ . Recall the inequality X T Y + Y T X ≤ 1 X T X + 2eT PLo M ϵ T ˜ o , and ϵ Y Y (Yan & Edwards, 2007); let X T = eT PLo , Y = M [ ]T Mo = MxT (x, f ) 01×kq MyT (x, f ) . From Assumption 1, Mo is Lipschitz and its Lipschitz constant γ is bounded by the triangle ˜ o = Mo (x, f ) − inequality ( ) (i.e., γ ≤ γ1 + γ2 ). Then from (8), M
V˙ ≤eT
(
PAo + ATo P +
1
e PLo LTo Pe + ϵγ 2 ∥e∥2 .
1 T
ϵ
ϵ
PLo LTo P + γ 2 ϵ In+kq
)
e + 2eT PQo ξ .
(11)
Θ
PAo + ATo P + ϵγ 2 In+kq − P˜
(7)
[
− 1ϵ In+kq+p
0
− µ1 In+kq+h
0
]
P˜ T ≤ 0, (12)
where P˜ = PLo CfT PQo . Then substitute Θ from (11) into (12), expand to get Θ + µ1 CfT Cf + µ1 PQo QoT P ≤ 0, and applying
[
]
the Schur complement yields Ω :=
⎤
⎡
[ ] [ ] Fx ˜ x, 2 L1 Qx M 0 ⎦ e − L2 ey + 0 ξ + ⎣ 0 ⎦ , L3 0 0 0
] −L
Next, substitute Ao , Qo from (8) into (10) and apply the Schur complement to obtain
x˙ˆ = Axˆ + Bu + Fx fˆ + Mx,2 (xˆ , fˆ ) + L1 ey ,
zT
Consider the following observer for system (5):
[ T x ] T T
[
Substitute this into V˙ to get
System (1) can therefore be re-expressed using (2)–(4) as
Let x¯ =
)
(
˜o ≤ Mo xˆ , fˆ , and therefore 2eT PLo M
⎣ ⎦
(
0p×(k−1)q
e˙ = A¯ 1 − LA¯ 3 e + In+kq
p×h
(2)
My (x, f ) = Fy f + My,2 (x, f ),
]
stituting for y from (5) and ey from (6) into (7) yields (1)
y = Cx + Du + My (x, f ) + Qy ξ ,
¯ Mx,2 (x, f ) − Mx,2 (xˆ , fˆ ). Next, ( define ) A3 ˜ y = My,2 (x, f ) − My,2 xˆ , fˆ . SubFy and M
=
[
implying eT
[ ]
] e ξT Ω ξ
[
Θ + µ1 CfT Cf QoT P
PQo −µIh
]
≤ 0,
≤ 0. Then substitute for e, Cf , and
V˙ from (7), (10), and (11), respectively, and apply the Schur complement to get µ1 eTf ef ≤ µξ T ξ − V˙ . Integrate both sides from
Please cite this article as: W.-S. Chua, J.C.L. Chan, C.P. Tan et al., Robust fault reconstruction for a class of nonlinear systems. Automatica (2019) 108718, https://doi.org/10.1016/j.automatica.2019.108718.
W.-S. Chua, J.C.L. Chan, C.P. Tan et al. / Automatica xxx (xxxx) xxx
0 to T , divide by T , and use V (T ) ≥ 0 to get − µ
∫T
1 µT
2
∫T
V (0) T
≤
V (T )−V (0) T
≤
2
∥ξ ∥ dt − ∥ef ∥ dt. Let T → ∞ and take square roots 0 T 0 ( ) to get R ef ≤ µR (ξ ). Divide both ( sides ) by R (ξ ), take sup over all ξ ∈ Ξ , and use (9) to get γ ξ , ef ≤ µ, thus completing the proof. □
Proof. For (10) to hold, A + A(T + ϵγ) 2 In+kq ≤ 0 must hold. This implies A is stable, and that A¯ 1 , A¯ 3 is detectable (i.e., their unobservable modes must be stable). From 1 in Chua ( Corollary ) ¯ 1 , A¯ 3 are the values et al. (2017), the unobservable modes of A [ ] sIn+kq − A¯ 1 of s that make R(s) = lose rank (i.e., the values of s A¯ 3 that cause rank (R(s)) < n + kq). Substituting for A¯ 1 from (7) and A¯ 3 from (8) yields
⎡
sIn − A
0
0
sI(k−1)q − A˜ 1
0
A˜ 2
sIq
0
Fy
⎢
C
(
−Fx
⎤
0 ⎥ ⎥
⎦.
(13)
)
Substituting for A˜ 1 , A˜ 2 from (4), it can be seen that if s ̸ = 0 is an unobservable mode )mode of (A, C ), or if s = 0 is an( unobservable ) ( of A, Fx , C , Fy , then R(s) loses rank. Hence A¯ 1 , A¯ 3 is detectable if and ( only if (A), C ) is detectable, and Fx and Fy are chosen such that A, Fx , C , Fy does not have zeros at the origin, thus proving the necessary condition for (10) to hold. □ Remark 5. This section focused on estimating faults modelled as polynomials. The results can be extended to the case of piecewise polynomial functions (where m0 , . . . , mk−1 are piecewise constant) provided that, for the case where k ≥ 2, f and its derivatives up to order k − 2 are continuous; this ensures that f (k) = 0 (almost everywhere). Such faults include constant and ramp incipient faults, or a combination of both. The continuity condition above is not practically restrictive, as the main challenge in fault estimation is dealing with slowly varying faults that are not easy to detect (Alwi, Edwards, & Tan, 2009). Finally, note that the bound in Proposition 3 holds regardless of initial conditions. ♯
˜ o . Since Remark 6. If ξ = 0, (8) becomes e˙ = Ao e + Lo M P > 0 and Θ < 0 from (10), in the absence of external disturbances, observer (6) is able to asymptotically reconstruct x¯ (and therefore f ). ♯ 3. Robust reconstruction of non-polynomial faults The scheme in Section 2 relies on the fault being (piecewise) polynomial and that its kth time derivative equal to zero for some finite k ∈ N+ as in (3). This section proposes an alternative method to relax this assumption. Remark 7. It can be seen that non-polynomial faults cover a larger class of faults than in (3), as it also includes periodic signals with non-zero kth time-derivatives for all k ∈ N+ , such as sinusoidal signals as in Chan, Tan, Trinh, and Kamal (2019), Chan, Tan, Trinh, Kamal, and Chiew (2019) and Delshad et al. (2016). ♯ Let the observer for system (1) and (2) have the structure:
[ e˙ =
A
Fx
0
0
]
)
(
)
ey = y − C xˆ − Du − Fy fˆ − My,2 xˆ , fˆ .
[
[ ] e−
L1 L2
A¯ 1
][ ] [ ] ˜ x,2 ξ M , + 0 Iq f˙ 0
[ ey +
L
[
Qx
0
Q¯ x
ξ˜
] T T
ef
.
(15)
˜x M
Define A¯ 3 = C Fy and Q¯ y = Qy 0p×q . Substituting for y ˜ y from (8), and ey from from (1), Mx (x, f ) and My (x, f ) from (2), M (14) into (15) results in
[
]
[
]
[ ] ˜x ) [ ] M ( ) ¯ ¯ e˙ = A1 − LA3 e + In+q −L + Q¯ x − LQ¯ y ξ˜ . ˜ My Ao Lo Qo (
(16)
˜o M
Proposition 8. Let P ∈ R(n+q)×(n+q) = P T > 0, Z = PL, and A = P A¯ 1 − Z A¯ 3 , and ϵ and µ be two positive scalars. Suppose there exists a set of values for P , Z , ϵ, γ , and µ that satisfies the following LMI:
⎡ ¯ A ⎢ PT ⎢ ⎢ T ⎢−Z ⎢ ⎣ Cf QT
P
−Z
CfT
−ϵ In+q
0
0
0
−ϵ Ip
0
0
0
−µIn+q
0
0
0
Q
⎤
0 ⎥ ⎥
0 ⎥ ≤ 0,
⎥ ⎥ 0 ⎦ −µIh
(17)
¯ = A + AT + ϵγ 2 In+q , and Q = P Q¯ x − Z Q¯ y . where Cf = 0 Iq , A ) ( Then the RMS gain from ξ˜ to ef is bounded by µ, i.e., γ ξ˜ , ef ≤ µ. ♯
[
]
Proof. Notice that (8) and (16) have identical structures. Therefore by using Assumption 1, the same arguments in Proposition 3 can be used on (16) to derive LMI ( (17),) which results in observer (14) being designed such that γ ξ˜ , ef ≤ µ, thus completing the proof. □ Proposition 9. For LMI (17) to be feasible, it is necessary that (A, Fx , C , Fy ) be minimum phase. ♯ T 2 Proof. For LMI (17) to be feasible,( A + A ) + ϵγ In+q ≤ 0 must hold. This implies A is stable and A¯ 1 , A¯ 3 is detectable. The de[ ] ( ) ¯ sI n+q − A1 ¯ ¯ tectability of A1 , A3 can be analysed using R(s) = . A¯ 3 ¯ ¯ By substituting for A1 from (15) and A3 from (16), it can be seen that R(s) loses rank if and only if s is a zero of (A, Fx , C , Fy ), thus completing the proof. □
Remark 10. This paper has shown how to estimate dynamic faults appearing in nonlinear functions; this has not been considered in the fault-estimation literature. Moreover, by exploiting weak assumptions about the fault (i.e., being expressed as a (piecewise) polynomial function of time as in (3) in Section 2), the necessary condition in Proposition 9 can be relaxed to the one in Proposition 4, i.e., that (A, Fx , C , Fy ) not have zeros at the origin and that (A, C ) be detectable. ♯ 4. Simulation example To demonstrate the efficacy of the proposed scheme, consider the following three-phase current motor model (Xiong & Saif, 2001): x˙ 1 = x2 , x˙ 3 = −0.3222x3 + 1.9 cos(x1 ) + u2 ,
˙ x˙ˆ = Axˆ + Bu + Fx fˆ + Mx,2 xˆ , fˆ + L1 ey , fˆ = L2 ey ,
(
3
]T
Let x¯ = xT f , xˆ¯ = xˆ T fˆ T , and e = x¯ − xˆ¯ = eTx Using (1), (2), (7), and (14), the following is obtained:
Proposition 4. For LMI (10) to hold, it is necessary that)(A, C ) be ( detectable, and that Fx , Fy be chosen such that A, Fx , C , Fy does not have zeros at the origin. ♯
R(s) = ⎢ ⎣
] T T
[
(14)
x˙ 2 = − 0.2703x2 − 12.01x3 sin(x1 ) + 48.04 sin(2x1 )
(18)
+ u1 + 12.01fx3 sin(x1 ) + ξ ,
Please cite this article as: W.-S. Chua, J.C.L. Chan, C.P. Tan et al., Robust fault reconstruction for a class of nonlinear systems. Automatica (2019) 108718, https://doi.org/10.1016/j.automatica.2019.108718.
4
W.-S. Chua, J.C.L. Chan, C.P. Tan et al. / Automatica xxx (xxxx) xxx
where x1 and f represent the rotor angle and fault, respectively. The inputs are given as u1 = 36.19, u2 = 1.9333, while the outputs are the rotor angle x1 and rotor acceleration x˙ 2 . The scheme by An et al. (2016) did not consider fault reconstruction, and is therefore not applicable. System (18) is clearly nonlinear, and thus the linear schemes (Chen et al., 2018; Gao et al., 2007; Ifqir et al., 2018) are inapplicable. The schemes in Saha et al. (2011, 2013) cannot be utilised without linearising (18). Furthermore, f is multiplicative and affects the output, and therefore the schemes in Delshad et al. (2016), Gao et al. (2019), Jiang and Chowdhury (2005), Laghrouche et al. (2015), Liu et al. (2016), Wu et al. (2018), Xiong and Saif (2001), Yan and Edwards (2007) and Zhao et al. (2019) are also not applicable. [The initial]T condition of the system was set as x(0) = 0.1 0 1 , while the observer was set to have zero initial conditions. Two fault scenarios were considered: (i) a polynomial fault f (t) = 0.1(t − 3) ∗ r(t − 3) − 0.1(t − 8) ∗ r(t − 8) − 0.05(t − 10) ∗ r(t − 10) + 0.05(t − 13) ∗ r(t − 13), and a non-polynomial fault f (t) = 0.2 sin(t) + 0.1(t − 3) ∗ r(t − 3) − 0.1(t − 10) ∗ r(t − 10), where r(t) is the Heaviside unit step function. Since f is not constant, the scheme in Chua et al. (2017) is also not applicable. Both fault scenarios were simulated with two cases: ξ = 0 for all t, and ξ (t) = 0.05 sin (0.2t + π /6) + 0.2. In scenario (i), f is a piecewise polynomial function of time, and therefore the method presented in Section 2 is applicable. The following design variables were chosen: k = 2, γ = 0.01, Fx = [ ]T [ ]T 0 1 0 , and Fy = 0 1 - it can be verified that the conditions in Proposition 4 are satisfied. LMI (10) was solved, yielding µ = 1.285, ϵ = 1359, and
⎡
732.8
⎢−425.6 ⎢
P =⎢ ⎢
0
−425.6 512.1
0 0
−1.240 −0.3894
0
584.2
0
0.0067
⎤
−0.5597⎥ ⎥ ⎥, 0 ⎥ −21.95 ⎦
⎣−1.240 −0.3984 0 242.6 0.0067 −0.5597 0 −21.95 4.005 ⎤ ⎡ ⎤ ⎡ −2.313 3.606 0.0003 ⎥ ⎢ ⎢ −1.293 ⎥ ⎢ 2.993 0.9993⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ L=⎢ 0 ⎥ ⎥ , λ (A0 ) = ⎢−0.1092⎥ . ⎢ 0 ⎥ ⎢ ⎣0.1161 0.0576⎦ ⎣−0.5275⎦ 1.085 0.6364 −0.3222
Fig. 2. Left: Non-polynomial fault f (solid) and its reconstructions - fˆn (dashed) for when ξ = 0, and fˆr (dash-dotted) for when ξ ̸ = 0. Right: The empirical RMS gain.
(19)
]T
0 1 - it can be verified that the condition in Proposition 9 is satisfied. LMI (17) was solved, yielding µ = 2.273, ϵ = 497.1, and
⎡
274.8
⎢−158.6
P =⎢ ⎣
0 0.1309
⎡
3.510
⎢ 2.962
L=⎢ ⎣
0
0.6213
−158.6 188.3
0
0
211.7
0
0.1309
⎤
−0.5885⎥ ⎥, ⎦ 0
−0.5885 0 2.003 ⎡ ⎤ ⎤ −2.096 −0.0009 ⎢ ⎥ ⎢ ⎥ 0.9948 ⎥ ⎥ , λ (A0 ) = ⎢ −1.416 ⎥ . ⎢ ⎥ ⎦ 0 ⎣ −1.128 ⎦ 1.129 −0.3222
sub-figure of Fig. 1 shows the empirical ( ) RMS gain ( ) (at time instant T ) from ξ to ef , defined as: γT ξ , ef = RT ef /RT (ξ ), where
√ ∫ T
Then in scenario (ii), f is clearly non-polynomial, and therefore the method in Section 3 is required.[ The following design ]T variables were chosen: γ = 0.01, Fx = 0 1 0 , and Fy =
[
Fig. 1. Left: Polynomial fault f (solid) and its reconstructions - fˆn (dashed) for when ξ = 0, and fˆr (dash-dotted) for when ξ ̸ = 0. Right: The empirical RMS gain.
(20)
The left sub-figure of Fig. 1 shows the polynomial fault f and its reconstructions. After the initial transients arising from the mismatch in the initial conditions, f is faithfully reconstructed when ξ = 0, demonstrating the efficacy of the scheme as per Remark 6. This is also observed in the case of ξ ̸ = 0. The right
RT (w) = T1 0 ∥w∥2 dt. We can see that γT ξ , ef is ultimately bounded by µ as calculated in (19), showing the robustness of the polynomial fault reconstruction method in Section 2. The same observations can also be drawn from Fig. 2 for the nonpolynomial fault reconstruction method in Section 3, thus also showing its efficacy.
(
)
5. Conclusion This paper presented two observer schemes to robustly reconstruct faults that enter the state and output equations of the system via nonlinear functions. The first scheme considers the case where the fault can be expressed as a polynomial function of time, while the second considers non-polynomial faults with unknown dynamics. LMI techniques were used to design the observer such that the RMS gain from the disturbances to the fault reconstruction error is bounded. Necessary conditions for the feasibility of the LMI were presented. Lastly, two simulation examples were performed and their results verify the efficacy of the proposed system. References Alwi, H., Edwards, C., & Tan, C. P. (2009). Sliding mode estimation schemes for incipient sensor faults. Automatica, 45(7), 1679–1685. An, H., Liu, J., Wang, C., & Wu, L. (2016). Disturbance observer-based antiwindup control for air-breathing hypersonic vehicles. IEEE Transactions on Industrial Electronics, 63(5), 3038–3049.
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Please cite this article as: W.-S. Chua, J.C.L. Chan, C.P. Tan et al., Robust fault reconstruction for a class of nonlinear systems. Automatica (2019) 108718, https://doi.org/10.1016/j.automatica.2019.108718.