连续时间系统中基于学习观察器的故障重建方法
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"《连续时间系统中的故障重构学习观察器》是一篇发表于2016年3月的亚洲控制期刊的研究论文，作者是Qingxian Jia、Wen Chen、Yingchun Zhang和Huayi Li。本文主要探讨了在连续时间系统中利用学习观察器（Learning Observers，LO）进行故障重构的问题。
论文的核心内容关注如何设计学习观察器来同时重建系统的状态和执行器故障。作者首先提出了学习观察器存在的必要条件，这些条件用于验证针对故障重构的学习观察器是否存在。通过使用线性矩阵不等式技术，作者开发了一种有效的LO设计算法，使得观察器增益矩阵可以方便地求解。此外，文章还扩展了对传感器故障重构的研究，进一步提升了观察器的适应性和鲁棒性。
论文的关键字包括：故障重构、学习观察器、自适应观察器以及线性矩阵不等式。论文引入时指出，随着系统复杂度的增加和对故障诊断要求的提高，对于连续时间系统中故障的准确识别和快速响应变得尤为重要。通过实际的仿真示例，作者展示了基于学习观察器的故障重构方法的有效性和实用性，这对于工业控制系统的健康监控和故障恢复策略具有重要意义。这篇论文为故障检测与隔离、自适应控制理论的发展以及实际应用提供了新的视角和工具。"
Q. Jia et al.: Fault Reconstruction for Continuous-time Systems via Learning Observers 551
Further, fault-reconstructing error e
f
(t) has a new form
e
f
(t)=f
a
(t)−K
1
f
a
(t − 𝜏)−K
2
Ce
x
(t)
− K
1
f
a
(t − 𝜏)+K
1
f
a
(t − 𝜏)
= K
1
e
f
(t − 𝜏)−K
2
Ce
x
(t)+
f
a
(t)
(6)
where
f
a
(t)∶=f
a
(t)−K
1
f
a
(t − 𝜏).
To derive the stability and convergence of the LO
such that accurate reconstruction of time-varying faults
can be guaranteed, the following assumption is needed.
Assumption 1. Assume that
f
a
(t)
∞
≤ k
f
,wherek
f
is a
small positive constant.
Note that Assumption 1 has a restriction on actu-
ator faults f
a
(t).GivenK
1
= I
m
and constant actuator
faults occur in the system (1), we have
f
a
(t) ≡ 0. There-
fore, it follows from (6) that
e
f
(t)=e
f
(t − 𝜏)−K
2
Ce
x
(t). (7)
Remark 1. In [8,9], fast AOs are proposed to reconstruct
time-varying faults under the assumption that the deriva-
tive of f
a
(t) is norm-bounded, i.e.
f
a
(t) ≤ f
1
where
0 ≤ f
1
≤ ∞. Comparatively, the fault-constraining
condition described in Assumption 1 is less restrictive
than that in [8,9] because the norm-bounded condition
on derivative of actuator fault vector is not required in
this paper. Under Assumption 1, the proposed LO-based
fault-reconstructing strategy can be applied to recon-
struct time-varying faults, especially fast-varying faults.
In addition, by selecting a gain matrix K
1
, which is close
to I
m
, and a sufciently small learning interval 𝜏, the pro-
posed LO can achieve accurate reconstruction of actua-
tor faults; no matter they are constant or time-varying.
3.2 Stability analysis of the learning observer
In this subsection, the stability and convergence
of the proposed LO will be proved and reconstruction
of actuator faults will be guaranteed. The following
theorem serves this purpose.
Theorem 1. Suppose that Assumption 1 holds, if there
exist positive-denite symmetric matrices P ∈ R
n×n
, Q ∈
R
m×m
and K
1
∈ R
m×m
, and matrices K
2
∈ R
m×p
and
Y ∈ R
n×p
such that the following relations hold
PA + A
T
P − YC − C
T
Y
T
< 0, (8)
E
T
P = 𝛼
1
K
2
C, (9)
and
𝛽
1
K
T
1
K
1
− Q ≤ 0, (10)
where 𝛼
1
=(1 + 𝜖
0
)𝜆
max
(Q),and𝛽
1
= 𝛼
1
(1 + 𝜖
1
) with
𝜖
0
≥ 0and𝜖
1
> 0, then the LO (4) can guarantee that
state-estimating and fault-reconstructing errors are uni-
formly ultimately bounded, and observer gain matrix L
can be determined by L = P
−1
Y.
Proof. Consider the following Lyapunov function candi-
date:
V(t)=e
T
x
(t)Pe
x
(t)+
∫
t
t−𝜏
e
T
f
(s)Qe
f
(s)ds (11)
where P and Q are positive-denite symmetric matrices.
The derivative of the Laypunov candidate with
respect to time can be derived as
V(t) ≤e
T
x
(t)
P(A − LC)+(A − LC)
T
P
e
x
(t)
+ 2e
T
x
(t)PEe
f
(t)+e
T
f
(t)Qe
f
(t)
− e
T
f
(t − 𝜏)Qe
f
(t − 𝜏)
≤e
T
x
(t)
P(A − LC)+(A − LC)
T
P
e
x
(t)
+ 2e
T
x
(t)PEe
f
(t)−𝜖
0
e
T
f
(t)Qe
f
(t)
+ 𝛼
1
e
T
f
(t)e
f
(t)−e
T
f
(t − 𝜏)Qe
f
(t − 𝜏)
(12)
where 𝛼
1
=(1 + 𝜖
0
)𝜆
max
(Q) with 𝜖
0
≥0.
Substituting (6) into (12) leads to
V(t) ≤e
T
x
(t)
P(A − LC)+(A − LC)
T
P
e
x
(t)
+ 2e
T
x
(t)
PE − 𝛼
1
(K
2
C)
T
K
1
e
f
(t − 𝜏)
− e
T
x
(t)
2PE − 𝛼
1
(K
2
C)
T
K
2
Ce
x
(t)
+ 2e
T
x
(t)
PE − 𝛼
1
(K
2
C)
T
f
a
(t)
+ 𝛼
1
e
T
f
(t − 𝜏)K
T
1
K
1
e
f
(t − 𝜏)
+ 2𝛼
1
e
T
f
(t − 𝜏)K
T
1
f
a
(t)+𝛼
1
f
T
a
(t)
f
a
(t)
− 𝜖
0
e
T
f
(t)Qe
f
(t)−e
T
f
(t − 𝜏)Qe
f
(t − 𝜏).
(13)
Further, with the aid of (9), the above inequality can
be simplied as
V(t) ≤e
T
x
(t)
P(A − LC)+(A − LC)
T
P
e
x
(t)
− 𝜖
0
e
T
f
(t)Qe
f
(t)+2𝛼
1
e
T
f
(t − 𝜏)K
T
1
f
a
(t)
+ 𝛼
1
f
T
a
(t)
f
a
(t)−e
T
f
(t − 𝜏)Qe
f
(t − 𝜏)
+ 𝛼
1
e
T
f
(t − 𝜏)K
T
1
K
1
e
f
(t − 𝜏).
(14)
© 2014 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd
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