多速率传感器融合下的联合最优滤波与故障检测
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"这篇研究论文探讨了在存在未知输入情况下的多速率传感器融合中的联合最优滤波与故障检测问题。在复杂环境中，由于各种因素如随机噪声、未知输入（UIs）以及故障的存在，确保所有传感器具有相同的采样率是困难的。作者提出了解决多速率传感器系统中这一问题的方法，旨在同时实现数据融合的最优滤波效果和有效的故障检测。"
文章深入研究了多速率传感器系统，在这个系统中，各个传感器可能由于分布式或异构性而具有不同的采样率。在实际应用中，这种差异可能导致数据处理和融合的挑战。同时，系统可能会受到随机噪声的干扰，还有可能出现未知输入，这些未知输入可能是外部干扰或系统内部的非建模动态。此外，系统的组件可能会出现故障，这会影响传感器的数据质量和系统的总体性能。
为了应对这些挑战，论文提出了一个联合最优滤波和故障检测的框架。最优滤波的目标是通过融合来自不同采样率传感器的数据，以最精确地估计系统状态，即使在存在未知输入的情况下也是如此。通过这种方式，可以提高系统对环境变化和不确定性因素的适应能力。
故障检测部分则关注于识别和隔离系统中的异常行为。论文中可能涉及了设计特定的故障检测器，用于监测系统状态的变化，以确定是否存在故障。这种检测器可能基于滤波器的残差分析或其他统计方法，以区分正常操作与故障状态。
论文经过多次修订，最终在2015年10月被接受，并于同年11月在线发布。关键词包括多速率传感器系统、故障检测、扰动解耦、最优滤波和未知输入观测器，表明该研究涵盖了从信号处理到系统诊断的多个关键领域。
这篇研究论文提供了对复杂系统中多速率传感器融合与故障检测问题的深刻洞察，为实际工程应用提供了解决方案，有助于提升系统的稳定性和可靠性。通过结合最优滤波理论和故障检测技术，该工作有望改进未来多传感器系统在不确定条件下的性能表现。
H. Geng et al. / Information Fusion 29 (2016) 57–67 59
Fig. 1. The joint optimal filtering and FD scheme for the proposed multi-rate sensor fusion systems.
claim the presence of possible faults as soon as any a measurement is
available. The dynamical model and sensor model are
x
k+1
= Ax
k
+ B
u
u
k
+ B
d
d
k
+ B
w
w
k
+ B
u
f
a
k
, (1)
y
j,n
j
k
= C
j
x
n
j
k
+ v
j,n
j
k
+ E
j
f
s
j,n
j
k
, j = 1, 2,...,p, (2)
where x
k
∈
n
is the plant state evolving at a basic period h ; y
j,n
j
k
∈
m
is the j-th sensor measurement sampled at the period n
j
h,where
n
j
is an integer; u
k
∈
r
is the known input and d
k
∈
q
is called UI
representing unexpected disturbances or modeling errors; f
a
k
∈
r
is
the fault occurring in the actuator; f
s
n
j
k
∈
m
represents the fault oc-
curring in the j-th sensor; A, B
l
(l ∈ {u, d, w}), C
j
and E
j
are known with
appropriate dimensions; p is the sensor number; w
k
and v
j,n
j
k
are
independent zero-mean, Gaussian, and white noises with covariance
being Q
k
and R
j,n
j
k
, respectively.
Here the UIs are a specific category of input data together with
noise and faults, i.e., UIs represent the unexpected inputs without any
a priori information about their value or evolution; noises represent
the unexpected inputs with known statistics; faults represents the in-
puts to be detected, which may appear at any a time with any value.
Specifically, the UI term B
d
d
k
can be used to depict a number of mod-
eling uncertainties, e.g., unknown interconnecting effect in the large
scale systems, linearization error of nonlinear systems, model reduc-
tion error and the unpredictable parameter variations. In general, UIs
and faults should be somewhat different. Otherwise, faults cannot be
reliably detected in the presence of both UIs and noises. Our proposed
UI-decoupling condition later can be regarded as the sufficient con-
dition of separating UIs and faults.
Remark 1. Let B
w
= 0, R
j,n
j
k
= 0, the stochastic model in (1) and (2)
is reduced to the deterministic one considered in [17] and [18].In[17],
the lifting technique was used to convert the multi-rate system into
a linear time-invariant (LTI) model with slow sampling rate, based on
which a state observer is constructed as in the single-rate case. Then a
residual signal is derived from the difference between the actual and
estimated measurements of the single-rate observer. However, as the
observer outputs at a slow rate, the residual signal is hence gener-
ated at the same slow rate, which is not ideal for fast-rate FD. In [18],
a bank of single-rate observers are derived and each observer is ded-
icated to each sensor. Such a one-to-one strategy is simple because it
avoids the design difficulty of causality constraint due to multi-rate
nature, and however it is not suitable to fast and reliable fault de-
tection by the fact that each fault report just utilizes the information
from the corresponding sensor, instead of all information from multi-
rate sensors. Let B
u
= [B
f
0], E
j
= [0 D
f
], R
j,n
j
k
= 0andreplace f
a
k
and
f
s
j,n
j
k
by f
k
= [ f
a
k
f
s
k
]
T
, the stochastic model in (1) and (2) is reduced
to the deterministic one in [21] where an observer-based FD filter was
designed through minimizing the influence of the norm-bounded UI
on the residual in the sense of H
∞
/H
∞
or H
∞
/H
−
optimization. In
general, the above models do not consider stochastic noise, which is
widely accepted in modeling state evolvement and sensor sampling,
especially in complex environment, moreover, they only concern the
fault detection and the state estimation is not involved in. It motivates
us to propose the problem of joint filtering and FD of multi-rate sen-
sor fusion with both deterministic and stochastic uncertainties, and
further present multi-rate FD scheme including multi-rate observer,
residual generation and hypothesis test as shown later.
Remark 2. To the best of our knowledge, this is the first attempt to
establish a MRO for FD of the multi-rate system under multiple uncer-
tainties. As mentioned in the introduction, there are three common
ideas for observer design in multi-rate FD with UIs.
•
Thefirstideaistotransformthemulti-ratesystemintoasingle-
rate one, based on which a single-rate observer is designed for
residual generation, and the residual is generated at a slow rate
[17]. If the stochastic noises are augmented as part of unknown in-
puts, then the dimension of the total unknown inputs will become
larger. It brings out two disadvantages. One is that the condition of
decoupling the residual with unknown inputs is more difficult to
meet. The other is that statistical information about noises is not
utilized, which is not desirable for fast and reliable fault detection.
•
The second idea is to divide the multi-rate system into a collection
of several subsystems so that only the UI-decoupling subsystem
is considered in the followed design. Then a bank of observers
running at different rates are designed where each observer is
dedicated to one sensor [18]. However, each observer just utilizes
its corresponding sensor’s measurements, instead of all measure-
ments from multi-rate sensors. It is not a good consideration for
fast and reliable fault detection. Moreover, when the stochastic
noise is involved in, it is impossible to decouple the noises from
both state equations and output equations of the subsystems. In
other words, this idea is hardly extended to the case of stochastic
noises.
•
Thethirdideaistotransformthemulti-ratesystemintoasingle-
rate one, based on which a single-rate observer is designed for
residual generation by maximizing the effect of the fault on the
residual while minimizing the influence of the UI on the residual
[21]. However, if this observer is used to deal with the stochastic
UI (i.e., the noises), the noises have to be regarded as part of the
deterministic UI and its effect on the residual is minimized by H
∞
optimization, and hence the statistics of the noise is not used in
observer design.
In general, the presence of stochastic noises will lead to different
observer design.
Denote N = LCM
{n
j
, j = 1, 2,...,p}. B
=
˚
A
B
, for ∈ {u, d,
w},
˚
A =
⎡
⎢
⎢
⎣
I
n
0 ··· 0
AI
n
··· 0
.
.
.
.
.
.
.
.
.
.
.
.
A
N−1
··· AI
n
⎤
⎥
⎥
⎦
.
Applying the lifting technique to (1) and (2),wehavethelifted
system with a slow period Nh:
x
k+1
= Ax
k
+ B
u
u
k
+ B
d
d
k
+ B
w
w
k
+ B
u
f
a
k
, (3)
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