非线性系统鲁棒辨识：处理缺失观测值的状态空间模型
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更新于2024-08-26
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"这篇研究论文探讨了在状态空间模型结构中，缺失观测值的非线性系统的鲁棒辨识问题。文章关注的是那些输出测量数据受到异常值污染，并且部分输出测量数据随机丢失的情况。在实际工业过程中，这些问题很常见，论文全面考虑了这两种挑战。通过采用基于稳健的t分布观测模型，旨在提高系统识别的健壮性，以应对不完整和噪声数据的挑战。"
正文:
在工业自动化和控制系统中，非线性系统的辨识是一个关键问题，尤其是在面对实际应用中的观测数据不完整和存在异常值的情况下。这篇被接受发表在《IEEE Transactions on Industrial Informatics》期刊的论文，由Xianqiang Yang、Xin Liu和Shen Yin共同撰写，深入研究了如何在这些复杂条件下进行鲁棒的系统识别。
首先，论文指出在处理工业过程数据时，异常值和随机缺失观测是普遍存在的问题。异常值可能源于传感器故障、测量误差或数据处理错误，而随机缺失观测可能是由于通信中断或其他系统不可预见的行为。这些问题会严重影响系统模型的准确性和稳定性。
论文的核心贡献在于提出了一种新的鲁棒辨识方法，该方法采用了基于t分布的观测模型。与传统的高斯分布假设不同，t分布能够更好地捕捉数据中的异常点，因为它具有更厚的尾部，对异常值具有更高的抵抗力。通过这种方式，即使在存在异常观测的情况下，也能得到更稳定和可靠的系统参数估计。
此外，对于随机缺失观测的问题，论文提出了一个综合处理框架。该框架考虑了观测数据的不完全性，并在状态空间模型的框架内进行了建模。这种建模方法允许系统在缺失数据的情况下仍然能够进行有效的参数学习，提高了模型的适应性和鲁棒性。
论文还可能包含了数值模拟和实际工业案例分析，以验证所提出方法的有效性。通过这些实验，作者们展示了他们的方法在处理噪声和缺失数据时相比传统方法的优势，可能包括更精确的系统模型，以及在有挑战性环境下的稳定性能。
总结来说，这篇论文对工业界和学术界都具有重要的价值，它提供了一种新的工具来处理非线性系统识别中的现实挑战。通过对异常值和随机缺失观测的全面考虑，该研究有望推动工业自动化和控制领域的理论发展和实践应用。
1551-3203 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TII.2018.2871194, IEEE
Transactions on Industrial Informatics
SUBMITTED TO IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS 3
Alogrithm 1 Particle filtering with missing outputs
1: Generate L samples {x
l
0
}
l=1,2,...,L
from the initial PDF
p(x
0
) and the weight of each sample is set to w
l
0
= 1/L.
Set k = 1.
2: repeat
3: Generate new particles {x
l
k
}
l=1,2,...,L
according to Eq.
(5).
4: When y
k
is available, calculate the corresponding
weights for all the new particles with Eq. (6); when y
k
is missing, calculate the corresponding weights with Eq.
(7).
5: Calculate N
eff
according to Eq. (8). If N
eff
< N
thd
,
execute resampling step and replace all the particles
generated in step 3 with resampled particles. The weights
of all the new particles are reset to 1/L. If N
eff
≥ N
thd
,
go to the next step.
6: Let k = k + 1 and go to repeat.
7: until k=N.
the real PDF p(x
1:k
|y
o
1
:o
γ
, Θ
s
), the principle of importance
sampling is used here [15]. Suppose q(x
1:k
|y
o
1
:o
γ
, Θ
s
) is
a proposal from which {x
l
1:k
, l = 1, 2, . . . , L} are easily
generated and q(·) is often called an importance density
[15]. The importance density is usually selected as [15], [24]
q(x
k
|y
o
1
:o
γ
, Θ
s
) = p(x
k
|x
k−1
, Θ
s
). (5)
where p(x
k
|x
k−1
, Θ
s
) represents the probability of state tran-
sition at instant k − 1. Based on this choice, the corresponding
weights for the new particles at time k are derived as [24]
w
l
k
∝ w
l
k−1
p(y
k
|x
l
k
, Θ
s
). (6)
When the output data are missing, the new particles are
generated from p(x
k
|x
l
k−1
, Θ
s
) and the corresponding weights
keep unchanged [15], [24]
w
l
k
= w
l
k−1
. (7)
Moreover, the resampling step is applied here to deal with
the degeneracy of the particles [15]. The weight of every
particle after resampling is then reset to w
l
k
= 1/L [24].
In order to judge whether the resampling step should be
performed, N
eff
is calculated to denote the effective particle
number [15]
N
eff
=
1
L
l=1
(w
l
k
)
2
. (8)
where w
l
k
is iteractively calculated based on Eqs. (6) or (7).
If N
eff
falls below the threshold N
thd
, a degeneracy problem
occurs and the resampling step is required [15], [27]. The main
steps of particle filter are concluded in Algorithm 1.
IV. THE DETAILED DERIVATIONS OF THE PROPOSED
METHOD
A. Brief introduction to EM algorithm
EM algorithm is an effective method in the area of system
identifications and shows superiority over conventional max-
imum likelihood estimates (MLE) in incomplete-data cases
[14], [15]. The main content of EM algorithm is to iteratively
execute the expectation step (E-step) and the maximization
step (M-step) until the desired parameters obtained [19], [21].
In the E-step, the expected value of log-likelihood of the
complete data (Q-function) is calculated based on the currently
estimated parameter Θ
s
as [19]
Q(Θ|Θ
s
) = E
C
mis
|C
obs
,Θ
s
{log p(C
mis
, C
obs
|Θ)}, (9)
In the M-step, the estimated parameter is updated through
maximizing the Q-function as [10], [22]
Θ = arg max
Θ
Q(Θ|Θ
s
). (10)
The EM algorithm is an iterative algorithm and guarantees
convergence with the Q-function keeping nondecreasing in
each iteration [28]. The initial choice Θ
1
is important for the
EM algorithm.
B. The derivations of Q-function
Based on Eqs. (1)-(3) and the probability chain rule, the
complete data log-likelihood log p(C
mis
, C
obs
|Θ), can be sim-
plified as
log p(u
1:N
, x
1:N
, y
1:N
, λ
1:N
|Θ)
= C
1
+
N
k=1
log p(y
k
|u
k
, x
k
, λ
k
, σ, θ
h
) +
N
k=1
log p(λ
k
|r)
+
N
k=2
log p(x
k
|x
k−1
, u
k−1
, δ, θ
f
) + log p(x
1
|Θ
1
). (11)
where C
1
= log p(u
1:N
, Θ) is a constant. Considering system
output data y
1:N
= {y
o
1
:o
α
, y
m
1
:m
β
}, the conditional expecta-
tions of log p(C
mis
, C
obs
|Θ) with respect to λ
1:N
, x
1:N
, and
y
m
1
:m
β
can be calculated as
Q(Θ|Θ
s
) =
o
α
k=o
1
p(x
k
|C
obs
, Θ
s
)
p(λ
k
|C
obs
, x
k
, Θ
s
)
log p(y
k
|u
k
, x
k
, λ
k
, σ, θ
h
)dλ
k
dx
k
+
m
β
k=m
1
p(x
k
, y
k
|C
obs
, Θ
s
)
p(λ
k
|C
obs
, x
k
, y
k
, Θ
s
) log p(y
k
|u
k
, x
k
, λ
k
, σ, θ
h
)
dλ
k
dx
k
dy
k
+
o
α
k=o
1
p(x
k
|C
obs
, Θ
s
)
p(λ
k
|C
obs
, x
k
,
Θ
s
) log p(λ
k
|r)dλ
k
dx
k
+
m
β
k=m
1
p(x
k
, y
k
|C
obs
, Θ
s
)
p(λ
k
|C
obs
, x
k
, y
k
, Θ
s
) log p(λ
k
|r)dλ
k
dx
k
dy
k
+
N
k=2
[p(x
k
, x
k−1
|C
obs
, Θ
s
) log p(x
k
|x
k−1
, u
k−1
, δ, θ
f
)]
dx
k
dx
k−1
+
p(x
1
|C
obs
, Θ
s
) log p(x
1
|Θ
1
)dx
1
+ C
1
. (12)
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