周期非线性系统分布式滤波：考虑跳跃不确定性和不可靠通道
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本文研究了一类在传感器网络中处理离散时间周期性非线性系统时面临的特殊挑战，即存在跳跃不确定性与不稳定的通信通道。针对这种问题，作者引入了一个受马尔可夫链随机过程控制的随机变量来模型化跳跃不确定性。在考虑了网络中节点间通信的不稳定性后，研究的核心在于设计一种分布式滤波算法，旨在保证在面临频繁变化的不确定性以及不可靠的通信条件时，仍能有效地估计系统的状态。
首先，研究的背景是随着物联网(IoT)的发展，传感器网络被广泛应用于工业控制和自动化领域，其中实时性和准确性成为关键。然而，这些系统的运行环境通常复杂多变，包括噪声、干扰和设备故障等，可能导致数据传输的不稳定性和系统模型的不确定性。马尔可夫链随机变量的引入允许系统动态地捕捉跳跃性不确定性，这在工业控制中可能源自设备的突然故障或环境参数的快速变化。
其次，文章探讨了分布式滤波技术在处理这类问题中的应用。传统的集中式滤波方法可能因单点故障而失效，分布式滤波则通过将计算任务分散到网络节点上，提高了系统的鲁棒性和可用性。通过利用节点间的协作和信息共享，即使在部分通信链路中断的情况下，也能实现对系统状态的相对准确估计。
为了设计有效的滤波器，研究者可能采用了诸如粒子滤波、滑动窗口滤波或者基于模型预测的方法。他们可能需要考虑如协方差矩阵更新、状态更新规则、以及如何在马尔可夫跳变模型下处理观测数据的一致性问题。同时，对于不可靠的通信通道，可能采用了错误检测和纠正机制，或者利用信道的统计特性来优化通信策略。
最后，该研究还可能包含了理论分析和仿真验证两部分。通过数学推导，作者可能会证明在给定条件下，提出的分布式滤波算法能够收敛并保持性能稳健。而通过数值实验，研究人员展示了该方法在实际应用中的性能，包括跟踪精度、响应速度以及对网络拓扑变化的适应性。
这篇研究论文关注了分布式滤波在处理具有跳跃不确定性和不稳定通信的周期非线性系统中的实际问题，旨在为复杂环境下的传感器网络提供有效的状态估计解决方案，具有重要的理论价值和实际工程意义。
H
¯
(k) = [H
¯
i j
(k)]
n × n
with H
¯
i j
(k) = a
i j
H
i j
(k)
△ H
¯
(k) = [ △ H
¯
i j
(k)]
n × n
with △ H
¯
i j
(k) = a
i j
△ H
i j
(k)
N
¯
△
(k) = [N
¯
△ i j
(k)]
n × n
with N
¯
△ i j
(k) = a
i j
N
△ i j
(k)
L
¯
(k) = [L
¯
i j
(k)]
n × n
with L
¯
i j
(k) = a
i j
L
i j
(k)
Evidently, △ H
¯
(k) = M
¯
△
(k)F
¯
△
(k)N
¯
△
(k). Moreover, it is not
difficult to get that a
i j
= 0 if j ∉ 𝒩
i
. Defining
𝒲
p × q
≜ {V
~
= [V
i j
] ∈ ℝ
np × nq
| V
i j
∈ ℝ
p × q
, U
i j
= 0 if j ∉ 𝒩
i
}
(7)
Then, H
¯
(k)
and L
¯
(k)
are two sparse matrices which meet
H
¯
(k) ∈ 𝒲
n
x
× n
x
and L
¯
(k) ∈ 𝒲
n
x
× n
y
. It is worth pointing out that the
special structure of 𝒲
p × q
will be helpful to analysis and design of
the filter in Section 3. The more detailed information about this
structure can be found in the following lemma.
Lemma 1 [35]: Let P = diag{P
1
, P
2
, …, P
n
} and non-singular
matrix P
i
∈ ℝ
p × p
for all i = 1, 2, …, n. If X = PW for W ∈ ℝ
np × nq
,
then we can readily obtain W ∈ 𝒲
p × q
⟺ X ∈ 𝒲
p × q
.
Let
η(k) =
x
¯
T
(k) x
F
T
(k)
T
∈ ℝ
(2n) × n
x
,
υ(k) =
ω
T
(k) ξ
T
(k)
T
∈ ℝ
n
y
+ n
ω
,
z
¯
(k) = z
(k) − z
F
(k). Combining
the system (1) and distributed filter (5), one immediately obtains
the following augmented system:
η
(k + 1) = {A
η
(k) + θ(k)ΔA
η
(k)}η(k)
+
∑
s = 1
l
γ
¯
s
C
ηs
η(k − s) + D
1η
(k)υ(k)
+
∑
s = 1
l
γ
¯
s
D
2ηs
(k)υ(k − s) + E
η
(k) f (x(k))
+
∑
s = 0
l
γ
~
s
(k)[C
ηs
(k)η(k − s) + D
2ηs
(k)υ(k − s)]
z
¯
(k) = U
η
(k)η(k)
(8)
where
A
η
(k) =
A
¯
(k) 0
γ
¯
0
L
¯
(k)C
¯
(k) H
¯
(k) + △ H
¯
(k)
,
ΔA
η
(k) =
ΔA
¯
(k) 0
0 0
C
ηs
(k) =
0 0
L
¯
(k)C
¯
(k − s) 0
,
E
η
(k) =
E
¯
(k)
0
, D
1η
(k) =
B
¯
(k) 0
γ
¯
0
L
¯
(k)D
¯
(k) L
¯
(k)1
n
D
2ηs
(k) =
0 0
L
¯
(k)D
¯
(k − s) 0
, U
η
(k) =
U
¯
(k) −U
¯
(k)
Note that (8) is a p-periodic stochastic system. To be more
specific,
A
η
(k)
,
B
η
(k)
,
C
ηs
(k)
,
E
η
(k)
,
D
1η
(k)
,
D
2ηs
(k)
,
U
η
(k)
are all p-
periodic matrices.
Assumption 2 [36, 37]: If each function
f
i
( ⋅ ):ℝ → ℝ in (8) is
continuous and bounded, and there exist corresponding constant
scalars δ
i
and ρ
i
such that
f
i
(0) = 0 and
δ
i
≤
f
i
(α)
α
≤ ρ
i
, a ≠ 0, i = 1, 2, …, n
x
(9)
then, Υ
1
⩽ 0 and Υ
2
⩽ 0.
From Assumption 2 and [38], one can obtain
[ f
i
(x
i
(k)) − δ
i
x
i
(k)][ f
i
(x
i
(k)) − ρ
i
x
i
(k)] ≤ 0, i = 1, 2, …, n
x
(10)
Denote e
^
k
= [e
^
k1
, e
^
k2
, …, e
^
kn
x
]
T
∈ ℝ
n
x
as the unit column vector
where e
^
ki
equals to 1 when i = k and 0 otherwise. Then, we have
f (x(k))
x(k)
T
e
^
k
e
^
k
T
−
δ
i
+ ρ
i
2
e
^
k
e
^
k
T
∗ δ
i
ρ
i
e
^
k
e
^
k
T
f (x(k))
x(k)
≤ 0
(11)
Letting p-periodic matrices
Q(k) = diag{q
1
(k), q
2
(k), …, q
n
x
(k)} > 0, we have
f (x(k))
x(k)
T
Q(k) −F
2
Q(k)
∗ F
1
Q(k)
f (x(k))
x(k)
≤ 0
(12)
where
F
1
= diag{δ
1
ρ
1
, δ
2
ρ
2
, …, δ
n
x
ρ
n
x
},
F
2
= diag
δ
1
+ ρ
1
2
,
δ
2
+ ρ
2
2
, …,
δ
n
x
+ ρ
n
x
2
Furthermore, it follows from (12) that
f (x(k))
x
¯
(k)
T
Q(k) −1
n
T
⊗ (F
¯
2
Q(k))
∗ I
n
⊗ (F
¯
1
Q(k))
f (x(k))
x
¯
(k)
≤ 0
(13)
where F
¯
1
= (1/n)F
1
, F
¯
2
= (1/n)F
2
.
Remark 3: To ease the later derivation, we can get the following
inequality from (13). By adding this inequality to ΔV
1
(η(k)) and
ΔV
2
(η(k)), we can get (21)
ΔV
12
= −
f (x(k))
η(k)
T
Q(k) −1
n
T
⊗ (F
¯
2
Q(k)) 0
∗ I
n
⊗ (F
¯
1
Q(k)) 0
∗ ∗ 0
f (x(k))
η(k)
≥ 0
(14)
Before deducing our main results in this paper, we give the
following definitions and lemma.
Definition 1 [39]: System (8) is said to be stochastically stable if
for υ(k − s) ≡ 0 ( k ⩾ 0 and s = 0, 1, …, l) and any initial condition
η(0), the following inequality holds:
E
∑
k = 0
∞
∥ η(k) ∥
2
|η(0) < ∞
(15)
Definition 2: Given a disturbance attenuation level β > 0, for any
υ(k) ∈ l
2
[0, ∞) and T > 0, if the filtering error z
¯
(k) satisfies the
following dissipation inequality:
∑
i = 0
T
E z
¯
T
(i)𝒬z
¯
(i) + 2
∑
s = 0
l
z
¯
T
(i)𝒮υ(i − s) +
∑
s = 0
l
υ
T
(i − s)ℛυ(i − s)
⩾ β
∑
i = 0
T
E
∑
s = 0
l
υ
T
(i − s)υ(i − s)
(16)
under the zero-initial condition, then the system (8) is said to be
strictly (𝒬, 𝒮, ℛ)- β-dissipative.
Assumption 3: The dissipative coefficient matrices 𝒬 ⩽ 0,
ℛ = ℛ
T
.
Remark 4: In fact, define
υ
supply
(i) =
z
¯
T
(i) υ
T
(i) υ
T
(i − 1) ⋯ υ
T
(i − l)
T
. Then, the
supply rate [19, 40] in (16) can be expressed as
848 IET Control Theory Appl., 2017, Vol. 11 Iss. 6, pp. 846-856
© The Institution of Engineering and Technology 2017
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