一类具有跳跃不确定性和不可靠通道的周期非线性系统的分布式滤波资源-CSDN文库

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根据提供的文件内容，我们可以提炼出以下知识点： 1. 分布式滤波：分布式滤波是一种信号处理技术，在多个处理节点上分布执行数据的滤波处理。本文讨论了在具有跳跃不确定性和不可靠通道的周期非线性系统中的分布式滤波问题。这种滤波方式能够在不集中处理数据的情况下，减少系统复杂性和运算成本。 2. 跳跃不确定性：在实际的控制系统中，系统的动态可能受到各种未预见因素的影响，导致其状态发生突变，这种现象被称为跳跃不确定性。在本研究中，使用马尔可夫链来描述这种跳跃不确定性，马尔可夫链是一种具有无记忆性质的随机过程，即下一状态仅依赖于当前状态，而与之前的状态历史无关。 3. 不可靠通道：在传感器网络等分布式系统中，传感器与过滤器之间的通信可能会因为各种外界干扰而变得不可靠。本文引入了离散时间衰落模型来描述这种通道衰落现象，即通道系数的离散概率分布是预先已知的。这允许设计者在设计分布式滤波器时考虑这些因素，以保证滤波过程的稳定性和鲁棒性。 4. 周期Lyapunov函数：为了确保系统的稳定性，本文建立了基于固定网络拓扑的周期Lyapunov函数。Lyapunov函数是一种能量函数，用于分析系统稳定性的一种方法。在周期性系统中，Lyapunov函数通常是时间的周期函数，可以用来证明系统状态是否能够最终趋于稳定。 5. 线性矩阵不等式（LMIs）：在分布式滤波器的设计中，通过求解一组线性矩阵不等式来得到滤波器的设计。线性矩阵不等式是控制理论中用于解决优化问题的一种工具，特别是在系统的稳定性分析和鲁棒控制设计中。 6. 离散时间系统：本文的研究对象是离散时间的周期非线性系统。在这些系统中，系统的动态行为是通过一系列时间点上状态的更新来描述的，区别于连续时间系统的连续动态行为描述。 7. 传感器网络：传感器网络是由大量传感器节点组成的一种特殊类型的复杂网络。这些节点通常具有有限的计算能力和通信能量。在过去的几十年里，由于其易于维护、安装和成本低廉等优点，传感器网络在许多实际情境中得到了广泛应用，如信息收集、无线网络和军事传感等。 8. 数值仿真：为了验证所提出的分布式滤波器设计技术的有效性和优越性，本文采用数值仿真的例子进行演示。仿真结果可以显示滤波器性能，并与未使用分布式滤波的情况进行对比分析。通过上述知识点，我们可以全面了解这篇文章所涉及的分布式滤波技术在具有跳跃不确定性和不可靠通道的周期非线性系统中的应用及其研究的前沿性。这项研究为处理复杂网络系统中的信号处理问题提供了理论依据和实践指导。

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IET Control Theory & Applications

Research Article

Distributed filtering for a class of periodic

non-linear systems with jumping

uncertainties and unreliable channels

ISSN 1751-8644

Received on 18th August 2016

Revised 17th December 2016

Accepted on 4th January 2017

E-First on 14th February 2017

doi: 10.1049/iet-cta.2016.1096

www.ietdl.org

Jie Tao

, Renquan Lu

, Hongye Su

, Zheng-Guang Wu

, Yong Xu

National Laboratory of Industrial Control Technology, Institute of Cyber-Systems and Control, Zhejiang University, Yuquan Campus, Hangzhou

310027, People's Republic of China

Guangdong Key Laboratory of IoT Information Processing, School of Automation, Guangdong University of Technology, Guangzhou 510006,

People's Republic of China

E-mail: rqlu@gdut.edu.cn

Abstract: The problem of distributed filtering for discrete-time periodic non-linear systems over sensor networks in the presence

of jumping uncertainties and unreliable channels is investigated in this study. A random variable which is subject to a Markov

chain is introduced to govern the jumping uncertainties. Considering the unreliable channels between the system and filters, a

discrete-time fading model is introduced to describe this channel fading phenomenon with the discrete probability distribution of

each channel coefficients known a priori. On the basis of a fixed network topology, by establishing the periodic Lyapunov

function, sufficient conditions on the existence of distributed filter are proposed to ensure the stability and dissipativity of the

filtering error system. The distributed filter for discrete-time periodic non-linear systems is designed via solving a set of linear

matrix inequalities. The merits and validity of the proposed design techniques are verified by means of a numerical simulation

example.

1 Introduction

As a special type of complex networks, sensor networks are usually

composed of a large number of sensor nodes which have limited

computation capabilities and communication energy. During the

past few decades, sensor networks have been a very hot topic due

to its advantages including easy maintenance, installation, low cost

etc. Therefore, sensor networks have been widely used in many

practical situations such as information collection, wireless

network and military sensing [1, 2]. Accordingly, a large spectrum

of theoretical achievements on filtering and fault detection

problems for sensor networks has been obtained [3–11]. For

example, the fault detection problem for discrete-time Markovian

jump systems in the presence of randomly varying non-linearities

and sensor saturation has been studied in [3]. In [10], to obtain less

conservative results, the small scale gain theorem has been utilised.

Furthermore, based on the model transformation method, the

desired distributed fuzzy filter parameters are solved for discrete-

time Takagi–Sugeno (T–S) fuzzy systems over a class of sensor

networks taking into account the joint effect of time-varying delays

and multiple probabilistic packet losses.

It should be pointed out that due primarily to the abrupt

environment disturbances or sudden structural changes, the

parameter uncertainties would appear in a probabilistic way in real

situations [12], which would unavoidably deteriorate the overall

system performance. As we all know, Markov model [13–22] is

very suitable for describing this phenomenon. Motivated by

Aberkane and Dragan [23], a random variable, which is subject to a

two-state Markov chain, is used to model this jumping

uncertainties. Furthermore, the network-induced phenomena

including the packet dropouts, communication delays and the

channel fading phenomenon has not attracted considerable research

interest despite its frequently occurrence in practical applications

[24]. In recent years, some initial research attention have been

attracted to deal with the problems of sensor networks with fading

channels, see [25]. Up to now, the design of filter to cope with the

randomly occurring uncertainties and fading channels has not been

thoroughly studied yet.

Periodic systems have been a focus of research in the past few

decades, since they can be applied to multiplexed systems,

communication scheduling, aerospace and so on [26]. Accordingly,

there is a rich and growing literature that established the study of

periodic systems, see [27–29]. It is worth mentioning that non-

linearity is ubiquitous in many practical situations and one of

sources for causing instability and unsatisfactory performance.

Very recently, some researchers consider periodic non-linear

systems. In [30], sufficient conditions for uniform local and global

asymptotic stabilisation of general non-linear systems with periodic

coefficients have been obtained by using a method for constructing

a damping control based on a generalisation of weak Jurdjevic–

Quinn conditions. In [31], by resorting to a mode-dependent

Lyapunov–Krasovskii functional, a dissipative filter has been

designed for discrete-time periodic MJSs subject to continuous and

bounded non-linearity.

Motivated by the aforementioned considerations, the problem

of distributed filtering of discrete-time periodic non-linear systems

in the presence of jumping uncertainties and unreliable channels is

studied in this paper. On the basis of a periodic Lyapunov

functional, the sufficient conditions are given to ensure the stability

and dissipativity of the filtering error system. By constructing a

special Lyapunov matrix P(k), the distributed filter gains are

derived via solving a set of linear matrix inequalities. A simulation

result is utilised to illustrate the validity of the developed design

techniques.

Notations: The notations used throughout this paper are fairly

standard.

ℝ

denotes the n-dimensional Euclidean space. The

notation P > 0 ( ⩾ 0) means that P is real symmetric and positive

definite (positive semi-definite). The symbol ⊗ denotes the

Kronecker product. I and 0 represent the identity matrix and the

zero matrix with appropriate dimensions, respectively. The

superscript ‘

T’ represents the transpose. l

[0, ∞) is the space of

square-summable vector functions over [0, ∞). ∥ ⋅ ∥ denotes the

Euclidean norm of a vector. Moreover, in symmetric block

matrices, ‘ ∗’ denotes the term that is induced by symmetry, and

diag{⋯} stands for a block-diagonal matrix. Besides, E{x} and

E{x | y} will, respectively, mean expectation of x and expectation of

x condition on y. Matrices, if their dimensions are not explicitly

stated, are assumed to have compatible dimensions for algebraic

operations.

IET Control Theory Appl., 2017, Vol. 11 Iss. 6, pp. 846-856

846

2 Problem formulation and preliminaries

In this paper, we assume that 𝒢 = (𝒱, ℰ, 𝒜) denotes a directed

graph with n sensor nodes. To be more specific,

𝒱 is a set

including all sensor nodes, i.e. 𝒱 = {1, 2, …, n}. ℰ is a set

including all edges which can be described by a ordered pair (i,j).

𝒜 = [a

i j

] ∈ ℝ

n × n

represents a connection weight matrix whose

elements take value on either 0 or 1. It is worth pointing out that

(i, j) ∈ ℰ ⇔ a

i j

= 1 always holds for all i, j ∈ 𝒱. Furthermore, due

to the fact that all nodes are self-connected, for all i ∈ 𝒱, a

= 1.

To facilitate the following derivation, we denote

𝒩

= {j ∈ 𝒱:(i, j) ∈ ℰ} as all nodes which sent information to

node i. It is evident that

i ∈ 𝒩

In this paper, we consider a class of periodic non-linear systems

with jumping uncertainties and unreliable channels which can be

described by the following model:

lx(k + 1) = (A(k) + θ(k)ΔA(k))x(k) + B(k)ω(k) + E(k) f (x(k))

(k) =

∑

s = 0

l(k)

(k)y

(k − s) + ξ(k), i = 1, 2, …, n

z(k) = U(k )x(k)

(1)

where

f (x(k)) = [ f

(k)), f

(k)), …, f

(k))]

∈ ℝ

denotes

the non-linear input, x(k) = [x

(k), x

(k), …, x

(k)]

∈ ℝ

represents the system state,

ω(k) ∈ ℝ

and

ξ(k) ∈ ℝ

are the

external disturbance signal belonging to l

[0, ∞),

(k) ∈ ℝ

means

the measured output received by the filter subject to unreliable

channels where

(k) = C

(k)x(k) + D

(k)ω(k)

, z(k) ∈ ℝ

stands for

the signal to be estimated. It is worth mentioning that

l(k) = min {l, k}, where l is a given position scalar.

(k), B(k), C

(k), D

(k), E(k), U(k) are known real matrices with

appropriate dimensions.

Assumption 1: For simplicity, let all variables in (1) equal zero,

when k < 0.

The uncertain matrix ΔA(k) in (1) represents the additive norm-

bounded disturbance which satisfies the following inequality:

ΔA(k) = M(k)F(k)N(k)

(2)

where M (k) and N(k) are p-periodic known matrices and F(k) is a

time-varying unknown matrix meeting F

(k)

F(k) ≤ I

. It should be

pointed out that a set of matrices

{

(k)}

k ≥ 0

are said to be p-

periodic if

(k) = W

(k + p), i ∈ 𝒱, k ≥ 0. Moreover, A(k), B(k ),

(k)

, E (k), U(k ) are also p-periodic matrix sequences in this

paper.

The stochastic variables

(k) (s = 0, 1, …, l) are the mutual

independent channel coefficients which have probability density

function

(γ

(k)) on the interval [0,1]. In addition, ∀s ∈ {0, 1, …, l},

the expectation and variance of

(k)

are given by

≜ E{γ

(k)} =

∑

(k)

q(γ

(k))γ

(k),

γs

≜ E{(γ

(k) − γ

)

}

(3)

In this paper, θ(k) is a stochastic process that governs the

randomly occurring uncertainties. The parameter

θ(k) ∈ {0, 1}(k > 0) has two different modes which are subject to a

Markov chain. The corresponding transition probability matrix

(TPM) for uncertain jumping parameters is of the following form:

ϖ =

1 − ϖ

(4)

Remark 1: In many practical situations, because of the abrupt

environment changes or sudden structural disturbances, the

parameter uncertainties may appear in a probabilistic way [12]. It is

well known that there are two most commonly used models to

describe the randomly occurring uncertainties. One is independent

and identically distributed Bernoulli model and the other one is

Gilbert–Elliott model (i.e. two-state Markov model). From [32], it

is easy to know that the two-state Markov model reduces to the

independent and identically distributed Bernoulli model when

= 1 − ϖ

. Therefore, in this paper, randomly occurring

uncertainties are more general and cover the existing ones in [33]

as a special case.

Remark 2: In fact, the signal

(k)

received by the ith filter (ith

sensor) is a weighted sum of the measurement output

(k)

from

different channels with different delays [34]. The channel

coefficients (channel weights) are independent and distributed

variables with different expectations and variances. Note that the

fading measurements are more general than some other network-

induced phenomena. For instance, when l = 0, fading

measurements reduce to probabilistically degraded measurement.

When

l > 0, fading measurements include time delays.

For the ith filter (the ith sensor), to make full use of available

information from 𝒩

which denotes all nodes which sent

information to node i, we consider the following filter structure:

(k + 1) =

∑

j ∈ 𝒩

i j

(k) + ΔH

i j

(k)]x

(k) + a

i j

(k)y

(k)

(k) = U(k)x

(k)

(5)

where

(k) = [x

Fi1

(k), x

Fi2

(k), …, x

Fin

(k)]

∈ ℝ

is the estimate of

state x(k) on sensor node i and

(k) ∈ ℝ

is the output of the ith

filter.

i j

(k)

, L

i j

(k)

are filter gains to be determined which are p-

periodic matrix sequences. The uncertain matrix

ΔH

i j

(k)

in (5)

stands for additive norm-bounded disturbance which satisfies the

following inequality:

ΔH

i j

(k) = M

(k)F

(k)N

Δi j

(k)

(6)

where M

(k)

and N

Δi j

(k)

are p-periodic known matrices and F

(k)

is a time-varying unknown matrix meeting

(k)

F(k) ≤ I

. The

initial values of filters

(0)(i = 1, 2, …, n) are supposed to be

known in advance.

To facilitate analysis, we denote

(k) = 1

⊗ x(k), z

(k) = 1

⊗ z(k),

= [1 … 1

]

, I

= diag{1, …, 1

}

(k) = [x

(k) x

(k) … x

(k)]

(k) = [z

(k) z

(k) … z

(k)]

(k) = I

⊗ A(k), △ A

(k) = I

⊗ △ A(k)

(k) = 1

⊗ B(k), C

(k) = diag{C

(k), C

(k), …, C

(k)}

(k) = [D

(k) D

(k) … D

(k)]

(k) = 1

⊗ E(k), U

(k) = I

⊗ U(k)

△

(k) = I

⊗ M

△

(k),

△

(k) = I

⊗ F

△

(k)

(k) = γ

(k) − γ

(s = 0, 1, …, l)

and

IET Control Theory Appl., 2017, Vol. 11 Iss. 6, pp. 846-856

847

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