analysis and synthesis of networked control systems …fhan/publication/pdf/thesis_cityu.pdfcontrol...

Analysis and Synthesis of NetworkedControl Systems With Limited

Communication Capacity

FEI HAN

DOCTOR OF PHILOSOPHY

CITY UNIVERSITY OF HONG KONG

MARCH 2014

CITY UNIVERSITY OF HONG KONGl¢½A

Analysis and Synthesis of Networked ControlSystems With Limited Communication

CapacitykÏ&Uåe]²XÁ

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Submitted toDepartment of Mechanical and Biomedical Engineering

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in Partial Fulfillment of the Requirementsfor the Degree of Doctor of Philosophy

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Fei HanVÛ

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Abstract

With the rapid development of modern industry, control systems are becoming

larger in scope and more decentralized in location, and are thus difficult to be

implemented in a traditional directly-connected way. Consequently, networked

control systems(NCSs) have attracted considerable research attention in recent

decades, where the various components of control systems are connected through

communication networks with benefits such as easy maintenance and low cost.

However, the introduction of communication networks intro control systems will

bring several challenging issues due to limited communication capacity, such as

packet dropouts, network-induced delays, quantization, data rate and media access

constraints. Due to these network-induced issues, the performance of NCSs will be

much degraded and control systems can even become unstable. Therefore, it is of

both theoretical and practical significance to develop novel approaches to analysis

and synthesis of NCSs in order to reduce the adverse effects of these network-induced

issues. In particular, this thesis will concentrate on the control and estimation

problems of NCSs with limited communication capacity.

At first, a novel output feedback controller design method for a class of discrete-

time linear NCSs is presented, where the issues of network-induced delays, packet

dropouts and quantization in both sensor-to-controller (S/C) and controller-to-

actuator (C/A) channels are addressed simultaneously. The packet dropouts and

network-induced delays are modeled together as the bounded time-delays in the

buffer of the receiving node. A new asynchronous quantization scheme is proposed,

where the dynamic quantization parameters at each node are updated locally so

that the synchronized quantization parameters between sending and receiving nodes

Abstract ii

are not needed. The corresponding quantization errors are converted into the

bounded system uncertainties. By constructing a Lyapunov-Krasovskii functional, a

sufficient condition for the asymptotical stability of the closed-loop NCSs is derived

in terms of a set of linear matrix inequalities. Moreover, the corresponding dynamic

output feedback controller gains are obtained by an algorithm based on the cone

complementarity linearization.

Then we study the H∞ state feedback control problem for a class of networked

nonlinear systems with packet dropouts and network-induced delays, where the

nonlinear systems are represented by T-S fuzzy dynamic models. The packet dropouts

and network-induced delays are modeled together as the time-delays at receiving node

governed by a transition probability matrix. A piecewise compensator is designed to

estimate the lost or delayed packet throughout the transmission in order to obtain

the better H∞ performance of the closed-loop NCSs. Based on a piecewise Lyapunov

functional, the piecewise compensator and controller parameters are derived by

introducing some slack matrices and solving a set of linear matrix inequalities.

We also investigate the network-based filter design method for a class of nonlinear

systems represented by T-S fuzzy dynamic models. A unified framework is proposed

to model the networked nonlinear filtering systems with network-induced delays,

packet dropouts and quantization. Dynamic quantizers are utilized to solve the

saturation and dead zone problems in comparison to traditional static quantizers,

and the delays and packet loss are modeled together as the time-delays in the buffer

at the receiving node. The attention is focused on the design of a piecewise filter so

that the overall filtering error system is asymptotically stable with a guaranteed H∞

performance. The corresponding filter parameters are determined by linear matrix

inequality techniques based on a piecewise Lyapunov functional.

Finally, the modeling and control of a network-based nonlinear quadrotor is

presented. The network-based nonlinear quadrotor is approximated by a T-S fuzzy

dynamic model. Both the network-induced delays and packet dropouts in S/C and

C/A channels are addressed. Based on a common Lyapunov functional, a fuzzy

controller is designed by solving a set of linear matrix inequalities so that the

Abstract iii

resulting closed-loop quadrotor system is asymptotically stable with a guaranteed

H∞ performance. Simulation results are provided to illustrate the effectiveness of

the proposed methods.

Acknowledgement

I would like to express my deepest appreciation to my supervisors Prof. Gang

Feng and Prof. Yong Wang, and I would not have completed this thesis without their

full support and invaluable guidance throughout my Ph.D. studies. Their rigorous

attitude of scholarship and great enthusiasm for research have always inspired me

throughout my study period. I admire and owe them for their profound insight and

broad knowledge, and my future career will benefit from both of them.

I would also like to thank Prof. Dong Sun, a member of my qualifying panel. He

has always given me constructive suggestions and insightful comments which have

contributed greatly to my research over the past four years. I further wish to thank

Prof. Chuangyin Dang and Prof. Youfu Li at City University of Hong Kong and

Dr. Qing Liang at University of Science and Technology of China for their valuable

advices.

I would like to express my sincere gratitude to Prof. Jianbin Qiu, Dr. Changzhu

Zhang and Dr. Qing Gao. Their insightful conversations and constructive advices

have given me a great deal of inspiration.

It is also my pleasure to thank my friends and colleagues at City University of

Hong Kong, University of Science and Technology of China and other universities.

They are Dr. Yuan Fan, Dr. Cheng Song, Dr. Yanyan Shen, Dr. Yan Zheng, Dr.

Weilin Yang, Miss. Enyu Zhuang, Mr. Shaobao Li, Mr. Feng Zhou, Miss. Xiaofang

Hu, Miss. Meichen Guo, Dr. Xiangpeng Li, Dr. Jianjun Wang, Zhengtian Wu, Dr.

Jianyu Yang, Dr. Tao Ju, Dr. Yanhua Wu, Dr. Benchi Li, Mr. Changqing Shen, Mr.

Hao Yang, Mr. Fuzhou Niu, Miss. Weicheng Ma, Mr. Mingyang Xie, Dr. Weiguang

Liang, Dr. Taike Yao, Miss Liyao Ma, Mr. Fan Zhou, Mr. Yongcheng Li, Mr. Ke

Acknowledgement v

Deng, Miss. Min Zhu, Mr. Haiqing Sun, Mr. Jianting Wang, Mr. Hengshu Zhu, Mr.

Jian Chen, Mr. Jianjun Zhu, Mr. Xudan Cao and Mr. Jun He for their kind help in

my studies.

For financial support, I am very grateful to the Research Grants Council of the

Hong Kong Special Administrative Region of China under project CityU 113311,

City University of Hong Kong for providing stipend scholarship and PETER HO

Conference Scholarship, and the University of Science and Technology of China for

Ph.D. International Conference Fund.

Finally, I would like to thank my parents deeply. Their love, encouragement, and

support kept my life on the right track, especially during the difficult periods of my

life, which is much like a perfect-designed feedback controller right for me. Moreover,

I am profoundly indebted to them for their guidance, unconditional sacrifice, and

everything that they have given to me. It is staunchly believed that they are and will

always be the reason of my striving to make progress. This thesis is dedicated to them.

I would also like express my appreciation to my girlfriend Dr. Xue Yang. Without

her unconditional love, encouragements and patience, I would not have completed

this study. Our love makes me greatly confident to face all kinds of difficulties and

challenges in our future life.

Table of Contents

Abstract i

Acknowledgement iv

List of Figures ix

Notations x

1 Introduction 1

1.1 Background and Literature Review . . . . . . . . . . . . . . . . . . . 1

1.1.1 Networked Control Systems . . . . . . . . . . . . . . . . . . . 3

1.1.2 Networked Nonlinear Systems via T-S Fuzzy Models . . . . . 11

1.2 Thesis Outline and Contributions Overview . . . . . . . . . . . . . . 14

2 A Novel Asynchronous Quantization Scheme for Output Feedback

Control of Networked Control Systems 17

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Physical Plant . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.2 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.3 Communication Links . . . . . . . . . . . . . . . . . . . . . . 23

2.2.4 Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.5 Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.6 Closed-loop System . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Table of Contents vii

2.4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 A Novel Dropout Compensation Scheme for Control of Networked

T-S Fuzzy Dynamic Systems 42

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2 Model Description and Problem Formulation . . . . . . . . . . . . . . 44

3.2.1 Physical Plant . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2.2 Dynamic Compensator . . . . . . . . . . . . . . . . . . . . . . 46

3.2.3 Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48


3.2.5 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 50

3.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.5 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4 H∞ Filter Design of Networked Nonlinear Systems With Commu-

nication Constraints via T-S Fuzzy Dynamic Models 70

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2.1 Physical Plant . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2.2 Quantization, Encoding and Decoding . . . . . . . . . . . . . 74

4.2.3 Communication Links . . . . . . . . . . . . . . . . . . . . . . 75

4.2.4 Filter Error System . . . . . . . . . . . . . . . . . . . . . . . . 76


4.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3.1 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.3.2 Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.4 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Table of Contents viii

5 Fuzzy Modeling and Control of A Nonlinear Quadrotor Under

Network Environment 86

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.2 Model Description and Problem Formulation . . . . . . . . . . . . . . 88

5.2.1 Description of the quadrotor . . . . . . . . . . . . . . . . . . . 89

5.2.2 Communication . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.2.3 Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93



5.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6 Concluding Remarks 104

6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.2 Potential Research Problems . . . . . . . . . . . . . . . . . . . . . . . 106

Bibliography 108

Curriculum Vitae 126

List of Figures

1.1 Typical NCS setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 A typical framework of NCSs . . . . . . . . . . . . . . . . . . . . . . 4

1.3 One structure of the event-triggered NCSs . . . . . . . . . . . . . . . 8

2.1 Structure of the closed-loop networked control system . . . . . . . . . 19

2.2 Simulation Results with the proposed quantization scheme . . . . . . 39

2.3 State response with traditional finite-level logarithmic quantizers . . . 40

3.1 Structure of the networked T-S fuzzy system . . . . . . . . . . . . . . 44

3.2 The time-sequence diagram of the signals in the closed-loop system . 50

3.3 Simulation Results of Example 5.1 . . . . . . . . . . . . . . . . . . . 63

3.4 Simulation Results of Example 5.2 . . . . . . . . . . . . . . . . . . . 67

4.1 Overall filtering error system . . . . . . . . . . . . . . . . . . . . . . . 72

4.2 Membership functions . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3 Filtering error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.1 Photo of the quadrotor . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2 Structure of the quadrotor system . . . . . . . . . . . . . . . . . . . . 89

5.3 Delays in the buffer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.4 Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Notations

Rn n-dimensional Euclidean space

Rm×n the set of m× n real matrices

(∗) a term that can be induced by symmetry in symmetric matrices

diag ... the block-diagonal matrix

P > 0(≥ 0) P being positive (nonnegative)

I the identity matrix

0 the zero matrix

Em×n the m by n matrix in which each element value equals 1

AT transpose of the matrix A

∥ · ∥ the standard Euclidean norm

Ex the expectation of x

Ex|y the expectation of x and x conditional on y

Chapter 1

Introduction

1.1 Background and Literature Review

It is well known that the actuators, controller and sensors are usually implemented

in the same physical area in traditional control systems, and the different system

nodes are connected by electrical wires directly. Control and communications were

two different areas with few intersection in those cases because there were no

limitations in the transmissions among each nodes [1]. However, the expanding

physical setups increase the limitations of the traditional point-to-point architecture,

which makes it hard to satisfy the demanding control requirements, such as

decentralized control, remote control, etc [2]. Therefore, networked control systems

(NCSs) have attracted great research attention in recent decades in order to solve

these problems [3].

NCSs are feedback control systems in which the control loops are closed through

a real-time network [4]- [5]; see Fig.1.1. The actuators, controller and sensors of

a physical plant are distributed in a large physical space in NCSs, so they have

significant advantages over traditional control systems, such as lower costs and more

convenience for installation, extension and remote control [6]- [7]. Additionally, with

the rapid development of the network access technologies, such as controller area

network (CAN), ethernet, wireless LAN, Internet, etc, NCSs are being employed in

various real applications these decades. Some typical applications are smart grids [8],

1.1 Background and Literature Review 2

Figure 1.1: Typical NCS setup

intelligent transportation systems [9], intelligent home [10], etc.

Obviously, cheap network access makes it practical to implement those network-

based systems. However, communication links are usually unreliable due to media

access constraint, network-induced delays, packet dropouts, etc, which will degrade

the performance of the closed-loop systems or even make the systems unstable in some

cases [4]. It is noted that different nodes of the closed-loop systems are connected

by electrical wires directly in traditional point-to-point wiring control systems, where

the signal will be transmitted perfectly without any constraints. Consequently, the

traditional analysis and synthesis methods cannot be applied to the control and/or

filtering problems of NCSs directly. Therefore, it is of both theoretical and practical

significance to investigate some novel methods for analysis and synthesis of NCSs

with limited communication capacity.

There are mainly two strategies to address those problems on NCSs. One is called

control of network [3], which concentrates on the investigations of communication

protocols and qualities such that the real-time network-based systems have better

network environments. The other is called control over network [3], which

concentrates on control strategies over present networks to minimize the effect of

unreliable transmissions. This thesis mainly focuses on “control over network” based

on existing network protocols and conditions.

Most research on NCSs consider linear plants, and many results have been

1.1.1 Networked Control Systems 3

reported [4]- [6]. However, most practical physical plants have nonlinear properties,

and the control of networked nonlinear systems (NNSs) is thus more significant. In

recent decades, great effort has been devoted to model-based fuzzy control systems

[11]- [12]. In particular, Takagai-Sugeno (T-S) fuzzy models [13] have been widely

studied. This model describes a nonlinear system by a group of local linear systems,

which are blended by several IF-THEN rules [14]- [15]. Therefore, T-S fuzzy models

provide a basis for systematic stability analysis and synthesis of nonlinear control

systems by applying conventional control theory. Thus, this thesis will investigate

approaches to analysis and synthesis of networked nonlinear systems via T-S dynamic

fuzzy models.

1.1.1 Networked Control Systems

Networked control systems involve real-time network links. Fig.1.2 illustrates a

typical framework of NCSs.

There are communication links in two channels, one is from sensors to the

controller (S/C), the other is from the controller to actuators (C/A). Many issues

arise due to the unreliable transmission caused by communication networks, such as

quantization, network-induced delays, packet dropouts, etc. All these issues would

occur in both channels. Therefore, the inputs of the controller are no longer the same

with the outputs of sensors, and the inputs of actuators are also different from the

outputs of the controller. We will discuss the different network issues separately in

detail as follows.

Network-induced Delays

The network-induced delay phenomenon is one of the most common issues of

NCSs. Before the package is transmitted from sensors to the controller through the

network, they have to be quantized and encoded first, and the packet carrying the

sensor signals will queue and wait for the transmission in routers or bus if the network

load is too heavy [16]. All of these issues will lead to the delays in the S/C channel.

Similarly, there also exist network-induced delays in the C/A channel. Systems with


Figure 1.2: A typical framework of NCSs

state delays have been widely studied during the past few decades [17]- [19].

The stabilization methods for NCSs with constant network-induced delays are

investigated at the early stage in order to simplify the problem [20]- [22]. To mention

a few, the authors in [20] propose a stabilization method by utilizing a time-delayed

state feedback controller. The authors in [21] introduce a buffer to convert the time-

varying delays to a constant value so as to simplify the problem, but this method will

degrade the system performance obviously.

In the last decades, many results addressing the time-varying network-induced

delays are reported. In most of the published results, the hold-scheme is utilized,

where the latest packet is stored and used directly if the current packet is delayed

during the transmission [23]- [28], which can be expressed as follows [23]:

pr(t) = θ(t)p(t) + (1− θ(t))pr(t− 1), (1.1)

where pr(t) denotes the packet utilized by the receiving node, p(t) represents the

packet at the sending node, and θ(t) is a scalar binary distributed random variable

which equals 1 when the packet is successfully transmitted in time, while it equals 0

otherwise.

Similarly, the following expression also denotes the hold-scheme [24]- [25]:

pr(t) = p(t− τ(t)), (1.2)

where τ(t) is the time-delay in the receiving node at time instant t.


Remark 1.1. It is noted that the packet p(t) in (1.1)-(1.2) can be system states

x(t), outputs y(t) and inputs u(t), representing different filtering/control cases and

communication channels.

Different from the hold strategy, the authors in [26] propose a predictive method

for the NCSs with random network delays in both forward and feedback channels.

In addition, the authors in [29] propose a compensation method for systems with

random delays in S/C channel based on the compensation strategy.

Packet Dropouts

Packet dropout is another critical problem for NCSs, which occurs when there

are packet collisions, buffer overflows and other network congestions [5]. Obviously,

packet dropout phenomena will degrade the performance of control systems or even

make them unstable in some cases. To deal with this problem, many results have

been presented [4]- [7], [30]- [32].

Many approaches to modeling NCSs with packet dropouts have been reported.

To mention a few, the authors in [30] consider the Markovian packet loss process, and

they model the packet dropouts as a discrete-time Markov chain with given transition

probability matrix. The authors in [31] model the systems with packet dropouts as

switching systems depending on whether the packets are successfully transmitted

or not. The authors in [7], [33]- [34] model the systems with packet dropouts as

asynchronous dynamic systems.

All these methods for modeling the packet dropout phenomena can be classified

into two categories. One can be called zero strategy [7], [32], and they have the

following mathematical expression:

pr(t) = α(t)p(t), (1.3)

where pr(t) denotes the packet utilized by the receiving node, p(t) represents the

packet at the sending node, and α(t) is a scalar binary distributed random variable

which equals 1 when the packet is successfully transmitted, while it equals 0 otherwise.


The other category can be called hold strategy [30], [31], in which the data at the

last time instant are held when the current packet is lost during the transmission.

They can be expressed as follows:

pr(t) = α(t)p(t) + (1− α(t))pr(t− 1). (1.4)

When the multiple packet dropouts are considered, the following model is also

applied [33]- [35].

pr(t) = α(t)p(t) + (1− α(t))α(t− 1)p(t− 1) + · · ·

+(1− α(t))(1− α(t− 1)) · · · (1− α(t− T + 2))α(t− T + 1)p(t− T + 1)

+(1− α(t))(1− α(t− 1)) · · · (1− α(t− T + 1))α(t− T )p(t− T ), (1.5)

where T is the maximum delay steps.

Quantization

The data need to be quantized before being transmitted through the

communication links because of the limited network bandwidth [36]- [38]. The

signals are converted into several discrete values selected from a finite set during

the quantization procedure. It is obvious that quantization errors arise due to the

finite number of bits, which will degrade the system performance.

There are numerous quantization approaches, however, just a few of them are

widely utilized for filtering and control tasks. Logarithmic quantizer and uniform

quantizer are two widely used quantizers. The logarithmic quantizer is expressed as

follows [36]- [37]:

qL(v) =

sgn(v)Vi if Vi

1+∆< |v| ≤ Vi

1−∆

0 if v = 0,(1.6)

where Vi = ρiV0, i = 0,±1,±2, · · · , 0 < ρ < 1, V0 is a positive scaling constant,

∆ = 1−ρ1+ρ

, and sgn(.) is a sign function satisfying

sgn(v) =

1 if v > 0

−1 if v < 0

0 if v = 0.

(1.7)


The corresponding quantization error satisfies:

qL(v)− v = δv, (1.8)

where δ ∈ [−∆,∆].

Logarithmic quantizaters provide a sector bound method to deal with the

quantization errors, and have been widely used in various control and filtering

problems [36]- [37].

The other typical quantizer is the uniform quantizer, wihch is described as follows

[39]:

qU(v) =

sgn(v)⌊2N−1v⌋+0.5

2N−1 if |v| < v;

sgn(v)(1− 0.5

2N−1 v)

if |v| = v(1.9)

where v > 0 is the given constant quantizer limitation, N is the number of given

quantization bits and ⌊x⌋ = maxz ∈ Z, z ≤ x.

The corresponding quantization error is obtained as |qU(v)− v| ≤ v2N−1 .

However, the uniform quantizer mentioned above cannot solve the saturation and

dead zone problems arising from quantization [37]. In order to solve those problems,

the authors in [37] present a dynamic quantization scheme, which is expressed as

follows:

qµ(z) = µq

(z

µ

), (1.10)

where µ > 0 is a ”zoom” variable, with which the quantizer can deal with both large

and small variables.

Media Access Constraints

Due to limited communication resources, the media access constraint problem is

another network-induced issue to be considered in NCSs [43]. It is unpractical to

transmit all the packets from different nodes sharing with one communication link at

each time instant t. Therefore, it is desirable to reduce the transmission frequency of

each sending node in order to guarantee the utilization of every node. It will be more

important when the cost of the network access is high, such as the wireless networks.


Figure 1.3: One structure of the event-triggered NCSs

Considering this issue, event-triggered and self-triggered systems have attracted much

research attention in recent years [43]- [52].

A typical structure of the event-triggered NCSs is illustrated in Figure 1.3, where

“ETM” part stands for the event-triggered mechanism. The basic idea of event-

triggered systems is to detect a designed triggering condition. If the condition is

satisfied at time instant t, then the ETM sends the packet through the network.

Otherwise, no packets will be transmitted. A commonly used event-triggering

condition is as follows [40],

∥es(t)∥2 < δ∥s(t)∥, t ∈ [tk, tk+1), (1.11)

where es(t) = s(tk) − s(t), t ∈ [tk, tk+1) is the error vector, and s(t) can be system

states x(t) and outputs y(t), representing different triggering mechanisms.

The triggering condition means that the ETM broadcasts the packet only if the

error between the current value and the last updated value is larger than a threshold.

By this means, the packets transmitted through the network could be greatly reduced.

It is claimed in [41] that the numbers of packets needed to be transmitted are less

than those utilizing periodic sampling.

Noting from (1.11), the ETMs are required to monitor the triggering condition

all the time, which needs a dedicated hardware for this purpose. However, this is

not always practical. In this case, self-triggered control is alternative [42]. The main


idea of self-triggered control is to compute the next sampling and broadcasting time

instant of the sending node at the current time instant t.

Most of the existing literatures consider the event-triggered and self-triggered

control without the considerations of the network-indueced issues [40], [43]- [51].

To mention a few, the authors in [51] examine a class of real-time control systems

in which each control task triggers its next release based on the value of the last

sampled state. Considering event-triggered control and self-triggered control, the

authors in [47] present a new technique for the computation of the execution instants

by exploiting the concept of isochronous manifolds. The authors in [48] propose a

method by using the current state of the plant to decide the next time instant, and this

technique is developed for two classes of nonlinear control systems, state-dependent

homogeneous systems and polynomial systems. The authors in [50] investigate the

observer-based controllers for linear systems and propose advanced event-triggering

mechanisms that will reduce communication in both S/C and C/A channels. The

main idea of ETMs is the utilization of a Luenberger observer at the sensor node,

and the use of a model-based open-loop predictor that runs both at the sensor and

controller node.

In the meantime, some literatures on event-triggered and self-triggered control

consider the analysis problems [44]- [49]. For example, the following linear time-

invariant system is considered in [46]

x(t) = Ax(t) +B1u(t) +B2w(t), (1.12)

and it is assumed that there exists an H∞ controller u(t) = −BT1 Px(t) that

asymptotically stabilizes the unforced system, where P is a symmetric positive semi-

definite matrix satisfying the H∞ algebraic Riccati equation (ARE):

0 = PA+ ATP − PB1BT1 P + I + 1

γ2PB2B

T2 P. (1.13)

Different from those results that use known control laws in advance, the authors

in [43] propose a co-design method for the event generator and controller for event-

triggered control systems. Applying the LMI techniques, a sufficient condition for


the existence of the event-triggered controller and event generator is established. A

similar co-design method is also used in [41].

Joint Network-induced Issues

The results reviewed above focus on the stabilization and filtering problems

considering one particular issue of network-based systems to simplify the problem to

be addressed. However, most of those issues will appear simultaneously in practice,

and their simultaneous consideration is thus warranted and more practical..

The authors in [53] consider the modelling and control of NCSs with both

network-induced delays and packet dropouts. A switched linear system model is

proposed to describe the NCS, and a sufficient condition is derived for the exponential

stability of the closed-loop system. The authors in [54] present the optimal estimation

results of NCS subject to random delays and packet dropouts. The authors in [55]

investigate the problem of H∞ filtering for network-based filtering systems subject

to quantization and packet dropouts. Based on a piecewise Lyapunov functional,

the approach to the design of H∞ piecewise filter is proposed such that the filtering

error system is stochastically stable with a guaranteed H∞ performance. The authors

in [56] address the filtering problems for linear NCSs with packet dropouts, network-

induced delays and quantization. The authors in [39] present a controller design

method for a class of linear network-based systems with communication constraints

in both S/C and C/A channels, where network-induced delays, packet dropouts and

quantization are considered simultaneously. The bounded stability of the closed-loop

control system is obtained via the uniform quantization method.

Some results are also presented for the event-triggered control methods with

network issues. The authors in [45] examine event-triggered dada transmission in

distributed networked control systems with packet loss and transmission delays. A

distributed event-triggering scheme is proposed, where a subsystem broadcasts its

state information to its neighbors only when the subsystems’s local state error exceeds

a specified threshold. Considering the packet dropouts and network-induced delays,

a maximal allowable number of successive dropouts (MANSD) and a bound on delays

1.1.2 Networked Nonlinear Systems via T-S Fuzzy Models 11

are provided. The authors in [41] propose a discrete event-triggered communication

scheme for a class of networked T-S fuzzy systems with bounded network-induced

delays. The main contribution is that the proposed scheme does not need a continuous

measurement and computation, and system states are only measured and checked at

a constant sampling period. However, it is noted that the method proposed in [41]

still need to sample and compute at constant time instants. Moreover, it is not

practical for the controller to obtain the premise variables of the physical plant.

The authors in [52] combine the model-based networked control systems and event-

triggered control, and the proposed framework is utilized for the stabilization of

uncertain dynamical systems and is extended to systems subject to quantization and

time-varying network delays.

1.1.2 Networked Nonlinear Systems via T-S Fuzzy Models

Fuzzy logic and fuzzy control have attracted great research attention since Zadeh

published his results on fuzzy sets [59]- [63]. However, those fuzzy controllers are

almost applied in a trial-and-error manner and lack systematic design methods at

the early stage of the research on fuzzy control.

In recent decades, great efforts on fuzzy logic have been devoted to model-based

fuzzy control systems, which provide a powerful platform for systematic stability

analysis and controller design. Among those different fuzzy models, dynamic T-S

fuzzy models have been widely studied for their great approximation capacity to

nonlinear systems. The dynamic T-S fuzzy model describes a nonlinear system by a

group of fuzzy IF-THEN rules in the form of local linear or affine models which are

smoothly connected by fuzzy membership functions, which is shown as follows:

Plant Rule Rl: IF θ1(t) is F l1 and θ2(t) is F l

2 and · · · and θv(t) is F lv, THEN x(t+ 1) = Alx(t) + Blu(t)

y(t) = Clx(t), l ∈ I := 1, 2, ..., r,(1.14)

where Rl denotes the lth fuzzy inference rule, r the number of inference rules, F lφ(φ =

1, 2, ..., v) the fuzzy sets, x(t) ∈ ℜnx the state vector, y(t) ∈ ℜnz the output vector,


u(t) ∈ ℜnw the control input vector, [θ1(t), θ2(t), · · · , θv(t)] the premise variables,

which are some measurable variables of the system such as the state variables, and

(Al, Bl, Cl) denotes the matrices of the system’s lth local model.

By using a standard fuzzy inference method [12], which includes a singleton

fuzzifier, product fuzzy inference and center-average defuzzier, the T-S fuzzy system

(1.14) can be rewritten as x(t+ 1) = A(µ)x(t) +B(µ)u(t)

y(t) = C(µ)x(t),(1.15)

where

A(µ) =r∑

l=1

µlAl, B(µ) =r∑

l=1

µlBl, C(µ) =r∑

l=1

µlCl, (1.16)

and µl is the normalized membership function satisfying

µl =ζl(θ)∑r

φ=1 ζφ(θ), ζl(θ) =

v∏φ=1

F lφ(θφ), µl ≥ 0,

r∑l=1

µl = 1 (1.17)

with F lφ(θφ) representing the grade of membership of θφ in the fuzzy set F l

φ.

At the early stage of the research on T-S fuzzy control systems, some basic

stability analysis and controller design results are proposed [64]- [66]. The basic

idea of these methods is to design a feedback controller for each local model and

construct a global controller from the local controllers to guarantee the closed-loop

performance, and the widely used controller is called parallel distributed compensator

(PDC) with the following form


2 and · · · and θv(t) is F lv, THEN

u(t) = Klx(t), l ∈ I := 1, 2, ..., r, (1.18)

where Kl denotes the controller gain in the lth local model to be determined.

However, the stability criterion is mostly based on common quadratic Lyapunov

functions, which is proved to be conservative when dealing with highly complex

nonlinear systems [67].

In order to reduce the conservativeness of the common Lyapunov functional

method, the piecewise quadratic Lyapunov functional method is proposed [68]- [69].


After partitioning the fuzzy system (1.15) into several polyhedral regions Sii∈I

according to the membership functions, the following subsystems in each region is

obtained x(t+ 1) = Aix(t) + Biu(t)

y(t) = Cix(t),(1.19)

where

Ai =∑

m∈ℵ(i)

µmAm,Bi =∑

m∈ℵ(i)

µmBm, Ci =∑

m∈ℵ(i)

µmCm, (1.20)

with 0 ≤ µm(θ(t)) ≤ 1,∑

m∈ℵ(i) µm(θ(t)) = 1, i ∈ I ,.

Different from the controller (1.18), the following piecewise controller is utilized

u(t) = Kix(t), (1.21)

where Ki denotes the controller gain in Region Si to be determined.

The other less conservative method is based on fuzzy quadratic Lyapunov

functions [70]. It is also able to deal with a wider class of fuzzy systems than

those based on common quadratic Lyapunov functions because common Lyapunov

functions are the special case of fuzzy ones.

For networked nonlinear systems via T-S fuzzy dynamic models, most of the

existing literatures assume that the premise variables of the physical plant are always

available at the controller node, and then utilize a parallel distributed compensator

(PDC) as the controller to stabilize the physical plant [32], [71]- [77]. Similarly,

for networked nonlinear filtering systems via T-S fuzzy dynamic models, the authors

in [55] also assume that the partition information of the premise variables are available

at the filter node. However, the authors in [75]- [76] claim that these assumptions are

unpractical and propose a new asynchronous scheme, which considers the situation

when premise variables of the physical plant and fuzzy observer are in different

partition regions. Applying the compensation strategy, the authors in [29] also utilize

the asynchronous premise variables between sending and receiving nodes.

1.2 Thesis Outline and Contributions Overview 14

1.2 Thesis Outline and Contributions Overview

It the previous section, we have introduced the fundamentals of networked

control systems with limited communication capacity, and a large number of existing

results focused on the NCSs with various network-induced issues have been reviewed.

However, it is noted that there are still many important problems to be addressed.

(i) To simply the communication constraint problems of networked control systems in

most existing works, the network communication is assumed to exist in one channel,

either form the sensor to the controller or from the controller to the actuator. (ii)

Most of the existing results on the stabilization or estimation problems of network-

based systems only consider one or two network issues to simplify the concerned

problems, and few of them address those typical network problems simultaneously.

(iii) To deal with the quantization issue, the logarithmic quantizers are widely used

and the corresponding sector bound method is utilized to treat the quantization

errors. It should be noted that this kind of quantization method needs infinite

network bandwidth, which is on the contrary to the objective of the quantization

procedure. Otherwise, only bounded stability of the closed-loop system can be

achieved. (iv) Most of the existing methods treat the network-induced delays and

packet dropouts with either zero or hold strategies, but the performance of the closed-

loop NCSs is unsatisfactory when either of them is adopted. Motivated by these

issues on the research of network-based systems, this thesis will focus on analysis

and synthesis problems of NCSs considering various communication constraints. The

main contributions of this thesis are outlined as follows.

Chapter 2 considers the problem of output feedback control for network-

based discrete-time systems under unreliable communications subject to packet

dropouts, network-induced delays, and quantization. The network issues in both

sensor-to-controller (S/C) and controller-to-actuator (C/A) channels are addressed

simultaneously in a unified framework. Different from the existing results, a new

asynchronous quantization scheme is proposed, which does not need synchronous

quantization parameters between sending and receiving nodes. A dynamic output


feedback controller with the new quantization scheme is then designed and it is shown

that the resulting networked closed-loop control system is asymptotically stable.

Chapter 3 considers the problem of H∞ state feedback control for networked

nonlinear systems under unreliable communication links with packet dropouts. The

nonlinear plant in this chapter is described by a Takagi-Sugeno (T-S) fuzzy model.

The packet dropouts in both S/C and C/A channels are considered, which are

modeled by Bernoulli processes. A new compensation scheme for the estimation of

missing packets is proposed, and a piecewise state feedback controller is designed

so that the resulting closed-loop control system is stochastically stable with a

guaranteed H∞ performance. The system performance is improved by the proposed

compensation scheme in comparison to the existing methods. Then the results are

extended to the case when both network-induced delays and packet dropouts exist

in communication links.

Chapter 4 addresses the problem of H∞ filter design for networked nonlinear

systems under unreliable communication links with packet dropouts, network-induced

delays and quantization. The nonlinear plant in this chapter is described by a T-S

fuzzy dynamic model, and these three network constraints are treated in a unified

framework simultaneously. The network-induced delays and packet dropouts are

modeled by the time-delays in the buffers at the receiving node. A piecewise filter

is designed without knowing the region information of the premise variables of the

physical plant. Based on a piecewise quadratic Lyapunov functional, the overall

filtering error system is proved to be asymptotically stable with a guaranteed H∞

performance.

Chapter 5 considers the fuzzy modeling and H∞ state feedback control for

network-based quadrotor under unreliable communications. Both the network-

induced delays and packet dropouts are addressed. The networked nonlinear

quadrotor in this chapter is firstly approximated by a T-S fuzzy dynamic model.

The network-induced delays and packet dropouts in both S/C and C/A channels

are modeled in a unified framework. Then a fuzzy controller is designed so that the

resulting closed-loop quadrotor system is asymptotically stable with a guaranteed


H∞ performance.

Chapter 2

A Novel Asynchronous

Quantization Scheme for Output

Feedback Control of Networked

Control Systems

2.1 Introduction

As mentioned in Chapter 1, most of the researchers focus on just one or two

network-induced issues of NCSs in order to simplify the problem to be addressed. The

authors in [39] consider three typical network issues simultaneously, which are packet

dropouts, network-induced delays and quantization, but only bounded stability of

the closed-loop control system is obtained via the uniform quantization method.

It is noted that quantization is one of the critical issues of NCSs because data

cannot be accurately transmitted due to the limited bandwidth of communication

links [99]. Uniform quantizers and logarithmic quantizers are two widely used

quantizers in control area. With the uniform quantizer, the solutions of the control

system are shown to converge to an ellipsoid rather than zero in [39], in other words,

the asymptotic stability of the system cannot be achieved. On the other hand,

the widely used logarithmic quantizer requires infinite network bandwidth when

2.1 Introduction 18

the system approaches its equilibrium [85] [94] [101]- [102], and this is contrary

to the original purpose of quantization. In addition, a finite-level logarithmic

quantizer is proposed in [104], but it is assumed that the ”zoom” variables should be

synchronized between the sending and receiving nodes. More recently, some results on

quantization with mismatched encoder/decoder are reported [105]- [106]. However,

all of the results on quantization mentioned above are under the assumption of perfect

transmissions, which is hard to be the case in network circumstances. It is thus of

both theoretical and practical significance to consider new quantization schemes in

order to achieve the asymptotical stability of the system under the unreliable network

environment in practice, which motivates our research.

In this chapter, we propose a new logarithmic quantization scheme for NCSs.

It is shown that the closed-loop control system is asymptotically stable with the

proposed quantizer and observer-based output feedback controller, and none of the

synchronized scaling parameters between the sending and receiving nodes are needed.

Moreover, those three typical network-induced issues in both S/C and C/A channels

are modeled in a unified framework simultaneously. The contributions of this chapter

can be summarized as follows: 1) a new asynchronous quantization scheme for

networked control systems is proposed; 2) three typical network issues in both S/C

and C/A channels are considered simultaneously; and 3) different from the existing

results, the asymptotical stability of the networked closed-loop control system can be

guaranteed by utilizing the proposed quantization scheme.

The remainder of the chapter is organized as follows. Section II is devoted to the

description of the new quantization scheme and the problem formulation. Section III

presents the analysis and synthesis results based on a quadratic Lyapunov-Krasovskii

functional. In Section IV, a simulation example is given to illustrate the effectiveness

of the proposed scheme. Finally, a conclusion is drawn in section V.

2.2 Model Description 19

Figure 2.1: Structure of the closed-loop networked control system

2.2 Model Description

In this chapter, we focus on discrete-time linear systems with quantization,

packet dropouts, and network-induced delays in communication links as illustrated

in Fig.2.1. Note that all the network issues exist in both S/C and C/A channels.

Therefore, the packet at the sending and receiving nodes in both channels are not

the same. Now, we model the physical plant, quantizers and observer-based output

feedback controller mathematically.

2.2.1 Physical Plant

The linear physical plant considered in this chapter is given by: x(t+ 1) = Ax(t) +Bu(t)

y(t) = Cx(t),(2.1)

where x(t) ∈ ℜnx represents the state vector, u(t) ∈ ℜnu the input vector, y(t) ∈ ℜnz

the output vector, and (A,B,C) denote the matrices of the system.

2.2.2 Quantization 20

2.2.2 Quantization

Due to the limited bandwidth of communication links, information needs to be

quantized at the sending node before it is transmitted through the network. The

widely used logarithmic quantizer is expressed as follows [94]:

Q(v) =

sgn(v)ρiV0 if ρiV0

1+∆< |v| ≤ ρiV0

1−∆

0 if v = 0,(2.2)

where 0 < ρ < 1, V0 is a positive scaling constant, i = 0,±1,±2, · · · , ∆ = 1−ρ1+ρ

, and

sgn(.) is a sign function satisfying

sgn(v) =

1 if v > 0

−1 if v < 0

0 if v = 0.

(2.3)

The corresponding quantization error satisfies:

Q(v)− v = δv, (2.4)

where δ ∈ [−∆,∆], and we have Q(v) = (δ + 1)v.

It is noted that the value of i approaches infinity in (2.2) as |v| decreases to zero,

which indicates that the infinite number of quantization levels is needed by using

the logarithmic quantizer. Obviously, it is unpractical to implement the infinite-

level logarithmic quantizer (2.2) in practical situations. To address this problem, the

logarithmic quantizer (2.2) is improved in [104] as follows:

q(v) =

sgn(v)V0 if |v| > V0

1−∆

sgn(v)ρiV0 if ρiV0

1+∆< |v| ≤ ρiV0

1−∆

sgn(v)ρNV0 if |v| ≤ ρNV0

1+∆,

(2.5)

where i = 1, 2, · · · ,N − 1.

Different from the traditional logarithmic quantizer (2.2), the maximum value of i

in (2.5) is finite, say, N −1, which makes the improved quantizer (2.5) practical to be

implemented. In order to analyze the quantization errors of the improved quantizer


(2.5), the relationship between the two quantizers is established as follows:

q(v) =

µvQ(v) if |v| > V0

1−∆

Q(v) if ρiV0

1+∆< |v| ≤ ρiV0

1−∆

Q(v) + ϵv if |v| ≤ ρNV0

1+∆,

(2.6)

where ϵv is a parameter satisfying 0 ≤ ϵv ≤ Vmin = ρNV0, and µv = V0

Q(v)∈ (0, 1].

Similar to [39], we consider the compact set L0 such that for any v ∈ L0, a lower

bound of µv is defined as follows

µ = minµv : v ∈ L0, (2.7)

so we have µv ∈ [µ, 1].

According to (2.4) and (2.6), we obtain

q(v) =

µv(δ + 1)v if |v| > V0

1−∆

(δ + 1)v if ρiV0

1+∆< |v| ≤ ρiV0

1−∆

(δ + 1)v + ϵv if |v| ≤ ρNV0

1+∆,

(2.8)

where δ ∈ [−∆,∆].

Remark 2.1. It is noted that there still exists a non-zero term ϵv in the improved

quantizer (2.8) when the concerned networked system approaches its equilibrium,

which makes the asymptotical stability of the concerned networked system hard to

be achieved. Similar issue will happen when a uniform quantizer is utilized in the

stabilization of a given networked system, and only bounded stability can be achieved

[39].

To overcome the aforementioned difficulty, the dynamic quantization method with

zooming variables as in [103] is resorted. However, different from the synchronous

quantizers in [103], the following asynchronous quantizers in S/C and C/A channels


are proposed respectively as follows,

gqy(g−1y) =

Syymax if |g−1y| > ymax

1−∆1

Syρi11 ymax if ρ

i11 ymax1+∆1

< |g−1y| ≤ ρi11 ymax1−∆1

SyρNy

1 ymax if |g−1y| ≤ ρNy1 ymax1+∆1

gqu(g−1uc) =

Suymax if |g−1uc| > umax

1−∆2

Suρi22 umax if ρ

i22 umax1+∆2

< |g−1uc| ≤ ρi22 umax1−∆2

SuρNu2 umax if |g−1uc| ≤ ρNu

2 umax1+∆2

,

(2.9)

where Sy = g · sgn(y), Su = g · sgn(uc), g and g denote the asynchronous scaling

parameters at the receiving and sending nodes, respectively, ymax and umax are positive

constants, 0 < ρ1, ρ2 < 1, i1 = 1, 2, · · · ,Ny − 1, i2 = 1, 2, · · · ,Nu− 1, ∆1 =1−ρ11+ρ1

, and

∆2 =1−ρ21+ρ2

.

Remark 2.2. It is noted that ymax and umax in (2.9) denote the maximum output

values of two quantizers in S/C and C/A channels, respectively, which are transmitted

through the communication links with network-induced delays and packet dropouts.

Obviously, they are not the maximum values of actual system outputs or controller

outputs.

Similar to (2.8), we have

gqy(g−1y) =

µδ+1 gg

−1y if |g−1y| > ymax1−∆1

δ+1 gg−1y if ρ

i11 ymax1+∆1

< |g−1y| ≤ ρi11 ymax1−∆1

δ+1 gg−1y + ϵy if |g−1y| ≤ ρ

Ny1 ymax1+∆1

gqu(g−1uc) =

µδ+2 gg−1uc if |g−1uc| > umax

1−∆2

δ+2 gg−1uc if ρ

i22 umax1+∆2

< |g−1uc|

≤ ρi22 umax1−∆2

δ+2 gg−1uc + ϵu if |g−1uc| ≤ ρNu

2 umax1+∆2

,

(2.10)

where δ+j = δj + 1, j = 1, 2, δ1 ∈ [−∆1,∆1], δ2 ∈ [−∆2,∆2] and µ ∈ [µ, 1].

Remark 2.3. Most results consider the dynamic quantization under the assumption

that the network communications are perfect so that the ”zoom” variables can

2.2.3 Communication Links 23

be synchronously obtained at the sending and receiving nodes [103]- [104] [107].

Nevertheless, this assumption is hard to be satisfied in practice because the network

channel is always imperfect, that is, there are network-induced delays and packet

dropouts throughout the communication, and the traditional synchronous dynamic

quantization method can not be utilized reliably. Therefore, we propose the new

asynchronous quantizers (2.9), where the scaling parameters g and g are generated

at receiving and sending nodes respectively, and their values can be different.

2.2.3 Communication Links

Note that the phenomena of packet dropouts and network-induced delays exist

both in S/C and C/A channels. Therefore, the inputs to the controller yc(t) are not

the same as the outputs of the controlled plant y(t), while the control inputs to the

plant u(t) are also different from the outputs of the controller uc(t).

It is standard to assume that there exist buffers in controller and actuator nodes,

respectively, which store the received historical packets [89]. We model the unreliable

transmission as follows:

yc(t) = gy(it)gu(jt)qy(g−1y (it)g

−1u (jt)y(it)

)u(t) = gy(it)gu(jt)qu

(g−1y (it)g

−1u (jt)uc(jt)

), (2.11)

where qy(g−1y (it)g

−1u (jt)y(it)) and qu(g−1

y (it)g−1u (jt)uc(jt)) denote the latest data stored

in the buffer of the plant and controller nodes at time instant t, respectively. gy(it),

gy(it), gu(jt) and gu(jt) are scaling parameters updated at each node satisfying gy(it+1) = gy(it)γy(it)it+1−it

gy(t) = gy(it)γy(it)t−it gu(jt+1) = gu(jt)γu(jt)

jt+1−jt

gu(t) = gu(jt)γu(jt)t−jt ,

(2.12)

where “it” and “jt” are the latest time instants when the corresponding ‘ACK’ is

received by two receiving nodes at time t, respectively.


The updating parameters γy(t) and γu(t) in (2.12) satisfy

γy(t) =

γy ∈ (1,∞) if

∣∣qy(g−1y (t)g−1

u (t)y(t))∣∣ = ymax

γy∈ (0, 1) if

∣∣qy(g−1y (t)g−1

u (t)y(t))∣∣ = ymin

1 Otherwise

γu(t) =

γu ∈ (1,∞) if

∣∣qu(g−1y (t)g−1

u (t)uc(t))∣∣ = umax

γu∈ (0, 1) if

∣∣qu(g−1y (t)g−1

u (t)uc(t))∣∣ = umin

1 Otherwise,

(2.13)

where ymin = ρNyymax and umin = ρNuumax.

We define

η1(t) = t− it, η2(t) = t− jt, (2.14)

where η1(t) and η2(t) are the time-delays of the packets in the controller and actuator

nodes due to the network-induced delays and packet dropouts, respectively.

The following assumption is needed on modeling the random time-delays in the

buffers caused by the unreliable transmission.

Assumption 2.1. The time-delays η1(t) and η2(t) are time varying and satisfy 0 ≤

η1(t) ≤ η1, 0 ≤ η2(t) ≤ η2, where η1 and η2 represent the upper bounds of the

time-delays in the buffers of these two different nodes, respectively.

Remark 2.4. A time stamp is added to the packet before it is transmitted through

the network links in both S/C and C/A channels, and the network delays η1(t)

and η2(t) are measurable by comparing the time stamp of the latest received packet

with the current time instant. An ”ACK” signal representing the acknowledgement

of the packet will be transmitted to the sending node once the packet arrives at the

receiving node, and then both nodes acknowledge that the latest package information

is transmitted successfully. It is natural to assume that the ”ACK” signal has a

high priority identifier so that it can be transmitted and received without delay and

loss [108].

Remark 2.5. It is noted from (2.12) that we are able to ”zoom” the corresponding

variables at the receiving and sending nodes separately, and they are updated

asynchronously.


Proposition 2.1. Consider the asynchronous zooming variables between the sending

and receiving nodes in (2.12), the following bounded condition holds:

g0≤ gη(t)

gη(t)≤ g0, (2.15)

where

gη(t) = gu(t− η2(t))gy(t− η1(t))

gη(t) = gu(t− η2(t))gy(t− η1(t))

g0=γ η1yγ η2u

γ η1y γη2u

, g0 = g−1

0. (2.16)

Proof. From (2.12) we have

gy(it) = gy(it−1)γy(it−1)it−it−1

= gy(it−2)γy(it−2)it−1−it−2γy(it−1)

it−it−1

= · · ·

= gy(it−s)γy(it−s)it−s+1−it−s · · · γy(ti−1)

it−it−1 ,

while

gy(it) = gy(it−s)γy(it−s)it−it−s , (2.17)

where it−s denotes the latest time instant when an “ACK” signal is received at time

it. Obviously, it − it−s ≤ η1.

Therefore, we obtain

gy(it)

gy(it)=

γy(it−s)it−s+1−it−s · · · γy(ti−1)

it−it−1

γy(it−s)it−it−s

∈

[(γy

γy

)η1

,

(γyγy

)η1]. (2.18)

Similarly, we also have

gu(jt)

gu(jt)=

γu(jt−s)jt−s+1−jt−s · · · γu(tj−1)

jt−jt−1

γu(jt−s)jt−jt−s

∈

[(γu

γu

)η2

,

(γuγu

)η2]. (2.19)

2.2.4 Observer 26

Noting it = t− η1(t) and jt = t− η2(t), we obtain(γy

γy

)η1

≤ gy(t− η1(t))

gy(t− η1(t))≤

(γyγy

)η1

(γu

γu

)η2

≤ gu(t− η2(t))

gu(t− η2(t))≤

(γuγu

)η2

, (2.20)

which implies

g0≤ gη(t)

gη(t)≤ g0, (2.21)

and thus the proof is completed.

2.2.4 Observer

Based on the system (2.1), we consider the following observer.

x(t+ 1) = Ax(t) +Ryc(t), (2.22)

where x(t) is the estimated state; A and R are observer gains to be determined.

2.2.5 Controller

Based on the observer (2.22), we consider the following controller

uc(t) = Kx(t), (2.23)

where K is the controller gain to be determined.

2.2.6 Closed-loop System

Then from (2.1), (2.11), (2.20)-(2.23), we have the following closed-loop system: x(t+ 1) = Ax(t) + gη(t)Bqu(g−1η (t)Kx(t− η2(t))

)x(t+ 1) = Ax(t) + gη(t)Rqy

(g−1η (t)Cx(t− η1(t))

).

(2.24)

It is noted that (2.24) can be rewritten as follows: x(t+ 1) = Ax(t) + gη(t)

gη(t)Bgη(t)qu

(g−1η (t)Kx(t− η2(t))

)x(t+ 1) = Ax(t) + gη(t)

gη(t)Rgη(t)qy

(g−1η (t)Cx(t− η1(t))

).

(2.25)

2.3 Main Results 27

The problem to be addressed in this chapter is described as follows:

Dynamic Output Feedback Controller Design Problem. Consider the

linear system (2.1) and suppose that the network parameters η1 and η2 are given.

Design the observer-based output feedback controller in the form of (2.22) and (2.23)

such that the augmented system (2.25) is asymptotically stable.

2.3 Main Results

In this section, the solutions to the problem described in the last section will

be given in the framework of the linear matrix inequality (LMI) approach based on

Lyapunov-Krasovskii functional.

Before proceeding further, the following lemmas are introduced.

Lemma 2.2. [109] For matrices H and E, and scalar ε > 0, the following inequality

holds:

HFE + ETF THT ≤ εHHT + ε−1ETE, (2.26)

where F satisfies F TF ≤ I.

Lemma 2.3. [110] Given appropriately dimensioned matrices Ω1,Ω2, and Ω3 with

Ω1 = ΩT1 , then

Ω1 + Ω3Υ(k)Ω2 + ΩT2Υ

T (k)ΩT3 < 0 (2.27)

holds for all Υ(k) satisfying ΥT (k)Υ(k) ≤ I if and only if for some ε > 0

Ω1 + ε−1Ω3ΩT3 + εΩT

2Ω2 < 0. (2.28)

Consider the closed-loop control system (2.25), where an improved quantization

scheme (2.9) with (2.12) is utilized.

We define z(t) = g−1y (t)g−1

u (t)x(t)

z(t) = g−1y (t)g−1

u (t)x(t),(2.29)

2.3 Main Results 28

and (2.25) can be rewritten as follows

z(t+ 1) = γ−η2(t)−1u

γ−η1(t)−1y

·[γη2(t)u

γη1(t)y

Az(t) + gη(t)

gη(t)gη(t)Bqu

(g−1η (t)Kx(t− η2(t))

)]z(t+ 1) = γ

−η2(t)−1u γ

−η1(t)−1y ·[

γη2(t)u γ

η1(t)y Az(t) + gη(t)

gη(t)gη(t)Rqy

(g−1η (t)Cx(t− η1(t))

)].

We consider the following Lyapunov-Krasovskii functional candidate

V (t) = V1(t) + V2(t) + V3(t), (2.30)

where

V1(t) = zT (t)P1z(t) + zT (t)P2z(t)

V2(t) =−1∑

q=−η1

t−1∑p=t+q

ea(p−t+1)zT (p)Q1z(p)

+−1∑

q=−η2

t−1∑p=t+q


+−1∑

q=−η2

t−1∑p=t+q


V3(t) =−1∑

q=−η1

t−1∑p=t+q

ea(p−t+1)dT (p)Z1d(p)

+−1∑

q=−η2

t−1∑p=t+q

ea(p−t+1)dT (p)Z2d(p)

+−1∑

q=−η2

t−1∑p=t+q

ea(p−t+1)dT (p)Z3d(p), (2.31)

d(t) = z(t+1)− z(t), d(t) = z(t+1)− z(t), z(t) = z(t)− z(t), and Pj = P Tj > 0, j =

1, 2, Qi = QTi > 0, Zi = ZT

i > 0, i = 1, 2, 3. Then we have the following result.

Lemma 2.4. Consider the system (2.1) and the improved quantizer (2.9). Then,

for any initial state x(0) and t ≥ 0, the following inequality (2.32) holds if there

exist matrices M,N, S, Pj = P Tj > 0, j = 1, 2, Qi = QT

i > 0, Zi = ZTi > 0, i =

2.3 Main Results 29

1, 2, 3, ε1, ε2, εB, εC > 0 satisfying Θ < 0,

V (t+ 1) <

e−aV (t) if the system is in S1

e−aV (t) + ϵ21 if the system is in S2

e−aV (t) + ϵ22 if the system is in S3

e−aV (t) + ϵ23 if the system is in S4,

(2.32)

where a is a positive constant, and

S1 :x(t), x(t)|ymin <

∣∣qy(g−1η (t)y(tη1))

∣∣ ≤ ymax

and umin <∣∣qu(g−1

η (t)uc(tη2))∣∣ ≤ umax

S2 :

x(t), x(t)|ymin <


∣∣ ≤ ymax and∣∣qu(g−1

η (t)uc(tη2))∣∣ = umin

S3 :

x(t), x(t)|umin <

∣∣qu(g−1η (t)uc(tη2))

∣∣ ≤ umax and∣∣qy(g−1

η (t)y(tη1))∣∣ = ymin

S4 :

x(t), x(t)|


∣∣ = ymin and∣∣qu(g−1

η (t)uc(tη2))∣∣ = umin

Θ =

Π1 Π2 0 R

∗ Ψ+ Ξ + ΞT KT 0

∗ ∗ −ε2 0

∗ ∗ ∗ ε1

< 0, (2.33)

with

Π1 = diag−P−1 + ε2(∆2),−Z−1 + ε2(∆2),−e−aη1Z1,−e−aη2Z2,−e−aη2Z3

Π2 =

[ΓT1 ΓT

2

√η1M

√η2N

√η2S

]TP = diag

P1, P2

, Pj = (1 + τ)Pj, j = 1, 2

Z = diagZ12, Z3

, Z12 = (1 + τ)(η1Z1 + η2Z2), Z3 = (1 + τ)η2Z3

Ψ = diag−e−aP1 + η1Q1 + η2Q2,−e−aP2 + η2Q3,

−e−aη1Q1 + ε1(∆1),−e−aη2Q2,−e−aη2Q3

2.3 Main Results 30

Ξ =[

e−aη1M + e−aη2N e−aη2S −e−aη1M −e−aη2N −e−aη2S]

ε2(∆2) = diag2ε2Bi∆2∆

T2B

Ti + 2εBEbi∆2∆

T2E

Tbi,

2ε2Bi∆2∆T2B

Ti + 2εBEbi∆2∆

T2E

Tbi

ε1(∆1) = 2ε1C

Ti ∆

T1∆1Ci + 2εCE

Tci∆

T1∆1Eci Eb1 Eb2 Eb3

Ec1 Ec2 Ec3

=

0 δB δB

0 δC δC

ε1 =

−ε1 0

∗ −εC

, ε2 = −ε2 0

∗ −εB

Γ1 =

Ai 0 0 BiK −BiK

Ai − γjA γjA −RCj BiK −BiK

Γ2 =

Ai − I 0 0 BiK −BiK

Ai − γjA γjA− I −RCj BiK −BiK

γ1 = 1, γ2 = γ−1

uγ−1

y, γ3 = γ−1

u γ−1y

K =

0 0 0 K K

0 0 0 K K

, R =

0 RT 0 RT 0 0 0

0 RT 0 RT 0 0 0

T

A1 A2 A3

B1 B2 B3

=

A γ−1uγ−1yA γ−1

u γ−1y A

B γ0B γ0B

γ0=γ−η1−1y

γ−η2−1u

g0 + γ−1yγ−1ug0

2

γ0 =µγ−η1−1

y γ−η2−1u g

0+ γ−1

y γ−1u g0

2

δ =γ−η1−1y

γ−η2−1u

g0 − γ−1yγ−1ug0

2

δ =−µγ−η1−1

y γ−η2−1u g

0+ γ−1

y γ−1u g0

2

ϵ21 = (1 + τ−1)(γ0+ δ)2BT (P1 + P2 + η1Z1 + η2Z2 + η2Z3)Bu

2min

ϵ22 = (1 + τ−1)RT (P2 + η2Z3)Ry2min

ϵ23 = (1 + τ−1)(γ0+ δ)2BT (P1 + 2P2 + η1Z1 + η2Z2 + 2η2Z3)Bu

2min

+ (1 + τ−1)RTP2Ry2min. (2.34)

Proof. The proof procedures in different cases are similar. Without loss of generality,

2.3 Main Results 31

we just consider the proof of the case when the system is in region S2, that is,

ymin <∣∣qy(g−1

η (t)y(it))∣∣ ≤ ymax and

∣∣qu(g−1η (t)uc(jt))

∣∣ = umin.

If ymin <∣∣qy(g−1

η (t)y(iy))∣∣ < ymax, qy(g−1

η (t)y(it)) is a standard logarithmic

quantizer as (2.2), and qy(g−1η (t)y(it)) = g−1

η (t)(δ1 + 1)Cx(t − η1(t)) according to

(2.9) and (2.14). Additionally, qy(g−1η (t)y(it)) = g−1

η (t)µ(δ1 + 1)Cx(t − η1(t)) if

qy(g−1η (t)y(it)) = ymax, where µ < µ < 1. Therefore, we have

qy(g−1η (t)y(it)) = g−1

η (t)µ(δ1 + 1)Cx(t− η1(t)), (2.35)

where µ < µ ≤ 1.

Based on (2.9), (2.14) and∣∣qu(g−1

η (t)uc(jt))∣∣ = umin, we have

qu(g−1η (t)uc(jt)) = g−1

η (t)(δ2 + 1)Kx(t− η2(t)) + εu(t− η2(t)), (2.36)

where |εu(t− η2(t))| ≤ umin.

Based on (2.35) and (2.36), the closed-loop system (2.30) can be rewritten asz(t+ 1) = γ2Az(t) + γ

η(t) gη(t)

gη(t)B(δ2 + 1)Kz(t− η2(t))

+ γη(t) gη(t)

gη(t)Bεu(t− η2(t))

z(t+ 1) = γ3Az(t) + µR(δ1 + 1)γη(t)gη(t)

gη(t)Cz(t− η1(t)),

(2.37)

where γη(t) = γ−η2(t)−1

uγ−η1(t)−1y

and γη(t) = γ−η2(t)−1u γ

−η1(t)−1y .

Note that (2.37) can be expressed as follows:

z(t+ 1) = γ2Az(t) + (γ0B + B)(δ2 + 1)Kz(t− η2(t))

− (γ0B + B)(δ2 + 1)Kz(t− η2(t))

+ (γ0B + B)εu(t− η2(t))

z(t+ 1) = (γ2A− γ3A)z(t) + γ3Az(t)

− µR(δ1 + 1)(γ0C + C)z(t− η1(t))

+ (γ0B + B)(δ2 + 1)Kz(t− η2(t))

− (γ0B + B)(δ2 + 1)Kz(t− η2(t))

+ (γ0B + B)εu(t− η2(t)),

(2.38)

where z(t) = z(t)− z(t).

Then the original stability analysis problem is converted to a robust control

problem with parameter uncertainties in the system matrices.

2.3 Main Results 32

Define ζ(t) =[zT (t) zT (t) zT (t− η1(t)) zT (t− η2(t)) zT (t− η2(t))

]T, and

we have

V1(t+ 1) − e−aV1(t)

= zT (t+ 1)P1z(t+ 1)− e−azT (t)P1z(t)

+zT (t+ 1)P2z(t+ 1)− e−azT (t)P2z(t)

=[A+ (γ

0B + B)εu(t− η2(t))

]TP1

[A+ (γ

0B + B)εu(t− η2(t))

]+[A+ (γ

0B + B)εu(t− η2(t))

]TP2

[A+ (γ

0B + B)εu(t− η2(t))

]−e−azT (t)P1z(t)− e−azT (t)P2z(t)

≤ (1 + τ)ATP1A+ (1 + τ−1)(γ0+ δ)2BTP1Bu

2min

+(1 + τ)ATP2A+ (1 + τ−1)(γ0+ δ)2BTP2Bu

2min

−e−azT (t)P1z(t)− e−azT (t)P2z(t)

= ζT (t)ΓT1 P Γ1ζ(t)− e−azT (t)P1z(t)− e−azT (t)P2z(t) + ϵ211, (2.39)

where

A = γ2Az(t) + (γ0B + B)(δ2 + 1)Kz(t− η2(t))

−(γ0B + B)(δ2 + 1)Kz(t− η2(t))

A = (γ2A− γ3A)z(t) + γ3Az(t)− µR(δ1 + 1)(γ0C + C)z(t− η1(t))

+(γ0B + B)(δ2 + 1)Kz(t− η2(t))− (γ

0B + B)(δ2 + 1)Kz(t− η2(t)),

P = diag (1 + τ)P1, (1 + τ)P2

ϵ21 = (1 + τ−1)(γ0+ δB)

2BT (P1 + P2)Bu2min

ϵ211 = (1 + τ−1)(γ0+ δ)2BT (P1 + P2)Bu

2min

Γ1 =

γ2A 0 0

γ2A− γ3A γ3A −R(δ1 + 1)(γ0C + C)

(γ0B + B)(δ2 + 1)K −(γ

0B + B)(δ2 + 1)K

(γ0B + B)(δ2 + 1)K −(γ

0B + B)(δ2 + 1)K

. (2.40)

2.3 Main Results 33

Additionally,

V2(t+ 1) − e−aV2(t)

≤ η1zT (t)Q1z(t)− e−aη1zT (t− η1(t))Q1z(t− η1(t))

+η2zT (t)Q2z(t)− e−aη2zT (t− η2(t))Q2z(t− η2(t))

+η2zT (t)Q3z(t)− e−aη2 zT (t− η2(t))Q3z(t− η2(t)),

V3(t+ 1) − e−aV3(t)

≤ η1dT (t)Z1d(t)− e−aη1

t−1∑α=t−η1

dT (α)Z1d(α)

+η2dT (t)Z2d(t)− e−aη2

t−1∑α=t−η2

dT (α)Z2d(α)

+η2dT (t)Z3d(t)− e−aη2

t−1∑α=t−η2

dT (α)Z3d(α)

≤ η1dT (t)Z1d(t) + e−aη1 η1ζ

T (t)MZ−11 MT ζ(t)

+2ξT (t)e−aη1M [z(t)− z(t− η1(t))]

+η2dT (t)Z2d(t) + e−aη2 η2ζ

T (t)NZ−12 NT ζ(t)

+2ξT (t)e−aη2N [z(t)− z(t− η2(t))]

+η2dT (t)Z3d(t) + e−aη2 η2ζ

T (t)SZ−13 ST ζ(t)

+2ξT (t)e−aη2S [z(t)− z(t− η2(t))]

≤ ζT (t)ΓT2 ZΓ2ζ(t) + e−aη1 η1ζ

T (t)MZ−11 MT ζ(t)

+e−aη2 η2ζT (t)NZ−1

2 NT ζ(t) + e−aη2 η2ζT (t)SZ−1

3 ST ζ(t)

+ζT (t)(Ξ + ΞT

)ζ(t) + ϵ212, (2.41)

where

Γ2 =

γ2A− I 0 0

γ2A− γ3A γ3A− I −µR(δ1 + 1)(γ0C + C)

(γ0B + B)(δ2 + 1)K −(γ

0B + B)(δ2 + 1)K

(γ0B + B)(δ2 + 1)K −(γ

0B + B)(δ2 + 1)K

Z = diag (1 + τ)(η1Z1 + η2Z2), (1 + τ)η2Z3

ϵ212 = (1 + τ−1)(γ0+ δ)2BT (η1Z1 + η2Z2 + η2Z3)Bu

2min. (2.42)

2.3 Main Results 34

It then follows that (2.33) implies V1(t+1)−e−aV1(t)−ϵ21 < 0 by applying Lemma

2.2, Lemma 2.3 and the Schur complement operation. Therefore, we have V1(t+1) <

e−aV1(t) + ϵ21 in the case when ymin < |qy(y(tη1))| ≤ ymax and |qu(uc(tη2))| = umin.

Similarly, we are able to prove the other cases, and thus the proof is completed.

The following corollary can be obtained from Lemma 2.4.

Corollary 2.5. Consider the closed-loop system (2.30). z(t) = g−1y (t)g−1

u (t)x(t)

converges to the following ellipsoid for any initial state x(0) = x0,

Z∞ =z|zT (t)P1z(t) ≤ V∞

, (2.43)

where V∞ = (1− α)−1β, α = e−a, β = maxϵ21, ϵ22, ϵ23.

Proof. From Lemma 2.4, it is noted that V (t) decreases and converges to a bounded

region because 0 < e−a < 1. We denote the region as V∞, which can be obtained by

solving V∞ = αV∞ + β, where α = e−a and β = maxϵ21, ϵ22, ϵ23. Then, we have

V∞ = (1− α)−1β. (2.44)

It is noted that zT (t)P1z(t) < V (t), thus the proof is completed.

Proposition 2.6. There exists a time instant ts so that the quantizers will not be

saturated after time ts if the quantization level 2Nu and 2Ny of the quantizers in

C/A and S/C channels satisfy the following conditions respectively

Nu >1

2logρ

(1− δ)−2y2maxαK

(P−11 + P−1

2

)KT (Bu2max +Ry2max)

Ny >1

2logρ

(1− δ)−2y2maxαCP−1

1 CT (Bu2max +Ry2max), (2.45)

where B = (1 + τ−1)(γ0+ δB)

2BT (P1 + 2P2 + η1Z1 + η2Z2 + 2η2Z3)B, R = (1 +

τ−1)RT (P2 + η2Z3)R and α = (1− α)−1.

Proof. Obviously, there exists a time instant ts so that V (t) < (1−α)−1β holds for all

t > ts, because V∞ < (1−α)−1β is satisfied according to Corollary 2.5. Additionally,

we have

zT (t)P1z(t) < V∞ < (1− α)−1β

zT (t)P2z(t) < V∞ < (1− α)−1β, (2.46)

2.3 Main Results 35

which imply that

z(t)zT (t) < (1− α)−1βP−11

z(t)zT (t) < (1− α)−1βP−12 . (2.47)

It is noted that both quantizers in S/C and C/A channels will not be saturated

if the following conditions are satisfied according to (2.9)

|Cz(t)| < ymax

1− δ, |Kz(t)| < umax

1− δ, (2.48)

and the following conditions imply (2.48)

Cz(t)zT (t)CT <y2max

(1− δ)2

Kz(t)zT (t)KT +Kz(t)zT (t)KT <u2max

(1− δ)2. (2.49)

By substituting (2.47) and (2.45) into (2.49), the quantizers will not be saturated

if (2.45) is satisfied. Therefore, there exists a time instant ts such that the quantizers

in both channels will not be saturated if the quantization level Ny and Nu satisfy

(2.45), and the proof is completed.

Now, we have the following main result.

Theorem 2.1. With the improved logarithmic quantizers (2.9) with (2.12) and

tranmission scheme (2.11), the state x(t) of the closed-loop system (2.25) converges

to zero asymptotically if (2.33) and (2.45) are satisfied.

Proof. It is noted that z(t) converges to the ellipsoid Z∞ exponentially from Corollary

2.5, which implies that both quantizers in S/C and C/A channels will no longer be

saturated after time instant ts according to Proposition 2.6. It means gu(t)gy(t) will

not increase for all t > ts from (2.12) with (2.13), that is, gu(t)gy(t) will decrease or

remain unchanged.

Considering t > ts, it is noted that whenever gu(t)gy(t) remains unchanged, V (t)

will decrease exponentially until |Cz(t)| is less than ymin and/or |Kz(t)| is less than

umin according to Lemma 2.4, forcing gu(t)gy(t) to decrease finally. Therefore, the

system will finally be located at region S2, S3 or S4, which implies that gu(t)gy(t)

2.3 Main Results 36

will decrease to infinitesimal by factor γu, γ

yor γ

uγy

according to the region where

the system stays, and we obtain g−1u (t)g−1

y (t) → +∞. It is noted from (2.29) that

z(t) = g−1y (t)g−1

u (t)x(t), and we conclude x(t) → 0 as t→ 0 since z(t) is bounded for

all t > ts according to Corollary 2.5. Thus the proof is completed.

Remark 2.6. It is noted that the number of levels of the quantizers (2.9) utilized in

this chapter is finite, but the infinite quantization accuracy is able to be achieved by

using the asynchronous scaling parameters gu(t)gy(t) and gu(t)gy(t), which result in

the asymptotical stability of the concerned networked system.

Now we present the output feedback controller design method based on the

improved logarithmic quantization scheme.

It is pointed that (2.33) is not strict LMI because of the existence of P−1j , j = 1, 2,

Z−112 and Z−1

3 . By utilizing a cone complementarity linearization (CCL) algorithm [80],

we can solve this nonconvex feasibility problem by converting it into an optimization

problem with LMI constraints.

Introducing new matrices Pj, j = 1, 2, Z12 and Z3 with the following definition,

Pj = P−1j , j = 1, 2, Z12 = Z−1

12 , Z3 = Z−13 , (2.50)

we obtain the strict LMI Θ < 0 as (2.33), where P−1j , j = 1, 2, Z−1

12 and Z−13 are

replaced by Pj, j = 1, 2, Z12 and Z3, respectively. Then the problem of observer-

based output feedback controller design can be converted to the following nonlinear

minimization problem with LMI constraints

minimize Trace(∑

j

PjPj + Z12Z12 + Z3Z3

), (2.51)

subject to Pj I

∗ Pj

> 0,

Z12 I

∗ Z12

> 0,

Z3 I

∗ Z3

> 0,

Θ < 0, j ∈ 1, 2. (2.52)

Then, the above nonlinear minimization problem can be sloved by the algorithm

described as follows:

2.4 Simulation 37

Algorithm 2.1.

Step 1. Find a set of feasible matrices P 0j , j = 1, 2, Z0

12, Z03 , P 0

j , Z012, Z0

3 , A0, R0 and

K0 that satisfies the conditions in (2.52). Set σ = 0.

Step 2. Solve the following optimization problem for the variables(Pj, Z12, Z3, Pj, Z, Z3, A, R,K

):

minimize Trace(∑

j

(P σj Pj + PjP

σj ) + Zσ

12Z12 + Z12Zσ12 + Zσ

3 Z3 + Z3Zσ3

)subject to (2.52).

Set P (σ+1)j = Pj, Z(σ+1)

12 = Z12, Z(σ+1)3 = Z3, P (σ+1)

j = Pj, Z(σ+1)12 = Z12, Z(σ+1)

3 = Z3,

A(σ+1) = A, K(σ+1) = K, and R(σ+1) = R.

Step 3. With the gains A, R and K obtained in Step 2, check whether (2.33) is

feasible with respect to the matrices M,N, S, Pj = P Tj > 0, j = 1, 2, Qi = QT

i >

0, Zi = ZTi > 0, i = 1, 2, 3, ε1, ε2, εB, εC > 0. If it is feasible, the obtained gains A,

R, K are the solutions and exit. Otherwise, set σ = σ + 1, and go to Step 2. If σ

reaches the specified number of iterations, print “no solutions” and exit.

2.4 Simulation

In this section, we consider an inverted pendulum on a cart under the network

environment, where the controller and the pendulum system are connected by network

communication links. The physical model can be found in [111] and the dynamics of

the inverted pendulum system is expressed as follows:

(M +m)x+mlθcosθ −mlθ2sinθ = u

mlxcosθ + 4

3ml2θ −mglsinθ = 0, (2.53)

where M and m are the masses of the cart and the pendulum, respectively, l denotes

the half length of the pendulum, θ is the angle of the pendulum from the vertical,

u denotes the force applied to the cart, and g is the gravity acceleration. Selecting

x = [ x1 x2 ]T = [ θ θ ]T as the state variables and linearizing the physical model

2.4 Simulation 38

at the equilibrium point x = [ 0 0 ]T , we obtain the following model: x(t) = Acx(t) +Bcu(t)

y(t) = Cx(t)

where

Ac =

0 1

3(M+m)gl(4M+m)

0

, Bc =

0

− 3l(4M+m)

, C =[0.01 0.01

]In this study, the model parameters are given as M = 8.0kg, m = 2.0kg, l =

0.75m, and g = 9.8m/s2. After discretizing the above pendulum system with sampling

period Ts = 10ms, we obtain the following discrete-time system x(t+ 1) = Ax(t) +Bu(t)

y(t) = Cx(t)

where

A =

1.0006 0.0100

0.1153 1.0006

, B =

0

−0.0012

, C =[0.01 0.01

]It is noted that the open-loop system is unstable. The objective is to design

a dynamic output feedback controller under the network environment so that the

closed-loop system is asymptotically stable.

The network and quantizer parameters are chosen as follows:

η1 = η2 = 1, µ = 0.95,

γu = γy = 1.01, γu= γ

y= 0.999, ρ = 0.95, N = 128.

Applying Theorem 2.1 and Algorithm 2.1, we obtain the following output

feedback controller gains

A =

0.8142 −0.1489

−0.3460 0.6559

, R =

23.1937

29.6205

,K =

[156.6671 85.0786

].

A number of simulations with different initial conditions and randomly generated

delays have been carried out. One particular case with the initial condition x(0) =

2.4 Simulation 39

0 2 4 6 8 10 12 14−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Time in seconds

stat

e re

spon

se

x

1

x2

(a) State response

0 2 4 6 8 10 12 14−4

−2

0

2

4

6

8

10

12x 10

−3

Time in seconds

qutiz

ed o

utpu

t y(t

) tr

ansm

itted

in th

e S

/C c

hann

el

(b) System output y transmitted in the S/C

channel

0 2 4 6 8 10 12 1440

60

80

100

120

140

160

Time in seconds

qutiz

ed in

put u

(t)

tran

smitt

ed in

the

C/A

cha

nnel

(c) System input uc transmitted in the C/A

channel

0 2 4 6 8 10 12 140

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

4

Time in seconds

zoom

var

iabl

e g y(t

)gu(t

)

(d) Zoom variable gy(t)gu(t)

Figure 2.2: Simulation Results with the proposed quantization scheme

x(0) =[

π4

π10

]Tis recorded in Fig.2.2. It can be observed that the system states

converge to zero asymptotically, which demonstrates the effectiveness of the proposed

method.

In order to illustrate the improvement of the proposed asynchronous quantization

scheme, the following finite-level logarithmic quantizers without the asynchronous

scaling variables are used for comparison, which are extremely similar to the uniform

quantizers in [39] because both of them have non-zero quantization errors when the

2.5 Conclusion 40

0 2 4 6 8 10 12 14−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Time in seconds

stat

e re

spon

se

x

1

x2

Figure 2.3: State response with traditional finite-level logarithmic quantizers

concerned networked system approaches the equilibrium,

qy(y) =

sgn(y)ymax if |y| > ymax

1−∆1

sgn(y)ρi11 ymax if ρi11 ymax1+∆1

< |y| ≤ ρi11 ymax1−∆1

sgn(y)ρNy

1 ymax if |y| ≤ ρNy1 ymax1+∆1

qu(uc) =

sgn(uc)ymax if |uc| > umax

1−∆2

sgn(uc)ρi22 umax if ρi22 umax1+∆2

< |uc| ≤ ρi22 umax1−∆2

sgn(uc)ρNu2 umax if |uc| ≤ ρNu

2 umax1+∆2

.

With the same network parameters, the system states are illustrated in Fig.2.3.

It is observed that the system states only converge to the bounded region instead of

zero. It is noted that better control results are achieved by applying the proposed

asynchronous quantization scheme in comparison to the existing quantization

methods.

2.5 Conclusion

The output feedback control problem of NCSs is studied in this chapter.

A new asynchronous quantization scheme is proposed, and an observer-based

output feedback controller is designed. It is shown that the closed-loop system

is asymptotically stable even with network-induced delays, packet dropouts and

quantization in both S/C and C/A channels. Moreover, the proposed quantization

2.5 Conclusion 41

scheme does not require synchronous quantization parameters between the sending

and receiving nodes, which is more practical than the existing results. The

effectiveness of the proposed method is illustrated by the simulation of the control of

an inverted pendulum finally.

Chapter 3

A Novel Dropout Compensation

Scheme for Control of Networked

T-S Fuzzy Dynamic Systems

3.1 Introduction

In this chapter, we are interested in designing a state feedback controller for a

class of nonlinear NCSs subject to packet dropouts and network-induced delays.

It is noted that packet dropout is one of the critical problems of controller design

for NCSs, which occurs if there exist packet collisions, buffer overflows or other

network congestions [5]. The methods for modeling packet dropout phenomenon

can be classified into two categories as mentioned in Chapter 1. One is called zero

strategy [7], [32], [114], in which the missing packet is set to be zero without any

compensation. This strategy has a relatively simple mathematical expression for

analysis and synthesis of control systems, but the overall system will be open-loop

for a period of time if packets are lost continuously. Obviously, it will degrade the

system performance or even make it unstable if the packet loss rate is high. The other

is called hold strategy [30]- [31], in which the data at last sampling time are held when

the current packet is lost during the transmission. Different from the zero strategy,

it does not result in the open-loop scenario. However, the controller performance

3.1 Introduction 43

in this case might not be satisfactory if multiple packet dropouts phenomena occur,

because the system control inputs are not updated frequently enough. Therefore,

to achieve better system performance, a better strategy is needed to deal with the

packet dropout phenomenon. Moreover, most of the results on NCSs with packet

dropouts reported in literature consider only linear plants [4]- [6], [30], [126]. Though

there are some works on fuzzy-model-based nonlinear NCSs, they are all based on

zero/hold strategy [7], [76]. It is thus significant for us to consider a new strategy in

order to achieve better performance of T-S fuzzy systems under the unsatisfactory

network environment in practice, which motives our research.

In this chapter, we propose a new compensation method for fuzzy-model-based

nonlinear NCSs to deal with packet dropouts in both S/C and C/A channels.

The nonlinear physical plant is described by a T-S fuzzy model. By utilizing a

piecewise Lyapunov function method, the H∞ controller is obtained. The result is

then extended to the case when both network-induced delay and packet dropout

phenomena exist. Finally, two examples are given to show that our strategy is

effective and able to achieve better performance in comparison to the zeor/hold

strategies. The contributions of this chapter can be summarized as follows: 1) a new

approach to solving the H∞ control design problem of the T-S fuzzy control system

with packet dropouts in both S/C and C/A channels is proposed; 2) the optimal

H∞ performance is achieved by utilizing the proposed compensation approach; 3) the

proposed approaches can deal with the case when both packet dropouts and network-

induced delay phenomena exist.

The remainder of the chapter is organized as follows. Section II is devoted to

problem formulation and the description of the novel compensation strategy. Section

III presents the H∞ analysis and synthesis results based on a piecewise quadratic

Lyapunov function. Section IV presents an extension to the case when both network-

induced delay and packet dropout phenomena exist. In Section V, two simulation

examples are given to illustrate the effectiveness of the proposed approaches and show

better performance in comparison with the existing methods. Finally, a conclusion

is drawn in section VI.

3.2 Model Description and Problem Formulation 44

Figure 3.1: Structure of the networked T-S fuzzy system

3.2 Model Description and Problem Formulation

In this chapter, we focus on a class of T-S fuzzy systems with packet dropouts

in communication links as illustrated in Fig.1. Note that the packet dropout

phenomenon exists both in S/C and C/A channels. Therefore, the inputs to the

controller x(t) are not the same as the states of the controlled plant x(t), and the

control inputs to the plant u(t) are also different from the outputs of the controller

u(t). Now, we model the physical plant, dynamic compensator and controller

mathematically.


The T-S fuzzy-model-based physical plant in this chapter is given by:


2 and · · · and θv(t) is F lv, THEN x(t+ 1) = Alx(t) +Blu(t) +Dlw(t)

z(t) = Llx(t) +Hlu(t), l ∈ I := 1, 2, ..., r,(3.1)


1, 2, ..., v) the fuzzy sets, x(t) ∈ ℜnx the state vector, u(t) ∈ ℜnu the input vector,

z(t) ∈ ℜnz the regulated output vector, w(t) ∈ ℜnw the disturbance input vector,

[θ1(t), θ2(t), · · · , θv(t)] the premise variables, which are some measurable variables of

3.2.1 Physical Plant 45

the system such as the state variables, and (Al, Bl, Dl, Ll, Hl) denotes the matrices

of the system’s lth local model.

By using a standard fuzzy inference method, which includes a singleton fuzzifier,

product fuzzy inference and center-average defuzzier, the T-S fuzzy system in (3.1)

can be inferred as, x(t+ 1) = A(µ)x(t) +B(µ)u(t) +D(µ)w(t)

z(t) = L(µ)x(t) +H(µ)u(t),(3.2)

where

A(µ) =r∑

l=1

µlAl, B(µ) =r∑

l=1

µlBl, D(µ) =r∑

l=1

µlDl,

L(µ) =r∑

l=1

µlLl, H(µ) =r∑

l=1

µlHl, (3.3)


µl =ζl(θ)∑r

φ=1 ζφ(θ), ζl(θ) =

v∏φ=1

F lφ(θφ), µl ≥ 0,

r∑l=1

µl = 1 (3.4)


φ.

In order to investigate the robust H∞ state feedback control problem for system

(3.1) based on piecewise Lyapunov functions, the premise variable space is partitioned

into a number of polyhedral regions Sii∈I ⊆ ℜv, which are divided into crisp and

fuzzy regions with the following definitions, respectively,

Sic = θ(t)|µm(θ(t)) = 1,m ∈ ℵ(ic), ic ∈ I , (3.5)

and

Sif = θ(t)|0 ≤ µm(θ(t)) < 1,m ∈ ℵ(if ), if ∈ I , (3.6)

where ℵ(ic),ℵ(if ) are sets containing the indices of rules in each region, and I is the

set of polyhedral regions. Local models act in crisp or fuzzy regions: in crisp regions

the dynamic system is described by one local model and in fuzzy regions the dynamic

system is determined by a blending of several local models. The partition method

is not unique, one feasible approach is to set crisp regions when the dynamic system

3.2.2 Dynamic Compensator 46

is governed by one local model, and set different fuzzy regions when the dynamics

are determined by different local models according to the membership functions.

Therefore, the rules of T-S fuzzy systems induce polyhedral partition regions. An

example to set polyhedral regions for a practical system will be described in Example

3.1 in Section V.

With such a partition method, the global fuzzy control system (3.2) is rewritten

in each region as x(t+ 1) = Aix(t) + Biu(t) +Diw(t)

z(t) = Lix(t) +Hiu(t), x(t) ∈ Si, i ∈ I ,(3.7)

where

Ai =∑

m∈ℵ(i)

µmAm,Bi =∑

m∈ℵ(i)

µmBm,Di =∑

m∈ℵ(i)

µmDm,

Li =∑

m∈ℵ(i)

µmLm,Hi =∑

m∈ℵ(i)

µmHm (3.8)

with 0 ≤ µm(θ(t)) ≤ 1,∑

m∈ℵ(i) µm(θ(t)) = 1.

In order to carry out the controller design based on piecewise Lyapunov functions,

we also define a set T that represents all possible transitions among regions as follows:

T :=(i, j)|θ(t) ∈ Si, θ(t+ 1) ∈ Sj,∀i, j ∈ I

. (3.9)

The states remain in the same region Si in the case of j = i, (i, j) ∈ T . Otherwise,

the states transit from region Si to Sj.

3.2.2 Dynamic Compensator

Before presenting the compensation results, it is assumed in this section that

there do not exist network-induced delay effects in communication links, and only

packet dropout phenomenon is considered in the unreliable transmission [6].

The dynamic compensator in S/C channel has the following form,

x(t+ 1) = α(t)Aix(t) + (1− α(t))Asx(t), i, s ∈ I , (3.10)

3.2.2 Dynamic Compensator 47

where x(t) is the compensated state, Ai, As, i, s ∈ I are compensator gains in

different regions to be determined, and α(t) is a Bernoulli process representing the

unreliable transmission in S/C channel. Moreover, α(t) = 0 when the data are lost,

while α(t) = 1 means a successful transmission.

A natural assumption on α(t) can be made as follows:

Prob α(t) = 1 = E α(t) = α,Prob α(t) = 0 = 1− α. (3.11)

For the compensated state x(t), we also define a set T to describe the region

transitions:

T =(s, k)|x(t) ∈ Ss, x(t+ 1) ∈ Sk, s, k ∈ I

. (3.12)

Remark 3.1. There is a memory in the compensator node such that the missing

states x(t+1) can be estimated as x(t+1) by historical states stored in the memory

according to (3.10).

Remark 3.2. The dynamic compensator is event-driven, that is, the compensated

state x(t + 1) will be estimated by x(t) if the packet carrying x(t) arrives at the

compensator node at sampling time t, otherwise, x(t) is utilized for the estimation.

Obviously, the compensator is still able to estimate the state when the multiple packet

dropouts phenomenon occurs.

Remark 3.3. It is noted that the states of a physical plant are available at the

physical plant node instantly, but not always available in the compensation node

due to the unreliable transmission. Therefore, the compensated states are utilized

for determining the region where the compensator stays when the packet dropout

phenomenon occurs, and the region information of x(t) is no longer needed at the

compensator node. In this case, the plant states and the compensated states might

not be in the same region, thus the region transitions of the physical plant and

compensator might not be synchronized, which are defined in (3.9) and (3.12),

respectively. Therefore, the compensator in (3.10) utilizes compensation gains

in different regions according to the plant states and compensated ones. More

specifically, Ai and As are compensation gains in regions Si and Ss, respectively.

3.2.3 Controller 48

Remark 3.4. It is noted from [127] that the compensator will be identical to a

linear compensator if we design a fuzzy compensator based on conventional parallel

distributed compensation (PDC) [117] when the premise variables between the

physical plant and the designed fuzzy filter are different. Therefore, the piecewise

compensator in (3.10) is designed, which is less conservative.

Remark 3.5. The proposed compensation scheme is more general than the

traditional hold and zero strategies. More specifically, hold strategy is a special case

when Ai = As = I in (3.10), and zero strategy is a special case when Ai = As = 0.

Remark 3.6. Methods utilizing the hold strategy need an assumption that the

maximum dropout step is bounded when dealing with multiple packet dropouts [30],

[55]. Actually, we are not always able to estimate the maximum packet-loss upper

bound beforehand in practical situation. However, this assumption is not needed in

our approach.

3.2.3 Controller

Based on the compensator given in (3.10), the following piecewise controller is

considered,

u(t) = α(t)Kix(t) + (1− α(t))Ksx(t), i, s ∈ I , (3.13)

where Ki, Ks, i, s ∈ I are controller gains in different regions to be determined.

Remark 3.7. Similar to Remark 2.3, it is also noted that the states of physical plant

are not always available in the controller node because of the unreliable transmission.

Therefore, the compensated state x(t) will be employed for the controller design if the

packet carrying x(t) is dropped during the transmission. Otherwise, x(t) is utilized.

It is also noted that the plant states and the compensated states might not be in the

same region. Therefore, the controller in (3.13) utilizes controller gains in different

regions according to the plant states and compensated ones, which is similar to the

compensator given in (3.10). More specifically, Ki and Ks are controller gains in

regions Si and Ss, respectively.

3.2.4 Closed-loop System 49

Remark 3.8. The controller in (3.13) is rather different from those given in some

existing results, such as [32, 77], which utilize a parallel distributed compensator

(PDC) with the assumption that the premise variables of the physical plant are

always available at the controller node. However, this assumption is unpractical

under network-based circumstances. Moreover, it is also assumed that there exists

a perfect communication link so that the region index information of the physical

plant can be sent to the receiving node in [55]. However, this assumption is also not

needed in this chapter, which is more practical.


We model the packet dropouts in C/A channel as follows,

u(t) = β(t)u(t), (3.14)

where β(t) is a Bernoulli process representing the unreliable transmission in C/A

channel, which is independent of α(t). More specifically, β(t) = 0 when the packet

is dropped, while β(t) = 1 indicates a successful transmission.

Similar to (3.11), β(t) is assumed to have the following stochastic properties:

Prob β(t) = 1 = E β(t) = β,Prob β(t) = 0 = 1− β. (3.15)

Additionally, we define α(t) = α(t)− α and β(t) = β(t)− β. It is clear that

E α(t) = 0,E α(t)α(t) = σ21,E

β(t)

= 0,E

β(t)β(t)

= σ2

2, (3.16)

where σ1 =√α(1− α), σ2 =

√β(1− β).

Then based on (3.10), (3.13) and (3.14), the physical plant (3.7) is rewritten as, x(t+ 1) = (Ai + α(t)β(t)BiKi)x(t) + (1− α(t))β(t)BiKsx(t) +Diw(t),

z(t) = (Li(t) + α(t)β(t)HiKi)x(t) + (1− α(t))β(t)HiKsx(t), i, s ∈ I .(3.17)

From (3.10) and (3.17), we have the following augmented closed-loop system, ξ(t+ 1) = Ais(t)ξ(t) + Diw(t),

z(t) = Lis(t)ξ(t), i, s ∈ I ,(3.18)

3.2.5 Problem Formulation 50 lost data Sensor

Controller

Actuator

( )x k ( 1)x k + ( 2)x k +

kt ˆ( )u k ˆ ˆ( 1) ( 1)su k K x k+ = + t1kt + 2kt +

ˆ( 1)u k +

ˆ( ) ( )iu k K x k=

Figure 3.2: The time-sequence diagram of the signals in the closed-loop system

where

ξ(t) =[xT (t) xT (t)

]T,Di =

[DT

i 0]T,

Ais(t) = α(t)Ais1 + β(t)Ais2 + α(t)β(t)Ais3 + Ais4,

Lis(t) = α(t)Lis1 + β(t)Lis2 + α(t)β(t)Lis3 + Lis4,

Ais1 =

βBiKi −βBiKs

Ai −As

,Ais2 =

αBiKi (1− α)BiKs

0 0

,Ais3 =

BiKi −BiKs

0 0

,Ais4 =

Ai + αβBiKi (1− α)βBiKs

αAi (1− α)As

,Lis1 =

[βHiKi −βHiKs

],Lis2 =

[αHiKi (1− α)HiKs

],

Lis3 =[HiKi −HiKs

],Lis4 =

[Li + αβHiKi (1− α)βHiKs

].(3.19)

The time-sequence diagram of the signals in the closed-loop system is illustrated

in Fig.2.

3.2.5 Problem Formulation

Before proceeding further, some basic definitions are introduced as follows.

Definition 3.1. [55] The closed-loop fuzzy system in (3.18) is said to be

stochastically stable in the mean square sense if, when w(t) ≡ 0 and for any initial

3.3 Main Results 51

condition ξ(0), there is a finite matrix Y > 0 such that

E

∞∑t=0

|ξ(t)|2|ξ(0)

< ξT (0)Y ξ(0). (3.20)

Definition 3.2. [55] The closed-loop fuzzy system in (3.18) is stochastically stable

with guaranteed H∞ performance γ if the following two conditions are satisfied:

Stochastic stability

The closed-loop fuzzy system (3.18) is stochastically stable in the sense of

Definition 2.1;

H∞ performance

Under zero-initial conditions, the controlled output z(t) satisfies

∥z∥E ≤ γ∥w∥2, (3.21)

where ∥z∥E := E√∑∞

t=0 |z(t)|2

, and γ > 0 is a prescribed scalar.

The problem to be addressed in this chapter is described as follows.

H∞ Dropout Compensator and Controller Design Problem. Consider the

fuzzy system in (3.1) and suppose that the network parameters α and β are given.

Design a compensator and controller in the form of (3.10) and (3.13) respectively

such that the augmented system (3.18) is stochastically stable with guaranteed H∞

performance γ by Definition 3.2.

3.3 Main Results

In this section, the piecewise compensator and H∞ state feedback controller

analysis and synthesis problem of the fuzzy system (3.7) is investigated, which is

solved by a linear matrix inequality (LMI) approach based on piecewise Lyapunov

functions.

The following lemma presents a condition to guarantee the stochastic stability

and H∞ performance of the closed-loop system (3.18).

3.3 Main Results 52

Lemma 3.1. Consider the system (3.1) and suppose that the compensator and

controller gain matrices As and Ks, s ∈ I of the local compensators in (3.10) and

controllers in (3.13) are given. The augmented system (3.18) is stochastically stable

with guaranteed H∞ performance γ, if there exist matrices Xis = XTis > 0, i, s ∈ I

satisfying Γ1 0 Γ2Xis 0

∗ Γ3 Γ4Xis Γ5

∗ ∗ −Xis 0

∗ ∗ ∗ −I

< 0, i, s ∈ I , (3.22)

where

Γ1 = diag−γ2I,−γ2I,−γ2I,−γ2I,

Γ2 =[σ1L T

is1 σ2L Tis2 σ1σ2L T

is3 L Tis4

]T,

Γ3 = diag−Xjk,−Xjk,−Xjk,−Xjk,

Γ4 =[σ1A T

is1 σ2A Tis2 σ1σ2A T

is3 A Tis4

]T,

Γ5 =[

0 0 0 DTi

]T. (3.23)

Proof. Consider the following piecewise Lyapunov function,

V (t) = ξT (t)X−1is ξ(t), i, s ∈ I , (3.24)

where XTis = Xis > 0 are Lyapunov matrices to be determined.

It is known that the closed-loop system in (3.18) can be demonstrated

stochastically stable in the mean square sense with H∞ performance γ under zero

initial conditions by proving the following index J is negative:

J = E V (t+ 1)|η(t)+ Eγ−2zT (t)z(t)|η(t)

− ξT (t)X−1

is ξ(t)− wT (t)w(t), (3.25)

where η(t) =[ξT (t) wT (t)

]T.

3.3 Main Results 53

From (3.19), we have

J = E

ηT (t) A T

i

DTi

X−1jk

[Ai Di

]η(t)|η(t)

+E

γ−2ξT (t)

(α(t)L T

is1 + β(t)L Tis2 + α(t)β(t)L T

is3 + L Tis4

)(α(t)Lis1 + β(t)Lis2 + α(t)β(t)Lis3 + L T

is4

)ξ(t)|η(t)

−ηT (t)diag

X−1

is , Iη(t)

= ηT (t)

−[Γ4 Γ5

]TΓ−13

[Γ4 Γ5

]−[Γ2 0

]TΓ−11

[Γ2 0

]−diag

X−1

is , I

η(t). (3.26)

Then by the Schur complement and (3.22), we have J < 0. The proof is thus

completed.

Remark 3.9. A piecewise quadratic Lyapunov function is utilized in Lemma 3.1,

which is defined in (3.24). It is noted that (3.24) will reduce to the common quadratic

Lyapunov function if Xis = X for any i, s ∈ I .

In terms of Lemma 3.1, now we present the controller synthesis result in the

following theorem.

Theorem 3.1. Consider system (3.1). The closed-loop system (3.18) is stochastically

stable with guaranteed H∞ performance γ, if there exist matrices Xis = XTis >

0, Qs, G, Rs, i, s ∈ I such that the following linear matrix inequalities are satisfied:Γ1 0 Γ2 0∗ Γ3 Γ4 Γ5

∗ ∗ Xis − G− GT 0∗ ∗ ∗ −I

< 0, i, s ∈ I , (3.27)

3.3 Main Results 54

where

Xis =

Xis11 Xis12

∗ Xis22

, G =

G G

G G

, Dm =

Dm

0

,

Γ2 =

σ1Πmis5

σ2Πmis6

σ1σ2Πmis7

Πmis8

, Γ4 =

σ1Πmis1

σ2Πmis2

σ1σ2Πmis3

Πmis4

, Γ5 =

000Dm

,

Πmis1 =

βBmQi − βBmQs βBmQi − βBmQs

Ri −Rs Ri −Rs

,Πmis3 =

BmQi −BmQs BmQi −BmQs

0 0

,Πmis2 =

αBmQi + (1− α)BmQs αBmQi + (1− α)BmQs

0 0

,Πmis4 =

AmG+ αβBmQi + (1− α)βBmQs

αRi + (1− α)Rs

AmG+ αβBmQi + (1− α)βBmQs

αRi + (1− α)Rs

Πmis5 =

[βHmQi − βHmQs βHmQi − βHmQs

],

Πmis6 =[αHmQi + (1− α)HmQs αHmQi + (1− α)HmQs

],

Πmis7 =[HmQi −HmQs HmQi −HmQs

],

Πmis8 =[LmG+ αβHmQi + (1− α)βHmQs

LmG+ αβHmQi + (1− α)βHmQs

]. (3.28)

Moreover, the controller and compensation gains are respectively given by

Ki = QiG−1, Ai = RiG

−1, i ∈ I . (3.29)

Proof. According to Lemma 3.1, if there exist matrices Xis > 0, i, s ∈ I satisfying

(3.22), the closed-loop fuzzy system (3.18) is stochastically stable with guaranteed

H∞ performance γ. Based on (3.8), the left hand side of (3.22) can be rewritten as

3.3 Main Results 55∑m∈ℵ(i) µmΨmjkis, j, k, i, s ∈ I , where

Ψmjkis =

Γ1 0 Γ2Xis 0

∗ Γ3 Γ4Xis Γ5

∗ ∗ −Xis 0

∗ ∗ ∗ −I

,

Γ2 =[σ1L T

mis1 σ2L Tmis2 σ1σ2L T

mis3 L Tmis4

]T,

Γ4 =[σ1A T

mis1 σ2A Tmis2 σ1σ2A T

mis3 A Tmis4

]T,

Amis1 =

βBmKi −βBmKs

Ai −As

, Amis2 =

αBmKi (1− α)BmKs

0 0

,Amis3 =

BmKi −BmKs

0 0

, Amis4 =

Am + αβBmKi (1− α)βBmKs

αAi (1− α)As

,Lmis1 =

[βHmKi −βHmKs

], Lmis2 =

[αHmKi (1− α)HmKs

],

Lmis3 =[HmKi −HmKs

], Lmis4 =

[Lm + αβHmKi (1− α)βHmKs

].

(3.30)

We can see from (3.30) that the compensation matrices are coupled with the

Lyapunov matrices, which is difficult for the controller synthesis. To facilitate the

controller design, we introduce an additional slack matrix G =

G G

G G

.

Then, post- and pre-multiplying Ψmjkis by diag

I, I, X−1is G, I

and its transpose,

respectively, lead toΓ1 0 Γ2G 0

∗ Γ3 Γ4G Γ5

∗ ∗ −GTX−1is G 0

∗ ∗ ∗ −I

< 0, i, s ∈ I ,m ∈ ℵ(i). (3.31)

Note that

Xis − G− GT + GTX−1is G = (Xis − GT )X−1

is (Xis − G) ≥ 0, (3.32)

which implies

− GTX−1is G ≤ Xis − G− GT . (3.33)

3.4 Extensions 56

Based on (3.33), the following inequality implies (3.31)Γ1 0 Γ2G 0

∗ Γ3 Γ4G Γ5

∗ ∗ Xis − G− GT 0

∗ ∗ ∗ −I

< 0, i, j ∈ I ,m ∈ ℵ(i). (3.34)

We define Qi = KiG, Ri = AiG. It then follows that (3.27) implies (3.34).

Based on Lemma 3.1, the fuzzy system (3.18) is stochastically stable with H∞

performance γ, and the proof is thus completed.

Remark 3.10. If we let Ai = As = I or Ai = As = 0, the proposed dropout

compensation results will reduce to hold or zero strategy-based ones, respectively. In

hold strategy, the last packet x(t − 1) stored in the buffer will be utilized for the

controller design when the packet carrying x(t) is lost. It is noted that x(t − 1)

and x(t) may be in different regions, thus the asynchronous control method is also

utilized [76]. In zero strategy, the inputs of the controller are assumed to be zero

when packet dropout phenomenon occurs. In this case, the asynchronous control

problem is not involved [32].

3.4 Extensions

Besides packet dropout phenomenon, network-induced delay is another important

issue in study of networked control systems [5]. In order to deal with this issue, the

following assumption is needed.

Assumption 3.1. The delays and consecutive steps of packet dropouts in S/C

channel are bounded, say, less than N [128].

Based on the compensator in (3.10), we have the following compensation scheme:

x(t) = x(t|td), td ≤ t, (3.35)

where x(t) is the compensated state at sampling time t if the current packet x(t)

does not arrive, and x(t|td) is the estimated state compensated by x(td), which is the

latest system state stored in the buffer.

3.4 Extensions 57

We define d(t) = t − td, which is the time-delay of the packet in the controller

caused by the network-induced delay and packet dropout phenomena in S/C channel,

and it is assumed to satisfy 0 ≤ d(t) ≤ N . Obviously, the time-delay in the buffer

d(t) is only related to d(t− 1) and the current packet received at time t.

Remark 3.11. A time stamp is added to the packet before it is transmitted into the

network links in both S/C and C/A channels, and network delay d(t) is measurable

by comparing the time stamp of the latest packet with the current time instant.

Remark 3.12. It is noted that any packet which does not arrive in time will be

compensated no matter it is delayed or dropped. When a delayed packet x(t− d(t))

arrives at sampling time t, it will be utilized for the estimation of x(t) if it is newer

than the packet stored in the buffer. Otherwise, the packet x(t−d(t)) will be dropped

and the original latest packet stored in the buffer will be utilized. Therefore, network-

induced delays, packet dropouts and the packets out of sequence can be treated in

the unified model simultaneously.

The dynamic compensator has the following form when the packet is delayed or

lost during the transmission:

x(t+ 1|t) = Ai0x(t),

x(t+ 1|t− 1) = Ai1x(t|t− 1),...

x(t+ 1|t−N) = AiN x(t|t−N),

(3.36)

where x(t|t− 1), · · · , x(t|t−N) are states of the dynamic compensator estimated by

the latest states stored in the buffer. Ai0 , · · · , AiN , i0, · · · , iN ∈ I are compensator

gains in different regions to be determined.

Remark 3.13. It is noted that the plant states and the estimated states compensated

by different states stored in the buffer might not be in the same region, thus the

compensator in (3.36) utilizes compensation gains in different regions according to the

plant states and compensated ones. More specifically, Ai1 , · · · , AiN are compensation

gains in region Si1 , · · · , SiN , respectively.

3.4 Extensions 58

The following piecewise controller is utilized:

u(t) = αp0(t)Ki0x(t) + αp1(t)Ki1x(t|t− 1) + · · ·+ αpN(t)KiN x(t|t−N), (3.37)

where

αpq(t) = α(d(t) = q, d(t− 1) = p),

αpq(t) =

1, d(t) = q, d(t− 1) = p,

0, d(t) = q, d(t− 1) = p.(3.38)

and Ki0 , · · · , KiN , i0, · · · , iN ∈ I are the controller gains in different regions to be

determined.

Remark 3.14. The proposed controller in (3.37) is similar to the model in [78].

However, [78] employs the hold strategy, where a delayed state x(t− d(t)) is utilized

for the controller design at sampling time t. If we let x(t) = x(t − d(t)) specifically,

our compensation scheme will reduce to the model in [78].

Remark 3.15. Similar to the model in the second section, the premise variables

are also not available at the compensator and controller nodes with the existence of

packet dropout and network-induced delay phenomena. Therefore, the asynchronous

approach is also employed.

We model the delays and packet dropouts in C/A channel as follows,

u(t) = β(t)u(t), (3.39)

where β(t) = 1 when the packet carrying u(t) arrives at the actuator node in time.

Otherwise, β(t) = 0.

Natural assumptions can be made as follows:

Prob αpq(t) = 1 = E αpq(t) = πpq,Prob β(t) = Eβ(t) = β, (3.40)

and the transition probability matrix is

T =

π00 π01 · · · π0N

π10 π11 · · · π1N... ... . . . ...

πN0 πN1 · · · πNN

. (3.41)

3.4 Extensions 59

Additionally, we define epq(t) = epq(t) − epq, p, q = 0, 1, · · · , N , where epq(t) =

αpq(t)β(t) and epq = Eepq(t) = πpqβ. It is clear that

E epq(t) = 0,E epq(t)epq(t) = σ2pq (3.42)

with σpq =√epq(1− epq).

Based on (3.1), (3.36), (3.37) and (3.39), the closed-loop fuzzy control system is

expressed as follows: ξ(t+ 1) = Ωi(t)ξ(t) + Diw(t),

z(t) = Φi(t)ξ(t),(3.43)

where

ξ(t) =[xT (t) xT (t|t− 1) · · · xT (t|t−N + 1) xT (t|t−N)

]T,

Ωi(t) = ep0(t)Ωi0 + · · ·+ epN(t)ΩiN + Ωia,

Φi(t) = ep0(t)Φi0 + · · ·+ epN(t)ΦiN + Φia, Di =[DT

i00 · · · 0 0

]T,

Ωi0 =

Bi0Ki0 · · · 0 0

0 · · · 0 0... . . . ... ...

0 · · · 0 0

, · · · ,ΩiN =

0 · · · 0 Bi0KiN

0 · · · 0 0... . . . ... ...

0 · · · 0 0

,

Ωia =

Ai0 + ep0Bi0Ki0 · · · epN−1Bi0KiN−1

epNBi0KiN

Ai0 · · · 0 0... . . . ... ...

0 · · · AiN−10

,

Φi0 =[Hi0Ki0 · · · 0 0

], · · · ,ΦiN =

[0 · · · 0 Hi0KiN

],

Φia =[Li0 + ep0Hi0Ki0 · · · epN−1Hi0KiN−1

epNHiKiN

]. (3.44)

Based on the model above, we have the following stability analysis result.

Lemma 3.2. Consider fuzzy system (3.7) and suppose that the compensator and

controller gain matrices Aiδ andKiδ , iδ ∈ I , δ = 0, 1, · · · , N of the local compensators

in (3.36) and controllers in (3.37) are given. The closed-loop system (3.43) is

3.4 Extensions 60

stochastically stable with guaranteed H∞ performance γ, if there exists a set of

matrices Xi0...iN−1iN = XTi0...iN−1iN

> 0, i0, · · · , iN−1, iN ∈ I satisfying:Λ1 0 Λ2Xi0...iN−1iN 0

∗ Λ3 Λ4Xi0...iN−1iN Λ5

∗ ∗ −Xi0...iN−1iN 0

∗ ∗ ∗ −I

< 0, i, s ∈ I , (3.45)

where

Λ1 = diag−γ2I, · · · ,−γ2I,−γ2I,Λ2 =[σp0Φ

Ti0 · · · σpNΦ

TiN ΦT

ia

]T,

Λ3 = diag−Xj0...jN−1jN , · · · ,−Xj0...jN−1jN ,−Xj0...jN−1jN,

Λ4 =[σp0Ω

Ti0 · · · σpNΩ

TiN ΩT

ia

]T,Λ5 =

[0 · · · 0 DT

i0

]T. (3.46)

Proof. The proof is omitted for its similarity to Lemma 3.1.

Remark 3.16. A piecewise quadratic Lyapunov function is also utilized in Lemma

3.2, which will reduce to the common quadratic Lyapunov function if Xi0...iN−1iN = X

for any i0, . . . , iN ∈ I in (3.45).

In terms of Lemma 3.2, now we present the controller synthesis result in the

following theorem.

Theorem 3.2. Consider fuzzy system (3.7). The closed-loop system (3.43) is

stochastically stable with guaranteed H∞ performance γ, if there exist matrices

Xi0...iN−1iN = XTi0...iN−1iN

> 0, Riδ , Qiδ , iδ ∈ I and G such that the following linear

matrix inequalities are satisfied for all δ, p = 0, 1, 2, · · · , N:Λ1 0 Λ2 0∗ Λ3 Λ4 Λ5

∗ ∗ Xi0...iN−1iN − G− GT 0

∗ ∗ ∗ −I

< 0,m ∈ ℵ(i0), i0, · · · , iN ∈ I ,

(3.47)

3.4 Extensions 61

where

G =

G · · · G G... . . . ... ...

G · · · G G

G · · · G G

, Dm =

Dm

...

00

,

Λ2 =

σp0Θmi0

...

σpNΘmiN

Θmia

, Λ4 =

σp0Ξmi0

...

σpNΞmiN

Ξmia

, Λ5 =

0...

0Dm

,

Ξmi0 =

BmQi0 · · · BmQi0 BmQi0

0 · · · 0 0... . . . ... ...

0 · · · 0 0

...

ΞmiN =

BmQiN · · · BmQiN BmQiN

0 · · · 0 0... . . . ... ...

0 · · · 0 0

,

Ξmia =

Ξmia(1) · · · Ξmia(1) Ξmia(1)

Ri0 · · · Ri0 Ri0

... . . . ... ...

RiN−1· · · RiN−1

RiN−1

,Ξmia(1) = AmG+ ep0BmQi0 + · · ·+ epNBmQiN ,

Θmi0 =[HmQi0 · · · HmQi0 HmQi0

]...

ΘmiN =[HmQiN · · · HmQiN HmQiN

],

Θmia =[Θpia(1) · · · Θpia(1) Θpia(1)

],

Θpia(1) = LpG+ em0HpQi0 + · · ·+ emNHpQiN . (3.48)

Moreover, the controller gains and the compensation matrices are respectively

3.5 Simulation Examples 62

given by

Kiδ = QiδG−1, Aiδ = RiδG

−1, δ = 0, 1, · · · , N − 1. (3.49)

Proof. The proof is omitted for its similarity to Theorem 3.1.

3.5 Simulation Examples

In this section, we use two examples to demonstrate the effectiveness of the

compensator and controller design methods proposed in this chapter.

Example 3.1. In this example, we use a nonlinear inverted pendulum on a cart [7] to

demonstrate the improvement of the proposed compensation method in comparison

with the existing results.

x1 = x2,

x2 =g sin(x1)− amlx22 sin(2x1)/2− a cos(x1)u

4l/3− aml cos2(x1)+ w,

where x1 denotes the angle of the pendulum from the vertical axis, x2 is the angular

velocity, g = 9.8m/s2 is the gravity constant, m is the mass of the pendulum, a =

1/(m+M), M is the mass of the cart, 2l is the length of the pendulum, u is the force

applied to the cart, and w is the external disturbance. In this example, we choose

m = 2kg, M = 8kg, and 2l = 1m.

Then, we linearize the plant around the origin, x = (±60, 0), and x = (±88, 0).

After choosing sampling period T = 0.01s, we obtain the discrete-time T-S fuzzy

system as follows:

Plant Rule Rl: IF |x1| is F l, THEN x(t+ 1) = Alx(t) + Blu(t) +Dlw(t)

z(t) = Llx(t) +Hlu(t), l ∈ I := 1, 2(3.50)


0

0

0.2

0.4

0.6

0.8

1

|x1(t)|

µ

Rule1Rule2

π

18

π

3

S2

S1

(a) Membership functions of the system in (3.50)

0 50 100 150 200

0

0.5

1

Time in samples

α(t)

0 50 100 150 200

0

0.5

1

Time in samples

β(t)

(b) Data-packet dropout

0 50 100 150 2000

0.2

0.4

0.6

0.8

Time in samples

stat

e x 1(t

) tr

ajec

tory

Zero−StrategyHold−StrategyTheorem 3.2

0 50 100 150 200−2

−1

0

1

Time in samples

stat

e x 2(t

) tr

ajec

tory


(c) State responses

0 50 100 150 2000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Time in samples

Res

pons

e of

the

ratio

ζ


(d) Response of the ratio ζ

Figure 3.3: Simulation Results of Example 5.1

where

A1 =

1 0.01

0.1729 1

, B1 =

0

−0.0018

, D1 =

0

0.01

,A2 =

1 0.01

0.0585 1

, B2 =

0

−0.0007792

, D2 =

0

0.01

,L1 = L2 =

[0.5 0.1

], H1 = 0.005, H2 = 0.01.

The membership functions are shown in Fig.3.3(a). It is noted that the fuzzy

system is governed by one linear dynamic when 0 ≤ |x1| < π18

according to Fig.3.3(a),

and a fuzzy blending of two local dynamics when π18

≤ |x1| ≤ π3. Applying the

partition method given in the second section, the physical plant can be partitioned


into two regions, which are described as follows:

S1 := x|0 ≤ |x1| <π

18,

S2 := x| π18

≤ |x1| ≤π

3.

Similar to the case in the physical plant node, the states in the compensator and

controller nodes are partitioned into two regions as follows:

S1 := x|0 ≤ |x1| <π

18,

S2 := x| π18

≤ |x1| ≤π

3.

With such a partition, the fuzzy system is rewritten in each region as follows: x(t+ 1) = Aix(t) + Biu(t) +Diw(t)

z(t) = Lix(t) +Hiu(t), x(t) ∈ Si, i ∈ 1, 2,

where

A1 = A1,B1 = B1,D1 = D1,L1 = L1,H1 = H1,

A2 =2∑

m=1

µmAm,B2 =2∑

m=1

µmBm,D2 =2∑

m=1

µmDm,

L2 =2∑

m=1

µmLm,H2 =2∑

m=1

µmHm.

It is noted that the open-loop system is unstable. The objective is to design

a piecewise fuzzy compensator and state feedback controller of the form (3.10)

and (3.13) so that the closed-loop system (3.18) is stochastically stable with H∞

performance γ.

When α = 0.2 and β = 0.2, there are no feasible solutions utilizing the zero or

hold strategies, while we obtain γmin = 0.5551 by applying Theorem 3.1. The feasible

controller and compensator gains are

K1 =[−2.3634 0.8538

], K2 =

[−2.3633 0.8538

],

A1 = 10−3 ×

−0.0882 0.3245

−0.2131 0.7617

, A2 = 10−3 ×

−0.0852 0.3237

−0.2088 0.7608

.


Table 3.1: minimum H∞ performance γmin in each situation of Example 5.1

Models α/β 0/0 0.2/0.2 0.3/0.3 0.4/0.4 0.5/0.5 0.6/0.6 1/1

Zero-Strategy γmin ∞ ∞ 7.985 0.737 0.437 0.314 0.096

Hold-Strategy γmin ∞ ∞ ∞ 0.811 0.392 0.278 0.140

Theorem 3.1 γmin ∞ 0.555 0.368 0.287 0.240 0.208 0.135

With the purpose of illustrating the state responses, α = 0.4 and β = 0.4 are

assumed such that there are feasible solutions by different strategies. We assume the

initial conditions of the system to be x(0) = x(0) =[

π5

0]T

. The data missing

is randomly generated according to the probability shown in Fig.3.3(b). The state

responses are illustrated in Fig.3.3(c), where the external disturbance w(t) = 0. It

can be observed that the states of the closed-loop system converge to zero.

In order to illustrate the H∞ performance, we assume the initial conditions to

be zero, and the external disturbance w(t) = e−0.1t cos (2t). Fig.3.3(d) shows the

response of the ratio ζ of the closed-loop system, where ζ =√∑k

i=0 zT (i)z(i)√∑k

i=0 wT (i)w(i)

.

From Fig.3.3 we can observe that better H∞ performance and system response can

be achieved by the proposed compensation method in comparison with other strategies.

The more detailed comparison between the existing strategies and the

compensation strategy proposed in this chapter on the minimum H∞ performance

γmin under various scenarios of α and β is illustrated in Table 3.1, which clearly

demonstrates that much better H∞ performance can be achieved by utilizing the

proposed strategy than those existing ones.

Remark 3.17. It is noted from Table 3.1 that the proposed strategy does not achieve

the best system performance in the special case when α = β = 1, which implies that

there are no packet dropouts throughout the transmission. That is because the

compensator incurs some unnecessary effects in that special case. In this case, a

normal fuzzy controller would be used instead if there are no packet dropouts, which

can be found in some existing results, see [12] and references therein.


Example 3.2. In this example, we consider both packet dropout and network-

induced delay problems, aiming to illustrate the effectiveness of Theorem 3.2.

Consider the following discrete-time fuzzy system of form (3.7) with two rules.

Rule Rl: IF x1 is F l1, THEN x(t+ 1) = Alx(t) + Blu(t) +Dlw(t)

z(t) = Llx(t) +Hlu(t), l ∈ I := 1, 2(3.51)

where

A1 =

1.5 −0.5

1 0

, B1 =

1

1

, D1 =

0.2

0.3

,A2 =

−1 −0.5

1 0

, B2 =

−2

1

, D2 =

0

0.01

,L1 = L2 =

[0.5 0.1

], H1 = 0.4, H2 = 0.2.

The membership functions are shown in Fig.3.4(a). According to the partition

method given in the second section, there are two regions of the physical plant, which

are illustrated as follows:

S1 := x1| −∞ < x1 ≤ −3,

S2 := x1| − 3 < x1 ≤ 1.

Similar to the case in the physical plant node, the states in the compensator and

controller nodes are partitioned into following two regions:

S1 := x1| −∞ < x1 ≤ −3,

S2 := x1| − 3 < x1 ≤ 1.

With such a partition, the fuzzy system is rewritten in each region as follows: x(t+ 1) = Aix(t) + Biu(t) +Diw(t)

z(t) = Lix(t) +Hiu(t), x(t) ∈ Si, i ∈ 1, 2,


0

0.2

0.4

0.6

0.8

1

x1(t)

µ

Rule1Rule2

1−3

S2

S1

(a) Membership functions of the system in (3.51)

0 20 40 60 80 100

0

0.5

1

1.5

2

Time in samples

time

dela

y d(

t)

0 20 40 60 80 100

0

0.5

1

Time in samples

β(t)

(b) Network performance

0 20 40 60 80 100−2

−1

0

1

Time in samples

stat

e x 1(t

) tr

ajec

tory

0 20 40 60 80 100−0.4

−0.2

0

0.2

0.4

Time in samples

stat

e x 2(t

) tr

ajec

tory

(c) State responses

0 20 40 60 80 1000

0.02

0.04

0.06

0.08

0.1

0.12

Time in samples

Res

pons

e of

the

ratio

ζ

(d) Response of the ratio ζ

Figure 3.4: Simulation Results of Example 5.2

where

A1 = A1,B1 = B1,D1 = D1,L1 = L1,H1 = H1,

A2 =2∑

m=1

µmAm,B2 =2∑

m=1

µmBm,D2 =2∑

m=1

µmDm,

L2 =2∑

m=1

µmLm,H2 =2∑

m=1

µmHm.

In this example, the largest consecutive steps of network-induced delays and

packet dropouts N = 2 is assumed. Additionally, β = 0.95, and the transition

3.6 Conclusion 68

probability matrix is given by

T =

0.95 0.03 0.02

0.95 0.03 0.02

0.95 0.03 0.02

.By applying Theorem 3.2, the corresponding controller and compensation gains

are obtained as follows,

K1 =[−0.8153 0.1178

], K2 =

[−0.8117 0.0779

],

A1 = 10−3 ×

1.3262 −0.8298

1.6967 4.4351

, A2 = 10−3 ×

−0.5474 1.5152

−4.3911 5.9701

,and the minimum H∞ performance γmin = 0.2040. However, it is also noted that for

this example there is no feasible solution by utilizing the hold strategy.

To illustrate the closed-loop performance, some simulations have been done with

the initial conditions of the system as x(0) = x(0) =[−2 0

]T. The network-

induced delays and data missing are randomly generated according to the probability

shown in Fig.3.4(b). The state responses of the closed-loop system are illustrated in

Fig.3.4(c), where the external disturbance w(t) = 0. It can be observed that the

states of the closed-loop system converge to zero.

In order to illustrate the H∞ performance, we assume the initial conditions to

be zero, and the external disturbance w(t) = e−0.1t cos (2t). Fig.3.4(d) shows the

response of the ratio ζ of the closed-loop system.

From Fig.3.4 we can observe that the system is stochastically stable with

guaranteed H∞ performance, which shows the effectiveness of Theorem 3.2.

3.6 Conclusion

The H∞ control problem of fuzzy-model-based nonlinear NCSs with packet

dropouts is discussed in this chapter. A new compensation method for packet

dropouts is proposed, and a piecewise state feedback controller is designed. It is shown

that the closed-loop system is stochastically stable with guaranteed H∞ performance

3.6 Conclusion 69

with the existence of packet dropouts in both S/C and C/A channels. Moreover, the

proposed compensation method can achieve better control performance in comparison

to the existing approaches such as zero and hold strategies. Moreover, the results

are extended to the case where both packet dropouts and network-induced delay

phenomena exist. The developed results are illustrated by two simulation examples.

Chapter 4

H∞ Filter Design of Networked

Nonlinear Systems With

Communication Constraints via

T-S Fuzzy Dynamic Models

4.1 Introduction

In recent decades, great effort has been devoted to filter design in the field of

signal processing and control application. Among various filtering approaches, H∞

filter has attracted great attention because a guaranteed noise attenuation level can

be obtained without requiring the knowledge of noises a priori, and various theoretical

and application results have been published [129]- [130].

Generally, the current system output is needed in the H∞ filtering approach.

However, the measured output received by filter yF (t) is different from the current

system output y(t) under the network circumstance. Three typical problems of

network are quantization, delays and packet dropouts. Therefore, great attention

has been paid to the investigation of the filtering problem with communication

constraints, and many results have been reported [131]- [134]. To mention a few, the

authors in [131] study the H∞ filtering problem for networked discrete-time systems

4.1 Introduction 71

with random packet losses, which is described by a two-state Markov chain. The

authors in [132] consider the problem of state estimation for discrete-time dynamic

systems with quantized measurements, in which a logarithmic quantizer is utilized for

the quantization process. The authors in [133] investigate the problem of robust H∞

estimation for discrete-time systems subject to communication limitations including

measurement quantization, signal transmission delay, and packet dropout, in which

the three typical communication constraints are studied simultaneously.

It is noted that the considered plants are linear systems in all of the

aforementioned filtering results. However, most of the industrial plants have

nonlinear properties, so the filter design of nonlinear systems under the network

circumstance is desirable. As mentioned in Chapter 1, T-S fuzzy models can describe

some complex nonlinear systems by a group of local linear systems, which are blended

by several IF-THEN rules [14]- [15]. More recently, some results on the filtering

of fuzzy dynamic systems under the network circumstance have been reported.

However, most of them consider one or two aspects of the communication constraints.

To mention a few, the authors in [135] consider the H∞ filtering for nonlinear systems

with time-varying delay via T-S fuzzy model approach. [136] is concerned with H∞

filter design for fuzzy-model-based systems with transfer delays and packet loss. The

authors in [55] investigate the H∞ filtering for nonlinear discrete-time systems subject

to quantization and packet dropouts via T-S dynamic models. It is worth nothing that

few papers address the filter design problem of nonlinear systems with simultaneous

consideration of the three typical communication constraints mentioned previously,

which motivates our research.

In this chapter, we investigate the filter design of nonlinear systems subject to

those three important constraints of communication simultaneously. The nonlinear

physical plant is described by a T-S fuzzy model. By utilizing a piecewise Lyapunov

function method, the H∞ filter is obtained. Finally, an example is given to show the

effectiveness of our approach. The contributions of this chapter can be summarized

as follows: 1) a new approach to H∞ filtering is proposed, which considers three

communication constraints simultaneously for the nonlinear system via T-S dynamic

4.2 Model Description 72

Figure 4.1: Overall filtering error system

models; and 2) the proposed piecewise filter is based on piecewise quadratic Lyapunov

functions without obtaining the region information where the current system states

stay, which is more useful in application.


problem formulation and the description of the novel filter design method. Section III

presents the H∞ filtering results based on a piecewise quadratic Lyapunov function.

In Section IV, a simulation example is given to illustrate the effectiveness of the

proposed approach. Finally, a conclusion is drawn in section V.

4.2 Model Description

In this chapter, we focus on a class of T-S fuzzy systems with communication

constraints as illustrated in Fig.4.2.


The T-S fuzzy-model-based physical plant is given by:

4.2.1 Physical Plant 73


2 and · · · and θv(t) is F lv, THEN

x(t+ 1) = Alx(t) +Dlw(t)

z(t) = Llx(t) +Hlu(t)

y(t) = Clx(t), l ∈ I := 1, 2, ..., r,

(4.1)


1, 2, ..., v) the fuzzy sets, x(t) ∈ ℜnx the state vector, y(t) ∈ ℜnz the regulated output

vector, z(t) ∈ ℜnz the signal vector to be estimated, w(t) ∈ ℜnw the disturbance

input vector, [θ1(t), θ2(t), · · · , θv(t)] the premise variables, which are some measurable

variables of the system such as the state variables, and (Al, Cl, Dl, Ll, Hl) denotes the

matrices of the system’s lth local model.

By using a standard fuzzy inference method, which includes a singleton fuzzifier,

product fuzzy inference and center-average defuzzier, the T-S fuzzy system in (4.1)

can be rewritten as, x(t+ 1) = A(µ)x(t) +D(µ)w(t)

z(t) = L(µ)x(t) +H(µ)u(t)

y(t) = C(µ)x(t),

(4.2)

where

A(µ) =r∑

l=1

µlAl, D(µ) =r∑

l=1

µlDl, C(µ) =r∑

l=1

µlCl,

L(µ) =r∑

l=1

µlLl, H(µ) =r∑

l=1

µlHl, (4.3)


µl =ζl(θ)∑r

φ=1 ζφ(θ), ζl(θ) =

v∏φ=1

F lφ(θφ),

µl ≥ 0,r∑

l=1

µl = 1 (4.4)


φ.

In order to investigate the robust H∞ filtering problem for system (4.1) based

on piecewise Lyapunov functions, the premise variable space is partitioned into a

4.2.2 Quantization, Encoding and Decoding 74

number of polyhedral regions Sii∈I ⊆ ℜv, which are divided into crisp and fuzzy

regions with the following definitions, respectively,

Sic = θ(t)|µm(θ(t)) = 1,m ∈ ℵ(ic), ic ∈ I , (4.5)

and

Sif = θ(t)|0 ≤ µm(θ(t)) < 1,m ∈ ℵ(if ), if ∈ I , (4.6)

where ℵ(ic),ℵ(if ) are sets containing the indices of rules in each region, and I is the

set of polyhedral regions.

With such a partition, the fuzzy system (4.2) is rewritten in each region asx(t+ 1) = Aix(t) +Diw(t)

z(t) = Lix(t) +Hiw(t)

y(t) = Cix(t), x(t) ∈ Si, i ∈ I ,

(4.7)

where

Ai =∑

m∈ℵ(i)

µmAm,Di =∑

m∈ℵ(i)

µmDm, Ci =∑

m∈ℵ(i)

µmCm,

Li =∑

m∈ℵ(i)

µmLm,Hi =∑

m∈ℵ(i)

µmHm, (4.8)

with 0 ≤ µm(θ(t)) ≤ 1,∑

m∈ℵ(i) µm(θ(t)) = 1.

In order to carry out the filter design based on piecewise Lyapunov functions, we

also define a set T that represents all possible transitions among regions as follows:

T :=(i, j)|θ(t) ∈ Si, θ(t+ 1) ∈ Sj,∀i, j ∈ I

. (4.9)

The states remain in the same region Si in the case of j = i, (i, j) ∈ T . Otherwise,

the states transit from region Si to Sj.

4.2.2 Quantization, Encoding and Decoding

Because of the limited bandwidth of communication links, data need to be

quantized and encoded at the sending node, while decoded at the receiving node.

There are numerous quantization approaches in the field of communication, however,


just a few of them are utilized for the filter design. A widely used quantization method

is logarithmic quantization [55]. The mathematical expression of the quantizer

is convenient for the filter design and controller synthesis. However, it has the

saturation problem. Moreover, the quantizer needs infinite network bandwidth when

the system near the equilibrium, which is unpractical in actual application.

Motivated by [137], we utilize the following dynamic quantizer:

qρ(v) = ρq

(v

ρ

), (4.10)

where ρ > 0 is the ”zoom” variable, and q(v) is a static quantizer satisfying

|q(v)− v| ≤ ∆, if |v| ≤M. (4.11)

Remark 4.1. The range of the dynamic quantizer in (4.10) is [−Mρ,Mρ], and

the quantization error is[−∆ρ, ∆ρ

]. Therefore, the saturation and deterioration of

performance near the equilibrium problems of many other quantization methods can

be solved by increasing and decreasing the ”zoom” variable ρ.

Remark 4.2. The exact form of the static quantizer q(v) considered in this chapter

is not specified, which means that any quantizer satisfying (4.11) can be utilized,

such as uniform quantization [39] and some non-uniform quantization methods.

It is noted that the measurement outputs need to be quantized by multiple

quantizers because the ranges of each output are different, which is denoted as follows:

q(v) =[q(1)Tρ1 (v1) q

(2)Tρ2 (v2) · · · q

(nv)Tρnv

(vnv)]T

= ∆q(v)N (4.12)

where vj is the jth component of v, q(j)ρj (vj) is a dynamic quantizer,

∆q(v) = diagq(1)ρ1 (v1), q

(2)ρ2 (v2), · · · , q

(nv)ρnv

(vnv)

, and N =[1 1 · · · 1

]T.

4.2.3 Communication Links

It is noted that packet dropout and network-induced delay phenomena exist in

the communication links. Therefore, the inputs to the filter yc(t) are not the same as

the outputs of the physical plant y(t).

4.2.4 Filter Error System 76

We assume that there exists a buffer in the filter node, which stores the received

historical data. Motivated by [138], we model the unreliable transmission as follows:

yc(t) = qρ(y(tτ )), (4.13)

where qρ(y(tτ )) denotes the latest data stored in the buffer of the filter node at

sampling time t. We use quantizers in (4.13) satisfying

qρ(y(tτ ))− y(tτ ) = ∆(y(tτ ))N, (4.14)

where ∆(y(tτ )) = diagq(1)ρ1 (y1(tτ )), · · · , q

(ny)ρny

(yny(tτ ))

with ∆(y(tτ )) ≤ ∆ = diagρ1M1, ρ2M2, · · · , ρnyMny

.

We define τ(t) = t − tτ , which is the time-delay of the packet in the filter node

caused by the network-induced delay and packet dropout phenomena.

The following assumption is needed on modeling the random time-delay in the

buffers caused by the unreliable transmission.

Assumption 4.1. The time-delay τ(t) is time varying and satisfy 0 ≤ τ1(t) ≤ τM ,

where τM represents the upper bound on the time-delay in the buffer.

4.2.4 Filter Error System

Based on the T-S fuzzy system in (4.7) in each region, we consider the following

piecewise filter. x(t+ 1) = Asx(t) +Rsyc(t),

z(t) = Lsx(t), s ∈ I(4.15)

where x(t) is the state vector of the filter, As, Ls, Rs, s ∈ I are filter matrices in each

local region to be determined.

For the filter state x(t), we also define a set T to describe the region transitions:

T =(s, k)|x(t) ∈ Ss, x(t+ 1) ∈ Sk, s, k ∈ I

. (4.16)

Remark 4.3. It is noted that the states of a physical plant are not always available

in the filter node due to the unreliable transmission. In this case, the plant states

and the filter states might not be in the same region, thus the region transitions of

4.2.5 Problem Formulation 77

the physical plant and filter might not be synchronized, which are defined in (4.9)

and (4.16), respectively. Therefore, the filter in (4.15) utilizes the gains according to

region of filter states.

Remark 4.4. It is noted from [127] that the filter will be identical to the linear filter

if we design a fuzzy filter based on conventional parallel distributed compensation

(PDC) [117] when the premise variables between the physical plant and the designed

fuzzy filter are different. Therefore, the piecewise filter in (4.15) is designed, which is

less conservative.

From (4.7), (4.13)-(4.15), we have the following filtering error dynamic system: ξ(t+ 1) = Aisξ(t) + Bshξ(t− τ(t)) + Diw(t) + Es,

e(t) = Lis(t)ξ(t) +Hiw(t), i, h, s ∈ I ,(4.17)

where

ξ(t) =[xT (t) xT (t)

]T,Lis =

[Li Ls

],

Es(t) =[

0 Rs∆(y(t− τ(t)))N],Di =

Di

0

T

,

Ais =

Ai 0

0 As

,Bsh =

0 0

−RsCh 0

. (4.18)


Before proceeding further, a basic definition is introduced as follows:

Definition 4.1. The filtering error system in (4.17) is said to be asymptotically

stable with an H∞ performance γ if it is asymptotically stable when w(t) = 0, and

satisfies

∥e∥2 ≤ γ∥w∥2, (4.19)

where ∥e∥2 :=√∑∞

t=0 eT (t)e(t), and γ > 0 is a prescribed scalar.

The problem to be addressed in this chapter is described as follows.

4.3 Main Results 78

H∞ Filter Design Problem. Consider the fuzzy system in (4.1). Design a

piecewise filter in the form of (4.15) such that the filtering error system (4.17) is

asymptotically stable with guaranteed H∞ performance γ.

Before presenting the main results, three lemmas are introduced as follows, which

will be utilized subsequently.

Lemma 4.1. [139] Given appropriately dimensioned matrices Ω1,Ω2, and Ω3 with

Ω1 = ΩT1 , then

Ω1 + Ω3Υ(k)Ω2 + ΩT2Υ

T (k)ΩT3 < 0 (4.20)

holds for all Υ(k) satisfying ΥT (k)Υ(k) ≤ I if and only if for some ε > 0

Ω1 + ε−1Ω3ΩT3 + εΩT

2Ω2 < 0 (4.21)

Lemma 4.2. [140] For any constant matrix M > 0, any scalars a and b with a < b,

the following inequation is satisfied:[b∑

i=a

x(i)

]TM

[b∑

i=a

x(i)

]≤ (b− a+ 1)

[b∑

i=a

xT (i)Mx(i)

]

(4.22)

Lemma 4.3. [55] For any constant matrix P > 0 and G, the following inequation

holds:

−GTP−1G ≤ P −G−GT (4.23)

4.3 Main Results

In this section, the piecewise filter design problem of the fuzzy system (4.1) is

investigated, which is solved by a linear matrix inequality (LMI) approach based on

piecewise Lyapunov functions.

4.3.1 Stability Analysis 79

4.3.1 Stability Analysis

The following lemma presents a condition to guarantee the stability and H∞

performance of the overall error system (4.17).

Theorem 4.1. Consider the system (4.1) and suppose that the filter matrices

As, Rs, Ls of the local filter in (4.15) are given. The overall error system (4.17)

is asymptotically stable with guaranteed H∞ performance γ, if there exist matrices

Pis = P Tis > 0, i, s ∈ I , Z, λ, g and ϵ satisfying

Πisjkh < 0, i, s, j, k, h ∈ I , (4.24)

where

Πisjkh =

Π11isjkh Π12

isjkh

∗ Π22isjkh

,Π11

isjkh = diag−Pis,−τ−1

M Z,−I, λ− g − gT,

Π12isjkh =

PjkΓ1 PjkBsh PjkDi 0 PjkΓ2

ZΓ3 ZBsh PjkDi 0 PjkΓ2

Lis 0 Hi 0 00 0 0 g 0

,Π22

isjkh = diag−Pjk,−τ−1

M Z,−γ2I,−λ+ ϵNTN,−ϵ,

Γ1 =

Ai 0RsCh As

,Γ2 =

0Rs∆

,Γ3 =

Ai − I 0RsCh As − I

. (4.25)

Proof. Consider the following piecewise Lyapunov function,

V (t) = V1 + V2, (4.26)

where

V1 = ξT (t)Pisξ(t), i, s ∈ I ,

V2 =−1∑

q=−τM

t−1∑p=t+q

ξT (p)Zξ(p), (4.27)

with P Tis = Pis > 0, Z are Lyapunov matrices to be determined, and ξ(t) = ξ(t+1)−

ξ(t).

4.3.1 Stability Analysis 80

It is known that the error system in (4.17) can be demonstrated as asymptotically

stable with H∞ performance γ under zero initial conditions by proving the following

index J is negative:

J = V (t+ 1) + γ−2zT (t)z(t)− V (t)− wT (t)w(t). (4.28)

From (4.17) and Lemma 4.2, we have

J ≤ ξT (t+ 1)Pjkξ(t+ 1)− ξT (t)XPisξ(t) + eT (t)e(t)− γ2wT (t)w(t)

+τM ξT (t)Zξ(t)− τ−1

M

t−1∑p=t−τ(t)

ξT (p)Zt−1∑

p=t−τ(t)

ξ(p) + λ− λ

= ηT (t)ΘTdiag Pjk, τMZ, I, λΘη(t)

+ηT (t)diag−Pis,−τ−1

M Z,−γ2I,−λη(t), (4.29)

where

η =[ξT (t)

∑t−1p=t−τ(t) ξ

T (p) wT (t) 1]T,

Θ =

Γ1 Bsh Di Es(t)

Γ3 Bsh Di Es(t)

Lis 0 Hi 0

0 0 0 1

, (4.30)

Then by the Schur complement, Lemma 4.1, Lemma 4.3 and (4.24), we have J < 0.

The proof is thus completed.

Remark 4.5. In order to achieve less conservative results, a piecewise quadratic

Lyapunov functional method is utilized in this work [141]. It is noted that common

quadratic Lyapunov function is a special case of the piecewise Lyapunov function

when the same Lyapunov function is utilized in different partitions. Another typical

less conservative Lyapunov function used in T-S fuzzy control is fuzzy Lyapunov

function, but it will be identical to the common Lyapunov function if the premise

variables of the physical plant and those of the designed fuzzy filter are different [127].

Obviously, (4.27) will reduce to the common quadratic Lyapunov function if Xis = X

for any i, s ∈ I .

4.3.2 Filter Design 81

4.3.2 Filter Design

In terms of Lemma 4.1, now we present the filter design result.

Theorem 4.2. Consider the system (4.1). The overall error system (4.17) is

asymptotically stable with guaranteed H∞ performance γ, if there exist matrices

Pis = P Tis > 0, Gs, Rs, Qs, i, s ∈ I , Z, λ, g and ϵ satisfying

Πisjkmn < 0, i, s, j, k ∈ I ,m ∈ ℵ(i), n ∈ ℵ(h), (4.31)

where

Πisjkmn =

Π11isjkmn Π12

isjkmn

∗ Π22isjkmn

,Π11

isjkmn = diagPis −Gs −GT

s , τMZ −Gs −GTs ,−I, λ− g − gT

,

Π12isjkmn =

Γ1 Γ4 Dm 0 Γ2

Γ3 Γ4 Dm 0 Γ2

Γ5 0 Hm 0 00 0 0 g 0

,Π22

isjkmn = diag−Pjk,−τ−1

M Z,−γ2I,−λ+ ϵNTN,−ϵ,

Gs =

Gs1 Gs2

Gs3 Gs2

, Γ1 =

Gs1Am + RsCn Qs

Gs3Am + RsCn Qs

, Γ2 =

Rs∆

Rs∆

,Γ3 =

Gs1(Am − I) + RsCn Qs −Gs2

Gs3Am + RsCn Qs −Gs2

, Γ4 =

−RsCn 0−RsCn 0

Γ5 =

[Lm −Ls

], Dm =

[DT

m DTm

]T. (4.32)

Moreover, the filter matrices are respectively given by

As = G−1s2 Qs, Rs = G−1

s2 Rs, s ∈ I . (4.33)

Proof. Based on (4.8), the left hand side of (4.24) can be rewritten as

4.4 Simulation Example 82∑m∈ℵ(i)

∑n∈ℵ(h) µmµnΠisjkmn, i, s, j, k ∈ I , where

Πisjkmn =

Π11isjkh Π12

isjkmn

∗ Π22isjkh

,

Π12isjkmn =

PjkΓ1 PjkBsn PjkDm 0 PjkΓ2

ZΓ3 ZBsn PjkDm 0 PjkΓ2

Lms 0 Hm 0 0

0 0 0 g 0

,

Γ1 =

Am 0

RsCn As

, Bsn =

0 0

−RsCn 0

,Γ3 =

Am − I 0

RsCn As − I

, Lms =[Lm −Ls

]. (4.34)

Then, post- and pre-multiplying Πisjkmn by diagGsP

−1jk , GsZ

−1, I, I, I, I, I, I, I

and its transpose, respectively. It is noted from Lemma 4.3 that (4.31) yields (4.24),

and the proof is thus completed.

4.4 Simulation Example

In this section, we use an example to demonstrate the effectiveness of the filter

design method proposed in this chapter.

Example 4.1. In this example, we use the discrete-time T-S fuzzy system as follows:

Plant Rule Rl: IF x1 is F l, THENx(t+ 1) = Alx(t) +Dlw(t)

z(t) = Llx(t) +Hlw(t)

y(t) = Cl, l ∈ I := 1, 2

4.4 Simulation Example 83

0

0

0.2

0.4

0.6

0.8

1

x1(t)

µ

Rule1Rule2

0.3 1

S2

S1

Figure 4.2: Membership functions

where

A1 =

0.512 0

0 0.575

, A2 =

0.412 0

0 0.432

,D1 =

0

0.1

, D2 =

0

0.1

, C1 = C2 =[1 1

],

L1 =[1 1

], L2 =

[0 1

], H1 = H2 = 0.1.

The membership functions are shown in Fig.4.2. According to the partition

method given in the second section, there are two regions, which are illustrated

as follows:

S1 := x|0 ≤ x1 < 0.3,

S2 := x|0.3 ≤ x1 < 1.

The objective of this example is to design a piecewise filter in the form of (4.15)

so that the overall error system is asymptotically stable with H∞ performance γ.

It is assumed that the maximum delay step τM = 5, and the quantization bound

4.4 Simulation Example 84

0 50 100 150 200−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time in samples

filte

ring

erro

r e(

t)

Figure 4.3: Filtering error

∆ = 4. By utilizing Theorem 4.1, we obtain the following filter matrices:

A1 =

0.408 −0.016

−0.021 0.418

, L1 =[−0.060 −0.194

],

A2 =

0.398 −0.015

−0.020 0.409

, L2 =[−0.048 −0.178

],

R1 = 10−14 ×

−0.134

−0.641

, R2 = 10−14 ×

−0.109

−0.758

with the minimum H∞ performance γmin = 0.2288.

The following disturbance w(t) is assumed

w(t) = e−0.1tcos(2t),

and the simulation results of the filtering error is shown in Fig.4.3. By calculation,

we have ∥w(t)∥22 = 2.0384, ∥e(t)∥22 = 0.0786, which yields√∥e(t)∥22∥w(t)∥22

= 0.1964 < γmin = 0.2288

showing the effectiveness of the H∞ filter design method.

4.5 Conclusion 85

4.5 Conclusion

The H∞ filtering problem of fuzzy-model-based nonlinear systems with

communication constraints is discussed in this chapter. A filter design method under

unreliable transmission is proposed, which considers quantization, network-induced

delays and packet dropouts simultaneously. It is shown that the filtering error system

is asymptotically stable with guaranteed H∞ performance. The developed results are

illustrated by a simulation example.

Chapter 5

Fuzzy Modeling and Control of A

Nonlinear Quadrotor Under

Network Environment

5.1 Introduction

In recent decades, great effort has been devoted to the control of quadrotor

aircrafts [142]- [144]. A quadrotor is a kind of Unmanned Aerial Vehicle (UAV) with

four propeller units which located in four symmetrical positions. They are able to

take off and land vertically without any airstrip, and they can hover in the air without

difficulties. All these features have attracted great research attention because of their

theoretical and practical significance, and they are applied in virous situations, such

as aerial photography, searching and rescuing people, forest fire detection and so

on [145].

Most of the existing results on the quadrotor are based on the traditional point-

to-point systems, where the sensors, controller and actuators are connected by wires

directly so that the actuators can update the control signals once the controller

obtains the information from the sensors and computes the control signals [143].

However, this is not true in actual applications, because the sensor information

are often delayed due to the procedure of measuring, sampling and transmitting.

5.1 Introduction 87

On the other hand, the embedded microprocessor utilized in the quadrotor is not

able to compute the control signals fast enough especially when some complex

control algorithms are implemented. Due to the huge computation and the limited

computational ability on the quadrotor aircraft, the computation time should be

considered. Additionally, the components of a quadrotor are often with an embedded

communication fieldbus [146], such as CAN network, which indicates the unperfect

transmission among the different nodes. Therefore, the control of network-based

systems is involved, which shows the great significance in practical situations [29].

In the meantime, the quadrotor dynamic is nonlinear, which makes it difficult

to design the corresponding controller to stabilize the quadrotor. However, most

of the existing results model the quadrotor system as a linear one, and design the

corresponding controller based on the linearized model [147]- [148]. The authors

in [149] utilize the conventional PID strategy to stabilize the quadrotor aircraft. It is

noted from Chapter 1 that T-S fuzzy models provide a powerful method for systematic

stability analysis and synthesis of some nonlinear systems [11]. It is proved that T-

S fuzzy systems are capable to approximate any smooth nonlinear system to any

accuracy on a compact set [12].

To our best knowledge, there are few results considering the nonlinear controller

design method of a nonlinear quadrotor under network environment, which motivates

our current research. In this chapter, we propose a novel analysis and synthesis

method of the network-based quadrotor aircraft. The controlled quadrotor is

approximiated by a T-S fuzzy model, and the corresponding fuzzy controller is

designed so that the overall system is asymptotically stable with guaranteed H∞

performance. Finally, a simulation is given to illustrate the effectiveness of our

method. The contribution of this chapter can be summarized as follows: 1) A

fuzzy modeling and fuzzy controller design method for the quadrotor under network

environment is proposed. 2) The delays caused by computation and the network are

considered.


model description and problem formulation. Section III presents the H∞ analysis

5.2 Model Description and Problem Formulation 88

Figure 5.1: Photo of the quadrotor

and synthesis results based on a common quadratic Lyapunov function. In Section

IV, a simulation is given to illustrate the effectiveness of the proposed approaches.

Finally, a conclusion is drawn in section V.

5.2 Model Description and Problem Formulation

In this chapter, we focus on the attitude modeling and control of a nonlinear

quadrotor under network environment, which is shown in Fig.5.1. Fig.5.2 illustrates

the system structure. It is noted that the delays and packet dropouts exist both in

S/C and C/A channels. Additionally, the delays caused by the computation of the

control unit are also considered. Therefore, the inputs to the controller xc(t) are not

the same as the states of the controlled aircraft x(t) at time instant t, and the control

inputs to the aircraft u(t) are also different from the outputs of the controller uc(t).

Now, we model the quadrotor aircraft and controller mathematically.

5.2.1 Description of the quadrotor 89

Figure 5.2: Structure of the quadrotor system

5.2.1 Description of the quadrotor

The attitude of the controlled quadrotor aircraft has three degrees of freedom

(3DOF). The quadrotor is able to move in roll, pitch and yaw freely by changing the

speed of the four symmetrically located propellers. The intelligent sensors installed in

the platform can measure the three axes ϕ, θ, ψ of the aircraft and obtain the angular

velocities ϕ, θ, ψ by differentiation. The nonlinear model of the quandrotor aircraft

considered in this chapter can be given as follows [143]:ϕ(t) = θψ

(Iy−IxIx

)+ Jr

IxθΩ + u1 + d1w(t)

θ(t) = ϕψ(

Iz−IxIy

)− Jr

IyϕΩ + u2 + d2w(t)

ψ(t) = ϕθ(

Ix−IyIz

)+ u3 + d3w(t)

(5.1)

where

Ω = Kv(−V2 − V4 + V1 + V3)

u1 =blK2

v (V22 − V 2

4 )

Ix, u2 =

blK2v (V

23 − V 2

1 )

Iy

u3 =blK2

v (V21 − V 2

2 + V 23 − V 2

4 )

Iz(5.2)

ϕ, θ, ψ denote the roll angle, pitch angle and yaw angle, respectively. w(t) is

the disturbance input vector. Vi is the voltage applied to propeller i, Kv is the

transformation constant, Jr is the rotator inertia, Ix, Iy, Iz are the inertia X, Y, Z

axis, respectively. b denotes thrust coefficient, and l is the distance from pivot to

motor.


We define the state vector as x = [ϕ, θ, ψ, ϕ, θ, ψ]T and the control input vector

as u = [u1, u2, u3]T . Then, the nonlinear dynamic (5.1) can be rewritten as x(t) = A(t)x(t) + Bu(t) +Dw(t)

z(t) = Cx(t) +Hw(t),(5.3)

where

A(t) =

0 IrIxΩ Iyzxx3 0 0 0

−JrIxΩ 0 Ixyzx1 0 0 0

0 Ixyzx1 0 0 0 0

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

,

B =

I3×3

03×3

, D =

D

03×1

, H =[

E3×1

],

C =[

03×3 I3×3

], D =

[d1 d2 d3

]T. (5.4)

It is noted that there are three nonlinear terms in the matrix A(t), which are

Ω, x1(t), x3(t). Each of these terms is bounded due to the bounded attitude angles

and input voltage to the propellers. Therefore, the nonlinear dynamic (5.3) can

be approximated by a T-S fuzzy dynamic model. The following sector nonlinearity

approach [150] is utilized.

The term Ω is bounded by [Ωmin,Ωmax] with Ωmin = 4KvVmin and Ωmax = 4KvVmax

according to (5.2). Then the weighting function of Ω can be chosen as

w1 =Ωmax − Ω(t)

Ωmax − Ωmin(5.5)

The term Ω can be rewritten as Ω(t) = w1Ωmin + (1 − w1)Ωmax. It is noted that

0 ≤ w1 ≤ 1.

Similarly, we have x1(t) = w2x1 + (1 − w2)x1 and x3(t) = w3x3 + (1 − w3)x3,


where

w2 =x1 − x1(t)

x1 − x1, w3 =

x3 − x3(t)

x3 − x3,

x1 = x3 = ϕmax = ψmax = αmax,

x1 = x3 = ϕmin = ψmin = αmin. (5.6)

Then we can rewrite the nonlinear dynamic in (5.3) as the following T-S fuzzy

model x(t) =∑8

l=1 µlAlx(t) +Bu(t) +Dw(t)

z(t) = Cx(t) +Hw(t),(5.7)

where

µ1 = w1(1− w2)(1− w3), µ2 = w1w2w3,

µ3 = w1(1− w2)w3, µ4 = w1w2(1− w3),

µ5 = (1− w1)(1− w2)(1− w3),

µ6 = (1− w1)w2w3,

µ7 = (1− w1)(1− w2)w3,

µ8 = (1− w1)w2(1− w3),

and

A1 =

0 IrIxΩmin Iyzxx3 0 0 0

−JrIxΩmin 0 Izxyx1 0 0 0

0 Ixyzx1 0 0 0 0

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

.

The expression of the corresponding Al, l = 2, ..., 8 can be obtained similarly, and

they are omitted for the limited space.

Remark 5.1. In fact, the fuzzy dynamic of the physical quadrotor aircraft (5.7) is

nonlinear due to the nonlinearity of the membership functions µl. Additionally, this

model will reduce to the existing linear one if we choose Al = A for all l = 1, ..., 8,

5.2.2 Communication 92

which can be found in [148]. Therefore, the existing linear model is the special case

of the proposed fuzzy model, which means that the proposed model is more general.

5.2.2 Communication

It is noted that the physical quadrotor dynamic is continuous-time. However,

the system state has to be sampled periodically in order to be transmitted to the

controller through the network. It is assumed the sampling period is T . Considering

the time-delay and packet dropout phenomena in both S/C and C/A channels, we

have the following expression

xc(t) = x(kT − η1(t))

= x(t− τ(t)− η1(t)),

u(t) = uc(t− τc − η2(t)), t ∈ [kT, (k + 1)T ) , (5.8)

where τ(t) = t − kT , and τc is the calculation time of the controller unit. η1(t)

and η2(t) are delays in the buffer of controller and actuator nodes caused by the

network-induced delay and packet dropout, respectively.

Remark 5.2. There are memories in the controller and actuator nodes, which can

store the latest received packet from the sending node. If there are network-induced

delays and/or packet dropouts, the latest packet stored in the buffer will be utilized

for the computation. Therefore, we are able to treat network-induced delays, packet

dropouts and the packets out of sequence in this unified model simultaneously.

It is standard to have the following assumption.

Assumption 5.1. The total time-delay η(t) throughout the overall system is

bounded, that is, 0 ≤ η(t) ≤ η, where η(t) = τ(t) + η1(t) + τc + η2(t).

5.2.3 Controller 93

5.2.3 Controller

In this chapter, the following fuzzy state feedback controller is considered based

on the fuzzy model given in (5.7).

uc(t) =8∑

j=1

µηjKjxc(t), (5.9)

where Kl are controller gains of each subsystem to be determined.

Remark 5.3. Similar to the fuzzy modeling method, the controller (5.9) is also

nonlinear because of the nonlinearity of µj, j = 1, ..., 8. It will reduce to the linear

controller as most of the existing results if Kj = K.


The closed-loop system is written as follows based on (5.7)-(5.9), x(t) =∑8

l=1 µlAlx(t) +∑8

j=1 µηjBKjx(t− η(t)) +Dw(t),

z(t) = Cx(t) +Hw(t)(5.10)


In order to present the main results, we have the following definition:

Definition 5.1. [151] The closed-loop fuzzy system (5.10) is called asymptotically

stable with guaranteed H∞ performance γ if the system is asymptotically stable when

w(t) = 0, and satisfies ∫ ∞

0

∥z(t)∥2dt ≤ γ2∫ ∞

0

∥w(t)∥2dt, (5.11)

where ∥z(t)∥2 = zT (t)z(t), ∥w(t)∥2 = wT (t)w(t), and γ > 0 is a prescribed scalar.

The problem addressed in this chapter can be described as follows.

H∞ Fuzzy State Feedback Controller Design Problem. Consider the fuzzy

dynamic of the quadrotor aircraft in (5.7). Design a fuzzy controller in the form of

(5.9) so that the closed-loop system (5.10) is asymptotically stable with guaranteed

H∞ performance γ by Definition 5.1.

5.3 Main Results 94

5.3 Main Results

In this section, analysis and synthesis of the fuzzy dynamic model of the quadrotor

aircraft in (5.7) is investigated, which is solved by a linear matrix inequality (LMI)

technique based on common Lyapunov function.

Firstly, we have the following analysis lemma.

Lemma 5.1. Consider the fuzzy system (5.7), and it is supposed that the controller

gain matrices Kj of the controller (5.9) are given. The closed-loop system (5.10)

is asymptotically stable with guaranteed H∞ performance γ if there exist matrices

P = P T > 0, Q = QT > 0, R = RT > 0, L1, L2,M,N satisfyingΓlj + Ξ + ΞT √

ηM√ηN C

∗ −R 0 0

∗ ∗ −R 0

∗ ∗ ∗ −γ2I

< 0, (5.12)

where

Γlj =

Π11 Π12 L2BKj 0 L2D

∗ Π22 L1BKj 0 L1D

∗ ∗ 0 0 0

∗ ∗ ∗ −Q 0

∗ ∗ ∗ ∗ −1

,

Ξ =[N 0 M −N −M 0

],C =

[C 0 0 0 H

]T,

Π11 = Q+ L2Al + ATl L

T2 ,Π12 = P − L2 + AT

l LT1 ,

Π22 = ηR− L1 − LT1 . (5.13)

Proof. Consider the following common Lyapunov functional candidate,

V (t) = V1(t) + V2(t) + V3(t), (5.14)

5.3 Main Results 95

where

V1(t) = xT (t)Px(t),

V2(t) =

∫ t

t−η

xT (s)Qx(s)ds,

V3(t) =

∫ 0

−η

∫ t

t+β

xT (s)Rx(s)dsdβ, (5.15)

P = P T > 0, Q = QT > 0, R = RT > 0 are Lyapunov matrices to be determined.

It is noted that the closed-loop system (5.10) can be demonstrated asymptotically

stable with an H∞ performance γ by proving the following index J is negative,

J = V (t) + γ−2zT (t)z(t)− wT (t)w(t). (5.16)

Differentiating V (t), we have

V1(t) = 2xT (t)Px(t),

V2(t) = xT (t)Qx(t)− xT (t− η)Qx(t− η),

V3(t) = ηxT (t)Rx(t)−∫ t

t−η

xT (s)Rx(s)ds

= ηxT (t)Rx(t)−∫ t−η(t)

t−η

xT (s)Rx(s)ds−∫ t

t−η(t)

xT (s)Rx(s)ds

≤ ηxT (t)Rx(t) + ξT (t)MR−1MT ξ(t) + 2ξT (t)M [x(t− η(t))− x(t− η)]

+ξT (t)NR−1NT ξ(t) + 2ξT (t)N [x(t)− x(t− η(t))] , (5.17)

where

M =[MT

1 MT2 MT

3 MT4 MT

5

]T,

N =[NT

1 NT2 NT

3 NT4 NT

5

]T,

ξ(t) =[xT (t) xT (t) xT (t− η(t)) xT (t− η) wT (t)

]T(5.18)

It is obvious that the following equation holds for any appropriate dimensioned

L1 and L2

Φ ,[xT (t)L1 + xTL2

][x(t) + Alx(t) +BKj(t− η(t)) +Dw(t)]

≡ 0 (5.19)

5.3 Main Results 96

Then we have the following inequation based on (5.17)-(5.19)

J ≤ 2xT (t)Px(t) + xT (t)Qx(t)− xT (t− η)Qx(t− η)

+ηxT (t)Rx(t) + ξT (t)MR−1MT ξ(t) + ξT (t)NR−1NT ξ(t)

+2ξT (t)M [x(t− η(t))− x(t− η)] + 2ξT (t)N [x(t)− x(t− η(t))]

+2Φ + γ−2zT (t)z(t)− wT (t)w(t) (5.20)

It is noted that (5.12) yields J < 0 in consideration of the Schur component

lemma. Thus the proof is completed.

In terms of Lemma 5.1, we present the controller design method for the quadrotor

aircraft by the following theorem now.

Theorem 5.1. Consider the fuzzy dynamic model (5.7). The closed-loop system

(5.10) is asymptotically stable with guaranteed H∞ performance γ if there exist

matrices P = P T > 0, Q = QT > 0, R = RT > 0, L(1), L(2),M,N, K so that the

following linear matrix inequations are satisfied:

Ψlj =

Γlj + Ξ + ΞT √

ηM√ηN C

∗ −R 0 0∗ ∗ −R 0∗ ∗ ∗ −γ2I

< 0, (5.21)

where

Γlj =

Π11 Π12 Kj 0 LD

∗ Π22 Kj 0 LD

∗ ∗ 0 0 0∗ ∗ ∗ −Q 0∗ ∗ ∗ ∗ −1

,

Π11 = Q+ LAl + ATl L

T , Π12 = P − L+ ATl L

T , Π22 = ηR− L− LT ,

Kj =

Kj

Kj

, L =

L(1) LT(1)

L(1) L(2)

. (5.22)

Moreover, the controller gain matrices are given by

Kj = L−1(1)Kj (5.23)

5.3 Main Results 97

Proof. It is noted from (5.4) that

B =

I3×3

03×3

, (5.24)

so we have

LBKj =

L(1) LT(1)

L(1) L(2)

I3×3

03×3

Kj =

L(1)

L(1)

Kj =

L(1)Kj

L(1)Kj

,

Kj

Kj

. (5.25)

If we define L1 = L2 = L, then it is obvious that (5.21) yields (5.12), which indicates

the stability of the closed-loop system according to Lemma 5.1. The proof is thus

completed.

The result of Theorem 5.1 can be improved by the following theorem.

Theorem 5.2. Consider the fuzzy dynamic model (5.7). The closed-loop system

(5.10) is asymptotically stable with guaranteed H∞ performance γ if there exist

matrices P = P T > 0, Q = QT > 0, R = RT > 0, L(1), L(2),M,N, K so that the

following linear matrix inequations are satisfied:

Ψll < 0, l = 1, ..., 8,

Ψlj +Ψjl < 0, l < j, (5.26)

where Ψlj are defined in (5.21)

Proof. It is noted from (5.21) that

8∑l=1

8∑j=1

µlµηjΨlj =

8∑l=1

µlµηl Ψll +

7∑l=1

8∑j=l+1

µlµηj (Ψlj +Ψjl). (5.27)

Then we have Ψlj < 0 if (5.26) is satisfied. Therefore, (5.26) yields (5.21), which

guaranteed the stability of the closed-loop system (5.10). Thus the proof of Theorem

5.2 is completed.

5.4 Simulation 98

5.4 Simulation

In this section, we use the simulation results of the attitude control of the

quadrotor aircraft to demonstrate the effectiveness of the fuzzy modeling and control

method proposed in this chapter.

We extract the parameters of the quadrotor from [144], which are shown as

follows:

Vi ∈ [−10V, 10V],

Kv = 54.945 rad s/V,

Jr = 6 · 10−5 kgm2,

Ix = 0.0552 kgm2,

Iy = 0.0552 kgm2,

Iz = 0.1104 kgm2,

b = 3.935139× 10−6 N/Volt,

l = 0.1969 m,

T = 0.005 s,

ϕmin = θmin = ψmin = −π/4 rad/s,

ϕmax = θmax = ψmax = π/4 rad/s,

ϕmax = θmax = π/2 rad/s, ψmax = π rad/s.

The objective of this simulation is to approximate the nonlinear quadrotor aircraft

and then design a corresponding fuzzy controller so that the closed-loop system is

asymptotically stable with H∞ performance γ.

Firstly, we have the following fuzzy system according to the approximation

method presented in Section II x(t) =∑8

l=1 µlAlx(t) +Bu(t) +Dw(t)

z(t) = Cx(t) +Hw(t),

5.4 Simulation 99

where

A1 =

0 2.39 −0.785 0 0 0

−2.39 0 0.785 0 0 0

0 0 0 0 0 0

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

,

A2 =

0 2.39 0.785 0 0 0

−2.39 0 −0.785 0 0 0

0 0 0 0 0 0

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

,

A3 =

0 2.39 0.785 0 0 0

−2.39 0 0.785 0 0 0

0 0 0 0 0 0

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

,

A4 =

0 2.39 −0.785 0 0 0

−2.39 0 −0.785 0 0 0

0 0 0 0 0 0

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

,

A5 =

0 −2.39 −0.785 0 0 0

2.39 0 0.785 0 0 0

0 0 0 0 0 0

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

,

5.4 Simulation 100

A6 =

0 −2.39 0.785 0 0 0

2.39 0 −0.785 0 0 0

0 0 0 0 0 0

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

,

A7 =

0 −2.39 0.785 0 0 0

2.39 0 0.785 0 0 0

0 0 0 0 0 0

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

,

A8 =

0 −2.39 −0.785 0 0 0

2.39 0 −0.785 0 0 0

0 0 0 0 0 0

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

,

B =

I3×3

03×3

, D =

E3×1

03×1

,C =

[03×3 I3×3

], H =

[E3×1

].

We assume that the total time-delay in the actuator buffer throughout the overall

system η(t) is bounded by η(t) ≤ η = 0.05 s.

5.4 Simulation 101

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Figure 5.3: Delays in the buffer

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time in seconds

angl

e

φθψ

Figure 5.4: Response

5.4 Simulation 102

Then by utilizing Theorem 5.2, we obtain the following controller gain matrices:

K1 =

−5.20 −2.74 1.31 −4.66 −1.96 −0.53

1.58 −5.17 0.05 −1.86 −4.65 −0.76

−0.76 −0.76 −3.89 −2.02 −2.03 −3.63

K2 =

−5.18 −2.72 −0.25 −4.67 −1.97 −0.45

1.55 −5.19 1.60 −1.85 −4.63 −0.84

−0.76 −0.76 −3.89 −2.03 −2.03 −3.63

K3 =

−5.18 −2.72 −0.25 −4.67 −1.97 −0.45

1.57 −5.17 0.05 −1.86 −4.64 −0.76

−0.76 −0.76 −3.89 −2.03 −2.03 −3.63

K4 =

−5.20 −2.74 1.30 −4.66 −1.96 −0.53

1.55 −5.19 1.60 −1.85 −4.63 −0.84

−0.76 −0.76 −3.89 −2.03 −2.03 −3.63

K5 =

−5.19 1.55 1.60 −4.63 −1.85 −0.84

−2.72 −5.18 −0.25 −1.97 −4.67 −0.45

−0.76 −0.76 −3.89 −2.03 −2.03 −3.63

K6 =

−5.17 1.58 0.05 −4.65 −1.86 −0.76

−2.74 −5.20 1.31 −1.96 −4.66 −0.53

−0.76 −0.76 −3.89 −2.03 −2.02 −3.63

K7 =

−5.17 1.57 0.05 −4.64 −1.86 −0.76

−2.72 −5.18 −0.25 −1.97 −4.67 −0.45

−0.76 −0.76 −3.89 −2.03 −2.03 −3.63

K8 =

−5.19 1.55 1.60 −4.63 −1.85 −0.84

−2.74 −5.205 1.31 −1.96 −4.66 −0.53

−0.76 −0.76 −3.89 −2.03 −2.03 −3.63

We also obtain the corresponding minimum H∞ performance γmin = 3.6371.

Fig.5.3 shows the delays in the buffer, which is generated stochastically. Applying

the designed fuzzy controller, the state responses without disturbance are shown in

Fig.5.4. It is noted that the responses are obtained by applying the nonlinear model

5.5 Conclusion 103

of the quandrotor in (5.1) directly. Fig.5.4 shows that all the angles converges to zero

asymptotically without the disturbance inputs.

It is assumed that w(t) = e−0.01tcos(2t). By calculation, we have∫∞0 ∥z(t)∥2dt∫∞0 ∥w(t)∥2dt =

2.1044 < γmin, showing the effectiveness of the H∞ controller design method.

5.5 Conclusion

The fuzzy modeling and H∞ state feedback controller design problem of the

nonlinear quadrotor aircraft with communication constraints is discussed in this

chapter. A fuzzy controller design method under unreliable transmission is proposed,

which considers computation delays, network-induced delays and packet dropouts

simultaneously. It is shown that the overall system is asymptotically stable with a

guaranteed H∞ performance. The developed results are illustrated by simulations.

Chapter 6

Concluding Remarks

Networked control systems have attracted considerable attention during the past

few decades due to their significant advantages over the traditional control systems.

Though a great deal of results have been reported on the estimation and control

problems of network-based systems, there are still many important problems to be

solved, such as analysis and synthesis of NCSs considering several typical network

issues simultaneously, filter design for network-based nonlinear physical plants, etc.

This thesis is devoted to some of these problems.

In this chapter, we summarize the major contributions of this thesis, and some

potential future investigations are also suggested.

6.1 Summary

In this thesis, we mainly focus on the estimation and control problems of linear

and nonlinear physical plants via T-S fuzzy models with various communication

constraints, such as packet dropouts, network-induced delays, and quantization.

Several novel results have been obtained and are described in detail as follows.

⋄ Control under multiple communication constraints

In chapter 2, we consider the control problem of network-based linear

discrete-time systems with network-induced delays, packet dropouts and

6.1 Summary 105

quantization in both S/C and C/A channels. Different from the existing

results, a novel asynchronous quantization method is proposed, and an observer-

based output feedback controller is designed. It has been proved that the

resulting closed-loop system is asymptotically stable. Moreover, the proposed

quantization method does not require synchronous quantization parameters

between the sending and receiving nodes, which is much more practical than

the existing results.

⋄ Compensation scheme for network-induced delays and packet

dropouts.

In contrast to most of the existing schemes for network-induced delays

and packet dropouts, which are either zero-strategy or hold-strategy, a novel

compensation scheme is proposed for nonlinear networked systems via T-S fuzzy

models to estimate the system states at each time instant when the packet

is delayed or lost in Chapter 3. The contributions can be summarized as

follows: (1) a new approach to solving the H∞ control design problem of the T-

S fuzzy control system with packet dropouts in both S/C and C/A channels is

proposed; (2) the optimal H∞ performance is achieved by utilizing the proposed

compensation approach; (3) the proposed approaches can deal with the case

when both packet dropouts and network-induced delay phenomena exist; and

(4) different from the most existing results, the synchronous premise variables

or its region information in different nodes are not needed in our method, which

is more practical.

⋄ Estimation under multiple communication constraints

The filter design method for networked nonlinear systems with multiple

communication constraints in both S/C and C/A channels is considered

in Chapter 4, including network-induced delays, packet dropouts and

quantization. These network issues are treated in a unified framework and

addressed simultaneously. Different from most of the existing results, the

region information is updated locally in different nodes, and the premise

6.2 Potential Research Problems 106

variables or the region information of the premise variables of the physical

plant are not needed at the filtering node. Based on a piecewise Lyapunov

functional, the piecewise filter parameters are derived by introducing some slack

matrices and solving a set of linear matrix inequalities, and it is shown that the

corresponding filtering error system is asymptotically stable with a guaranteed

H∞ performance.

⋄ Modeling and control of a quadrotor.

In chapter 5, the fuzzy modeling and controller design method is presented

for the nonlinear quadrotor, which is a kind of unmanned aerial vehicles.

The model of the quadrotor aircraft is typically nonlinear. However, most of

existing works utilize a linearized model at the operating point and design the

corresponding linear controller, which will lead to large model errors and lower

system performance when the system evolves at the other operating points. To

address this problem under the network environment, we have built the T-S

fuzzy model of the quadrotor, and the corresponding fuzzy controller is designed

by solving a set of linear matrix inequality techniques. It is shown that the

closed-loop quadrotor system is asymptotically stable with a guaranteed H∞

performance.

6.2 Potential Research Problems

Although some results on analysis and synthesis of NCSs with limited

communication capacity have been presented in this thesis, there are still many

relevant problems to be investigated. In this section, some related open problems

are introduced.

⋄ Control of NCSs with uncertain or partially unknown network

parameters

As in chapter 3, many researchers model packet dropouts and network-

induced delays by stochastic processes, such as the Bernoulli distribution

6.2 Potential Research Problems 107

and Markov chain. However, the statistics knowledge, e.g., the packet loss

probabilities and transition probabilities are assumed to be certain or exactly

known, which is not always true in practice. Further investigations on analysis

and synthesis problems of NCSs with uncertain or partially unknown statistics

knowledge would be interesting and warranted.

⋄ Compensation of Multiple Packet Transmission

It is assumed in most of the existing results that the sensors of NCSs are

located together so that the sensor signals can be packaged into one packet to

transmit through network links. However, this is not always true in practice.

The multiple sensors located in different physical space will transmit their

signals in separate packets to the controller. In this case, multiple packets

should be compensated separately for their loss or delays. This thesis has

considered the compensation of one packet transmission, but the compensation

problem of multiple packet transmission is still open and unsolved, which will

motivates our further investigations.

⋄ Co-design of event-triggered mechanism and controller.

It is noted that some network accesses are expensive and energy intensive,

so the reduction of transmission frequency is of great significance in practice.

Though some works have considered this problem and have proposed event-

triggered mechanisms, few of them address the co-design of the mechanism and

controllers of NCSs with limited communication capacity, which is definitely

one of the interesting and challenging research topics.

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vol. 18, no. 1, pp. 201-208, Feb. 2010

Curriculum Vitae

Fei Han received his Bachelor’s Degree in Automation from the University

of Science and Technology of China, Hefei, China, in 2009. He is currently

working toward the Ph.D. degree in the Department of Mechanical and Biomedical

Engineering at the University of Science and Technology of China & City University

of Hong Kong Joint Advanced Research Center, Suzhou, China.

His main research interests include networked control systems, nonlinear control

via T-S dynamic models and robotics.

His recent publication is listed as follows.

1. Fei Han, Gang Feng, Yong Wang, Jianbin Qiu, and Changzhu Zhang, “A

novel dropout compensation scheme for control of networked T-S fuzzy dynamic

Systems,” Fuzzy Sets and Systems, vol. 235, pp. 44-61, January 2014

2. Fei Han, Gang Feng, and Yong Wang, “H∞ filter design of networked nonlinear

systems with communication constraints via T-S fuzzy dynamic models,” in

Proceedings of American Control Conference (ACC2013), Washington DC,

USA, June 2013, pp. 6421-6426

3. Fei Han, Gang Feng, and Yong Wang, “Dynamic output feedback control

of networked control systems with limited communication capacity,” IEEE

Transactions on Industrial Electronics, Under review

4. Fei Han, Gang Feng, and Yong Wang, “Fuzzy modeling and control for a

nonlinear quadrotor under network environment,” 2014 IEEE International

Conference on CYBER Technology (IEEE-CYBER 2014), Under review

Curriculum Vitae 127

5. Zhiyu Xi, Qing Gao, Gang Feng, and Fei Han, “Sliding mode control design

for networked systems with packet loss,” IET Control Theory & Applications,

Under review.

6. Yong Wang, Weiguang Liang, Fei Han, and Jianliang Zhou, “Monitoring

method of vibration divergence fault based on evidence theory,” Journal of

Vibration, Measurement & Diagnosis, vol. 31, no. 4, pp. 424-428, August 2011

7. Weiguang Liang, Yong Wang, Fei Han, Jianliang Zhou, “Comparative research

of single fault diagnosis methods based on evidence theory,” Journal of

Southeast University, vol.39, pp. 183-188, September 2009

analysis and synthesis of networked control systems …fhan/publication/pdf/thesis_cityu.pdfcontrol...

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