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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley-IEEE Press Book Series on Control Systems Theory and Applications
- Sprache: Englisch
Control over Communication Networks Advanced and systematic examination of the design and analysis of networked control systems and multi-agent systems Control Over Communication Networks provides a systematic and nearly self-contained description of the analysis and design of networked control systems (NCSs) and multi-agent systems (MASs) over imperfect communication networks, with a primary focus on fading channels and delayed channels. The text characterizes the effect of communication channels on the stability and performance of NCSs, and further studies the joint impact of communication channels and network topology on the consensus of MASs. By integrating communication and control theory, the four highly-qualified authors present fundamental results concerning the stabilization of NCSs over power-constrained fading channels and Gaussian finite-state Markov channels, linear-quadratic optimal control of NCSs with random input gains, optimal state estimation with intermittent observations, consensus of MASs with communication delay and packet dropouts, and synchronization of delayed Vicsek models. Simulation results are given in each chapter to demonstrate the developed analysis and synthesis approaches. The references are comprehensive and up-to-date, enabling further study for readers. Topics covered in Control Over Communication Networks include: * Basic foundational knowledge, including control theory, communication theory, and graph theory, to enable readers to understand more complex topics * The stabilization, optimal control, and remote state estimation problems of linear systems over channels with fading, signal-to-noise constraints, or intermittent measurements * Consensus problems of MASs over fading/delayed channels, with directed and undirected communication graphs Control Over Communication Networks provides a valuable unified platform for understanding the analysis and design of NCSs and MASs for researchers, control engineers working on control systems over communication networks, and mechanical engineers working on unmanned systems. Preliminary knowledge of linear system theory and matrix analysis is required.
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Veröffentlichungsjahr: 2023
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Table of Contents
Cover
Title Page
Copyright
Dedication
About the Authors
Preface
Acknowledgments
Acronyms
List of Symbols
1 Introduction
1.1 Introduction and Motivation
1.2 Literature Review
1.3 Preview of the Book
1.4 Preliminaries
Bibliography
2 Stabilization over Power Constrained Fading Channels
2.1 Introduction
2.2 Problem Formulation
2.3 Fundamental Limitations
2.4 Mean‐Square Stabilizability
2.5 Numerical Illustrations
2.6 Conclusions
Bibliography
Note
3 Stabilization over Gaussian Finite‐State Markov Channels
3.1 Introduction
3.2 Problem Formulation
3.3 Fundamental Limitation
3.4 Stabilization over Finite‐State Markov Channels
3.5 Stabilization over Markov Lossy Channels
3.6 Conclusions
Bibliography
4 Linear‐Quadratic Optimal Control of NCSs with Random Input Gains
4.1 Introduction
4.2 Problem Formulation
4.3 Finite‐Horizon LQ Optimal Control
4.4 Solvability of Modified Algebraic Riccati Equation
4.5 LQ Optimal Control
4.6 Conclusion
Bibliography
5 Multisensor Kalman Filtering with Intermittent Measurements
5.1 Introduction
5.2 Problem Formulation
5.3 Stability Analysis
5.4 Examples
5.5 Conclusions
Bibliography
6 Remote State Estimation with Stochastic Event‐Triggered Sensor Schedule and Packet Drops
6.1 Introduction
6.2 Problem Formulation
6.3 Optimal Estimator
6.4 Suboptimal Estimators
6.5 Simulations
6.6 Conclusions
Bibliography
7 Distributed Consensus over Undirected Fading Networks
7.1 Introduction
7.2 Problem Formulation
7.3 Identical Fading Networks
7.4 Nonidentical Fading Networks
7.5 Simulations
7.6 Conclusions
Bibliography
8 Distributed Consensus over Directed Fading Networks
8.1 Introduction
8.2 Problem Formulation
8.3 Identical Fading Networks
8.4 Definitions and Properties of CIIM, CIM, and CEL
8.5 Nonidentical Fading Networks
8.6 Simulations
8.7 Conclusions
Bibliography
Notes
9 Distributed Consensus over Networks with Communication Delay and Packet Dropouts
9.1 Introduction
9.2 Problem Formulation
9.3 Consensusability with Delay and Identical Packet Dropouts
9.4 Consensusability with Delay and Nonidentical Packet Dropouts
9.5 Illustrative Examples
9.6 Conclusions
Bibliography
10 Distributed Consensus over Markovian Packet Loss Channels
10.1 Introduction
10.2 Problem Formulation
10.3 Identical Markovian Packet Loss
10.4 Nonidentical Markovian Packet Loss
10.5 Numerical Simulations
10.6 Conclusions
Bibliography
11 Synchronization of the Delayed Vicsek Model
11.1 Introduction
11.2 Directed Graphs
11.3 Problem Formulation
11.4 Synchronization of Delayed Linear Vicsek Model
11.5 Synchronization of Delayed Nonlinear Vicsek Model
11.6 Simulations
11.7 Conclusions
Bibliography
Index
Books in the IEEE Press Series on Control Systems Theory and Applications
End User License Agreement
List of Tables
Chapter 3
Table 3.1 Lists of transmission‐related definitions.
Chapter 9
Table 9.1 The tolerable ensuring mean‐square consensus.
List of Illustrations
Chapter 1
Figure 1.1 Networked control systems.
Figure 1.2 Fading phenomenon in wireless communications.
Figure 1.3 Point‐to‐point communication system.
Figure 1.4 Intuitive explanations of the data rate theorem.
Chapter 2
Figure 2.1 Networked control over a power‐constrained fading channel.
Figure 2.2 Transmission protocol configuration.
Figure 2.3 Scheduled transmissions with Algorithm 2.1.
Figure 2.4 Comparison of different channel capacities for scalar systems....
Figure 2.5 Comparison of stabilizability regions for two‐dimensional systems...
Chapter 3
Figure 3.1 Networked control over Gaussian Markov channels.
Figure 3.2 Communication structure.
Figure 3.3 Channel encoder/decoder/scheduler structure.
Figure 3.4 Scheduled transmissions with Algorithm 3.2.
Figure 3.5 The first round transmission with Algorithm 3.2.
Figure 3.6 Transmissions with the adaptive TDMA scheduler.
Figure 3.7 Stabilizability regions for .
Chapter 4
Figure 4.1 Linear time‐invariant (LTI) systems with random input gains.
Figure 4.2 Discrete‐time LTI systems with random input gains.
Figure 4.3 Parallel transmission strategy.
Chapter 5
Figure 5.1 The structure of the centralized sensor network.
Figure 5.2 The lower and upper bounds of .
Chapter 6
Figure 6.1 Remote state estimation with stochastic event‐triggered sensor sc...
Figure 6.2 Relative sum of MSE of different estimator under different packet...
Figure 6.3 Execution time of the optimal estimator and the suboptimal estima...
Chapter 7
Figure 7.1 Information transmission from agent to agent .
Figure 7.2 Communication topology for an undirected graph.
Figure 7.3 Mean‐square consensus error for agent 1 under an undirected commu...
Figure 7.4 Mean‐square consensus error for agent 1 under an undirected tree ...
Chapter 8
Figure 8.1 (a) A directed graph with a bidirectional edge. (b) Treat the bid...
Figure 8.2 (a) A star graph, (b) a directed graph with a cycle in its underl...
Figure 8.3 Communication graphs used in simulations: (a) a directed graph, (...
Figure 8.4 Mean‐square consensus error for agent 1 under a directed topology...
Figure 8.5 Mean‐square consensus error for agent 1 under a directed topology...
Chapter 9
Figure 9.1 The communication graph .
Figure 9.2 Upper bounds of tolerable from Corollary 1 and via simulations ...
Figure 9.3 Upper bounds of tolerable from Corollary 1 and via simulations ...
Chapter 10
Figure 10.1 Tolerable failure rate and recovery rate.
Figure 10.2 Communication graphs used in simulations: (a) an undirected grap...
Figure 10.3 Mean‐square consensus error for agent 1 under identical packet l...
Figure 10.4 Mean‐square consensus error for agent 1 under nonidentical packe...
Chapter 11
Figure 11.1 Trajectories of the heading angles with .
Figure 11.2 Trajectories of the heading angles with .
Figure 11.3 Trajectories of the heading angles with .
Figure 11.4 Trajectories of the heading angles with .
Figure 11.5 Trajectories of the heading angles with .
Guide
Cover
Table of Contents
Title Page
Copyright
Dedication
About the Authors
Preface
Acknowledgments
Acronyms
List of Symbols
Begin Reading
Index
Books in the IEEE Press Series on Control Systems Theory and Applications
End User License Agreement
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IEEE Press
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Desineni Subbaram Naidu
Ahmet Murat Tekalp
Control over Communication Networks
Modeling, Analysis, and Design of Networked Control Systems and Multi‐Agent Systems over Imperfect Communication Channels
Jianying ZhengBeihang University, Beijing, China
Liang XuShanghai University, Shangai, China
Qinglei HuBeihang University, Beijing, China
Lihua XieNanyang Technological University, Singapore, Singapore
IEEE Press Series on Control Systems Theory and Applications
Maria Domenica Di Benedetto, Series Editor
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Library of Congress Cataloging‐in‐Publication Data
Names: Zheng, Jianying, author. | Xu, Liang, author. | Hu, Qinglei, author. | Xie, Lihua, author.
Title: Control over communication networks : modeling, analysis, and design of networked control systems and multi-agent systems over imperfect communication channels / Jianying Zheng, Beihang University, Beijing, China, Liang Xu, Shanghai University, Shanghai, China, Qinglei Hu, Beihang University, Beijing, China, Lihua Xie, Nanyang Technological University, Singapore, Singapore.
Description: Hoboken, New Jersey : Wiley, [2023] | Includes bibliographical references and index.
Identifiers: LCCN 2023002566 (print) | LCCN 2023002567 (ebook) | ISBN 9781119885795 (hardback) | ISBN 9781119885801 (adobe pdf) | ISBN 9781119885818 (epub)
Subjects: LCSH: Supervisory control systems–Reliability. | Remote control. | Robust control. | Uncertainty (Information theory) | Fault tolerance (Engineering) | Multi‐agent systems.
Classification: LCC TJ222 .Z46 2023 (print) | LCC TJ222 (ebook) | DDC 620/.46–dc23/eng/20230213
LC record available at https://lccn.loc.gov/2023002566LC ebook record available at https://lccn.loc.gov/2023002567
Cover Design: WileyCover Image: © ProStockStudio/Shutterstock
To my parents, my husband Pengfei, and my daughter Coco (Jianying Zheng)
To my parents, my wife Xiaoxue, and my son Ze (Liang Xu)
About the Authors
Jianying Zheng received the BE degree in Automation from University of Science and Technology of China in 2010, and the PhD degree in Electronic and Computer Engineering from the Hong Kong University of Science and Technology, Hong Kong, in 2016. Between September 2014 and February 2015, she was a visiting student in the University of Newcastle, Australia. From 2016 to 2018, she worked as a Research Fellow in the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. Since 2018, she joined Beihang University, where she is currently an associate professor. Her current research interests include networked control systems, multi‐agent systems, and spacecraft formation control.
Liang Xu is a professor at Shanghai University, Shanghai, China. He received his PhD degree in Electrical and Electronic Engineering from Nanyang Technological University, Singapore, in 2018 and the MS and BSc degrees in Automation from Harbin Institute of Technology, China, in 2013 and 2011, respectively. Prior to his current position, he was a postdoctoral scholar at Nanyang Technological University, Singapore from 2018 to 2019, and a postdoctoral scholar at École polytechnique fédérale de Lausanne (EPFL), Switzerland from 2019 to 2022. His research interests include learning to control and networked control systems.
Qinglei Hu obtained his BEng degree in 2001 from the Department of Electrical and Electronic Engineering at the Zhengzhou University, P.R. China, and M. Eng. and PhD degree from the Department of Control Science and Engineering at the Harbin Institute of Technology with specialization in controls, P.R. China, in 2003 and 2006, respectively. From 2003 to 2014, he was with the Department of Control Science and Engineering at Harbin Institute of Technology and was promoted to the rank of professor with tenure in 2012. He worked as a postdoctoral research fellow in Nanyang Technological University from 2006 to 2007 and from 2008 to 2009, he visited University of Bristol as Senior Research Fellow supported by Royal Society Fellowship, and from 2010 to 2014, he visited Concordia University, Lakehead University, and Nanyang Technological University again as visiting professor. He joined Beihang University in 2014 as “Outstanding Bairen Plan” Professor, and vice director of Institute of Science and Technology.
His research interests include the design and control of spacecraft attitude system and fault tolerant control to aerospace engineering. He has published significantly on the subjects with over 100 technical papers while enjoying the application of the theory through astronautic consulting. He has been actively involved in various technical professional societies such American Institute of Aeronautics and Astronautics (AIAA), Institute of Electrical and Electronics Engineers (IEEE) and American Society of Mechanical Engineers (ASME), as reflected by AIAA Associate Fellow, IEEE Senior Member and general chair of many international conferences. He also previously served as associate editor for Aerospace Science and Technology, etc.
Lihua Xie received the PhD degree in electrical engineering from the University of Newcastle, Australia, in 1992. Since 1992, he has been with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, where he is currently a professor and the director, Centre for Advanced Robotics Innovation Technology (CARTIN). He served as the head of Division of Control and Instrumentation from 2011 to 2014, codirector of Delta‐NTU Corporate Laboratory for Cyber‐Physical Systems from 2016 to 2021 and director, Center for E‐City from July 2011 to June 2013. He is a Fellow of Institute of Electrical and Electronics Engineers (IEEE), International Federation of Automatic Control (IFAC), Chinese Automation Association (CAA), and Academy of Engineering Singapore.
Dr. Xie's research interests include robust control and estimation, networked control systems, multi‐agent networks, and unmanned systems. In these areas, he has published 9 books, over 500 journal papers, and over 20 patents/technical disclosures. He was listed as a highly cited researcher by Thomson Routers and Clarivate Analytics. In addition to fundamental research contributions, he has developed a universal navigation system for AGVs in manufacturing and logistics, patented WiFi‐based technologies for indoor positioning and human activity recognitions, and reliable and accurate positioning systems for UAV‐based structure inspection. He has received many awards for his research.
He is an editor‐in‐chief for Unmanned Systems and an associate editor for Sciences China – Information Science. He has served as an editor of IET Book Series in Control and an associate editor of a number of journals including IEEE Transactions on Automatic Control, Automatica, IEEE Transactions on Control Systems Technology, IEEE Transactions on Network Control Systems, and IEEE Transactions on Circuits and Systems‐II, etc. He was an IEEE Distinguished Lecturer from 2011 to 2014.
Preface
In networked control systems (NCSs), wired or wireless communication channels are used to link components among plants, sensors, and controllers to achieve control objectives. While there are many advantages, NCSs also introduce a series of challenging problems that arise from the limited resources and unreliability of the communication networks used for information transmission. For example, due to congestion, data losses and transmission delays may occur in digital communication channels. Besides, in wireless communication networks, which are widely used in sensor networks and multi‐agent systems (MASs), communication channels naturally suffer from inference, fading, and transmission noises. Since control is often used in safety or mission‐critical applications, we must take the uncertainties in communication networks into consideration and investigate how they affect the stability and performance of NCSs and MASs.
The book gives a systematical and self‐contained description for the analysis and design of NCSs and MASs over imperfect communication networks. Specifically, the book considers fading channels and delayed channels and includes two main parts. In the first part, the stabilization, optimal control, and remote state estimation of linear systems over channels with fading, signal‐to‐noise constraints, or intermittent measurements are considered. The channel requirements for the mean‐square stabilization and optimal control are characterized and the optimal estimator designs and performance analysis are conducted. In the second part, the joint impact of communication channels and network topology on the consensusability of MASs is analyzed. By integrating communication and control theory, we present several fundamental results on the stabilization, optimal control, and estimation of NCSs and the consensus of MASs over imperfect channels. The book intends to provide a unified platform for introducing the analysis and design of NCSs and MASs for researchers working in related areas.
January 2023
Jianying Zheng
Liang Xu
Qinglei Hu
Lihua Xie
Acknowledgments
The contents included in this book are the outgrowth and summary of the authors' academic research achievements on the analysis and design of networked control systems and multi‐agent systems over imperfect communication channels in the past several years. Some of the materials contained herein arose from the joint work with our collaborators, and the book would not have been possible without their efforts and support. In particular, the first author is indebted to Prof. Li Qiu from the Hong Kong University of Science and Technology, who led her into the gate of scientific research. We would like to thank our collaborators, including Prof. Jiu‐Gang Dong, Prof. Yilin Mo, Prof. Keyou You, Prof. Chao Yang, and Dr. Nan Xiao, for their helpful discussions, and the students Roudan Zhou and Xiao Wang for their assistance in preparing the manuscript.
We gratefully thank the financial support from the National Natural Science Foundation of China under Grants 61903018, 62273017, and 61960206011, the National Key Basic Research Program “Gravitational Wave Detection” Project under Grant 2021YFC2202600, the Beijing Natural Science Foundation under Grant JQ19017, and the Zhejiang Provincial Natural Science Foundation under Grant LD22E050004.
Jianying Zheng
Liang Xu
Qinglei Hu
Lihua Xie
Acronyms
a.e.
almost everywhere
ARE
algebraic Riccati equation
AWGN
additive white Gaussian noise
BMI
bilinear matrix inequality
CEL
compressed edge Laplacian
CIIM
compressed in‐incidence matrix
CIM
compressed incidence matrix
DEL
directed edge Laplacian
FSMC
finite‐state Markov channel(s)
i.i.d.
independent and identically distributed
IIM
in‐incidence matrix
IM
incidence matrix
LMI
linear matrix inequality(‐ies)
LTI
linear time‐invariant
LQ
linear quadratic
LQG
linear quadratic Gaussian
MARE
modified algebraic Riccati equation(s)
MAS
multi‐agent system(s)
MIMO
multi‐input‐multi‐output
MJLS
Markov jump linear system
MSE
mean‐square error
MMSE
minimum mean‐square error
NCS
networked control system(s)
SNR
signal‐to‐noise ratio
TCP
transport control protocol
TDMA
time division multiple access
List of Symbols
the set of natural numbers
the set of positive natural numbers
the set of real (complex) numbers
the set of ‐dimensional real (complex) column vectors
the set of ‐dimensional real (complex) matrices
the set of symmetric matrices
the set of positive semidefinite matrices
the real part of
the magnitude of
the cardinality of set
a column vector of ones
an matrix with all entries being zero
the ‐by‐ identity matrix
the transpose of matrix
the conjugate transpose of matrix
the th element of matrix
the th row of matrix
the th column of matrix
the inverse of matrix
the Moore–Penrose pseudoinverse of matrix
the spectral radius of matrix
the trace of matrix
the determinant of matrix
the null space of matrix
a diagonal matrix with diagonal entries and
the abbreviation of the symmetric matrix
the minimal eigenvalue of a real symmetric matrix
()
positive definite (semidefinite) matrix
the sequence
the Kronecker product
the Hadamard product
the expectation operator
the expectation conditioned on the event
a random variable with Gaussian distribution of mean and covariance matrix
()
the probability density function (probability) of the random variable
()
the probability density function (probability) of the random variable conditioned on the event that
the Euler's number
the natural logarithm
the logarithm to base 2
the logarithm to base
1Introduction
1.1 Introduction and Motivation
1.1.1 Networked Control Systems
Due to the flexible architecture and ease of installation and maintenance, communication networks are widely used in control systems, which result in networked control systems (NCSs), where the plants, actuators, sensors, and controllers are spatially distributed and interconnected by communication channels [Schenato et al., 2007, Hespanha et al.]. NCSs are ubiquitous in industry and daily life, such as teleoperation [Arcara and Melchiorri, 2002], power systems [Wang et al., 2012], and transportation systems [Seiler and Sengupta, 2001].
Even though NCSs have the advantages of low cost, easy implementation, and expansion to large‐scale applications, they also introduce new challenging problems arising from the limited resources and unreliability of the communication networks used for information transmission (see Figure 1.1). For example, the time delay may occur in digital communication channels due to data processing and transmission [Tse and Viswanath, 2005, Goldsmith, 2005]. Notably, in wireless communication networks, communication channels naturally suffer from interference, fading, and transmission noises [Tse and Viswanath, 2005, Goldsmith, 2005]. There into, fading is the time variation of channel strengths and is usually caused by two factors: one is the shadowing from obstacles; the other one is the multipath propagation [Tse and Viswanath, 2005, Goldsmith, 2005]. Packet drops can also be modeled as a special case of channel fading. Take Figure 1.2 as an illustration. The wireless signal may transmit through the car and undergo several paths before arriving at the receiver. If the phases of the received signals from different paths are the same, the signal strength is enhanced. Otherwise, the signal strength is reduced as a result of the cancellation of radio waves. Besides, the signal strength at the receiver side might be reduced due to the shadowing from the car. Since control is often used in safety‐ or mission‐critical applications, we must take the uncertainties in communication networks into consideration and investigate how they affect the stability and performance of control systems.
The classical control theory mainly deals with the systems with nearly perfect point‐to‐point connections and focuses on the design of control laws to achieve the given control performance. It can't be applied directly to the NCSs when the uncertainties in the communication network must be considered. A new control paradigm is required to deal with the interplay between control and communication. In this book, one of the main objectives is to study the stabilization, estimation, and optimal control of NCSs over channels with fading, packet drops, or delay.
1.1.2 Multi‐Agent Systems
Motivated by the collective behavior in nature, such as schooling fish, flocking birds, and marching locusts, multi‐agent systems (MASs) have attracted considerable research interest from the control community [Jadbabaie et al., 2003, Olfati‐Saber and Murray, 2004, Olfati‐Saber et al., 2007, Bliman and Ferrari‐Trecate, 2008, Cao et al., 2008, Ren and Beard, 2008, You and Xie, 2010, Cao et al., 2012, Trentelman et al., 2013, Qi et al., 2016, Qiu et al., 2017, Xu et al., 2018, Zheng et al., 2018]. With the rapid development of wireless communication networks, MASs have been applied in many industrial and military applications. Such systems usually involve large numbers of autonomous agents (e.g. robots, unmanned aerial vehicles, satellites), which share information via local interactions and work together to achieve collective objectives.
For MASs, each agent can have the same or different system dynamics, resulting in different types of MASs, e.g. first‐ and second‐order MASs, linear and nonlinear MASs, homogeneous and heterogeneous MASs. The interactions among the agents form the interaction topology, which can be fixed or time‐varying. Then the cooperative control of MASs is based on the system dynamics and the interaction topology to design the control laws, which can be centralized or distributed, to fulfill a task. Typical cooperative control tasks include consensus, formation, swarming/flocking, rendezvous, etc. There into, the consensus problem, which requires all agents to agree on a certain quantity of common interests, builds the foundation of other cooperative tasks.
Existing research on consensus assumes that the communication networks among agents are perfect. However, as mentioned earlier, in practical applications, communication channels naturally suffer from fading, signal‐to‐noise ratio (SNR) constraints, time delay, etc. Hence, it is of great significance to study how the uncertainties in communication networks influence the consensus of MASs. The other main objective of this book is to analyze the consensus problem of MASs over channels with fading, packet drops, and delay.
Figure 1.1 Networked control systems.
Figure 1.2 Fading phenomenon in wireless communications.
1.2 Literature Review
Control over communication channels/networks has been a hot research topic in the past decades [Matveev and Savkin, 2009, Como et al., 2014, You et al., 2015], motivated by the rapid developments of wireless communication technologies that enable the wide connection of geographically distributed devices and systems. However, the inclusion of wireless communication channels/networks also introduces challenges in the analysis and design of control systems due to constraints and uncertainties in wireless communications. We must take the communication channels/networks into consideration and study their impact on the stability and performance of control systems. This section briefly reviews existing results on the analysis and design of NCSs and MASs over imperfect communication channels.
1.2.1 Basics of Communication Theory
One of the main focuses of this book is to characterize the critical channel requirement such that the NCS can be mean‐square stabilized. Since the communication channel is used to transmit information about the system state, as illustrated in Figure 1.1, it is expected that if the channel capacity is large enough, the feedback connected system can be mean‐square stable. From this perspective, the communication channel capacity might be critical for the mean‐square stabilization of control systems.
The channel capacity problem is fundamental in communication theory since it dictates the maximum data rates that can be transmitted over channels with asymptotically small error probability [Tse and Viswanath, 2005, Goldsmith, 2005]. In this subsection, we briefly review the communication channel capacity definitions and discuss why the communication theoretic channel capacity is not the critical characterization of the capacity required for controls. We only discuss discrete memoryless channels, and most of the definitions are borrowed from Cover and Thomas [2006].
A discrete memoryless channel consists of three parts: an input alphabet , an output alphabet , and a probability transition matrix that describes the probability of observing the output symbol given the input symbol . The channel is memoryless if the probability distribution of the current channel output conditioned on the current channel input is independent of previous channel inputs or outputs. The configuration of the point‐to‐point communication system is depicted in Figure 1.3. We want to transmit a message reliably through the communication channel with appropriately designed channel encoders and decoders. The code in a communication system is defined as follows.
Figure 1.3 Point‐to‐point communication system.
Definition 1.1 (code)
An code for the channel consists of three parts:
A message index set .
An encoding function , generating codewords , , , .
A decoding function , generating an estimate for the transmitted message index.
The performance of the code is measured by the decoding error.
Definition 1.2 (Decoding error)
The maximal probability of error for an code is defined as .
The communication channel capacity which measures the maximal capacity for reliably transmitting the information is defined below.
Definition 1.3 (Channel capacity)
The rate of the code is defined as
A rate is achievable if there exists a sequence of codes such that tends to 0 as . The channel capacity is then defined as the supremum of all achievable rates.
The channel capacity in Definition 1.3 is called the Shannon channel capacity since C. E. Shannon proved in the channel coding theorem that this channel capacity equals the mutual information of the channel maximized over all possible input distributions [Shannon, 2001, Cover and Thomas, 2006]:
where the mutual information is defined as
The Shannon capacity of fading channels has been studied under various scenarios in Goldsmith and Varaiya [1997], Biglieri et al. [1998], Sadeghi et al. [2008], Abou‐Faycal et al. [2001], and Caire et al. [1999]. For example it is proved in Goldsmith and Varaiya [1997] that if the channel state information is available at the receiver side, the Shannon channel capacity of a fading channel is
where is the probability distribution function of the channel fading .
The Shannon channel capacity in Definition 1.3 assumes that the capacity‐achieving code can be sufficiently long, which would inevitably result in a large delay. Since delay is critical in control systems, we may expect that the communication theoretic Shannon channel capacity is not the right choice for controls. This has been confirmed by Sahai and Mitter [2006], where another kind of channel capacity is defined, named the anytime capacity, and showed that the anytime capacity should be the critical characterization of channel capacities for controls when moment stability is concerned. However, there is no systemic method to calculate the anytime capacity. In the following, we will briefly review the existing results on the control and estimation of NCSs, and the consensus of MASs over communication networks.
1.2.2 Stabilization of NCSs
1.2.2.1 Control over Noiseless Digital Channels
For control systems, with components connected through noiseless digital communication channels, the celebrated data rate theorem [Nair and Evans, 2004] is an important result in the past decades. The data rate theorem states that to keep the state of a scalar unstable discrete‐time linear system
mean‐square bounded, the data rate for the digital communication channel that connects the sensor to the controller should satisfy that
Intuitively, this result has the following explanation, see Figure 1.4. The controller wants to compensate for the expansion of uncertainties in the state estimation during the communication process. To ensure the boundedness of the system state, should be smaller than one, which gives the data rate theorem.
Figure 1.4 Intuitive explanations of the data rate theorem.
The result in (1.2) resembles the Shannon's source‐channel coding theorem [Cover and Thomas, 2006], with the left‐hand side being the Shannon channel capacity and the right‐hand side the source's uncertainty measure. Indeed, the right‐hand side of (1.2) denotes the information generating speed of the linear time‐invariant (lti) system [Elia, 2004, Nair et al., 2004], which is generating information about the unknown initial system state. This resemblance also motivates the researchers to study the control systems from the perspective of information theory, e.g. see Touchette and Lloyd [2000], Zang and Iglesias [2003], Martins et al. [2007], Martins and Dahleh [2008], Nair [2013], Silva [2013], Ranade and Sahai [2011, 2013, 2015], Ramnarayan et al. [2014], and Ranade [2014].
1.2.2.2 Control over Stochastic Digital Channels
For noisy channels, the stability problem is more complex because different stability definitions require different channel capacities. Matveev and Savkin [2007] prove that for almost sure stability, the Shannon capacity in relation to the unstable dynamics of the system establishes the critical condition for its stabilizability. For moment stability, Sahai and Mitter [2006] show that the Shannon capacity is too optimistic, while the zero‐error capacity is too pessimistic, and the anytime capacity is introduced to characterize the stabilizability conditions. Essentially, to keep the ‐moment of the state of an unstable scalar plant bounded, it is necessary and sufficient for the feedback channel's anytime capacity corresponding to anytime‐reliability to be greater than , where is the unstable eigenvalue of the plant. The anytime capacity has a more stringent reliability requirement than the Shannon capacity. However, it is worth noting that there exists no systematic method to calculate the anytime capacities of channels. In the control community, the anytime capacity is usually studied under the mean‐square stability requirement and is also named the mean‐square capacity. In the following, we survey the related results that aim to determine the requirements on noisy channels to ensure that the feedback‐connected linear systems can be mean‐square stabilized, which, on the other hand, reveals the mean‐square capacities for the channels studied.
One important kind of communication channel is the time‐varying digital channel. Minero et al. [2009] assumes that the data rate of the time‐varying digital channel under consideration is stochastic and iid and gives the mean‐square stabilizability condition for a connected discrete‐time lti system. For the scalar systems to ensure the mean‐square stabilizability, the following condition should be satisfied:
Similar to the explanation of the data rate theorem for noiseless channels, the inequality (1.3) intuitively implies that to ensure mean‐square stabilizability, it is necessary and sufficient for the average expanding factor of the system state during one iteration to be smaller than one. For vector systems, necessary and sufficient conditions are provided in the form of stability regions or characterized by rate vectors [Minero et al., 2009].
For a stochastic rate‐limited channel, You and Xie [2010] further show that the minimum data rate for the stabilization of a single‐input vector system is explicitly given in terms of unstable eigenvalues of the open‐loop matrix and the packet dropout rate, which reveals the amount of the additional bit rate required to counter the effect of packet dropouts on stabilization. Sufficient data rate conditions for mean‐square stabilization of multiple‐input vector systems are also derived there. When the packet drop is correlated over time, the problem becomes much more complicated. You and Xie [2011b] study mean‐square stabilization of linear systems over networks with Markovian packet drops. Since the sojourn time of the time‐homogeneous Markovian process that models the two‐state packet drop process is iid [Xie and Xie, 2009], a randomly sampled system approach is developed in You and Xie [2011b] to derive the mean‐square stabilizability condition. The same method is also adopted when deriving the data rate theorem with the additional consideration of system uncertainties in Okano and Ishii [2014]. Borrowing results from Markov jump linear systems, the mean‐square stabilizability results for a more general ‐state Markovian packet drop process are given in Minero et al. [2013], which contains the two‐state Markovian packet drop process as a special case. The existing results in Minero et al. [2009], You and Xie [2010, 2011b], and Minero et al. [2013] are both necessary and sufficient for scalar systems. However, for vector systems, generally, there exists a gap between the derived sufficient conditions and necessary conditions. The main difficulty for deriving conditions that are both necessary and sufficient is how to optimally allocate the bits to each unstable subsystem.
1.2.2.3 Control over Analog Channels
The above results focus on digital channels. As to analog channels, Elia [2005] considers the mean‐square stabilization problem over a pure multiplicative noise channel and derives the mean‐square capacity of such channels. Since the iid packet drop channel is one special kind of pure multiplicative noise channels, the results obtained in Elia [2005] can be easily used to derive the results for iid packet drop channels. Xiao et al. [2012] further derive sufficient and necessary conditions for mean‐square stabilization of multi‐input‐multi‐output (MIMO) systems controlled over parallel multiplicative noise channels. Qiu et al. [2013] propose a channel/controller codesign approach with channel resource allocations to stabilize lti systems controlled with imperfect input channels when the total input channel capacity is fixed. When the subchannel capacities are fixed a priori, Chen et al. [2014] derive the stabilizability condition with a majorization approach. The joint effect of the quantization and multiplicative noise on the mean‐square stabilizability is studied in Gu et al. [2015], whereas the case of both time‐delay and multiplicative noise is studied in Qi et al. [2017], Su et al. [2017], and Tan et al. [2015]. Chiuso et al. [2014] consider linear quadratic Gaussian (LQG)‐like control of scalar systems over communication channels suffering from data losses, delays, and SNR limitations. The stability of the closed‐loop system depends on a tradeoff among the snr constraint, packet loss probability, and time delay.
Braslavsky et al. [2007] study the mean‐square stabilization problem over an additive white Gaussian noise (awgn) channel and characterize the critical capacity to ensure mean‐square stabilizability. To ensure the mean‐square stabilization of a networked scalar system, the channel parameters should satisfy the following relation:
with denoting the SNR of the awgn channel. They also show that for the output feedback case, the capacity required for the awgn channel is generally larger than that of the state feedback case, unless the plant is minimum phase. They further show that the extension from linear encoders/decoders to more general causal encoders/decoders cannot provide additional benefits of increasing the channel capacity [Freudenberg et al., 2010].
Specifically, the results stated above deal with multiplicative noise channels or awgn channels separately. While in wireless communications, it is practical to consider them as a whole. Xiao and Xie [2011] have derived the necessary and sufficient conditions for such kinds of channels to ensure the mean‐square stabilizability under a linear encoder/decoder. It is still unknown whether we can achieve a larger stabilizability region with a more general causal encoder/decoder. We provide a positive answer to this question in Chapters 2 and 3.
1.2.3 LQ Optimal Control of NCSs over Fading Channels
As one of the most fundamental problems in control theory, linear‐quadratic (LQ) optimal control has attracted great attention and has been extensively studied for deterministic and stochastic linear systems. See Dragan et al. [2010], Fragoso et al. [1998], Freiling and Hochhaus [2003], Kalman [1960], Wonham [1968], and Zhou et al. [
