1
Introduction

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.

The performance of the code is measured by the decoding error.

The communication channel capacity which measures the maximal capacity for reliably transmitting the information is defined below.

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...