### cs代写|复杂网络代写complex network代考|CONSENSUS OF MASS WITH SWITCHING TOPOLOGIES

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## cs代写|复杂网络代写complex network代考|CONSENSUS OF MASS WITH SWITCHING TOPOLOGIES

Since the pioneer works $[65]$ in which heading consensus of the linearized Vicsek’s model was analyzed, consensus of MASs with switching topologies has attracted increasing attention from a wide range of scientific interests.

Consensus of first-order MASs with switching topologies: In the year of 2004, consensus problem of continuous time first-order (integrator-type) MASs with directed switching and balanced topology was formulated and studied in [116]. Due to the balanced property of each possible topology candidate, a common Lyapunov function was constructed in $[116]$ for analyzing the convergence behaviors of disagreement vector. Consensus of both continuous- and discrete time first-order MASs with directed switching topologies was further studied in [128] where each possible topology candidate is not required to be balanced. By using graphical approaches, some interesting issues on consensus of a class of first-order MASs with switching topologies were further addressed in [13]. By employing a CLFL based approach, it was proven in [83] that average consensus in continuous time first-order MASs with time delayed protocol can be achieved if each topology candidate is strongly connected and balanced, and some linear matrix inequalities hold. Note that most of the aforementioned results are concerned with consensus of first-order MASs with deterministically switching topologies. However, considering the underlying topology may randomly switch among a set of topology candidates in some practical applications, there have been a number of results focusing on consensus of first-order MASs with randomly switching topologies $[54,155,156]$.

Consensus of second-order MASs with switching topologies: Based on the stability results for switched systems provided in [108], some dwell time (DT) based criteria for consensus of continuous time second-order MASs under directed switching topologies were established in [129] where it was revealed that consensus can be achieved if each topology candidate contains a directed spanning tree and the DT for switchings among different topology candidates is larger than a threshold value. When the graph describing the communication topology among followers is undirected, it was proven in [59] by constructing a CLF that leader-following consensus could be achieved if the topology jointly contains a directed spanning tree. Later, leader-following consensus problem of MASs with switching jointly reachable interconnection and transmission delays was solved in [234] by designing the switching laws among topology candidates, where the dynamics of the leader are described by first-order integrator. Note that the switching mode for topology evolution of the MASs studied in [234] is a kind of state-dependent switching. By constructing a CLFL, Lin and Jia [85] showed that leaderless consensus of MASs with time-delayed protocol could be achieved if the underlying topology is undirected and jointly connected. Leaderless consensus of MASs with time-delayed protocols under directed switching topologies was further studied in [124]. Note that there is no specific restriction for the value of the DT for switching signals in the consensus criteria provided in $[59,85,124]$ as CLF- and CLFL-based approaches were respectively adopted. By constructing a CLF, Wen et al. [181] obtained some sufficient criteria for achieving consensus in MASs with intermittent communication. Note that the underlying communication topology of the closed-loop MASs with intermittent communication can be seen as a directed switching topologies with two topology candidates: A strong connected graph and the null graph. More recently, pulse-modulated intermittent control which unifies impulsive control and sampled control was proposed in [93] to solve the consensus problem of MASs under intermittent communications.

## cs代写|复杂网络代写complex network代考|EXTENSIONS AND APPLICATIONS OF CNSS WITH SWITCHING TOPOLOGIES

In the above sections, we have surveyed some recent developments in the analysis and synthesis of CNSs with switching topologies, mainly focusing on the synchronization and consensus behaviors and comparison to complex networks and MASs’ scenarios. The above survey is by no means complete. However, it depicts the whole general framework of coordination control for CNSs with dynamic communication networks and lays the fundamental basis for other exciting and yet critical issues concerning CNSs with switching topologies. These extensions still deserve further study, although a variety of efficient tools have been successfully developed to solve various challenging problems in those active research fields. Next, we elaborate on several state-of-the-art extensions and applications of CNSs with dynamic topologies.

Resilience analysis and control of complex cyber-physical networks. Most of the units in various network infrastructures are cyber-physical systems in the Internet of Things era. One of the essential and significant features of the cyber-physical system is integrating and interacting with its physical and cyber layers. As a new generation of CNS, the complex cyber-physical network has received drastic attention in recent years. Specifically, the CNSs’ paradigm provides an excellent way to model various large-scale crucial infrastructure systems, such as power grid systems, transportation systems, water supply networks, and many others [4]. These systems all capture the basic features that large numbers of interconnected individuals through wired or wireless communication links, and many essential functions of these large-scale

infrastructure systems fall under the purview of coordination of CNSs. Disruption of these critical networked infrastructures could be a real-world effect across an entire country and even further, significantly impacting public health and safety and leading to massive economic losses. The alarming historical events urgently remind us to seek solutions for maintaining certain functionality of CNSs against malicious cyberattacks (i.e., resilience or cybersecurity). It is critically essential to exploit security threats during the initial design and development phase.

Noteworthily, any successful cyber or physical attack mentioned above on complex cyber-physical networks may introduce undesired switching dynamics (e.g., loss of links due to DoS attacks or human-made physical damages) to the operation of these networks [194]. Inspired by the pioneering work [194], [168] further investigated the distributed observer-based cyber-security control of complex dynamical networks. This work considered the scenario that the communication channels for controllers and observers might both subject to malicious cyber attacks, which aim to block the information exchanges and result in disconnected topologies of the communication networks. New security control strategies are proposed, and an algorithm to properly select the feedback gain matrices and coupling strengths has been given. The asynchronous attacks in these two communication channels were explored in [169], where the attacks can be launched independently and may occur at different time intervals. Recently, [69] studied the distributed cooperative control for DC cyber-physical microgrids under communication delays and slow switching topologies would destruct the system’s transient behaviors at the switching time instants. The average switching dwell-time-dependent control conditions were given to ensure the exponential stability of the considered cyber-physical systems. For the event-triggered communication scenario, [26] studied the distributed consensus for general linear MASs subjected to DoS attacks. By the switched and time-delay system approaches, one constraint was provided to illustrate the convergence rate of consensus errors and uniform lower bound of non-attacking intervals of DoS attacks.

## cs代写|复杂网络代写complex network代考|Preliminaries

This chapter presents some preliminaries used in this book. In Section 2.1, notations are presented. Section $2.2$ begins by introducing the matrix theory that includes Schur complement lemma, Finsler’s lemma, Gershgorin’s dise theorem, and some other Lemmas. Then the Barbălat lemma and the $\mathcal{K}$ function are presented. In Section 2.3, algebraic graph theory is presented that includes directed (undirected) graph, connected graph, strongly connected graph, directed spanning tree, adjacency matrix, Laplacian matrix. Specifically, the nonsingular M-matrix theory is presented which will play a crucial role in constructing the MLFs. In Section 2.4, stability theory of switched systems is given. This section begins by introducing the Carathéodory’s solution of switched systems. Then the MLFs-based methods are presented, both dwell time and average dwell time stability analysis methods of switched systems are given. Note that this chapter provides some necessary tools for understanding the subsequent chapters of this book, which are especially important for a fresh graduate.

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## MATLAB代写

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