### cs代写|复杂网络代写complex network代考|Consensus of linear CNSs with directed switching topologies

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

In the past decade, the consensus problem of general linear CNSs has received a lot of attention $[76,146,162,185,186,224]$. Specifically, the consensus problem of linear CNSs under a directed fixed communication topology has been addressed in [76,224]. In [162], the robust consensus of linear CNSs with additive perturbations of the transfer matrices of the nominal dynamics was studied. In [163] and a number of subsequent papers, the robust consensus was analyzed from the viewpoint of the $\mathcal{H}_{\infty}$ control theory. Among other relevant references, we mention [146] where, while assuming that the open loop systems are Lyapunov stable, the consensus problem of linear CNSs with undirected switching topologies has been investigated. In the situation where the CNS is equipped with a leader and the topology of the system

belongs to the class of directed switching topologies, the consensus tracking problem has been studied in $[185,186]$. One feature of the results in these references is that the open loop agents’ dynamics do not have to be Lyapunov stable. Note that the presence of the leader in the CNSs considered in these references facilitate the derivations and the direct analyses of the consensus error system. However, when the open loop systems are not Lyapunov stable and/or there is no designated leader in the group, the consensus problem for linear CNSs with directed switching topologies remains challenging.

Motivated by the above discussion, this section aims to study the consensus problem for linear CNSs with directed switching topologies. Several aspects of the current study are worth mentioning. Firstly, some of the assumptions in the existing works are dismissed, e.g., the open loop dynamics of the agents do not have to be Lyapunov stable in this chapter. Furthermore, the CNSs under consideration are not required to have a leader. Compared with the consensus problems for linear CNSs with a designated leader, the point of difference here concerns the assumption on the system’s communication topology. In the previous work on the consensus tracking of linear CNSs such as [185], each possible augmented system graph was required to contain a directed spanning tree rooted at the leader. Compared with that work, the switching topologies in this section are allowed to have spanning trees rooted at different nodes. This is a significant relaxation of the previous conditions since it enables the system to be reconfigured if necessary (e.g., to allow different nodes to serve as the formation leader). This also has a potential to make the system more reliable.

## cs代写|复杂网络代写complex network代考|Problem formulation

Consider a CNS consists of $N$ agents that are labelled as agents $1, \ldots, N$. The dynamics of agent $i$ are described by
$$\dot{x}{i}(t)=A x{i}(t)+B u_{i}(t),$$
where $x_{i}(t) \in \mathbb{R}^{n}$ is the state, $u_{i}(t) \in \mathbb{R}^{m}$ is the control input, $A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{n \times m}$ are, respectively, the state matrix and control input matrix. It is assumed that the matrix pair $(A, B)$ is stabilizable. And it is assumed that the communication topology of the CNS under consideration switches dynamically over a graph set $\widehat{\mathcal{G}}$. where $\widehat{\mathcal{G}}=\left{\mathcal{G}^{1}, \ldots, \mathcal{G}^{\kappa}\right}, \kappa \geq 1$, denotes the set of all possible directed topologies.
Suppose that $\mathcal{G}(t) \in \widehat{\mathcal{G}}$ for all $t$. To describe the time-varying property of communication topology, assume that there exists an infinite sequence of non-overlapping time intervals $\left[t_{k}, t_{k+1}\right), k=0,1, \ldots .$ with $t_{0}=0,0<\tau_{m} \leq t_{k+1}-t_{k} \leq \tau_{M}<+\infty$, over which the communication topology is fixed. Here, $\tau_{M}>\tau_{m}>0$ and $\tau_{m}$ is called the dwell time. The introduction of the switching signal $\sigma(t):[0,+\infty) \mapsto{1, \ldots, \kappa}$ makes the communication topology of CNS (3.1) well defined at every time instant $t \geq 0$. For notational convenience, we will describe this communication topology using the time-varying graph $\mathcal{G}^{\sigma(t)}$.

Within the context of CNSs, only relative information among neighboring agents can be used for coordination. For each agent $i$, the following distributed consensus

protocol is proposed
$$u_{i}(t)=\alpha K \sum_{j=1}^{N} a_{i j}^{\sigma(t)}\left[x_{j}(t)-x_{i}(t)\right], \quad i=1, \ldots, N,$$
where $\alpha>0$ represents the coupling strength, $K \in \mathbb{R}^{m \times n}$ is the feedback gain matrix to be designed, and $\mathcal{A}^{\sigma(t)}=\left[a_{i j}^{\sigma(t)}\right]{N \times N}$ is the adjacency matrix of graph $\mathcal{G}^{\sigma(t)}$. Then, it follows from (3.1) and (3.2) that $$\dot{x}{i}(t)=A x_{i}(t)+\alpha B K \sum_{j=1}^{N} a_{i j}^{\sigma(t)}\left[x_{j}(t)-x_{i}(t)\right],$$
where $i=1, \ldots, N$.
Let $x(t)=\left[x_{1}^{T}(t), \ldots, x_{N}^{T}(t)\right]^{T}$, it thus follows from (3.3) that
$$\dot{x}(t)=\left[\left(I_{N} \otimes A\right)-\alpha\left(\mathcal{L}^{\sigma(t)} \otimes B K\right)\right] x(t),$$
where $\mathcal{L}^{\sigma(t)}$ is the Laplacian matrix of communication topology $\mathcal{G}^{\sigma(t)}$.
Before concluding this section, the following assumption is presented which will be used in the derivation of the main results.

## cs代写|复杂网络代写complex network代考|Numerical simulations

Consider the CNS (3.3) consisting of five agents, whose topology switches between the graphs $\mathcal{G}^{1}$ and $\mathcal{G}^{2}$ shown in Figure 3.1. For convenience, the weight of each edge is 1. Each agent represents a vertical take-off and landing (VTOL) aircraft. According to [109], the dynamics of the $i$ th VTOL aircraft for a typical loading and

flight condition at the air speed of $135 \mathrm{kt}$ can be described by the system (3.1), with $x_{i}(t)=\left[x_{i 1}(t), x_{i 2}(t), x_{i 3}(t), x_{i 4}(t)\right]^{T} \in \mathbb{R}^{4}$,
$$A=\left[\begin{array}{cccc} -0.0366 & 0.0271 & 0.0188 & -0.4555 \ 0.0482 & -1.01 & 0.0024 & -4.0208 \ 0.1002 & 0.3681 & -0.707 & 1.420 \ 0.0 & 0.0 & 1.0 & 0.0 \end{array}\right], B=\left[\begin{array}{cc} 0.4422 & 0.1761 \ 3.5446 & -7.5922 \ -5.52 & 4.49 \ 0.0 & 0.0 \end{array}\right] \text {, }$$
where the state variables are defined as: $x_{i 1}(t)$ is the horizontal velocity, $x_{i 2}(t)$ is the vertical velocity, $x_{i 3}(t)$ is the pitch rate, and $x_{i 4}(t)$ is the pitch angle [109]. It can be seen from Figure $3.1$ that $\mathcal{G}^{1}$ contains a directed spanning tree with node 2 as the leader, while $\mathcal{G}^{2}$ contains a directed spanning tree rooted at node $5 .$
The transformed Laplacian matrices $\widehat{\mathcal{L}}^{1}, \widehat{\mathcal{L}}^{2}$ in this example are
$$\widehat{\mathcal{L}}^{1}=\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ -1 & 1 & 0 & 1 \end{array}\right], \quad \widehat{\mathcal{L}}^{2}=\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ -1 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{array}\right]$$
Set $c_{1}=c_{2}=0.5$. Solving the LMI (3.8) gives that $\bar{\lambda}{\max }=2.5612$, where $\bar{\lambda}{\max }$ is defined in Corollary 3.1. Let $\beta=3$, solving LMI (3.9) gives that
$$K=\left[\begin{array}{cccc} 5.8206 & 0.2978 & -0.2615 & -2.7967 \ -1.1646 & -0.4522 & 0.0530 & 2.0420 \end{array}\right] \text {. }$$
Set $\alpha=4.1>2 / c_{0}=4.0$. Then, according to Corollary $3.1$, one knows that consensus in the closed-loop CNS (3.3) can be achieved if the dwell time $\tau_{m}>0.3135 \mathrm{~s}$. In simulations, let the topology switches between graph $\mathcal{G}^{1}$ and $\mathcal{G}^{2}$ every $0.32 \mathrm{~s}$. The state trajectories of the closed-loop CNS (3.3) are shown in Figs. $3.2$ and 3.3. The evolution of $|e(t)|$ is shown in Figure 3.4, which confirms that the CNS (3.3) achieves consensus.

## cs代写|复杂网络代写complex network代考|Problem formulation

X˙一世(吨)=一个X一世(吨)+乙在一世(吨),

X˙一世(吨)=一个X一世(吨)+一个乙ķ∑j=1ñ一个一世jσ(吨)[Xj(吨)−X一世(吨)],

X˙(吨)=[(我ñ⊗一个)−一个(大号σ(吨)⊗乙ķ)]X(吨),

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