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 $x_{i}(t)=\left[x_{i 1}(t), x_{i 2}(t), x_{i 3}(t), x_{i 4}(t)\right]^{T} \in \mathbb{R}^{4}$,
-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
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
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
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|Consensus of linear CNSs with directed switching topologies



在过去十年中,一般线性中枢神经系统的共识问题受到了很多关注[76,146,162,185,186,224]. 具体来说,[76,224] 已经解决了有向固定通信拓扑下线性 CNS 的共识问题。在 [162] 中,研究了线性 CNS 与标称动力学的传递矩阵的加性扰动的稳健共识。在 [163] 和随后的一些论文中,稳健的共识是从H∞控制理论。在其他相关参考文献中,我们提到了[146],其中假设开环系统是 Lyapunov 稳定的,研究了具有无向切换拓扑的线性 CNS 的共识问题。在CNS配备leader和系统拓扑的情况下

属于有向交换拓扑类,共识跟踪问题已经在[185,186]. 这些参考文献中结果的一个特点是开环代理的动力学不必是李雅普诺夫稳定的。请注意,这些参考文献中考虑的 CNS 中领导者的存在有助于推导和直接分析共识错误系统。然而,当开环系统不是李雅普诺夫稳定和/或组中没有指定的领导者时,具有定向切换拓扑的线性 CNS 的共识问题仍然具有挑战性。

受上述讨论的启发,本节旨在研究具有定向切换拓扑的线性 CNS 的共识问题。当前研究的几个方面值得一提。首先,现有工作中的一些假设被驳回,例如,在本章中,代理的开环动力学不必是 Lyapunov 稳定的。此外,正在考虑的 CNS 不需要有领导者。与具有指定领导者的线性 CNS 的共识问题相比,这里的不同之处在于对系统通信拓扑的假设。在之前关于线性 CNS 一致性跟踪的工作(如 [185])中,每个可能的增强系统图都需要包含一个以领导者为根的有向生成树。与那部作品相比,本节中的交换拓扑允许具有植根于不同节点的生成树。这是对先前条件的显着放宽,因为它使系统能够在必要时重新配置(例如,允许不同的节点充当编队领导者)。这也有可能使系统更可靠。

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

考虑一个 CNS 由ñ被标记为代理的代理1,…,ñ. 代理的动态一世被描述为

在哪里X一世(吨)∈Rn是状态,在一世(吨)∈R米是控制输入,一个∈Rn×n和乙∈Rn×米分别是状态矩阵和控制输入矩阵。假设矩阵对(一个,乙)是稳定的。并且假设所考虑的 CNS 的通信拓扑在图集上动态切换G^. 在哪里\widehat{\mathcal{G}}=\left{\mathcal{G}^{1},\ldots,\mathcal{G}^{\kappa}\right},\kappa\geq 1\widehat{\mathcal{G}}=\left{\mathcal{G}^{1},\ldots,\mathcal{G}^{\kappa}\right},\kappa\geq 1, 表示所有可能的有向拓扑的集合。
假设G(吨)∈G^对所有人吨. 为了描述通信拓扑的时变特性,假设存在无限序列的非重叠时间间隔[吨ķ,吨ķ+1),ķ=0,1,….和吨0=0,0<τ米≤吨ķ+1−吨ķ≤τ米<+∞,其上的通信拓扑是固定的。这里,τ米>τ米>0和τ米称为停留时间。开关信号的引入σ(吨):[0,+∞)↦1,…,ķ使CNS(3.1)的通信拓扑在每个时刻都得到很好的定义吨≥0. 为了符号方便,我们将使用时变图来描述这种通信拓扑Gσ(吨).

在 CNS 的上下文中,只有相邻代理之间的相关信息可用于协调。对于每个代理一世,以下分布式共识


在哪里一个>0表示耦合强度,ķ∈R米×n是要设计的反馈增益矩阵,并且一个σ(吨)=[一个一世jσ(吨)]ñ×ñ是图的邻接矩阵Gσ(吨). 然后,从(3.1)和(3.2)得出

让X(吨)=[X1吨(吨),…,Xñ吨(吨)]吨, 因此从 (3.3) 得出


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

考虑由五个代理组成的CNS(3.3),其拓扑在图之间切换G1和G2如图 3.1 所示。为方便起见,每条边的权重为 1。每个代理代表一架垂直起降 (VTOL) 飞机。根据[109],动态一世用于典型装载和


一个=[−0.03660.02710.0188−0.4555 0.0482−1.010.0024−4.0208 0.10020.3681−0.7071.420],乙=[0.44220.1761 3.5446−7.5922 −5.524.49 0.00.0], 
其中状态变量定义为:X一世1(吨)是水平速度,X一世2(吨)是垂直速度,X一世3(吨)是俯仰速率,并且X一世4(吨)是俯仰角[109]。从图中可以看出3.1那G1包含一个以节点 2 为领导者的有向生成树,而G2包含一个以节点为根的有向生成树5.

大号^1=[1000 0100 0010 −1101],大号^2=[1000 0100 −1010 0001]
放C1=C2=0.5. 求解 LMI (3.8) 得到 $\bar{\lambda} {\max }=2.5612,在H和r和\bar{\lambda} {\max }一世sd和F一世n和d一世nC○r○ll一个r是3.1.大号和吨\beta=3,s○l在一世nG大号米我(3.9)G一世在和s吨H一个吨ķ=[5.82060.2978−0.2615−2.7967 −1.1646−0.45220.05302.0420]. 小号和吨\alpha=4.1>2 / c_{0}=4.0.吨H和n,一个CC○rd一世nG吨○C○r○ll一个r是3.1,○n和ķn○在s吨H一个吨C○ns和ns在s一世n吨H和Cl○s和d−l○○pCñ小号(3.3)C一个nb和一个CH一世和在和d一世F吨H和d在和ll吨一世米和\tau_{m}>0.3135\mathrm~s.我ns一世米在l一个吨一世○ns,l和吨吨H和吨○p○l○G是s在一世吨CH和sb和吨在和和nGr一个pH\数学{G}^{1}一个nd\数学{G}^{2}和在和r是0.32 \mathrm ~ s.吨H和s吨一个吨和吨r一个j和C吨○r一世和s○F吨H和Cl○s和d−l○○pCñ小号(3.3)一个r和sH○在n一世nF一世Gs.3.2一个nd3.3.吨H和和在○l在吨一世○n○F|e(t)|$ 如图 3.4 所示,证实了 CNS (3.3) 达成共识。

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