## cs代写|复杂网络代写complex network代考|CONSENSUS TRACKING

Consider a CNS consisting of a leader and $N$ followers, where the leader is labelled as agent 0 and the followers are labelled as agents $1, \ldots, N$. The dynamics of agent $i, i=0,1, \ldots, N$, are given by
$$\dot{x}i(t)=A x_i(t)+C f\left(x_i(t), t\right)+B u_i(t),$$ where $x_i(t) \in \mathbb{R}^n$ represent the states of agent $i, f(\cdot, \cdot): \mathbb{R}^n \times[0,+\infty) \mapsto \mathbb{R}^p$ is a continuously differentiable vector-valued function representing the intrinsic nonlinear dynamics, $u_i(t) \in \mathbb{R}^m$ is the control input to be designed, $A \in \mathbb{R}^{n \times n}, B \in \mathbb{R}^{n \times m}$, and $C \in \mathbb{R}^{n \times p}$ are constant real matrices. It is assumed that the matrix pair $(A, B)$ is stabilizable. In this section, it is assumed that the leader will not being affected by any followers, i.e. $u_0(t) \equiv \mathbf{0}_m$ in CNS (6.1). Before moving on, the following assumption is made. Assumption 6.1 There exists a nonnegative constant $\varrho$, such that $$|f(y, t)-f(z, t)| \leq \varrho|y-z|, \forall y, z \in \mathbb{R}^n, t \geq 0$$ To achieve consensus tracking, the following distributed consensus tracking protocol is proposed for each follower $i$ : $$u_i(t)=\alpha F \sum{j=0}^N a_{i j}(t)\left(x_j(t)-x_i(t)\right), \quad i=1, \ldots, N,$$
where $\alpha>0$ represents the coupling strength, $F \in \mathbb{R}^{m \times n}$ is the feedback gain matrix to be designed, and $\mathcal{A}(t)=\left[a_{i j}(t)\right]_{(N+1) \times(N+1)}$ is the adjacency matrix of graph $\mathcal{G}(t)$. Here, $\mathcal{G}(t)$ describes the underlying communication topology among the $N+1$ agents at time $t$.

## cs代写|复杂网络代写complex network代考|Main results for fixed topology containing a directed spanning tree

In this section, distributed consensus tracking is addressed for CNS (6.1) with a fixed communication topology containing a directed spanning tree.

Without loss of generality, let $\mathcal{G}(t)=\mathcal{G}$ for all $t \geq 0$ since the communication topology is assumed to be fixed in this subsection. To derive the main results, the following assumption is needed.

Assumption 6.2 The communication topology $\mathcal{G}$ contains a directed spanning tree with agent 0 (i.e. the leader) as the root.

Under Assumption 6.2, the Laplacian matrix of directed graph $\mathcal{G}$ can be written as
$$\mathcal{L}=\left[\begin{array}{cc} 0 & \mathbf{0}N^T \ \mathbf{P} & \overline{\mathcal{L}} \end{array}\right], \quad \overline{\mathcal{L}}=\left[\begin{array}{cccc} \sum{j \in \mathcal{N}1} a{1 j} & -a_{12} & \ldots & -a_{1 N} \ -a_{21} & \sum_{j \in \mathcal{N}2} a{2 j} & \ldots & -a_{2 N} \ \vdots & \vdots & \ddots & \vdots \ -a_{N 1} & -a_{N 2} & \ldots & \sum_{j \in \mathcal{N}N} a{N j} \end{array}\right]$$
where $\mathbf{P}=-\left[a_{10}, \ldots, a_{N 0}\right]^T$. It can be thus obtained from Lemma $2.15$ that there exists a positive definite diagonal matrix $\Phi=\operatorname{diag}\left{\phi_1, \ldots, \phi_N\right}$ such that $\overline{\mathcal{L}}^T \Phi+$ $\Phi \overline{\mathcal{L}}>0$. One such $\phi=\left[\phi_1, \ldots, \phi_N\right]^T$ can be obtained by solving the matrix equation $\overline{\mathcal{L}}^T \phi=\mathbf{1}_N$
Since $u_0(t) \equiv \mathbf{0}_m$, one has
$$\dot{x}_0(t)=A x_0(t)+C f\left(x_0(t), t\right)$$

## cs代写|复杂网络代写complex network代考|CONSENSUS TRACKING

$$\dot{x} i(t)=A x_i(t)+C f\left(x_i(t), t\right)+B u_i(t),$$

## cs代写|复杂网络代写complex network代考|Main results for fixed topology containing a directed spanning tree

$$\mathcal{L}=\left[\begin{array}{lll} 0 & \mathbf{0} N^T \mathbf{P} & \overline{\mathcal{L}} \end{array}\right], \quad \overline{\mathcal{L}}=\left[\begin{array}{llllllll} \sum j \in \mathcal{N} 1 a 1 j & -a_{12} & \ldots & -a_{1 N} & -a_{21} & \sum_{j \in \mathcal{N} 2} a 2 j & \ldots & -a_{2 N} \end{array}\right.$$

$$\dot{x}_0(t)=A x_0(t)+C f\left(x_0(t), t\right)$$

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