## 5 Simulation
### Example 1  
Four follower agents are labeled as agents 1-4 and two leaders are labeled as agents 5-6, i.e., \(\mathcal{F}=\{1,2,3,4\}\) and \(\mathcal{W}=\{5,6\}\). The communication network is fixed in this case as shown in Figure 1, in which the topology among followers is strongly connected. Moreover, the information from agents 5-6 can only be received by agent 1 and agent 4, respectively.  
The dynamics of four followers is described as  
\[Agent\ 1:y_{1}(k+1) =0.7y_{1}(k)+u_{1}(k)\] \[Agent\ 2:y_{2}(k+1) =0.6 cos(y_{2}(k))+0.3y_{2}(k)+u_{2}(k)\] \[Agent\ 3:y_{3}(k+1) =\frac{y_{3}(k)}{1+y_{3}^{2}(k)}+u_{3}(k)\] \[Agent\ 4:y_{4}(k+1) =\frac{y_{4}(k)}{1+y_{4}^{2}(k)}+u_{4}^{3}(k)\]  
It is worth noting that the followers are heterogeneous and include linear and nonlinear dynamics. Agent 1 is a linear agent, which is widely used in autonomous vehicle formation control [2]. The cosine nonlinear dynamics of agent 2 is taken from [46]. The nonaffine nonlinear descriptions of agents 3-4 are picked from [36]. The heterogeneity and nonlinearity of followers become a great challenge to the controller design.  
The trajectories of two leaders are chosen as piece-wise step signals:  
\[Agent\ 5:w_{5}(k) =\left\{\begin{array}{ll}1.4,&\quad\text{if}\ k<165\\ 1.6,&\quad\text{if}\ 165\leq k<330\\ 1.3,&\quad\text{if}\ k\geq 330\end{array}\right.\] \[Agent\ 6:w_{6}(k) =\left\{\begin{array}{ll}0.7,&\quad\text{if}\ k<165\\ 1.2,&\quad\text{if}\ 165\leq k<330\\ 1.1,&\quad\text{if}\ k\geq 330\end{array}\right.\]  
\begin{table}
\begin{tabular}{c c c c} \hline  & **Example 1** & **Example 2** & **Example 3** \\ \hline Leader number & 2 & 3 & 3 \\ Leader trajectories & stationary & time-varying & stationary and time-varying \\ Topology & fixed & time-switching & time-switching \\ Theorem validation & Theorem 1 & Theorem 2 & Theorems 1–2 \\ Disturbance & none & none & additive channel \\ Parameters & \(\rho_{1}=\rho_{2}=\rho_{3}=\rho_{4}=0.3,\lambda\in[0.5\,5],\eta=0.5,\mu=1, \varepsilon=0.0001\) \\ \hline \end{tabular}
\end{table}
Table 1: Example settings and parameter selections.  
Figure 1: Communication graph in Example 1.  
The initial conditions \(y_{i}(0)\)\((i\in\mathcal{F})\) are selected as \(y_{1}(0)=1.2\), \(y_{2}(0)=1.8\), \(y_{3}(0)=0.6\), \(y_{4}(0)=0.4\). Initial values of the control inputs and estimated PPD are set as \(u_{i}(0)=0\)\((i\in\mathcal{F})\) and \(\widehat{\phi}_{i}(0)=0.5\)\((i\in\mathcal{F})\). Other parameters are listed in Table 1. Observing the communication network in Figure 1, the in-degree matrix is \(\mathcal{D}=diag(2,1,2,2,0,0)\) and the reciprocal of its largest diagonal entry is 0.5. Thus, \(\rho_{i}=0.3<1/d_{i}\)\((i\in\mathcal{F})\) satisfies the convergence condition in Theorem 1.  
The outputs and inputs of agents are shown in Figures 2 and 3 (with \(\lambda=2\)), respectively. They show that the follower agents are driven into the convex hull of leaders gradually. At the time instant \(k=165\) and \(k=330\), the convex hull moves to another location due to the change of the leaders' outputs. The followers can still converge to the new convex hull, which verifies the result in Theorem 1. To further study the influence of the key parameter, the tracking performance with different \(\lambda\) (i.e., \(\lambda=5\), \(2\), \(1\), and \(0.5\)) is presented in the sub-figures in Figure 2. The results show that, within a certain range, the smaller the \(\lambda\) value, the faster the tracking speed of follower agents. However, one should also note that the overshoots and oscillations may be intensified if the \(\lambda\) value is too small. The PPD estimation (with \(\lambda=2\)) is shown in Figure 4. Due to the heterogeneous dynamics of followers, the PPD estimations are quite different from each other even though they have the same initial value.  
Figure 3: Inputs of agents in Example 1 (\(\lambda=2\)).