# Formation Control of Nonlinear Multi-Agent Systems with Nested Input Saturation
## 4 Comparative Simulation Results  
In this section we present comparative simulation results in order to illustrate the superiority of the proposed control scheme against a well-established formation control algorithm described in ref. [35]. In this simulation scenario, we consider the formation control problem of five planar agents described by the following nonlinear dynamical model:  
\[ \begin{bmatrix}\dot{x}_{i}\\ \dot{y}_{i}\end{bmatrix} = \begin{bmatrix}0.5x_{i}y_{i}+\exp{(-x_{i}^{2}-y_{i}^{2} )}\\ 0.1x_{i}^{2}-0.05y_{i}^{2}+\sin(x_{i}y_{i})\end{bmatrix} + \begin{bmatrix}1+0.3\sin {(x_{i})}\\ 1+0.45\cos{(x_{i})}\end{bmatrix} u_{i}+d_{i}(t)\]  
with \(u_{i}=[u_{x,i},u_{y,i}]^{T}\in[-1.2,1.2]\times[-1.2,1.2]\) and \(d_{i}(t)=[\cos{(2t)},-\sin{(2.5t)}]^{T}\) for all \(i=1,\ldots,5\). The aim of this paradigm is to achieve a regular pentagon formation while the swarm dynamically tracks a reference trajectory set by a virtual leader centered within the formation. The reference trajectory is circular and it is given as \(x_{r}=\cos{(0.2t)}\), \(y_{r}=\sin{(0.2t)}\) for \(t\in[0,25]\). The control parameters for the proposed scheme were chosen as \(k_{l,i}^{1}=0.5,\;k_{l,i}^{2}=2,\lambda_{l,i}^{1}=0.5,\lambda_{l,i}^{2}=1,\rho _{l,i}^{1,\infty}=0.1,\;\rho_{l,i}^{2,\infty}=0.1,\;l\in\{x,y\},\;i=1,\ldots,5\) with \(\dot{u}_{i}\in[-5,5]\times[-5,5],\;i=1,\ldots,5\). For the scheme of [35] the gains were chosen with fine tuning as \(k_{i}=1.15,\;i=1,\ldots 5\). Moreover, the adjacency matrix \(\mathcal{A}\) of the communication graph and the matrix \(B\) are given by:  
\[\mathcal{A} = \begin{bmatrix}0&1&1&0&0\\ 1&0&0&1&0\\ 1&0&0&0&1\\ 0&1&0&0&1\\ 0&0&1&1&0\end{bmatrix}  \tag{18}\] \[B = \begin{bmatrix}1&0&0&0&0\end{bmatrix} . \tag{19}\]  
The left subfigures of Figures 2 and 3 display the neighborhood tracking error for each agent concerning the \(x\) and \(y\) axes, respectively. A clear observation emerges: the proposed control (red line) outperforms the control scheme from ref. [35] (blue line), showcasing a notably improved closed-loop response while also respecting both amplitude and rate input constraints. Conversely, the right subfigures of Figures 2 and 3, representing the control signals with respect to the \(x\) and \(y\) axes, respectively, reveal higher control effort in the proposed control scheme compared to that in ref. [35]. This increased effort aligns with the pursuit of the best output performance respecting the system's input limitations. To quantify these observations and to offer a comprehensive evaluation of the schemes' performance, we employ three performance indices, each providing a unique perspective on the system's behavior. The Average Squared Error index (\(\mu ASE\)) emphasizes larger errors, reflecting a faster convergence rate with lower values and the Average Absolute Error index (\(\mu AAE\)), in contrast to ASE, indicates a slower convergence rate but with reduced persistent oscillations. Additionally, the Total Energy Consumption index (\(\mu TEC\)) signifies energy utilization efficiency and reduced energy loss in the control process, quantifying the system's energy efficiency. It is important to note that smaller values for these indices  
\begin{table}
\begin{tabular}{c} \hline \hline
**Control Protocol** \\ \hline \(u_{d_{i}}(t)=-k_{i}^{1}\bigg{[}T\bigg{(}\frac{\dot{e}_{\lambda 1}(t)}{\dot{\rho}_{1,i}^{1}(t)}\bigg{)},\ldots,T\bigg{(}\frac{\dot{e}_{\lambda 1}(t)}{\dot{\rho}_{\lambda 1}(t)}\bigg{)}\bigg{]}^{T}\), \(k_{l}^{1}>0\) \\ \(a_{i}(t)=-k_{i}^{2}\bigg{[}T\bigg{(}\frac{u_{\lambda 1}(t)-\sigma(u_{\lambda 1}(t),\mu_{\lambda 1})}{\dot{\rho}_  
indicate superior control performance and efficiency in the tracking process. Table 2 presents these performance indices for both control schemes, providing a direct validation of the conclusions drawn from the figures. Figure 3: The evolution of tracking errors w.r.t. axis \(y\) on the left; The evolution of control inputs \(u_{y,i},\ i=1,\ldots,5\) on the right.