# Formation Control of Nonlinear Multi-Agent Systems with Nested Input Saturation
## 2 Problem Formulation and Preliminaries  
The system considered consists of N following agents and a leading virtual agent that generates a reference trajectory for the MAS. We assume that the agents obey the following first-order nonlinear dynamical model:  
Figure 1: The agents perform the desired pentagon formation in our lab’s arena. \[\dot{x}_{i} =f_{i}(x_{i})+g_{i}(x_{i})u_{i}+d_{i}(t) \tag{1}\] \[\dot{u}_{i} =\text{sat}_{r_{i}}\big{(}a_{i}(t)\big{)},\ i=1,\ldots,N \tag{2}\]  
where \(x_{i}\in\mathbb{R},i=1,\ldots,N\) denotes the state of each agent, \(f_{i},g_{i}:\mathbb{R}\rightarrow\mathbb{R},i=1,\ldots,N\) are unknown locally Lipschitz functions and \(d_{i}:\mathbb{R}_{+}\rightarrow\mathbb{R},i=1,\ldots,N\) models the piecewise continuous and bounded external disturbances. The dynamic control input \(u_{i},i=1,\ldots,N\) will be generated through (2), which utilizes nested saturation functions to impose magnitude and rate constraints on the control signals. Hence, by properly designing the control commands \(a_{i}(t),i=1,\ldots,N\) the control input of each agent may satisfy \(|u_{i}|\leq m_{i}\), \(|u_{i}|\leq r_{i},i=1,\ldots,N\) for some \(m_{i},r_{i}>0\) denoting the symmetrical magnitude and rate saturation levels, respectively. It should be noted that the results of the paper are extendable to arbitrary dimensions as dictated in Section 3.3. The communication architecture of the wireless network is denoted by an undirected graph \(\mathcal{G}=(\mathcal{V},\mathcal{E})\), where \(\mathcal{V}=(v_{1},v_{2},\ldots,v_{N})\) represents the vertices, and \(\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}\) symbolizes the edges. Each edge indicates bidirectional information exchange between neighboring agents in the group. The neighborhood of agent \(i\) is denoted by \(\mathcal{N}_{i}\) and the graph's adjacency matrix \(A=[A_{ij}]\in\mathbb{R}^{N\times N}\) assumes \(A_{ij}=1\) if \(j\in\mathcal{N}_{i}\), or \(A_{ij}=0\) otherwise. Furthermore, let the degree matrix \(D=\mathbf{diag}([D_{i}])\in\mathbb{R}^{N\times N}\) be the diagonal matrix with elements that satisfy \(D_{i}=\sum_{j\in\mathcal{N}_{i}}A_{ij}\). Consequently, the graph's Laplacian is obtained by \(L=D-A\in\mathbb{R}^{N\times N}\). Moreover, the state of the leader node, denoted by \(v_{0}\), is given by \(x_{0}:\mathbb{R}_{+}\rightarrow\mathbb{R}\) and it is assumed to be bounded and smooth. Note, that information about the reference trajectory is exclusively provided to a subset of the \(N\) agents. The access of the followers to the leader's state is represented by a diagonal matrix \(B=\mathbf{diag}([b_{1},b_{2},\ldots,b_{N}])\in\mathbb{R}^{N\times N}\). If \(b_{i}=1\), then the \(i\)-th agent receives state information from the leader node; conversely, if \(b_{i}=0\), then the agent has access to the leader's state information. Consequently, we define the augmented graph as \(\overline{\mathcal{V}}=(\overline{\mathcal{V}},\overline{\mathcal{E}})\), with \(\overline{\mathcal{V}}=\mathcal{V}\bigcup\{v_{0}\}\) and \(\overline{\mathcal{E}}=\mathcal{E}\bigcup\{(v_{i},v_{0}):b_{i}=1\}\subseteq \overline{\mathcal{V}}\times\overline{\mathcal{V}}\), along with the augmented set of neighbors \(\overline{\mathcal{N}}_{i}=\{v_{j}:(v_{i},v_{j})\in\overline{\mathcal{E}}\}, i=1,2,\ldots,N\). We now formulate the formation control problem for the aforementioned MAS subject to limited actuation capacity, encompassing adaptive performance regarding transient and steady-state response that is confronted in this paper. Our principal goal is to devise a decentralized control strategy for the following agents of the system, assuming unknown system dynamics and external disturbances. This work aims to establish and maintain a constantly feasible formation, outlined by the desired relative offsets \(d_{ij},j\in\mathcal{N}_{i},\ i=1,\ldots,N\), with adaptive performance characteristics.