# Formation Control of Nonlinear Multi-Agent Systems with Nested Input Saturation
## Main Results  
### Sufficient Conditions  
Owing to the incorporation of decentralized control protocols reliant on relative state information, each agent's control law should be derived from its neighborhood error feedback:  
\[e_{i}=\sum_{j\in\mathcal{N}_{i}}\left[A_{i,j}(x_{i}-x_{j}+d_{ij})\right]+b_{i }(x_{i}-x_{0}+d_{i0}) \tag{3}\]  
for \(i=1,\ldots,N\) with \(d_{ij},\ j\in\mathcal{N}_{i}\) denoting the relative inter-agent offsets that define the desired swarm formation. In addition, we define the collective neighborhood error vector as \(\mathbf{e}=[e_{1},\ldots,e_{N}]^{T}\in\mathbb{R}^{N}\). Leveraging on the graph topology and through straightforward algebraic manipulations, this vector can be expressed as:  
\[\mathbf{e}=(L+B)(\mathbf{x}-\mathbf{x}_{0}+\mathbf{d}) \tag{4}\]  
where \(\mathbf{x}=[x_{1},\ldots,x_{N}]\in\mathbb{R}^{N}\) represents the collective state vector of the MAS, with \(\mathbf{x}_{0}=[x_{0},\ldots,x_{0}]^{T}\in\mathbb{R}^{N}\) and  
\[\mathbf{d}=(L+B)^{-1}\Bigg{[}\sum_{j\in\mathcal{N}_{1}}\left(A_{1,j}d_{1j} \right)+b_{1}d_{10},\ldots,\sum_{j\in\mathcal{N}_{N}}\left(A_{N,j}d_{Nj} \right)+b_{N}d_{N0}\Bigg{]}^{T}\]  
corresponds to the relative offset of the \(i\)-th agent with respect to the leader within the desired formation configuration. Consequently, the formation is aligned with the leader's state, hence it is achieved when each agent's state \(x_{i}\) with respect to the leader state \(x_{0}\) maintains its respective offset \(\mathbf{d}_{i}\), \(i=1,\ldots,N\). Introducing the disagreement formation variable as \(\mathbf{q}=[q_{1},\ldots,q_{N}]^{T}=\mathbf{x}-\mathbf{x}_{0}-\mathbf{d}\), the formation control objective is met when the disagreement errors \(s_{i}\) approach an arbitrarily small set around the origin for all \(i=1,\ldots,N\). However, \(q_{i}\) represents global quantities and cannot be measured distributedly based on the local intercourse specifications, as they utilize information directly sourced from the leader. However, due to the non-singularity of \(L+B\) as a result of Assumption 1, we arrive at:  
\[\left|\mathbf{q}\right|\leq\frac{|\mathbf{e}|}{\lambda_{\min}(L+B)} \tag{5}\]  
where \(\lambda_{\min}(L+B)\) corresponds to the smallest singular value of \(L+B\).  
Remark 2: _From (5) it is concluded that the error vector \(\mathbf{e}\) can serve as a reliable measure of the formation control problem. Thus, transient and steady-state bounds on the neighborhood errors \(e_{i}\), \(i=1,\ldots,N\) can be directly mapped into constraints on the disagreement formation metrics \(q_{i}\), \(i=1,\ldots,N\). This leads to the conclusion that solving the adaptive performance control problem for all neighborhood errors \(e_{i}\) directly resolves the formation control problem of (1). Specifically, as dictated by (5), ensuring \(|e_{i}(t)|<\rho_{i}(t)\), for all \(t\geq 0\) and \(i=1,\ldots,N\), with \(\rho_{i}(t)\) denoting aperformance function to be designed, explicitly constrain the disagreement errors \(q_{i}\), \(i=1,\ldots,N\) within the compact sets \(\mathcal{Q}_{i}\coloneqq\{q_{i}\in\mathbb{R}:|q_{i}|\leq\frac{\beta_{i}}{\lambda_ {\min}(L+B)}\}\) where \(\tilde{p}_{i}\) denotes an upper bound on \(\rho_{i}(t)\). Yet, \(\lambda_{\min}(L+B)\) remains a global parameter, rendering it impractical for exploitation within decentralized control methodologies to directly enforce specific limits on \(\mathbf{q}\) through Equation (5). To address this challenge, a decentralized estimation algorithm [34] could be initially employed to estimate \(\lambda_{\min}(L+B)\).