# Formation Control of Nonlinear Multi-Agent Systems with Nested Input Saturation
## Main Results
### Decentralized Controller Design
By differentiating \(L\) with respect to time, we get:  
\[g(\mathbf{x}) =\mathbf{diag}([g_{1}(x_{1}),\ldots,g_{N}(x_{N})])\]6 \[g(\mathbf{x}) =\mathbf{diag}([g_{1}(x_{1}),\ldots,g_{N}(x_{N})])\]7 \[g(\mathbf{x}) =\mathbf{diag}([g_{1}(x_{1}),\ldots,g_{N}(x_{N})])\]8  
with \(\
abla_{\xi_{2}}\epsilon_{2}=\textbf{diag}\bigg{(}\bigg{[}\frac{1}{1-\xi_{2,1 }^{2}},\ldots,\frac{1}{1-\xi_{2,N}^{2}}\bigg{]}\bigg{)}\). Note, that \(\bar{u}(\xi_{1})\) is bounded due to the magnitude saturation, decoupling the boundedness of \(\xi_{1}\) and \(\xi_{2}\). Next owing to the continuity of the unknown dynamics of (1), the boundedness of the external disturbances \(d(t)\) and \(\textbf{x}_{0}\), the saturation limits \(m_{i},r_{i},\ i=1,\ldots,N\) as well as exploiting Assumption 2 which implies the boundedness of \(\textbf{x},\rho_{1}(t),\rho_{2}(t)\)[30] we conclude the existence of some positive constants \(F_{1},F_{2}\) such that:  
\[g(\mathbf{x}) =\mathbf{diag}([g_{1}(x_{1}),\ldots,g_{N}(x_{N})])\]9 \[\tilde{u}(\xi_{1}) =[u_{1}(t)-\sigma(u_{d_{1}}(t),m_{1}),\ldots,u_{N}(t)-\sigma(u_{ d_{N}}(t),m_{1})]^{T}\]0 \[\tilde{u}(\xi_{1}) =[u_{1}(t)-\sigma(u_{d_{1}}(t),m_{1}),\ldots,u_{N}(t)-\sigma(u_{ d_{N}}(t),m_{1})]^{T}\]1  
Additionally, the matrices \(\
abla_{\xi_{1}}\epsilon_{1},\ (\rho_{1}(t))^{-1},\ \
abla_{\xi_{2}}\epsilon_{2},\ (\rho_{2}(t))^{-1},\ (L+B)\) are positive definite thus \(\dot{L}\) becomes negative whenever \(\|\epsilon_{1}\|>\frac{F_{1}}{\min\{k_{1}\}}\) and \(\|\epsilon_{2}\|>\frac{F_{2}}{\min\{k_{2}\}}\). Moreover, considering the initial fulfillment of performance constraints, the initial values of \(\epsilon_{1},\ \epsilon_{2}\) are well-defined. Thus, the transformed errors \(\epsilon_{1},\ \epsilon_{2}\) are uniformly ultimately bounded. Consequently, the performance criteria, as dictated by inequality (11), are consistently met over time, ensuring adaptive performance specifications. Furthermore, all signals within the closed-loop system remain bounded, thereby concluding the proof. Theorem 2.1 ensures the boundedness of the neighborhood error **e** by establishing a compromise between output specifications and input limitations. However, in the absence of saturation, the system rapidly reverts to its predetermined performance characteristics exponentially. Therefore, it can be deduced that when amplitude saturation is inactive (i.e., \(\dot{\rho}_{i,1}(t)=-\lambda_{i,1}(\rho_{i,1}(t)-\rho_{i,1}^{\infty})\)), the disagreement formation variable **q** converges to the compact set \(\mathcal{Q}_{i}^{\infty}:=\Big{\{}\dot{q}_{i}\in\mathbb{R}:|q_{i}|\leq\frac{ \rho_{i}^{\infty}}{\lambda_{\min}(L+B)}\Big{\}}_{i,i}\ i=1,\ldots,N\) with an exponential rate of at least \(\exp(-\lambda_{i,1}t)\). Consequently, in this scenario, the performance bound on \(q_{i}(t)\)\(i=1,\ldots,N\), i.e., \(\bar{q}_{i}\), can be imposed by selecting \(\rho_{i}^{\infty}=\hat{\lambda}_{\min}(L+B)\bar{q}_{i}\), \(i=1,\ldots,N\), where \(\hat{\lambda}_{\min}(L+B)\) is the result of a decentralized connectivity estimation algorithm [34]. Note, that the performance specifications for the closed-loop MAS are determined by the evolution of \(\rho_{i,1}(t)\)\(i=1,\ldots,N\). Nevertheless, in the presence of magnitude and rate saturation, the performance constraints are inevitably relaxed to ensure the boundedness of the closed-loop signals. The extent of relaxation is contingent on the control gains \(k_{i,1},\ i=1,\ldots,N\). Moreover, the fluctuation of the performance envelope relies on the control tracking error \(u_{i}(t)-\sigma(u_{d_{i}}(t),m_{i})\), as articulated in Theorem 1. Therefore, a quick convergence of \(u_{i}(t)\rightarrow\sigma(u_{d_{i}}(t),m_{i})\) is desirable and can be achieved by opting for relatively higher values for \(\lambda_{i,2}\). However, large gains \(k_{i,j},\ i=1,\ldots,N\), \(j=1,2\) might lead to an excessive relaxation of the performance functions (7) and (9), resulting in an unnecessary deterioration in tracking performance owing to faster saturation of (6) and (10).