# Formation Control of Nonlinear Multi-Agent Systems with Nested Input Saturation
## Main Results
### Decentralized Controller Design  
In this section, we present an approximation-free decentralized dynamic control scheme that guarantees \(|e_{i}(t)|<\rho_{i}(t)\), \(i=1,\ldots,N\) and appropriately adapts the magnitude of \(\rho_{i}(t)\) to incorporate both the output (performance) and input (actuation) constraints. This eventually leads to the solution of the robust formation control problem with adaptive performance for the input-constrained MAS. To account for system constraints, we introduce a smooth saturation function \(\sigma(\chi,\bar{\sigma}):\mathbb{R}\rightarrow[-\bar{\sigma},\bar{\sigma}]\), where \(\bar{\sigma}>0\) represents the saturation level. Within this context, we select the saturation function as:  
\[\sigma(\chi,\bar{\sigma})=\left\{ \begin{array}{ll}\chi&\text{if }|\chi|<\bar{ \sigma}-\beta\\ p(\chi)&\text{if }|\chi|\in[\bar{\sigma}-\beta,\bar{\sigma}+\beta]\\ s_{\bar{\chi}}\bar{\sigma}&\text{if }|\chi|>\bar{\sigma}+\beta\end{array} \right.\]  
where:  
\[p(\chi)=-\frac{1}{4\beta}\Big{(}\chi^{2}-2s_{\chi}(\bar{\sigma}+\beta)\chi+(s _{\chi}\bar{\sigma}-s_{\chi}\beta)^{2}\Big{)}\]  
with \(s_{\chi}\) denoting the sign of \(\chi\) and \(\beta=10^{-6}\) serves as a small smoothing parameter. The design methodology of the proposed controller is elaborated as follows:  
**Step 1. ** The reference control signal is given by:  
\[u_{d_{i}}(t)=-k_{i,1}T\bigg{(}\frac{e_{i}(t)}{\rho_{i,1}(t)}\bigg{)},\;i=1, \ldots,N \tag{6}\]  
with \(k_{i,1}>0\) and \(T:(-1,1)\rightarrow(-\infty,\infty)\) denoting a smooth error transformation function. In this work, we select the mapping \(T(\chi)=\frac{1}{2}\ln\Bigl{(}\frac{1+\chi}{1-\chi}\Bigr{)}\). Owing to the definition of \(T\) the initial value of the performance \(\rho_{i,1}(t)\) should be selected to satisfy \(|e_{i}(0)|<\rho_{1}(0)\). **Step 2. ** The adaptive performance law that incorporates the output performance requirements and the input magnitude saturation is obtained by:  
\[\dot{\rho}_{i,1}=\bigg{[}\frac{\sigma(u_{d_{i}}(t),m_{i})-u_{d_{i}}(t)}{e_{i} (t)}-\lambda_{i,1}\bigg{(}1-\frac{\rho_{i,1}^{\infty}}{\rho_{i,1}(t)}\bigg{)} \bigg{]}\rho_{i,1}(t),\;i=1,\ldots,N \tag{7}\]  
where \(\lambda_{i,1}\), \(\rho_{i,1}^{\infty}>0\) correspond to the desired convergence rate and the maximum allowable steady-state error, respectively. **Step 3. ** The reference control rate signal is given by:  
\[a(t)=-k_{i,2}T\bigg{(}\frac{u_{i}(t)-\sigma(u_{d_{i}}(t),m_{i})}{\rho_{i,2}(t )}\bigg{)},\;i=1,\ldots,N \tag{8}\]  
with \(k_{i,2}>0\). **Step 4. ** Similar to (7), the adaptive performance law that encompasses rate saturation on the control signal is obtained by:  
\[\dot{\rho}_{i,2}=\bigg{[}\frac{\sigma(a_{i}(t),r_{i})-a_{i}(t)}{u_{i}(t)-\sigma (u_{d_{i}}(t),m_{i})}-\lambda_{i,2}\bigg{(}1-\frac{\rho_{i,2}^{\infty}}{\rho_ {i,2}(t)}\bigg{)}\bigg{]}\rho_{i,2}(t),\;i=1,\ldots,N \tag{9}\]  
with \(\lambda_{i,2}\), \(\rho_{i,2}^{\infty}>0\) and \(|u_{i}(t)-\sigma(u_{d_{i}}(t),m_{i})|<\rho_{2}(0)\). Thus, the dynamic decentralized controller, subjected to both magnitude and slew-rate saturation, is obtained by:  
\[\dot{u}_{i}=\sigma(a_{i}(t),r_{i}),\;u_{i}(0)\in[-m_{i},m_{i}] \tag{10}\]for \(i=1,\ldots,N\). **Remark 3**. : _Both adaptive laws (7) and (9) consist of two components. The second term, negative in nature, incorporates the predefined performance attributes and represents the conventional performance function \(\rho(t)=(\rho(0)-\rho_{\infty})\exp{(-\lambda t)}+\rho_{\infty}\). Conversely, the first term, non-negative and inactive under normal plant operation, i.e., when the control signal is away from saturation, triggers when control signals \(u_{d_{i}}(t),a_{i}(t),\ i=1,\ldots,N\) reach their saturation level. Its activation widens the corresponding performance boundaries based on the difference between the desired and saturated control signals, ensuring the boundedness of all loop signals.