# Formation Control of Nonlinear Multi-Agent Systems with Nested Input Saturation
## Main Results
### Decentralized Controller Design
Notably, as the second term in (7) and (9) becomes dominant, the relaxation of performance boundaries halts, and once the saturation ceases then the performance envelope retrieves its predefined form._  
**Remark 4**. : _The proposed control protocol operates in a decentralized manner, where each agent solely relies on local relative state information within its neighborhood, expressed in a common frame, to compute its individual control signal. Notably, this protocol does not utilize any knowledge of system dynamics or disturbances. Moreover, it does not employ approximation structures such as neural networks or fuzzy systems to acquire such insights. Moreover, generating the control signal does not necessitate intricate calculations, simplifying its implementation. Thus, the proposed formation protocol not only remains decentralized but also showcases low structural complexity._  
**Theorem 1**. : _Consider system (1)) obeying Assumptions 1 and 2. The proposed adaptive decentralized control protocol (6)-(10) ensures the boundedness of all loop signals and guarantees:_  
\[|e_{i}(t)|<\rho_{i,1}(t),\ i=1,\ldots,N \tag{11}\]  
_for all \(t\geq 0\)._  
Proof. : Let us first define the normalized tracking error vectors:  
\[\xi_{1} :=\left[\frac{\varepsilon_{1}(t)}{\rho_{1,1}(t)},\ldots,\frac{ \varepsilon_{N}(t)}{\rho_{N,1}(t)}\right]^{T} \tag{12}\] \[\xi_{2} :=\left[\frac{u_{1}(t)-\sigma(u_{d_{1}}(t),m_{1})}{\rho_{1,2}(t)},\ldots,\frac{u_{N}(t)-\sigma(u_{d_{N}}(t),m_{N})}{\rho_{N,2}(t)}\right]^{T} \tag{13}\]  
Notice from (4) and (12) that the state vector can be written as \(\mathbf{x}=(L+B)^{-1}\rho_{1}(t)\xi_{1}(t)+\mathbf{x}_{0}-\mathbf{d}\) with \(\rho_{1}(t)=\mathbf{diag}([\rho_{1,1}(t),\ldots,\rho_{N,1}(t)])\). Differentiating \(\xi_{1}(t)\) with respect to time and substituting (1), (2), (4) and (6)-(10) as well as adding and subtracting \((L+B)g(\mathbf{x})\sigma(u_{d}(t),m)\) we get:  
\[ \begin{split}\xi_{1}=&(\rho_{1}(t))^{-1}\big{[}(L+ B)(f(\mathbf{x})+g(\mathbf{x})\tilde{u}(\xi_{1})+d(t)-\dot{\mathbf{x}}_{0})\\ &+\xi_{1}(t)\lambda_{1}(\rho_{1}(t)-\rho_{1}^{\infty})+((L+B)g( \mathbf{x})-1)\sigma(u_{d_{i}}(t),m_{i})-k_{1}\epsilon_{1}(\xi_{1})\big{]} \end{split}  \tag{14}\]  
where:  
\[f(\mathbf{x}) =[f_{1}(x_{1}),\ldots,f_{N}(x_{N})]^{T}\] \[g(\mathbf{x}) =\mathbf{diag}([g_{1}(x_{1}),\ldots,g_{N}(x_{N})])\] \[\tilde{u}(\xi_{1}) =[u_{1}(t)-\sigma(u_{d_{1}}(t),m_{1}),\ldots,u_{N}(t)-\sigma(u_{ d_{N}}(t),m_{1})]^{T}\] \[\sigma(u_{d}(t),m) =[\sigma(u_{d_{1}}(t),m_{1}),\ldots,\sigma(u_{d_{N}}(t),m_{N})]^{T}\] \[\epsilon_{1}(\xi_{1}) =\left[T\bigg{(}\frac{\epsilon_{1}(t)}{\rho_{1,1}(t)}\bigg{)}, \ldots,T\bigg{(}\frac{\epsilon_{N}(t)}{\rho_{N,1}(t)}\bigg{)}\right]^{T}\] \[\lambda_{1} =\mathbf{diag}([\lambda_{1,1},\ldots,\lambda_{N,1}])\] \[\rho_{1}^{\infty} =\mathbf{diag}([\rho_{1,1}^{\infty},\ldots,\rho_{N,1}^{\infty}])\] \[k_{1} =\mathbf{diag}([k_{1,1},\ldots,k_{N,1}]).\]Following the same reasoning the dynamics of \(\xi_{2}\) may be written as:  
\[\dot{\xi}_{2}= (\rho_{2}(t))^{-1}\big{(}-k_{1}\sigma^{\prime}(u_{d}(t),m)\dot{\xi}_ {1}\
abla_{\xi_{1}}\epsilon_{1}+\xi_{2}(t)\lambda_{2}(\rho_{2}(t)-\rho_{2}^{ \infty})-k_{2}\epsilon_{2}(\xi_{2})\big{)} \tag{15}\]  
where:  
\[\sigma^{\prime}(u_{d}(t),m) =\bigg{[}\frac{d\sigma(u_{d_{1}}(t),m_{1})}{dt},\ldots,\frac{d \sigma(u_{d_{N}}(t),m_{N})}{dt}\bigg{]}^{T}\] \[g(\mathbf{x}) =\mathbf{diag}([g_{1}(x_{1}),\ldots,g_{N}(x_{N})])\]0 \[g(\mathbf{x}) =\mathbf{diag}([g_{1}(x_{1}),\ldots,g_{N}(x_{N})])\]1 \[g(\mathbf{x}) =\mathbf{diag}([g_{1}(x_{1}),\ldots,g_{N}(x_{N})])\]2 \[g(\mathbf{x}) =\mathbf{diag}([g_{1}(x_{1}),\ldots,g_{N}(x_{N})])\]3 \[g(\mathbf{x}) =\mathbf{diag}([g_{1}(x_{1}),\ldots,g_{N}(x_{N})])\]4 \[g(\mathbf{x}) =\mathbf{diag}([g_{1}(x_{1}),\ldots,g_{N}(x_{N})])\]5  
Next consider the positive definite and radially unbounded Lyapunov function candidate \(L=\frac{1}{2}\epsilon_{1}^{T}\epsilon_{1}+\frac{1}{2}\epsilon_{2}^{T}\epsilon _{2}\).