## 2 Preliminaries
### Numerical Models and Input Saturation  
The dynamic model of agents is given by  
\[\left\{ \begin{array}{rl}\dot{\mathbf{X}}_{t}^{i}&=f\big{(}\mathbf{q}_{t}^{i}\big{)} \\ f\big{(}\mathbf{q}_{t}^{i}\big{)}&=\big{[}v_{t}^{i}cos(\psi_{t}^{i}),v_{t}^{i}sin( \psi_{t}^{i})\big{]}\\ \dot{\mathbf{q}}_{t}^{i}&=\mathbf{u}_{t}^{i}\\ \dot{\mathbf{u}}_{t}^{i}&=\big{[}a_{t}^{i},\omega_{t}^{i}\big{]}^{\text{T}}\end{array} \right. \tag{1}\]  
where \(\mathbf{X}_{t}^{i}=\big{[}x_{t}^{i},\ y_{t}^{i}\big{]}\) denotes positions, \(\mathbf{q}_{t}^{i}=\big{[}v_{t}^{i},\psi_{t}^{i}\big{]}^{\text{T}}\) represents the velocity expressed with the speed \(v_{t}^{i}\geq 0\) and the direction \(\psi_{t}^{i}\in[-2\pi,2\pi]\), \(\mathbf{u}_{t}^{i}=\big{[}a_{t}^{i},\omega_{t}^{i}\big{]}^{\text{T}}\) indicates the accelerations that will be designed for formation control. The model is nonlinear while considering the maneuvering. Thus, the estimation of \(\mathbf{u}_{t}^{i}\) is vital. In this paper, CA is applied for controller design, and the distributed feedback law can be indicated as:  
\[\mathbf{u}_{*}^{i}=\sum a_{ij}\Big{(}\big{(}\mathbf{k}_{ij}\mathbf{F}_{j}\big{(}\mathbf{\Theta }_{j}\big{)}-\mathbf{k}_{ii}\mathbf{F}_{i}\big{(}\mathbf{\Theta}_{i}\big{)}\big{)}+\mathbf{w}_{ ij}\mathbf{p}_{ij}\Big{)} \tag{2}\]  
where \(\mathbf{u}_{*}^{i}\) indicates the desired acceleration, \(\mathbf{F}_{i}\) represents the processing functions of interacted information, \(\mathbf{\Theta}_{i}=\Big{(}\mathbf{X}_{t}^{i},\mathbf{q}_{t}^{i},\mathbf{u}_{t}^{i}\Big{)}\) denotes the interacted information, \(\mathbf{p}_{ij}\in\mathbb{R}^{n\times 1}\) indicates the expected relative position, \(\mathbf{k}_{ij}\in\mathbb{R}^{n\times m}\), \(\mathbf{w}_{ij}\in\mathbb{R}^{n}\) represent the weight matrix, \(n\) indicates the number of the controlled variables, and \(m\) represents the dimension of \(\mathbf{F}_{i}(\mathbf{\Theta}_{i})\). Combining \(\mathbf{\mathcal{L}}\), Equation (2) can be rewritten as:  
\[\left\{ \begin{array}{l}\mathbf{u}_{*}=\big{(}\mathbf{K}\circ\big{(}\mathbf{\mathcal{L}} \otimes\mathbf{I}_{n\times m}\big{)}\big{)}\mathbf{F}+\big{(}\mathbf{P}\circ(\mathbf{A}\otimes \mathbf{1}_{n\times 1})\big{)}\mathbf{1}_{nN\times 1}\\ \mathbf{F}=\Big{[}\mathbf{F}_{1}(\mathbf{\Theta}_{1})^{\text{T}},\cdots\mathbf{F}_{i}(\mathbf{\Theta }_{i})^{\text{T}},\cdots,\mathbf{F}_{N}(\mathbf{\Theta}_{N})^{\text{T}}\Big{]}^{\text{T} }\\ \mathbf{P}=\Big{[}\mathbf{p}_{1}^{\text{T}},\mathbf{p}_{2}^{\text{T}},\cdots\mathbf{p}_{i}^{ \text{T}},\cdots,\mathbf{p}_{N}^{\text{T}}\Big{]}^{\text{T}}\end{array} \right. \tag{3}\]  
where \(\mathbf{K}=\big{[}\mathbf{k}_{ij}\big{]}_{Nn}^{Nm}\), \(\mathbf{p}_{i}=\Big{[}\mathbf{w}_{ij}\mathbf{p}_{ij}\Big{]}_{n}^{N}\), \(\mathbf{I}_{n\times m}=\big{[}\mathbf{I}_{n}\big{]}_{n}^{m}\). Where \(\mathbf{I}_{n}\) represents the identity matrix, \(1_{n\times 1}\) and \(1_{nN\times 1}\) are constant matrices with value 1. Considering the input saturation, the desired output \(\mathbf{u}_{*}\) in (3) might not be feasible, and agents cannot realize the desired accelerations provided by the controller. Generally, constraints are placed behind the controller, thus the real inputs \(\mathbf{u}_{f}\) of the actuators are restricted. Defining the velocity boundary as \(\mathbf{q}_{lim}\), then the constraints can be indicated with (4), where \(k\) denotes the \(k\)-th element. \[\left\{ \begin{array}{l}\mathbf{u}_{f}^{i}(k)=\mathbf{u}_{*}^{i}(k),\left|\mathbf{u}_{*}^{i} (k)+\mathbf{q}_{l}^{i}(k)\right|\leq\mathbf{q}_{lim}^{i}(k)\\ \mathbf{u}_{f}^{i}(k)=\mathbf{u}_{*}^{i}(k),\mathbf{u}_{*}^{i}(k)+\mathbf{q}_{l}^{i}(k)>\mathbf{q} _{lim}^{i}(k)\\ \mathbf{u}_{f}^{i}(k)=-\mathbf{u}_{*}^{i}(k),\mathbf{u}_{*}^{i}(k)+\mathbf{q}_{t}^{i}(k)<-\mathbf{ q}_{lim}^{i}(k)\end{array} \right. \tag{4}\]  
Constraints in (4) help restrict the outputs, but the parameters of the controller cannot be adjusted timely without adaptive rules, which will hinder the controller from making correct decisions.