# Compensation Observer-Based Adaptive Output Feedback Control for Multi-Agent Systems
## 2 Establishing the Problem Model
### Controller Design  
In this section, a controller is constructed by utilizing the adaptive backstepping method with a gain compensation observer. Subsystems are recursively designed. Step 1: \(i=1\)  
The tracking error is defined as \(e_{1j}=y_{1}-y_{r}\). The origin of the phrase is  
\[\dot{e}_{1j}=x_{2j}+\varphi_{0,1,j}(y_{i})+\sum_{k=1}^{r_{i}}\varphi_{k,1,j}(y_ {i})a_{k,1,j}+F_{1,j}(y_{i})x_{1j}-\dot{y}_{r}. \tag{30}\]  
The following changes were made:  
\[\Theta_{i}=\left[ \begin{array}{ccccccccc}a_{1,1,j}&a_{2,1,j}&\cdots&a_{r_{1 },1,j}&a_{1,2,j}&a_{2,2,j}&\cdots&a_{r_{2},2,j}&b_{i}\end{array} \right] \tag{31}\]  
\[\Gamma=\left[\varphi_{1,1,j}(y_{i})&\varphi_{2,1,j}(y_{i})&\cdots&\varphi_{r_{ 1},1,j}(y_{i})&\varepsilon_{12j}&\varepsilon_{22j}&\cdots&\varepsilon_{r_{2}2j }&\varepsilon_{22j}\end{array}\right]^{T}. \tag{32}\]  
Substituting Equations (31) and (32) into (30) yields  
\[\dot{e}_{1j}=\varepsilon_{0j}+b_{i}\sigma_{i}\varepsilon_{s2j}+\varepsilon_{2 j}+\varphi_{0,1,j}(y_{i})+\Theta_{i}\Gamma+F_{1,j}(y_{i})x_{1j}-\dot{y}_{r} \tag{33}\]  
where \(\gamma_{\theta}\) and \(\gamma_{p}\) are positive design parameters. Then, we synchronously set \(\theta=\max\left\{\parallel\Theta_{i}\parallel^{2},\parallel\sigma_{i} \parallel^{2}\right\}\). Here, \(\theta\) is an unknown parameter and \(\rho\) is a positive design parameter. Next, we apply Formula (34):  
\[\varepsilon_{sij}=e_{ij}+\mathbf{\alpha}_{i-1}. \tag{34}\]  
At the same time, we set the auxiliary controller as  
\[\mathbf{\alpha}_{1}=\hat{p}\mathbf{\alpha}_{1} \tag{35}\]  
\[\mathbf{\alpha}_{1}=-c_{1j}e_{1j}-\varepsilon_{(\alpha j)}-\varphi_{0,1,j}(y_{i}) +\dot{y}_{r}-\frac{\rho}{2}\hat{\theta}\parallel\Gamma\parallel^{2}e_{1j}- \rho l_{i}^{\,1+2\Delta}e_{1j} \tag{36}\]  
where \(c_{1j}\) is the known normal number; \(\hat{\theta}\) is the estimated value; and \(\widetilde{\theta}=\theta-\hat{\theta}\), \(\hat{\theta}\) is the estimated value of the unknown parameter \(\theta\). Additionally, \(\widetilde{p}=p-\hat{p}\), \(\hat{p}\) are estimates of unknown parameter \(p\). According to inequality Theorem 1, the two parameters are scaled as follows:  
\[\left\{ \begin{array}{l}\Theta_{i}\Gamma e_{1j}\leq\frac{\rho}{4}\parallel \Gamma\parallel^{2}e_{1j}^{2}+\frac{1}{2\rho}\\ e_{1j}\varepsilon_{2j}\leq\rho l_{i}^{1+2\Delta}e_{1j}^{2}+\frac{1}{4\rho}l \varepsilon_{2j}^{2}\end{array} \right.. \tag{37}\]  
The Lyapunov equation can be changed into the following shape by defining the new parameter \(\Xi_{1}\). The expression of this function is defined below:  
\[\dot{V}_{1}\leq-c_{1j}e_{1j}^{\,2}+\hat{\theta}e_{1j}e_{2j}+\frac{1}{2\rho}+ \frac{1}{4\rho}l_{i}\overline{\varepsilon_{2j}}^{\,2}-\frac{1}{\gamma_{\theta} }\widetilde{\theta}\left(\hat{\theta}-\Xi_{1}\right)-\frac{b_{i}\sigma_{i}}{ \gamma_{p}}\widetilde{p}\left(\hat{p}+\overline{\pi}_{1}e_{1j}\right)+\dot{V }_{\overline{\varepsilon_{ij}}} \tag{38}\]  
Step 2: \(i=2\)  
As above, we set the Lyapunov function under the condition of \(i=2\) and obtain the following expression according to Formula (34):  
\[ \begin{array}{rl}\dot{V}_{2j}&=\dot{V}_{1j}+e_{2j}\dot{e}_{2j}\\ &=\dot{V}_{1j}+e_{2j}(\dot{e}_{s2j}-\dot{\mathbf{\alpha}}_{1})\dot{V}_{1j}+e_{2j}( \varepsilon_{s3j}-l_{i}L_{i}q_{i}\varepsilon_{s1j}+F_{2,j}(y_{i})\varepsilon_{ s2j}-\dot{\mathbf{\alpha}}_{1})\end{array}  \tag{39}\]One can then derive the following equation from the auxiliary controller:  
\[