# Compensation Observer-Based Adaptive Output Feedback Control for Multi-Agent Systems
## 2 Establishing the Problem Model
### Controller Design
\begin{array}{ll}\dot{\mathbf{\dot{\alpha}}}_{1}&=\frac{\partial\alpha_{1}}{ \partial\theta}\dot{\hat{\theta}}+\frac{\partial\alpha_{1}}{\partial\dot{p}}\dot {\hat{p}}+\frac{\partial\alpha_{1}}{\partial y_{i}}\dot{y}_{r}+\frac{\partial \alpha_{1}}{\partial y_{i}}\dot{y}_{r}+\frac{\partial\alpha_{1}}{\partial t_{ i}}\dot{l}_{i}+\frac{\partial\alpha_{1}}{\partial\varphi_{0,1,j}}\dot{\varphi}_{0,1,j}+ \frac{\partial\alpha_{1}}{\partial x_{02j}}\dot{\dot{\epsilon}}_{02j}\\ &+\frac{\partial\alpha_{1}}{\partial\dot{\epsilon}_{12j}}\dot{\dot{\epsilon}}_{ 2j}+\frac{\partial\alpha_{1}}{\partial\dot{\epsilon}}\Gamma+\frac{\partial \alpha_{1}}{\partial y_{i}}\left(\varphi_{0,1,j}(y_{i})+\varepsilon_{02j}+ \varepsilon_{2j}+\Theta_{i}\Gamma\right)\end{array}  \tag{40}\]  
In addition, we define the expression of the following parameters:  
\[ \begin{array}{ll}\mathcal{B}_{2}&=-l_{i}l_{i}q_{i}\varepsilon_{1j}+F_{2,j}(y _{i})\varepsilon_{s2j}-\frac{\partial\alpha_{1}}{\partial y_{r}}\dot{y}_{r}- \frac{\partial\alpha_{1}}{\partial y_{r}}\dot{y}_{r}-\frac{\partial\alpha_{1 }}{\partial y_{r}}\dot{\varphi}_{r}-\frac{\partial\alpha_{1}}{\partial t_{i}} \dot{l}_{i}-\frac{\partial\alpha_{1}}{\partial\varphi_{0,1,j}}\dot{\varphi}_{ 0,1,j}\\ &-\frac{\partial\alpha_{1}}{\partial\varepsilon_{02j}}\dot{\epsilon}_{02j}- \frac{\partial\alpha_{1}}{\partial\Gamma}\left(\dot{\varepsilon}_{k2j}+\dot{ \varepsilon}_{22j}+\dot{\varphi}_{k2j}\right)-\frac{\partial\alpha_{1}}{ \partial y}\left(\varphi_{0,1,j}(y_{i})+\varepsilon_{02j}\right)-\frac{ \partial\alpha_{1}}{\partial\dot{p}}\dot{\hat{p}}\\ \end{array}  \tag{41}\]  
Then, we can obtain the following expression:  
\[\dot{V}_{2j}=\dot{V}_{1j}+e_{2j}\bigg{(}e_{3j}+\mathbf{\alpha}_{2}+\mathcal{B}_{2 }-\frac{\partial\alpha_{1}}{\partial\dot{\theta}}\dot{\theta}-\frac{\partial \alpha_{1}}{\partial y_{i}}\left(e_{2j}+\Theta_{i}\Gamma\right)\bigg{)}.\]  
At this time, we use inequality Theorem 1 again:  
\[\left\{ \begin{array}{ll}\frac{\partial\alpha_{1}}{\partial y_{i}}\dot{c}_{ 2j}e_{2j}\leq\rho l_{i}^{1+2\Delta}\left(\frac{\partial\alpha_{1}}{\partial y _{i}}\right)^{2}e_{2j}^{2}+\frac{1}{4\rho}l_{i}\bar{\varepsilon}_{2j}{}^{2} \\ \frac{\partial\alpha_{1}}{\partial y_{i}}\dot{\Theta}_{j}\Gamma e_{2j}\leq \frac{\rho}{2}\left(\frac{\partial\alpha_{1}}{\partial y_{i}}\right)^{2}\theta \parallel\Gamma\parallel^{2}e_{2j}{}^{2}+\frac{1}{2\rho}\end{array} \right.. \tag{42}\]  
This theorem is then substituted into the Lyapunov equation, and the following parameters are defined:  
\[\left\{ \begin{array}{ll}\Xi_{1}=\gamma_{\theta}\left(\frac{\rho}{2}\parallel \Gamma\parallel^{2}e_{1j}{}^{2}\right)\\ \Xi_{2}=\Xi_{1}+\gamma_{\theta}e\frac{\rho}{2}\left(\frac{\partial\alpha_{1}}{ \partial y_{i}}\right)^{2}\parallel\Gamma\parallel^{2}e_{2j}+\gamma_{\theta}e _{1j}e_{2j}\end{array} \right.. \tag{43}\]  
The expression of the auxiliary controller can be obtained with Formula (44):  
\[ \begin{array}{ll}\mathbf{\alpha}_{2}&=-c_{2j}e_{2j}-\mathcal{B}_{2}-\rho l_{1}^{ 1+2\Delta}\left(\frac{\partial\alpha_{1}}{\partial y_{i}}\right)^{2}e_{2j}- \hat{\theta}\epsilon_{1j}-\frac{\rho}{2}\left(\frac{\partial\alpha_{1}}{ \partial y_{i}}\right)^{2}\hat{\theta}\parallel\Gamma\parallel^{2}e_{2j}\\ &+\frac{\partial\alpha_{1}}{\partial\theta}\left(\Xi_{2}-\gamma_{\theta}\gamma _{\tilde{\theta}}\hat{\theta}\right)\end{array}  \tag{44}\]  
where \(\gamma_{\tilde{\theta}}\) is positive design parameter. Substituting Formula (44) into (39), we can obtain the expression of the Lyapunov equation as  
\[ \begin{array}{ll}\dot{V}_{2j}&\leq-c_{2j}e_{2j}^{2}-c_{1j}e_{1j}^{2}+e_{2j}e _{3j}+\frac{2}{2\rho}+\frac{2}{4\rho}l_{i}\bar{\varepsilon}_{2j}^{2}-\frac{1} {7\rho}\tilde{\theta}\bigg{(}\dot{\hat{\theta}}-\Xi_{2}\bigg{)}\\ &-\frac{b\bar{\varepsilon}\alpha_{1}}{\gamma_{\rho}}\tilde{p}\Big{(}\dot{\hat{ p}}+\bar{\alpha}_{1}e_{1j}\Big{)}-e_{2j}\frac{\partial\alpha_{1}}{\partial\dot{ \theta}}\bigg{(}\dot{\hat{\theta}}-\Xi_{2}+\gamma_{\theta}\gamma_{\tilde{ \theta}}\hat{\theta}\bigg{)}+\dot{V}_{\bar{\varepsilon}_{ij}}\end{array}