# Compensation Observer-Based Adaptive Output Feedback Control for Multi-Agent Systems
## 2 Establishing the Problem Model
### Controller Design
 \tag{45}\]  
Step 3: \(i\in[3,\,z)\). The following formula can be obtained using the same inversion calculation method:  
\[ \begin{array}{ll}\mathbf{\alpha}_{i}&=-c_{ij}e_{ij}-e_{(i-1)j}-\mathcal{B}_{i}- \rho l_{i}^{1+2\Delta}\bigg{(}\frac{\partial\alpha_{i-1}}{\partial y_{i}} \bigg{)}^{2}e_{zj}-\frac{\rho}{2}\bigg{(}\frac{\partial\alpha_{i-1}}{\partial y _{i}}\bigg{)}^{2}\hat{\theta}\parallel\Gamma\parallel^{2}e_{ij}{}^{2}\\ &+\frac{\partial\alpha_{i-1}}{\partial\dot{\theta}}\big{(}\Xi_{i}-\gamma_{ \theta}\gamma_{\tilde{\theta}}\hat{\theta}\big{)}+\gamma_{\theta}\sum\limits_{q= 2}^{i-1}e_{q}\frac{\partial\alpha_{q-1}}{\partial y_{i}}\bigg{[}\frac{\rho}{2} \bigg{(}\frac{\partial\alpha_{i-1}}{\partial y_{i}}\bigg{)}^{2}\parallel \Gamma\parallel^{2}e_{ij}\bigg{]}\end{array}  \tag{46}\]  
\[\Xi_{i}=\Xi_{i-1}+\gamma_{\theta}\frac{\rho}{2}\bigg{(}\frac{\partial\alpha_{1}}{ \partial y_{i}}\bigg{)}^{2}\parallel\Gamma\parallel^{2}e_{ij}{}^{2} \tag{47}\]\[\dot{V}_{ij} \leq-\sum\limits_{a=1}^{i}c_{i}^{2}e_{a}^{2}+e_{(i-1)i}e_{ij}+\frac{i} {2\rho}+\frac{i}{4\rho}l_{i}\overline{\epsilon}_{2j}^{2}-\frac{1}{\gamma_{\theta }}\widetilde{\theta}\Big{(}\dot{\theta}-\Xi_{i}\Big{)}-\frac{b_{i}\sigma_{i}} {\gamma_{p}}\widetilde{p}\Big{(}\dot{\dot{p}}+\overline{\pi}_{1}e_{1j}\Big{)} \tag{48}\] \[-\sum\limits_{b=2}^{i}e_{b}\frac{\partial\alpha_{b-1}}{\partial \dot{\theta}}\bigg{(}\dot{\theta}-\Xi_{b}+\gamma_{\theta}\gamma_{\bar{\theta}} \dot{\theta}\bigg{)}+\dot{V}\overline{\epsilon}_{ij}\] Step 4: \(i\in[z,n]\). This section adds the \(H_{i,j}\) function and hysteresis input environment. The input is defined as follows: \[u_{i}(t)=\frac{1}{H_{i,j}(y_{i})}\Big{(}\alpha_{i}-\varepsilon_{s(i+1)j}\Big{)}. \tag{49}\]  
The expression of \(\alpha_{i}\) is the same as that in Formula (46). Next, we set the Lyapunov function to (50)  
\[V_{ij}=\frac{1}{2}e_{ij}^{2}+V_{(i-1)j}. \tag{50}\]  
The expression can be obtained by substituting the above parameters into the following calculation:  
\[\dot{V}_{nj} \leq-\sum\limits_{i=1}^{n}c_{i}^{2}\overline{e}_{i}^{2}+\frac{n} {2\rho}+\frac{n}{4\rho}l_{i}\overline{\epsilon}_{2j}^{2}-\frac{b_{i}\sigma_{ i}}{\gamma_{p}}\widetilde{p}\Big{(}\dot{\dot{p}}+\overline{\pi}_{1}e_{1j} \Big{)}-\frac{1}{\gamma_{\theta}}\widetilde{\theta}\bigg{(}\dot{\dot{\theta}}- \Xi_{n}\bigg{)} \tag{51}\] \[-\sum\limits_{i=2}^{n}e_{ij}\frac{\partial\alpha_{i}}{\partial \dot{\theta}}\bigg{(}\dot{\theta}-\Xi_{n}+\gamma_{\theta}\gamma_{\bar{\theta} }\dot{\theta}\bigg{)}+\dot{V}\overline{\epsilon}_{ij}\]  
Let the parameter value be  
\[\dot{\dot{\theta}} =\Xi_{n}-\gamma_{\theta}\gamma_{\bar{\theta}}\dot{\theta}\ . \tag{52}\] \[\dot{\dot{p}} =-\overline{\pi}_{1}e_{1j}\]  
In combination with Equation (51), the final expression of the Lyapunov equation can be obtained as follows:  
\[\dot{V}_{nj}\leq-\sum\limits_{i=1}^{n}c_{i}^{2}\overline{e}_{i}^{2}+\frac{n} {2\rho}+\frac{n}{4\rho}l_{i}\overline{\epsilon}_{2j}^{2}+\gamma_{\bar{\theta} }\widetilde{\theta}\dot{\theta}-\bigg{(}\frac{\dot{l}_{i}}{l_{i}}\sigma_{1}+l_ {i}-2\|P\|\zeta(y_{i})\|\bigg{)}\overline{\epsilon}_{ij}^{2}+M. \tag{53}\]