# Compensation Observer-Based Adaptive Output Feedback Control for Multi-Agent Systems
## 4 Stability Analysis  
This section is dedicated to analyzing the stability and tracking performance of the proposed control scheme. The globally stable operation and precise tracking capabilities of the closed-loop, decentralized, adaptive control system are ensured by incorporating a specially designed high-gain compensation observer. **Remark 7**. : _Consider a closed-loop system composed of the controlled system (7), the hysteresis signal (11), the compensation filter (14), the update law (24), and Equations (31) and (33). Next, select appropriate values for the gain's positive design parameters \(c_{i}\), \(\gamma_{\bar{\theta}}\), \(\Lambda_{i}\), \(l_{0}\), and \(\zeta(y_{i})\). The tracking error of each subsystem within the closed-loop decentralized control system converges to an arbitrarily small residual set, while all signals remain globally uniformly bounded._  
**Remark 8**. : _Notably, the proposed approach can effectively handle a wider range of interference signals, and system (7) encompasses more interference factors than systems in the existing literature. Thus, this system represents a broader category of multi-agent systems. The following are the key certification elements._  
To summarize, let the Lyapunov function of the whole be \(V_{nj}=V_{(n-1)j}+\frac{1}{2}\overline{e}_{nj}^{2}\), where the value is \([2,\cdots,n]\). Next, use [PERSON]'s inequality, inequality Theorem 1, to extend and reduce Formula (55), as shown below:  
\[\gamma_{\bar{\theta}}\widetilde{\theta}\widetilde{\theta}\leq\frac{\gamma_{\bar{ \theta}}}{2}\theta^{2}-\frac{\gamma_{\bar{\theta}}}{2}\widetilde{\theta}^{2}. \tag{54}\]  
Because of the universal existence of \(\frac{n}{4\rho}l_{i}\varepsilon_{2j}{}^{2}\leq\frac{n}{4\rho}l_{i}\| \overline{\varepsilon}_{ij}\|^{2}\), parameter \(\rho=n\) and the following equations are designed:  
\[LR=\frac{\dot{l}_{i}}{l_{i}}\sigma_{1}+l_{i}-\frac{1}{4}l_{i}-2\|P\|\|\zeta(y_ {i})\| \tag{55}\]  
\[\overset{\circ}{M}=M+\frac{1}{2}+\frac{\gamma_{\bar{\theta}}}{2}\theta^{2}. \tag{56}\]  
Combining (55)-(58), we can obtain Formula (59):  
\[\dot{V}_{nj}\leq-\overset{\circ}{\sum}_{i=1}^{n}c_{i}^{2}e_{i}^{2}-\frac{ \gamma_{\bar{\theta}}}{2}\widetilde{\theta}^{2}-(LR)\overline{\varepsilon}_{ ij}{}^{2}+\overset{\circ}{M}. \tag{57}\]  
Then, set the parameters in dynamic gain \(\dot{l}_{i}=-\Lambda_{i}l_{i}^{2}+\Lambda_{i}l_{i}l_{0}\sigma_{2}+l_{i}\zeta(y _{i})\) as follows:  
\[\sigma_{1}=2\|P\|,\sigma_{2}=\sigma_{1}+0.2,\Lambda_{i}=\frac{1}{2\sigma_{2} },l_{0}=1 \tag{58}\]  
where \(\|P\|\) is set to 0.5 in the simulation. The following outcomes can be obtained using this setting technique, which effectively reduces the interference of parameter variables:  
\[LR=\bigg{(}\frac{3}{4}-\frac{\sigma_{1}}{2(\sigma_{1}+0.2)}\bigg{)}l_{i}+\|P\| \|l_{0}\|. \tag{59}\]  
For Formula (59), we selected the design parameters to meet the standard requirements:  
\[c_{i}\geq\Omega\gamma_{\bar{\theta}}\geq\frac{2\Omega}{\gamma_{\theta}}LR\geq\Omega \tag{60}\]  
where \(\Omega\) is the positive term parameter designed in this study. Combining this formula with Formula (59), we can obtain  
\[\dot{V}_{nj}\leq-2\Omega V_{nj}+\overset{\circ}{M}. \tag{61}\]  
Next, solve the integral inequality:  
\[0\leq V_{nj}(t)\leq\frac{\overset{\circ}{M}}{2\Omega}+\left(V_{nj}(0)-\frac{ \overset{\circ}{M}}{2\Omega}\right)e^{-2\Omega T}. \tag{62}\]  
In addition, when \(t\rightarrow\infty\), the limit value can be obtained:  
\[\underset{t\rightarrow\infty}{\lim}V_{nj}(t)=\frac{\overset{\circ}{M}}{2 \Omega}. \tag{63}\]Here, the parameters \(\epsilon_{ij}\), \(e_{ij}\), \(\bar{\theta}\), and \(\widetilde{p}\) are bounded. Through the design process of Formulas (22)-(34), we can obtain expressions of the following main variables:  
\[ \begin{array}{l}\dot{\epsilon}_{ij}=x_{ij}-\dot{\pi}_{ij}\\ e_{1j}=y_{1}-y_{r}\\ e_{2j}=\varepsilon_{s2j}-\alpha_{1}\\ \vdots\\ e_{ij}=\varepsilon_{sij}-\alpha_{i-1}\end{array} .