# Compensation Observer-Based Adaptive Output Feedback Control for Multi-Agent Systems
## 5 Simulation Examples and a System Analysis of Control Strategies
### Pratical Example  
In this section, we employ a dual-motor coupled control system as an example of the simulation. The instance model of this control system is illustrated in Figure 9. Here, we treat the two motors as independent agents within a multi-agent system. Each agent is independently controlled, and overall stability is achieved by coordinating the two agents via controllers or information transmission. The physical meanings of the parameters are as follows: \(\theta_{2}\) is the drive rotation angle, and \(\theta_{3}\) is the load rotation angle, which are represented by \(x_{21}\) and \(x_{31}\), respectively. \(J_{1}\) is the moment of inertia of the load, \(J_{d}\) is the moment of inertia of the driver, \(g_{r}\) is the gear tooth ratio, \(k_{1}\) is the damping of the driver, \(k_{2}\) is the load damping, and \(k\) is the torsional elastic constant. The state equation of the system is  
\[\left\{ \begin{array}{l}\dot{x}_{21}=x_{22}+\varphi_{0,2,1}\\ \dot{x}_{22}=\varphi_{0,2,2}+f_{2,2}x_{2}+a_{k,2,2}\varphi_{k,2,2}+b_{2}H_{2,2 }w_{2}\\ y_{2}=x_{22}\end{array} \right. \tag{68}\]  
\[\left\{ \begin{array}{l}\dot{x}_{31}=x_{32}+\varphi_{0,3,1}\\ \dot{x}_{32}=\varphi_{0,3,2}+f_{3,2}x_{2}+a_{k,3,2}\varphi_{k,3,2}+b_{3}H_{3,2 }w_{3}\\ y_{3}=x_{32}\end{array} \right.. \tag{69}\]  
In Formula (68) of subsystem 2,  
\[ \begin{array}{l}f_{2,2}=-\frac{k_{1}}{J_{d}},b_{2}=1,H_{22}=0.1,a_{k,2,2}=1 \\ \varphi_{0,2,1}=0.05\cos(3y_{2})-0.02\sin(y_{2}-0.5)\\ \varphi_{0,2,2}=0.5\sin(2y_{2}-0.5),\varphi_{k,2,2}=\frac{kA}{J_{d}}(Ay_{2}-y_ {3}).\end{array}  \tag{70}\]  
In Formula (69) in subsystem 3,  
\[ \begin{array}{l}f_{3,2}=-\frac{k_{1}}{J_{d}},b_{3}=1,H_{32}=0.1,a_{k,3,2}=1 \\ \varphi_{0,3,1}=0.02\cos(y_{3})-0.02\sin(y_{3}-0.5)\\ \varphi_{0,3,2}=0.5\tan(2y_{3}),\varphi_{k,3,2}=\frac{k}{J_{1}}(Ay_{2}-By_{3}) \end{array} . \tag{71}\]  
Next, we set the parameter values as follows:  
\[ \begin{array}{l}A=\frac{1}{S^{\prime}},k=1,k_{1}=0.8,k_{2}=0.6,J_{1}=1.2,\\ J_{d}=1,gr=0.8,B=1,\Delta=1,\gamma_{\widehat{g}}=2,\gamma_{\theta}=4\\ c_{1}=c_{2}=20,q_{1}=2,q_{2}=1,\rho=2,.\end{array} \]  
Figure 9: A dual-motor coupled control system. To summarize, the simulation operation effect of the system can be obtained as follows:  
Comparing Figure 10 with Figure 11 yields a consistent conclusion, indicating that employing the compensation observer control strategy can further reduce tracking errors. This finding underscores the efficacy of the compensation observer control strategy in enhancing the system's tracking performance. A comparative analysis of Figures 12 and 13 similarly reveals an inherent flaw in the compensation observer control strategy: the exacerbation of input oscillations. This observation highlights a limitation of the compensational observer control strategy, indicating its tendency to amplify input oscillations. However, the degree of oscillation amplification is relatively mild and remains within an acceptable range. Figure 14 graphically depicts the trajectories of Systems 2 and 3 within the state space, indicating a nearly complete overlap between them. This significant overlap not only underscores the high effectiveness of the control strategy in tracking the desired trajectory but also confirms the existence of a stable periodic response in both systems. The results from the comparative experiments further support these findings. Figure 11: Results under the traditional control strategy. (**a**) Trajectory tracking diagram of System 2 and 3. (**b**) System 2 and 3 output error. Figure 10: Results under the compensation control strategy. (**a**) Trajectory tracking diagram of System 2 and 3. (**b**) System 2 and 3 output error. Figure 14: State space tracking performance of system 2 and 3. Figure 12: Input to System 2 and 3 under the compensation control strategy. Figure 13: Input to System 2 and 3 under the traditional control strategy. Figure 15 presents the actual and estimated values of the first state variable.