# Compensation Observer-Based Adaptive Output Feedback Control for Multi-Agent Systems
## 5 Simulation Examples and a System Analysis of Control Strategies
### Numerical Example  
We consider the following second-order numerical system:  
\[\left\{ \begin{array}{l}\dot{x}_{11}=x_{12}\\ \dot{x}_{12}=0.2\sin(2y_{1})+f_{1,2}x_{2}+b_{1}H_{12}w_{1}\\ y_{1}=x_{11}\end{array} \right. \tag{66}\]  
where \(f_{1,2}=-2,b_{1}=1,H_{12}=0.1\). The remaining parameters are all set to zero. To validate the effectiveness of the control strategy, we set the initial state of the control system to zero and compare the simulation results between the compensation observer control scheme and the traditional observer control scheme. The compensatory observer control scheme proposed in this design introduces a compensation term \(\frac{TH}{R^{2}+\sigma_{\alpha}z}\) into the traditional observer. Consequently, the traditional observer control scheme follows the observer structure shown in Equation (67), but without the compensation term. To ensure a fair comparison, all other parameters remain identical:  
\[\dot{\epsilon}_{0 ij} =\big{(}A_{i}-l_{i}L_{i}q_{i}c_{i}T\big{)}\epsilon_{0 ij}+F_{i,j}(y_ {i})\epsilon_{0 ij}+\varphi_{0,i,j}(y_{i})+l_{i}L_{i}q_{i}y_{i} \tag{67}\] \[\dot{\epsilon}_{kij} =\big{(}A_{i}-l_{i}L_{i}q_{i}c_{i}T\big{)}\epsilon_{kij}+F_{i,j}(y _{i})\epsilon_{kij}+\varphi_{k,j}(y_{i})\] \[\dot{\epsilon}_{sij} =\big{(}A_{i}-l_{i}L_{i}q_{i}c_{i}T\big{)}\epsilon_{sij}+F_{i,j}(y _{i})\epsilon_{sij}+H_{ij}(y_{i})u_{i}(t)\] \[\dot{\epsilon}_{zij} =\big{(}A_{i}-l_{i}L_{i}q_{i}c_{i}T\big{)}\epsilon_{cij}+F_{i,j}( y_{i})\epsilon_{cij}\]  
Figure 2 presents the compensation observer control strategy, while Figure 3 illustrates the traditional controller control strategy. The error range in Figure 2b is smaller, indicating better tracking performance. This result further validates the superiority of the compensation observer control strategy. Comparing the input waveforms between Figures 4 and 5 indicates that under both control strategies, the input waveform of the overall system exhibits periodic variations. However, under the compensation strategy, the system input presents localized oscillations. This result represents a drawback of the control scheme, indicating a limitation in its ability to mitigate localized oscillations in the system input. The dynamic behavior depicted in Figure 6, showing periodic states of the system output in three-dimensional state space, indicates the stable periodic responses of the system to certain inputs or initial conditions, indirectly suggesting a degree of stability in the system. Figure 3: Results under the traditional control strategy. (**a**) Trajectory tracking diagram of System 1. (**b**) System 1 output error. Figure 2: Results under the compensation control strategy. (**a**) Trajectory tracking diagram of System 1. (**b**) System 1 output error. Figure 4: Input to System 1 under the compensation control strategy. Figure 5: Input to System 1 under the traditional control strategy. Figure 6: State space tracking performance of System 1. As shown in Figure 7a, the derivative of the dynamic gain exhibits periodic oscillations. This phenomenon is similar to observations in Figure 7b. Additionally, the waveforms of \(\dot{\theta}_{i}\) and \(\dot{\dot{p}}\) trend towards zero, indicating that the integrated values of these parameters are bounded. This characteristic may reflect the stability of the system dynamics, as the integrated values of the parameters are constrained, leading to limited variations in the system state within a finite range. Based on the results shown in Figure 8, the waveforms of \(x_{i}\) and \(\dot{x}_{i}\) are nearly identical, indicating that the observer can accurately track and predict the system's state variables. This result demonstrates that the observer effectively estimates the system state in dynamic environments, thereby providing a necessary foundation for the performance of the stringent controller. Figure 8: Comparison of observer performance. (**a**) \(x_{1}\) and \(\dot{x}_{1}\) in System 1; (**b**) \(x_{2}\) and \(\dot{x}_{2}\) in System 1.