# Stability Analysis of Multi-Agent Tracking Systems with Quasi-Cyclic Switching Topologies
## 4 A Numerical Example  
In this section, a numerical example is provided to validate the effectiveness of the aforementioned theoretical results. Consider a multi-agent system with one target agent followed by three tracking agents under the quasi-cyclic switched topologies, which is shown in Figure 2, where \(\{\phi_{1},\phi_{2},\phi_{3}\}\in\mathcal{N}_{s}^{-}\) (\(\phi_{1}=\{\phi(1),\phi(2),\phi(3)\}\), \(\phi_{2}=\{\phi(4),\phi(5),\phi(6),\phi(7)\}\), \(\phi_{3}=\{\phi(8),\phi(9)\}\)) and \(\{\phi_{4},\phi_{5}\}\in\mathcal{N}_{us}^{+}\) (\(\phi_{4}=\{\phi(10),\phi(11),\phi(12)\}\), \(\phi_{5}=\{\phi(13),\phi(14),\phi(15)\}\)). According to the dynamic feedback linearization technology [39] that has adopted in distributed control of multi-agent systems with nonholonomic mobile robots [40; 41], the dynamics of the system (1) can be linearized with the following matrices  
\[A= \begin{bmatrix}0.6&0.8\\ -1.2&-0.5\end{bmatrix} ,\quad B= \begin{bmatrix}0.5\\ 0.5\end{bmatrix}  \tag{21}\]  
The controller (4) of each tracking agent is designed with parameters  
\[K_{1}=[3.61\quad 1.27],\quad K_{2}=[5.22\quad 1.08] \tag{22}\]  
which have been verified that guarantee the stability of the system when the agents are topologically connected and meet the requirements of Theorem 1. Consider a scenario where a team of three agents track a maneuvering target in a two-dimensional space. The initial position of the target agent is \(x_{t}(0)=[6,6]\). The initial positions of the three tracking agents are \(x_{1}(0)=[0,8]\), \(x_{2}(0)=[0,4]\) and \(x_{3}(0)=[4,0]\). Tracking trajectories of the agents are shown in Figure 3, from which we can see that successful tracking is eventually achieved. Errors of position between target agent and each tracking agent are plotted in Figure 4. The errors of tracking agents decreased to less than 0.01 within 5 s. Apparently every tracking agent successfully tracks the target agent under the quasi-cyclic topologies with the proposed tracking strategy (4). Figure 2: An example of quasi-cyclic switching topologies. The switching signal in the simulation is depicted in Figure 5. The switching sequence is: \(\phi(1)\rightarrow\phi(5)\rightarrow\phi(9)\rightarrow\phi(1)\rightarrow\phi(4) \rightarrow\phi(8)\rightarrow\phi(2)\rightarrow\phi(6)\rightarrow\phi(11) \rightarrow\phi(14)\rightarrow\phi(11)\rightarrow\phi(6)\rightarrow\phi(8) \rightarrow\phi(3)\rightarrow\phi(10)\rightarrow\phi(14)\rightarrow\phi(11) \rightarrow\phi(15)\rightarrow\phi(1)\rightarrow\phi(7)\rightarrow\phi(9) \rightarrow\phi(2)\rightarrow\phi(5)\rightarrow\phi(11) \rightarrow\phi(13)\rightarrow\phi(11)\rightarrow\phi(2)\rightarrow\phi(7) \rightarrow\phi(12)\rightarrow\phi(2)\rightarrow\phi(5)\rightarrow\phi(9) \rightarrow\phi(2)\) which can also be represented as \(\phi_{1}\rightarrow\phi_{2}\rightarrow\phi_{3}\rightarrow\phi_{1} \rightarrow\phi_{2}\rightarrow\phi_{3}\rightarrow\phi_{1}\rightarrow\phi_{2} \rightarrow\phi_{4}\rightarrow\phi_{5}\rightarrow\phi_{4}\rightarrow\phi_{1} \rightarrow\phi_{2}\rightarrow\phi_{3}\rightarrow\phi_{1}\rightarrow\phi_{2} \rightarrow\phi_{4}\rightarrow\phi_{5}\rightarrow\phi_{4}\rightarrow\phi_{1} \rightarrow\phi_{2}\rightarrow\phi_{4}\rightarrow\phi_{5}\rightarrow\phi_{4} \rightarrow\phi_{1}\rightarrow\phi_{2}\rightarrow\phi_{4}\rightarrow\phi_{1} \rightarrow\phi_{2}\rightarrow\phi_{4}\rightarrow\phi_{1}\rightarrow\phi_{2} \rightarrow\phi_{4}\rightarrow\phi_{1}\rightarrow\phi_{2}\rightarrow\phi_{3} \rightarrow\phi_{1}\)It obviously conforms to the quasi-cyclic law compared to Figure 2.  
The topology of the multi-agent system switches 33 times during the tracking process. The profiles of the norm of tracking errors are plotted in Figure 6. It can be clearly seen that the tracking error eventually converges to zero even though it increases at some instants due to the occurrence of unstable subsystems. From the above results in Figures 3-6, we can see that the states of tracking agents can track the target's state successfully with the quasi-cyclic switched topologies.