# Stability Analysis of Multi-Agent Tracking Systems with Quasi-Cyclic Switching Topologies
## 2 Preliminaries and Problem Formulation
### Problem Formulation  
Consider a set of agents committing the task of tracking a target agent, the dynamics of the \(i\)-th tracking agent is described as follows:  
\[\dot{x}_{i}(t)=Ax_{i}(t)+Bu_{i}(t),t\geq t_{0} \tag{1}\]  
where \(i=1,2,...,N\), \(x_{i}(t)\in\mathbb{R}^{m}\) is state vector of the \(i\)-th agent, \(A\in\mathbb{R}^{m\times m}\) and \(B\in\mathbb{R}^{m\times p}\) are dynamic matrices of the system, and \(u_{i}(t)\in\mathbb{R}^{p\times m}\) is the control input of the \(i\)-th tracking agent. The dynamics of the target agent is described by  
\[\dot{x}_{t}(t)=Ax_{t}(t),t\geq t_{0} \tag{2}\]  
where \(x_{t}(t)\in\mathbb{R}^{m}\) is the state vector of the target agent. If for any initial condition, the tracking agents in the system presented by Equation (1) can accommodate themselves to the target agent's state described by Equation (2) with the effect of designed controller such that  
\[\lim_{t\rightarrow\infty}\|x_{i}(t)-x_{t}(t)\|\to 0 \tag{3}\]  
then it is said that the tracking agents successfully track the target agent. \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Description** \\ \hline \(x_{i}(t)\) & state vector of the \(i\)-th agent \\ \(u_{i}(t)\) & control input of the \(i\)-th tracking agent \\ \(\sigma(t)\) & switching signal \\ \(N_{s}\) & period length of stable subsystems \\ \(N_{us}\) & period length of unstable subsystems \\ \(e_{ij}(t)\) & tracking error between the \(i\)-th and the \(j\)-th tracking agent \\ \(e_{i}(t)\) & tracking error between the \(i\)-th tracking agent and the target agent \\ \(\mathcal{N}_{s}^{-}\) & the set of stable subsystems in the quasi-cyclic switching process \\ \(\mathcal{N}_{us}^{-}\) & the set of unstable subsystems in the quasi-cyclic switching process \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Nomenclature. In the practical tracking process, due to obstacles or the interference of external communication signals, the communication topology of multi-agent systems is likely to be different over different time intervals. If the topology is connected, the whole interactive tracking system is a stable subsystem over this corresponding time interval, otherwise, it is called unstable subsystems. Additionally, the stable subsystems and the unstable subsystems appear alternately, the switching law of which does not conform to the complete periodicity. Therefore, this paper proposes a quasi-cyclic method to solve this practical problem. The switching signal \(\sigma(t):[t_{0},\infty)\mapsto\mathcal{I}_{\mathcal{N}}=\{1,2,...,n\}\) is piecewise constant, where \(n>1\) is the number of subsystems. \(\sigma(t)\) is continuous from right everywhere and may be either autonomous or controlled in general. Generally, the more information we know about the switching signals, the more refined conclusions we could obtain. In practice, switching topologies of the multi-agent tracking systems are usually not arbitrary due to some restrictions of physical entities, such as unknowable environmental impact, component failure and sensors. Therefore, in this section, we assume the switching signals satisfy the following assumption. **Assumption 1**. : _For a given set \(S=\{1,2,...,N_{T}\},n\geq N_{T}\in\mathbb{N}\), \(N_{T}\) is the length of the period of quasi-switching, and a mapping \(\phi:\mathcal{I}_{\mathcal{N}}\mapsto S\) satisfy_  
\[\phi(i)\in S,\forall i\in\mathcal{I}_{\mathcal{N}};\phi(\mathcal{I}_{ \mathcal{N}})=S.\]  
_Set \(\phi_{j}=\{i\in\mathcal{I}_{\mathcal{N}}:\phi(i)=j\},j\in S\). The switching law satisfies the following conditions, let \(\sigma(t_{i})\in\phi_{r},r\in S\)._  
* _If_ \(r<m,\sigma(t_{i+1})\in\{\phi_{r+1},\phi_{m+1}\}\)_;_
* _If_ \(r=m,\sigma(t_{i+1})\in\{\phi_{1},\phi_{m+1}\}\)_;_
* _If_ \(m<r<N_{T},\sigma(t_{i+1})\in\{\phi_{r+1},\phi_{m+1},\phi_{1}\}\)_;_
* _If_ \(r=N_{T},\sigma(t_{i+1})\in\{\phi_{1},\phi_{m+1}\}\)_._  
The switching signal that satisfies Assumption 1 is called quasi-cyclic switching signal.