# Stability Analysis of Multi-Agent Tracking Systems with Quasi-Cyclic Switching Topologies
## 3 Controller Design and Stability Analysis  
### Controller Design  
The distributed controller of \(i\)-th tracking agent is designed as  
\[u_{i}(t)=-K_{1}\sum_{j\in\mathcal{N}_{i}(t)}a_{ij}(\sigma(t))e_{ij}(t)-K_{2}h_{i} (\sigma(t))e_{i}(t) \tag{4}\]  
where \(K_{1},K_{2}\in\mathbb{R}^{p\times m}\) are the control matrices. \(e_{ij}(t)=x_{i}(t)-x_{j}(t)\) is the tracking error between the \(i\)-th tracking agent and the \(j\)-th tracking agent, \(e_{i}(t)=x_{i}(t)-x_{t}(t)\) is the tracking error between the \(i\)-th tracking agent and the target agent.  
Define the collective tracking error \(\mathcal{E}(t)=[e_{1}^{\mathcal{T}}(t),e_{2}^{\mathcal{T}}(t),...,e_{N}^{ \mathcal{T}}(t)]\). Then the error dynamics can be obtained through combining the tracking agent's dynamics (1), the target agent's dynamics (2), and the controller (4), which is designed with tracking errors based on distributed control protocol.  
\[\begin{split}\dot{\mathcal{E}}(t)&=(I_{N}\otimes A )\mathcal{E}(t)-(\mathcal{L}_{\sigma(t)}\otimes BK_{1}+\mathcal{H}_{\sigma(t) }\otimes BK_{2})\mathcal{E}(t)\\ &=((I_{N}\otimes A)-(\mathcal{L}_{\sigma(t)}\otimes BK_{1}+ \mathcal{H}_{\sigma(t)}\otimes BK_{2}))\mathcal{E}(t)\end{split} \tag{5}\]  
where \(\mathcal{L}_{\sigma(t)}\in\mathbb{R}^{N\times N}\), \(\mathcal{H}_{\sigma(t)}\in\mathbb{R}^{N\times N}\) are the corresponding Laplacian matrices of dynamic graph at tracking time \(t\) related to switching signal \(\sigma(t)\). Denote \(\mathcal{A}_{\sigma(t)}=(I_{N}\otimes A)-(\mathcal{L}_{\sigma(t)}\otimes BK_{ 1}+\mathcal{H}_{\sigma(t)}\otimes BK_{2})\), tracking error system (5) is reorganized as  
\[\dot{\mathcal{E}}(t)=\mathcal{A}_{\sigma(t)}\mathcal{E}(t) \tag{6}\]  
Figure 1: Illustration of quasi-cyclic switching times.  
Since \(\mathcal{A}_{\sigma(t)}=(I_{N}\otimes A)-(\mathcal{L}_{\sigma(t)}\otimes BK_{1}+ \mathcal{H}_{\sigma(t)}\otimes BK_{2})\), \(\mathcal{A}_{\sigma(t)}\) is related to control matrices \(K_{1}\), \(K_{2}\), and the Laplacian matrix of tracking topology with target agent \(\mathcal{L}_{\sigma(t)}\), \(\mathcal{H}_{\sigma(t)}\). Thus, according to the definition of successful tracking, the tracking problem can be elaborated as the stability of the above tracking error system.  
Then the collective tracking error of the multi-agent system is obtained:  
\[\mathcal{E}(t)=e^{\mathcal{A}_{\sigma(t)}(t-t_{0})}\mathcal{E}(t_{0}) \tag{7}\]  
Let \(a=max_{k\in I_{N}}a_{k}\), where \(a_{\sigma(t_{i})}\geq 0\), \(a_{\sigma(t_{j})}\geq 0\) such that [38]  
\[\begin{cases}\|e^{\mathcal{A}_{\sigma(t_{i})}t}\|\leq e^{a_{\sigma(t_{i})}- \lambda_{\sigma(t_{j})}t},\sigma(t_{i})\in\mathcal{N}_{s}^{-}\\ \|e^{\mathcal{A}_{\sigma(t_{j})}t}\|\leq e^{a_{\sigma(t_{j})}+\lambda_{\sigma( t_{j})}t},\sigma(t_{j})\in\mathcal{N}_{us}^{+}\end{cases} \tag{8}\]  
where \(\mathcal{N}_{s}^{-}=\{1,2,...,N_{s}\}\), \(\mathcal{N}_{us}^{+}=\{N_{T}-N_{us}+1,N_{T}-N_{us}+2,...,N_{T}\}\) is the set of a quasi-cyclic switching sets of stable subsystems and unstable subsystems respectively. \(\lambda_{\sigma(t_{i})}=-max\{Re(\lambda_{\mathcal{A}_{\sigma(t_{j})}})\}>0\) and \(\lambda_{\sigma(t_{j})}=max\{Re(\lambda_{\mathcal{A}_{\sigma(t_{j})}})\}>0\), \(\lambda_{\mathcal{A}_{\sigma(t_{j})}},\lambda_{\mathcal{A}_{\sigma(t_{j})}}\) are the eigenvalues of \(\mathcal{A}_{\sigma(t_{i})}\) and \(\mathcal{A}_{\sigma(t_{j})}\) respectively. \(\lambda_{\sigma(t_{i})}\) and \(\lambda_{\sigma(t_{j})}\) are related to the Laplacian matrix \(\mathcal{L}_{\sigma(t)}\), \(\mathcal{H}_{\sigma(t)}\) and control matrix \(K_{1}\), \(K_{2}\).  
**Remark 1**.: _According to the Equations (4)-(8), the eigenvalue of the subsystem is related to the Laplacian matrices of topology among multiple agents. The positive and negative eigenvalues correspond to the unstable and stable subsystem respectively. Different from cyclic switching in [37], quasi-cyclic switching system does not follow a strict cycle, and could jump from any subsystem in stable quasi-cyclic period to the first subsystem in unstable quasi-cyclic period and vice versa._