# Stability Analysis of Multi-Agent Tracking Systems with Quasi-Cyclic Switching Topologies
## 3 Controller Design and Stability Analysis
### Stability Analysis  
**Theorem 1**. : _Consider the multi-agent system (6), if there exist positive numbers \(\bar{\lambda}>0\), \(\bar{\lambda}>0\), \(\lambda^{\bigstar}>0\) and \(\lambda^{*}>0\) such that_  
\[ \begin{cases}\bar{\lambda}=\frac{(\sum_{i=1}^{N_{s}}\lambda_{\phi_{i}})}{N_{s} },\phi_{i}\in\mathcal{N}_{s}^{-}\\ \bar{\lambda}=\frac{(\sum_{j=1}^{N_{us}}\lambda_{\phi_{j}})}{N_{us}},\phi_{j} \in\mathcal{N}_{us}^{+}\end{cases}  \tag{9}\]  
\[ \begin{cases}\lambda^{\bigstar}=max(\lambda_{\phi_{i}}),\phi_{i}\in\mathcal{ N}_{s}^{-}\\ \lambda^{*}=max(\lambda_{\phi_{j}}),\phi_{j}\in\mathcal{N}_{us}^{+}\end{cases}  \tag{10}\]  
_where \(\lambda_{\phi_{i}}=min\{\lambda_{\phi(d_{i-1}+1)},\lambda_{\phi(d_{i-1}+2)},...,\lambda_{\phi(d_{i})}\}\), \(\lambda_{\phi_{j}}=max\{\lambda_{\phi(d_{j-1}+1)},\lambda_{\phi(d_{j-1}+2)},...,\lambda_{\phi(d_{j})}\}\). \(\tau_{max}^{us}\) and \(\tau_{min}^{s}\) refer to the maximum dwell time of unstable subsystems and the minimum dwell time of stable subsystems, respectively. Then if there exists \(0<\gamma<\bar{\lambda}\), which satisfies_  
\[\frac{\tau_{max}^{us}}{\tau_{min}^{s}}\leq\frac{((\bar{\lambda}-\gamma)\left\lceil \frac{N_{c}^{-}(t_{0},t)}{N_{s}}\right\rceil-\lambda^{\bigstar})N_{s}}{((\bar{ \lambda}+\gamma)\left\lfloor\frac{N_{c}^{-}(t_{0},t)}{N_{us}}\right\rfloor+ \lambda^{*})N_{us}} \tag{11}\]  
_the tracking multi-agent system (6) with quasi-cyclic switching signals is exponentially stable, and the rate of stability equals \(\Gamma\) which is a positive number related to \(\gamma\)._  
Proof. : Select the Lyapunov candidate for the tracking system (6) as  
\[V(t)=\mathcal{E}^{T}(t)\mathcal{E}(t) \tag{12}\]According to the collective tracking error of the multi-agent system (7), it has  
\[ \begin{split} V(t)&=[e^{\mathcal{A}_{\sigma(t)}(t-t_{0} )}\mathcal{E}(t_{0})]^{T}[e^{\mathcal{A}_{\sigma(t)}(t-t_{0})}\mathcal{E}(t_{0} )]\\ &=\mathcal{E}^{T}(t_{0})(e^{\mathcal{A}_{\sigma(t)}(t-t_{0})})^{T} e^{\mathcal{A}_{\sigma(t)}(t-t_{0})}\mathcal{E}(t_{0})\end{split}  \tag{13}\]  
Without loss of generality, suppose the subsystem is stable at the first time interval. Then, over the time interval \([p_{0},p_{1})=[t_{0},t_{s_{1}})\), according to (8), it has  
\[ \begin{split} V(t_{1})&\leq e^{2(a_{\sigma(t_{1})} -\lambda_{\sigma(t_{1})}(t_{1}-t_{0}))}V(t_{0})\\ V(t_{2})&\leq e^{2(a_{\sigma(t_{2})}-\lambda_{\sigma(t _{2})}(t_{2}-t_{1}))}V(t_{1})\\ &\leq e^{2(a_{\sigma(t_{2})}-\lambda_{\sigma(t_{2})}(t_{2}-t_{1} )+a_{\sigma(t_{2})}-\lambda_{\sigma(t_{1})}(t_{1}-t_{0}))}V(t_{0})\\...\\ V(t_{s_{1}})&\leq e^{2(a_{\sigma(t_{1})}-\lambda_{ \sigma(t_{1})}(t_{1}-t_{s_{1}}-1))}V(t_{s_{1}-1})\\ &\leq e^{2(a_{\sigma(t_{1})}-\lambda_{\sigma(t_{1})}(t_{1}-t_{s_ {1}}-1)+...+a_{\sigma(t_{2})}+\lambda_{\sigma(t_{2})}(t_{2}-t_{1})}\\ &\quad+a_{\sigma(t_{1})}-\lambda_{\sigma(t_{1})}(t_{1}-t_{0}))V (t_{0})\\ &\leq e^{2(\sum_{i=1}^{s_{1}}a_{\sigma(t_{i})}-\sum_{i=1}^{s_{1} }\lambda_{\sigma(t_{i})}T_{\sigma(t_{i})}^{-}(t_{0}t_{s_{1}}))}V(t_{0})\\ &\leq e^{2(\sum_{i=1}^{s_{1}}a_{\sigma(t_{i})}-\sum_{i=1}^{s_{1} }\lambda_{\sigma(t_{i})})}V(t_{0})\\ &\leq e^{2(\sum_{i=1}^{s_{1}}a_{\sigma(t_{i})}-\sum_{i=1}^{s_{ 1}}\lambda_{\sigma(t_{i})}\Big{[}\frac{N_{\sigma(t_{0}t_{s_{1}})}}{N_{\rm s}} \Big{]}-\lambda^{\star}N_{s}))}V(t_{0})\\ &\leq e^{2(a(N_{\sigma}(t_{0}t_{s_{1}})+1)-(\bar{\lambda}N_{s} \Big{[}\frac{N_{\sigma(t_{0}t_{s_{1}})}}{N_{\rm s}}\Big{]}-\lambda^{\star}N_{ s})\tau_{\rm min}^{s})}V(t_{0})\end{split}  \tag{15}\]  
At time \(p_{1}=t_{s_{1}}\), the multi-agent tracking system switches from a stable subsystem to an unstable subsystem. \[\begin{split} V(t_{s_{1}+1})&\leq e^{2(a_{\sigma(t _{s_{1}+1})}+\lambda_{\sigma(t_{s_{1}+1})}(t_{s_{1}+1}-t_{s_{1}}))}V(t_{s_{1}} )\\ &\leq e^{2(a_{\sigma(t_{s_{1}+1})}+\lambda_{\sigma(t_{s_{1}+1})}( t_{s_{1}+1}-t_{s_{1}}))}V(t_{s_{1}})\\ V(t_{  
With iterations and derivations, at time \(t\in[t_{k},t_{k+1})\), we have  
\[