Cooperative Control of Multi-Agent Systems

where ${\lambda }_i,i=2,\cdots ,N$, are the nonzero eigenvalues of $ℒ$. Let $\overline{\xi }\triangleq {\left[{\overline{\xi }}_1^T,\cdots ,{\overline{\xi }}_N^T\right]}^T=(U^T\otimes Q^{-1})\xi$. By the definitions of ξ and $\overline{\xi }$, it is easy to see that $\overline{\xi }=\left(\frac{1^T}{\sqrt{N}}\otimes Q^{-1}\right)\xi =0$. Then, by using (5.60), it follows that

$\begin{array}{l}{\xi }^T[\mathcal{X}+(\delta I_N+M)\otimes Q^{-1}]\xi ={\displaystyle \sum _{i=2}^N{\overline{\xi }}_i^T[AQ+Q{A}^T+(\delta +1)Q-2\alpha {\text{λ}}_{i}B{B}^T]{\overline{\xi }}_i}\\ \le {\displaystyle \sum _{i=2}^N{\overline{\xi }}_i^T[AQ+Q{A}^T+\epsilon Q-2B{B}^T]{\overline{\xi }}_i}\le 0,\end{array}$

(5.61)

where we have used the fact that $\alpha {\text{λ}}_i\ge 1,i=2,\cdots ,N$, and $\delta \le \epsilon -1$. Then, it follows from (5.59) and (5.61) that

${\dot{V}}_8\le -\delta V_8+\frac{1}{2{\text{λ}}_{\min }(Q)}{\displaystyle \sum _{i=1}^N{\theta }_i^2}+\frac{{\alpha }^2}{4}{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N{\varphi }_{ij}{a}_{ij}}}.$

(5.62)

By using Lemma 16 (the Comparison lemma), we can obtain from (5.62) that

$V_8\le \left[V_8(0)-\frac{1}{2\delta {\text{λ}}_{\min }(Q)}{\displaystyle \sum _{i=1}^N{\theta }_i^2}-\frac{{\alpha }^2}{4\delta }{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N{\varphi }_{ij}{a}_{ij}}}\right]e^{-\delta t}+\frac{1}{2\delta {\text{λ}}_{\min }(Q)}{\displaystyle \sum _{i=1}^N{\theta }_i^2}+\frac{{\alpha }^2}{4\delta }{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N{\varphi }_{ij}{a}_{ij}}}.$

(5.63)

Therefore, V₈ exponentially converges to the residual ...