# Redundancy-Reduction-Based Hierarchical Design in Synchronization of Multi-Agent Systems
## 3 Synchronization of Complex Dynamic Networks  
Before beginning our systematic process, the integrity of information transmission should be ensured. Throughout this paper, the synchronization algorithm is based on the following Lemma [14]. **Lemma 2**. : _For a given scalar \(h\), matrices \(P>0\), \( \begin{bmatrix}U_{1}&U_{2}\\ *&U_{3}\end{bmatrix} >0\), \(X\), \(X_{1}\), \(H= \begin{bmatrix}H_{1}&H_{2}&H_{3}\end{bmatrix} \) exist, and a diagonal matrix \(\Delta>0\); under Assumption 1, we have_  
\[\left[ \begin{array}{ccc}P+h\frac{X+X^{\text{T}}}{2}+U_{1}&-hX+hX_{1}-U_{1}&U _{2}\\ *&\Pi&-U_{2}\\ *&*&U_{3}\end{array} \right]>0 \tag{9}\]  
\[\Psi_{1}= \begin{bmatrix}\Psi_{11}&\Psi_{12}&\Psi_{13}\\ *&\Psi_{22}&-H_{2}^{\text{T}}\\ *&*&\Psi_{33}\end{bmatrix} <0 \tag{10}\]  
\[\Psi_{2}= \begin{bmatrix}\Psi_{11}&\Psi_{12}&\Psi_{13}&hH_{1}^{\text{T}}\\ *&\Psi_{22}&-H_{2}^{\text{T}}&hH_{2}^{\text{T}}\\ *&*&\Psi_{33}-hU_{3}&hH_{3}^{\text{T}}\\ *&*&*&-hU_{1}\end{bmatrix} <0 \tag{11}\]  
\[\Psi_{3}= \begin{bmatrix}\Psi_{11}+\text{Y}_{11}&\Psi_{12}+\text{Y}_{12}&\Psi _{13}+\text{Y}_{13}\\ *&\Psi_{22}+hU_{1}&-H_{2}^{\text{T}}+\text{Y}_{23}\\ *&*&\Psi_{33}+\text{Y}_{33}\end{bmatrix} <0 \tag{12}\]  
_where_  
\[\Psi_{11}=P(I_{N-1}\otimes A)+(I_{N-1}\otimes A)^{\text{T}}P- \frac{X+X^{\text{T}}}{2}\] \[\qquad+(\Delta\otimes I_{n})(I_{N-1}\otimes\Lambda)+(I_{N-1} \otimes\Lambda)(\Delta\otimes I_{n})\] \[\qquad+H_{1}+H_{1}^{\text{T}}\] \[\Psi_{12}=P+H_{2}\] \[\Psi_{13}=cPM(G\Phi\otimes\Gamma)-U_{2}+X-X_{1}+H_{3}-H_{1}^{ \text{T}}\] \[\Psi_{22}= -2(\Delta\otimes I_{n})\] \[\Psi_{33}=U_{2}+U_{2}^{\text{T}}+X_{1}+X_{1}^{\text{T}}-\frac{X+ X^{\text{T}}}{2}-H_{3}-H_{3}^{\text{T}}\] \[\text{Y}_{11}=h\frac{X+X^{\text{T}}}{2}(I_{N-1}\otimes A)+h(I_{N -1}\otimes A)^{\text{T}}\frac{X+X^{\text{T}}}{2}\] \[\qquad+h(I_{N-1}\otimes A)^{\text{T}}U_{1}(I_{N-1}\otimes A)\]\[ \begin{split}\text{Y}_{12}&=h\frac{X+X^{\text{T}}}{2}+h(I_{ N-1}\otimes A)^{\text{T}}U_{1}\\ \text{Y}_{13}&=h(I_{N-1}\otimes A)^{\text{T}}U_{2}+ ch\frac{X+X^{\text{T}}}{2}M([PERSON]\otimes\Gamma)\\ &\quad\quad+h(I_{N-1}\otimes A)^{\text{T}}(-X+X_{1})\\ &\quad\quad+ch(I_{N-1}\otimes A)^{\text{T}}U_{1}M([PERSON]\otimes \Gamma)\\ \text{Y}_{23}&=hU_{2}-hX+hX_{1}+chU_{1}M([PERSON] \otimes\Gamma)\\ \text{Y}_{33}&=hU_{3}+chU_{2}^{\text{T}}M([PERSON] \otimes\Gamma)+ch([PERSON]\otimes\Gamma)^{\text{T}}M^{\text{T}}U_{2}\\ &\quad\quad+ch(-X^{\text{T}}+X_{1}^{\text{T}})M([PERSON]\otimes \Gamma)\\ &\quad\quad+ch([PERSON]\otimes\Gamma)^{\text{T}}M^{\text{T}}(-X+X_{1 })\\ &\quad\quad+c^{2}h([PERSON]\otimes\Gamma)^{\text{T}}M^{\text{T}}U_{1}M ([PERSON]\otimes\Gamma)\\ \Pi&=\ -hX_{1}-hX_{1}^{\text{T}}+h\frac{X+X^{\text{T}}}{2}+U_ {1}\\ M&=\tilde{M}\otimes I_{n}\end{split} \]  
_Then, CDN (2) is globally exponentially synchronized._  
The proof process would be omitted here due to the limited space. The reader can see the whole proof process from the reference [14]. We partition the network into finite chains, of which no cycle is included in the sub-networks. Then, the process of CDN synchronization could be reduced to the finite chains and the central note.With Lemma 2, the following theorem is proposed. Theorem 3.1: _Consider a class of multi-agent systems with a known structural connection weight. Partition the CDN into multi-layers of sub-networks with according sub-central nodes and stay full-connected. Assuming that each layer of the sub-network asymptotically synchronizes to the corresponding sub-central node; then, the whole CDN achieves asymptotical synchronization._  
Proof: Partition the CDN with \(i=1,2,3,\ldots,n_{l}\) layers. Assuming that there would be \(s_{i}\) sub-networks in the \(i\)th layer, therefore there would be \(s_{i}\) sub-central nodes in the \(i\)th layer, \(s_{i}^{j}\) denotes the \(j\)th single sub-central node. By using the prior definition, the first sub-central node is named by the 1 st node, such that all nodes in the CDN could be described as \(x_{i,s_{i}^{j},k}\). With that, we have that \(x_{1,1,1}\) is the central node for all of the multi-agent systems.