# Consensus Algorithms Based Multi-Robot Formation Control under Noise and Time Delay Conditions
## 3 Consensus Analysis of Multi-Robot with Various Delays and Noise Conditions
Formula (17) left multiplied by \(\alpha^{H}\) is:  
\[\alpha^{H}\Bigg{[}jw\mathbf{I}_{2n}-(\mathbf{I}_{n}\otimes\mathbf{A})+\sum\limits _{\mathrm{d}=1}^{D}{(\mathbf{L}_{d}\otimes\mathbf{B})\zeta(t)e^{-jw\tau_{d}}} \Bigg{]}\alpha=0 \tag{18}\]  
Because each line of the left matrix of Formula (17) is zero, so \(jwa_{1}=\alpha_{2}\), and substituting it into Formula (18):  
\[\sum\limits_{\mathrm{d}=1}^{D}{\beta_{d}\zeta(t)e^{-jw\tau_{d}}}=\frac{w^{2}} {k_{1}+jwk_{2}} \tag{19}\]  
where \(\beta_{d}=\frac{\alpha^{H}(\mathbf{L}_{d}\otimes\mathbf{I}_{2})\alpha}{ \alpha^{H}\alpha}\). Replace A with B in Formula (19):  
\[F(w)=\sum\limits_{\mathrm{d}=1}^{D}{\beta_{d}\zeta(t)e^{jw\tau_{d}}}=\frac{w^{ 2}}{k_{1}-jwk_{2}} \tag{20}\]  
Take module operation on both sides of the upper equal sign:  
\[\|F(w)\|=\|\sum\limits_{\mathrm{d}=1}^{D}{\beta_{d}\zeta(t)e^{-jw\tau_{d}}}\| <\|\sum\limits_{\mathrm{d}=1}^{D}{\beta_{d}\zeta(t)}\|=\frac{\alpha^{H}( \mathbf{L}\otimes\mathbf{I}_{2})\alpha}{\alpha^{H}\alpha}\zeta(t)\leq\lambda_{ \mathrm{max}}\zeta(t) \tag{21}\]  
Let \(w_{\mathrm{max}}=\sqrt{\frac{\lambda_{\mathrm{max}}^{2}k_{2}^{2}\zeta^{2}(t)+ \zeta(t)\sqrt{\zeta^{2}(t)(\lambda_{\mathrm{max}}^{2}k_{2}^{2})^{2}+4\lambda_{ \mathrm{max}}^{2}k_{1}^{2}}}{2}}}\) get \(w\leq w_{\mathrm{max}}\), upper formula establishment. From Formula (20):  
\[\theta(w)=\mathrm{arg}[F(w)]=\mathrm{arctan}(\frac{k_{2}}{k_{1}}w) \tag{22}\]  
where \(\theta(w)\in[0,2\pi)\). Let \(\tau(w)=\frac{\theta(w)}{w},a=\frac{k_{2}}{k_{1}}\), deriving for \(\tau(w)\), we can obtain:  
\[M_{1}(w)=\frac{d\tau(w)}{dw}=\frac{1}{w^{2}}M_{2}(w)=\frac{1}{w^{2}}\bigg{[} \frac{aw}{a^{2}w^{2}+1}-\arctan(aw)\bigg{]} \tag{23}\]  
Deriving for \(M_{2}(w)\) we can obtain:  
\[\frac{dM_{2}(w)}{dw}=-\frac{2a^{3}w^{2}}{\left(a^{2}w^{2}+1\right)^{2}}<0 \tag{24}\]  
\(M_{2}(w)\) is decreasing, so when \(w>0\), \(M_{2}(w)<M_{2}(0)=0\), so \(M_{1}(w)<0\); that is, \(\tau(w)\) is also decreasing. So \(\tau(w)\geq\tau(w_{\mathrm{max}})=\tau_{\mathrm{max}}\). When \(\tau_{d}<\tau_{\mathrm{max}}\), when \(\tau_{d}<\tau_{\mathrm{max}}\), we can get:  
\[\tau(w)=\frac{\theta(w)}{w}=\frac{\mathrm{arg}\bigg{(}\sum\limits_{\mathrm{d} =1}^{D}{\beta_{d}\zeta(t)e^{jw\tau_{d}}}\bigg{)}}{w}\leq\frac{\mathrm{max}[w \tau_{d}]}{w}<\frac{w\tau_{\mathrm{max}}}{w}<\tau_{\mathrm{max}} \tag{25}\]  
That is to say, it contradicts \(\tau(w)\geq\tau(w_{\mathrm{max}})\). Therefore, when \(\tau_{d}<\tau_{\mathrm{max}}\), the eigenvalues of the system can be maintained in the left half plane, and the consensus of system (14) can be achieved. Theorem 1 is proved. **Theorem 2**. : _If system (14) is a directed graph and there is a spanning tree, the system can achieve consensus when the system delay \(\tau_{d}\) is smaller than the \(\tau_{\max}\) under the action of noise \(\zeta(t)\), and \(k_{1}\in(0,k_{0}k_{2}^{2})\). Among them:_  
\[\left\{ \begin{array}{l}\tau_{\max}=\min\limits_{\|\lambda_{i}\|_{j}\
eq 0} \left[\left[\arctan\left(\frac{k_{2}}{k_{1}}w_{i}\right)-\arg z(\lambda_{i}) \right]/w_{i}\right]\\ w_{i}=\sqrt{\frac{\lambda_{i}^{2}k_{2}^{2}\zeta^{2}(t)+\zeta(t)\sqrt{\zeta^{2 }(t)\left(\lambda_{i}^{2}k_{2}^{2}\right)^{2}+4\lambda_{i}^{2}k_{1}^{2}}}{2}} \end{array} \right. \tag{26}\]  
_where \(\arg z(\lambda_{i})\in(-\frac{\pi}{2},\frac{\pi}{2})\)._  
Proof. : Lemma 2 proved that, when the communication topology of multi-robot system (4) is directed graph and contains spanning tree, \(k_{1}\in(0,k_{0}k_{2}^{2})\), the coefficient matrix \(\mathbf{\Psi}\) has a double zero root, and the real part of the other eigenvalues is negative. The same analysis is performed using the frequency domain analysis method. Similar to the proof of Theorem 1, only when the system delay \(\tau\) increases to the value under the action of noise \(\zeta(t)\) does a non-zero eigenvalue of the system first appear on the imaginary axis, while the delay \(\tau\) is the critical value for the system to maintain stability.