# 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  
In the previous section, we analyzed the conditions under which second-order systems achieve consensus in an ideal environment. However, in real environments, due to noise interference and communication differences between different robotic hardware, the above control protocols need to be improved. Assuming that there are \(D\) kinds of different delays in the system, the multi-agent system (4) can be changed to:  
\[\dot{\mathbf{S}}(t)=(\mathbf{I}\otimes\mathbf{A})\cdot\mathbf{S}(t)-\sum \limits_{{\rm d}=1}^{D}\left(\mathbf{L}_{d}\otimes\mathbf{B}\right)\cdot \zeta(t)\cdot\mathbf{S}(t-\tau_{d}) \tag{14}\]  
where \(\zeta(t)\) is the communication noise or measurement noise between robots, \(\tau_{ij}\) is the transmission delay, which represents the time taken by \(i\) robot to receive and process information transmitted by \(j\) robot, \(\mathbf{L}_{d}\) is the Laplacian matrix corresponding to the sub-topological graph of the robot node when the delay is \(\tau_{d}\), and \(\sum\limits_{{\rm d}=1}^{D}\mathbf{L}_{d}=\mathbf{L}\). **Theorem 1**. : _If system (14) is a connected graph, the system can achieve consensus when the system delay \(\tau_{d}\) is less than \(\tau_{\rm max}\) under the action of noise \(\zeta(t)\). Among them:_  
\[\left\{ \begin{array}{l}\tau_{\rm max}=\left[\arctan\left(\frac{k_{2}}{k_{1 }^{2}}w_{\rm max}\right)\right]/w_{\rm max}\\ w_{\rm max}=\sqrt{\frac{\lambda_{\rm max}^{2}k_{2}^{2}\zeta^{2}(t)+\zeta(t) \sqrt{\zeta^{2}(t)\left(\lambda_{\rm max}^{2}k_{2}^{2}\right)^{2}+4\lambda_{ \rm max}^{2}k_{1}^{2}}}{2}}\end{array} \right. \tag{15}\]  
**Proof. ** Using the frequency domain analysis method for analysis, the Laplace transform of Equation (14) can be obtained:  
\[\mathbf{S}(s)=(s\mathbf{I}_{2n}-(\mathbf{I}_{n}\otimes\mathbf{A})+\sum \limits_{{\rm d}=1}^{D}\left(\mathbf{L}_{d}\otimes\mathbf{B}\right)\zeta(t)e^ {-\tau_{d}s})^{-1}\mathbf{S}(0) \tag{16}\]  
Let \(G_{\tau}(s)=s\mathbf{I}_{2n}-(\mathbf{I}_{n}\otimes\mathbf{A})+\sum\limits_{{ \rm d}=1}^{D}\left(\mathbf{L}_{d}\otimes\mathbf{B}\right)\zeta(t)e^{-\tau_{d}s}\), so the eigenvalues of the determinant \(|G_{\tau}(s)|\) are the eigenvalues of the system. Lemma 1 proves that multi-robot system (4) achieves the conditions of consensus. According to Lemma 1, how can the eigenvalues of system (14) be kept in the negative half-plane under the interference of time delay \(\tau\) and noise \(\zeta(t)\) relative to the system (4)? Because the measurement noise and communication noise are uncertainties in real environments, it is impossible to carry out accurate quantitative analysis. Therefore, only when the system delay \(\tau\) increases to a value under the action of noise \(\zeta(t)\) does a non-zero eigenvalue of the system appear on the virtual axis for the first time, while the time delay \(\tau\) is the critical value for the system to maintain stability. Assuming that the eigenvalue of the system is on the imaginary axis, let \(s=jw\) be the eigenvalue; then \(\alpha=\alpha_{1}\otimes[1,0]^{T}+\alpha_{2}\otimes[0,1]^{T}\) is the eigenvector corresponding to the eigenvalue, and if \(\|\alpha\|=1\), \(\alpha_{1},\alpha_{2}\in\mathbb{C}^{n}\), then:  
\[\left[jw\mathbf{I}_{2n}-(\mathbf{I}_{n}\otimes\mathbf{A})+\sum\limits_{\mathrm{d} =1}^{D}{(\mathbf{L}_{d}\otimes\mathbf{B})\zeta(t)e^{-jw\tau_{d}}}\right]\alpha=0 \tag{17}\]  
The imaginary eigenvalues of the system appear in pairs in conjugate form. This paper only analyzes the case where \(w>0\).