# Robustness Analysis for Multi-Agent Consensus Systems with Application to DC Motor Synchronization
## 4 Robust Consensus Analysis
### Robust Consensus for the Multi-Agent System  
In this section, the Lyapunov method is going to use to find the robustness margins of the multi-agent system. We will start by defining the following Lyapunov function taking into account the representation of the multi-agent system shown in (30):  
\[V(\xi)=\xi^{T}P\xi \tag{31}\]  
where, as stated in the Lyapunov method, the matrix \(P\) is a real symmetric positive definite matrix. Taking \(V(\xi)\) as in (31), the derivative of \(V\) along the trajectories of the multi-agent system is given by  
\[\dot{V}(\xi) = \xi^{T}P\xi+\xi^{T}P\xi\] \[= \xi^{T}P\left[\mathcal{L}(\mathbf{q^{0}})\xi+Bu_{1}+\mathcal{L}( \mathbf{r})\xi\right]\] \[+\left[\mathcal{L}(\mathbf{q^{0}})\xi+Bu_{1}+\mathcal{L}(\mathbf{ r})\xi\right]^{T}P\xi\]Now, considering the control law as defined in (17), it is possible to obtain the following equation:  
\[\dot{V}(\xi) = \xi^{T}P\left[\mathcal{L}(\mathbf{q^{0}})\xi+Bu_{1}+\mathcal{L}( \mathbf{r})\xi\right] \tag{32}\] \[+\left[\mathcal{L}(\mathbf{q^{0}})\xi+Bu_{1}+\mathcal{L}(\mathbf{ r})\xi\right]^{T}P\xi\] \[= \xi^{T}\left[P\left(\mathcal{L}(\mathbf{q^{0}})-Bk\right)+\left( \mathcal{L}(\mathbf{q^{0}})-Bk\right)^{T}P\right]\xi\] \[+2\xi^{T}P\mathcal{L}(\mathbf{r})\xi.\]  
The term in Equation (32) represents an inner product so the Cauchy-Schwarz inequality can be applied to obtain the following expression, see [30]  
\[2\xi^{T}P\mathcal{L}(\mathbf{r})\xi = 2\lambda(P)\xi^{T}\mathcal{L}(\mathbf{r})\xi\] \[\leq 2\lambda_{max}(P)\xi^{T}\mathcal{L}(\mathbf{r})\xi\] \[\leq 2\lambda_{max}(P)\gamma\|\xi\|^{2}\]  
Here, \(\gamma\) is the upper bound of uncertainty \(\|\mathcal{L}(\mathbf{r})\|\leq\gamma\) and \(\lambda(P)\) are the eigenvalues of the matrix \(P\). When we replace the previous terms in (32), the following condition is obtained:  
\[\dot{V}(\xi) \leq \xi^{T}\left[P\left(\mathcal{L}(\mathbf{q^{0}})-Bk\right)+\left( \mathcal{L}(\mathbf{q^{0}})-Bk\right)^{T}P\right]\xi\] \[+2\lambda_{max}(P)\gamma\|\xi\|^{2}\] \[\leq -\xi^{T}Q\xi+2\lambda_{max}(P)\gamma\|\xi\|^{2}\]  
where \(Q\) is a real symmetric positive definite matrix and satisfies the following equation:  
\[P\left(\mathcal{L}(\mathbf{q^{0}})-Bk\right)+\left(\mathcal{L}(\mathbf{q^{0}} )-Bk\right)^{T}P=-Q \tag{33}\]  
Finally,  
\[\dot{V}(\xi) \leq -\xi^{T}Q\xi+2\lambda_{max}(P)\gamma\|\xi\|^{2}\] \[\leq (-\lambda_{min}(Q)+2\lambda_{max}(P)\gamma)\left\|\xi\right\|^{2}\]  
Therefore, the condition to ensure the stability property of the uncertain multi-agent system (30) is the following:  
\[\gamma\leq\frac{\lambda_{min}(Q)}{2\lambda_{max}(P)} \tag{34}\]  
In the previous equation, it can be seen that the robustness margin is a function of the \(Q\) and \(P\) matrices that solve the Equation (33). Now, considering the closed-loop multi-agent system from (19)  
\[\mathcal{L}(\mathbf{q^{0}})-Bk=\left[\begin{array}{ccc}-2&0&-1\\ 0&-1&1\\ 1&0&-1\end{array}\right] \tag{35}\]  
and defining \(Q=I\), the solution of Equation (33) is as follows:  
\[P=\left[\begin{array}{ccc}0.3095&0.0714&0.1190\\ 0.0714&0.5000&0.2143\\ 0.1190&0.2143&0.5952\end{array}\right] \tag{36}\]Therefore, it can be noted that the uncertainty tolerated by the closed-loop multi-agent system must be \(\|\mathcal{L}(\mathbf{r})\|<0.6208\). This implies that the equivalent uncertainty supported by the system is equal to 70% in each parameter of the _Laplacian matrix_, that is,  
\[\mathcal{L}(\mathbf{q})=\left[\begin{array}{ccc}q_{11}&q_{12}&0\\ 0&q_{22}&q_{23}\\ q_{31}&0&q_{33}\end{array}\right] \tag{37}\]  
where \(q_{11}=q_{22}=q_{33}\in[-1.35,-0.65]\) and \(q_{12}=q_{23}=q_{31}\in[0.65,1.35]\). It is possible to appreciate this robustness margin in the experimental tests that will be presented later.