# Robustness Analysis for Multi-Agent Consensus Systems with Application to DC Motor Synchronization
## 4 Robust Consensus Analysis
### A Practical Example of Computing the Time Delay Margin in Communication between Agents
This relation that exists between \(p(s,e^{-\tau s})\) and \(p(s,T)\) is obtained from a transformation that is carried out in the exponential function that represents the time delay; see [32], while the relation between \(T\) and \(\tau\) is determined as follows (see [33]):  
\[\det(sI-A_{0}-A_{1}e^{-\tau s}) = p(s,e^{-\tau s}) \tag{41}\]1  
where \(\omega_{i}\in W\); the set of \(W\) is defined as follows:  
\[\det(sI-A_{0}-A_{1}e^{-\tau s}) = p(s,e^{-\tau s}) \tag{41}\]2  
in the case of the multi-agent system (39), the associated polynomial is the following:  
\[\det(sI-A_{0}-A_{1}e^{-\tau s}) = p(s,e^{-\tau s}) \tag{41}\]3  
where  
\[\det(sI-A_{0}-A_{1}e^{-\tau s}) = p(s,e^{-\tau s}) \tag{41}\]4 \[\det(sI-A_{0}-A_{1}e^{-\tau s}) = p(s,e^{-\tau s}) \tag{41}\]5 \[\det(sI-A_{0}-A_{1}e^{-\tau s}) = p(s,e^{-\tau s}) \tag{41}\]6 \[\det(sI-A_{0}-A_{1}e^{-\tau s}) = p(s,e^{-\tau s}) \tag{41}\]7 \[\det(sI-A_{0}-A_{1}e^{-\tau s}) = p(s,e^{-\tau s}) \tag{41}\]8 \[\det(sI-A_{0}-A_{1}e^{-\tau s}) = p(s,e^{-\tau s}) \tag{41}\]9 \[p(s,e^{-\tau s}) = \sum_{k=0}^{3}p_{k}(s)e^{-k\tau s}\]0 \[p(s,e^{-\tau s}) = \sum_{k=0}^{3}p_{k}(s)e^{-k\tau s}\]1Now, define the polynomial associated Hurwitz matrix as follows:  
\[p(s,e^{-\tau s}) = \sum_{k=0}^{3}p_{k}(s)e^{-k\tau s}\]2 \[H\left[p(s,T)\right] = \begin{bmatrix}a_{6}(T)&a_{4}(T)&a_{2}(T)&a_{0}(T)&0&0&0\\ a_{7}(T)&a_{5}(T)&a_{3}(T)&a_{1}(T)&0&0&0\\ 0&a_{6}(T)&a_{4}(T)&a_{2}(T)&a_{0}(T)&0&0\\ 0&a_{7}(T)&a_{5}(T)&a_{3}(T)&a_{1}(T)&0&0\\ 0&0&a_{6}(T)&a_{4}(T)&a_{2}(T)&a_{0}(T)&0\\ 0&0&0&a_{6}(T)&a_{4}(T)&a_{2}(T)&a_{0}(T)\end{bmatrix}  \tag{51}\]  
and let \(p(s,T)\) the polynomial associated with \(p(s,e^{-\tau s})\) and consider \(H[p(s,T)]\) as the Hurwitz matrix in the functioning of \(T\) of the associate polynomial \(p(s,T)\); if \(p(s,e^{-\tau s})\) is stable for \(\tau=0\), then \(p(s,e^{-\tau s})\) is stable for all \(\tau\in[0,\infty)\) if and only if  
\[\det\{H[p(s,T)]\
eq 0\}\qquad\forall\ T>0 \tag{52}\]  
This result shows stability conditions for any time delay; when this happens, it is said that the system is stable independent of the delay. However, our goal is to determine the maximum delay that the multi-agent system can support without losing the stability property. To do this, it is necessary to compute the values of \(T\) where the determinant of the Hurwitz matrix \(H[p(s,T)]\) becomes zero and using (47) and (48) compute the time delays for which the system loses the stability property. Carrying out the corresponding computation, we get the values of \(T\) that make the determinant of the Hurwitz matrix (51) zero; these are \(T_{1}=0.828427\) and \(T_{2}=2.774851\) and their corresponding \(\omega_{1}=\omega_{2}=0.5\). These, in turn, are used to compute \(\tau_{1}=3.1416\) and \(\tau_{2}=7.5702\); thus, the maximum delay supported by the multi-agent system is \(\tau_{max}=3.1416\) s. The robustness analysis performed in this section guarantees that adequate performance is obtained in experimental testing of the DC motor synchronization.