# 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  
A fundamental aspect to achieve the consensus of multi-agent systems is communication between agents, which is why it is important to consider all the possible phenomena that occur in this process. An important point is the time delay that exists when one agent sends messages to another; see [5; 6], this can be represented by  
\[\dot{\xi}(t) = A_{0}\xi(t)+A_{r}\xi(t-\tau)+Bu_{1}(t) \tag{38}\]  
where  
\[A_{0}=\left[ \begin{array}{ccc}-1&0&0\\ 0&-1&0\\ 0&0&-1\end{array} \right]\qquad\quad A_{r}=\left[ \begin{array}{ccc}0&1&0\\ 0&0&1\\ 1.25&0&0\end{array} \right]\qquad\quad B=\left[ \begin{array}{c}3\\ 0\\ 0\end{array} \right]\]  
The structure of the multi-agent system previously presented is due to the topology selected in Section 3.1, and it is important to note that the element of the matrix Ar(3, 1) is equal to 1.25 because uncertainty was considered in that parameter. However, it could have been considered in all the matrix terms (37) without affecting the methodology. The calculations would be similar to those made for this particular case of the DC motor synchronization. Now, considering the control law from (17), the following closed-loop multi-agent system is obtained:  
\[\dot{\xi}(t) = A_{0}\xi(t)+A_{1}\xi(t-\tau)+B\xi_{cm}^{d} \tag{39}\]  
where  
\[A_{1}=A_{r}-Bk=\left[ \begin{array}{ccc}-1&0&-1\\ 0&0&1\\ 1.25&0&0\end{array} \right]\]  
Once the structure of the system is well defined, the fundamental problem will be to calculate the maximum time delay that the multi-agent system supports without the stability property being lost. It is well known that the property of stability of (39) is determined by the roots of the following characteristic equation; see [31]  
\[\det(sI-A_{0}-A_{1}e^{-\tau s}) \tag{40}\]  
Once this equation is developed, it represents functions called quasipolynomials that have the following form:  
\[\det(sI-A_{0}-A_{1}e^{-\tau s}) = p(s,e^{-\tau s}) \tag{41}\] \[p(s,e^{-\tau s}) = \sum_{k=0}^{3}p_{k}(s)e^{-k\tau s}\] (42) \[p(s,e^{-\tau s}) = p_{0}(s)+p_{1}(s)e^{-\tau s}+p_{2}(s)e^{-2\tau s}+p_{3}(s)e^{-3 \tau s} \tag{43}\]where  
\[p_{0}(s) = (s+1)^{3}\] \[p_{1}(s) = (s+1)^{2}\] \[p_{2}(s) = 1.25(s+1)\] \[p_{3}(s) = 0\]  
The multi-agent system (39) is stable if and only if the following condition is satisfied:  
\[\det(sI-A_{0}-A_{1}e^{-\tau s})\
eq 0\qquad\forall\;s\in\mathbb{C}_{+} \tag{44}\]  
where \(\mathbb{C}_{+}\) denotes the set of complex numbers with real part greater or equal to zero. Like an extension of the stability concept, the quasipolynomials that satisfy the previous condition are called stable quasipolynomials. Due to difficulty in determining the roots of quasipolinomial functions, we will define the following associated polynomials:  
\[p(s,T) = \sum_{k=0}^{n}p_{k}(s)(1-T_{\rm S})^{2k}(1+T_{\rm S})^{2n-2k} \tag{45}\] \[\det(sI-A_{0}-A_{1}e^{-\tau s}) = p(s,e^{-\tau s}) \tag{41}\]0  
These associate polynomials have an important relation between their roots: suppose that \(s^{0}=j\omega\) for some value of \(\omega^{0}\geq 0\); then, \(s^{0}=j\omega^{0}\) is a root of the characteristic equation \(p(s,e^{-\tau s})\) for some value of \(\tau\geq 0\) if and only if \(s^{0}=j\omega^{0}\) is a root of \(p(s,T)\) for some value of \(T\geq 0\).