# A Theoretical Framework to Determine RHP Zero Dynamics in Sequential Interacting Sub-Systems
## 4 Analysis of the Transmission Zero Polynomial  
Following the proposed method, the analysis of \(n\)-interconnected sub-systems depends on the sign of the zero degree coefficient from the transmission zero polynomial. Replacing \(s=0\) in (9) gives:  
\[\zeta_{0}=\prod_{i=1}^{n}c_{i}\left(\prod_{i=1}^{n}\gamma_{i}-\prod_{i=1}^{n}( 1-\gamma_{i})\right) \tag{12}\]  
Defining with \(\Gamma=\{\gamma_{1},\gamma_{2},\cdots,\gamma_{n}\}\) the set that contains all possible values of \(\gamma\) and using the binomial coefficient, (12) can be written as:  
\[\zeta_{0} = \prod_{i=1}^{n}c_{i}\left(-1+\sum_{i=1}^{n-1}(-1)^{i}\binom{n}{i }\Gamma\right)=\] \[= \prod_{i=1}^{n}c_{i}\left(-1+\underbrace{\gamma_{1}+\gamma_{2}+ \cdots+\gamma_{n}}_{n}\underbrace{-\gamma_{1}\gamma_{2}-\gamma_{2}\gamma_{2}- \cdots}_{\frac{n(n-1)}{2}}\right)\]Next, the expected value of the set \(\Gamma\) (i.e., \(\hat{\gamma}=\mathbb{E}[\Gamma]=\frac{1}{n}\sum\limits_{i=1}^{n}\gamma_{i}\)) is used to perform a variable change in (13), leading to:  
\[\zeta_{0}(\hat{\gamma}) = n\hat{\gamma}^{n-1}-\frac{n(n-1)}{2}\hat{\gamma}^{n-2}+\cdots \tag{14}\] \[- \frac{n(n-1)}{2}\hat{\gamma}^{2}+n\hat{\gamma}-1\] \[= \hat{\gamma}^{n}-(\hat{\gamma}-1)^{n}\]  
**Proposition 5**.: _Let \(n\in\mathbb{N}\) be an even number and \(\hat{\gamma}\in\mathbb{R}\) be defined in the interval \(I=(0,1)\), then:_  
\[\zeta_{0}(\hat{\gamma})=\hat{\gamma}^{n}-(\hat{\gamma}-1)^{n}\ \ \ \forall n=2,4,\cdots\]  
_has only one real solution for \(\hat{\gamma}=\frac{1}{2}\)._  
The proof is given in Appendix A.  
For a better understanding, a case with \(n=4\) is further exemplified. The set of all possible values of \(\gamma\) is \(\Gamma=\{\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4}\}\), while the coefficient \(\zeta_{0}\) from (13) and the polynomial \(\zeta_{0}(\hat{\gamma})\) given in (14) are:  
\[\zeta_{0}=\prod_{i=1}^{n}c_{i}\left(-1+\underbrace{\gamma_{1}+\cdots+\gamma_{ 4}}_{4}-\underbrace{\gamma_{1}\gamma_{2}-\gamma_{1}\gamma_{3}-\cdots-\gamma_{3 }\gamma_{4}}_{6}+\underbrace{\gamma_{1}\gamma_{2}\gamma_{3}+\cdots+\gamma_{4} \gamma_{1}\gamma_{2}}_{4}\right), \tag{15}\]  
\[\zeta_{0}(\hat{\gamma})=\prod_{i=1}^{n}c_{i}\left(4\hat{\gamma}^{3}-6\hat{ \gamma}^{2}+4\hat{\gamma}-1\right) \tag{16}\]  
The solutions of (16) are:  
\[\zeta_{0}(\hat{\gamma})=0\Rightarrow\begin{cases}\hat{\gamma}_{1}^{*}=\frac{ 1}{2}\\ \hat{\gamma}_{2}^{*}=\frac{1}{2}+\frac{1}{2}i\\ \hat{\gamma}_{3}^{*}=\frac{1}{2}-\frac{1}{2}i\end{cases}\]  
Since \(\gamma\in\mathcal{R}\), we are only interested in the real solution. Therefore, all \(\gamma_{i}\), \(i=\overline{1,4}\) combinations with \(\hat{\gamma}=\frac{1}{2}\) will bring the coefficient \(\zeta_{0}\) to zero. If we consider \(\Gamma=\{0.3,0.6,0.4,0.7\}\), the expected value is \(\hat{\gamma}=\frac{1}{2}\), and substituting the values in (15), we have:  
\[\zeta_{0}=\prod_{i=1}^{n}c_{i}\left(-1+2-1.45+0.45\right)=0.\]  
For the particular study case proposed in this work, when there is an even number of modules, the interacting system has MP behavior, _if and only if_ the average value of the interaction coefficients: \(\mathbb{E}[\gamma_{i}]<0.5\). Otherwise, the process has NMP behavior. If the average value is \(\mathbb{E}[\gamma_{i}]=0.5\), a transmission zero appears in the origin. The above analysis stands for all systems composed of an even number of sub-systems. However, a process with an odd number of sub-systems has always MP dynamics, because all the roots of the polynomial (9) are negative.  
Moreover, the proposed method was successfully applied on the sextuple tank process available at Ghent University, consisting of three coupled sub-systems, which always has an MP dynamics. After that, the interacting process was controlled with a distributed predictive control technique, with the results available in [14].