# A Theoretical Framework to Determine RHP Zero Dynamics in Sequential Interacting Sub-Systems
## 2 The Polynomial Form to Determine Transmission Zeros  
The textbook definition indicates that in multivariable systems, the transmission zeros are defined when the system matrix is rank deficient [16]. This rule is a basis for the remainder of this work. Let \(G(s)\in\mathbb{R}[s]^{n\times n}\) be the transfer function matrix of the \(n\)-interacting sub-systems:  
\[G(s)= \begin{bmatrix}G_{1,1}&0&\cdots&0&G_{1,n}\\ G_{2,1}&G_{2,2}&0&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&G_{n,n-1}&G_{n,n}\end{bmatrix}  \tag{1}\]  
where each sub-system is described by two transfer functions with three distinct time constants placed in a row in (1) (i.e., a first order and a second order transfer function, respectively). Hence, one may, without loss of generality, assume first order transfer functions on the main diagonal:  
\[G_{i,i}(s)=\frac{c_{i}\gamma_{i}}{(T_{3i}s+1)}\]  
and second order transfer functions below the main diagonal:  
\[G_{i,i-1}(s)=\frac{c_{i}(1-\gamma_{i-1})}{(T_{3i-2}s+1)(T_{3i-1}s+1)}\]  
where \(i=\overline{2,n}\) and \(n\) is the number of sub-systems. Note that for \(i=1\), \(G_{1,1}=c_{1}\gamma_{1}/(T_{1}s+1)\) and \(G_{1,n}=c_{1}(1-\gamma_{n})/(T_{2}s+1)(T_{3}s+1)\) are the transfer functions corresponding to the first module coupled circularly with the last module (e.g., as in recirculation loops). Each transfer function has a gain depending on two constant values determined by the physical properties of the assembly: \(\gamma_{i}\) and \(c_{i}\), \(i=\overline{1,n}\). The value \(0<\gamma_{i}<1\) represents the interaction coefficient, which characterizes the coupling degree among the modules. The value \(c_{i}\) is a constant depending on system specification. The remaining variable \(T_{i}\) is a time constant dependent on \(\gamma_{i}\) and \(c_{i}\). Using the Smith-McMillan form [17], the zero polynomial can be computed from (1) as:  
\[z(s)=\prod_{i=1}^{n}\epsilon_{i}(s) \tag{2}\]  
where \(\epsilon_{i}\), \(i=\overline{1,n}\) are the diagonal elements of the Smith matrix \(S(s)=G(s)h(s)\), with \(h(s)\) a polynomial that is the LCM of the denominators of all the elements in \(G(s)\). Using the property \(\overline{1,n-1}\), it results that the solutions of (2) can be determined only through the evaluation of the last element:  
\[\epsilon_{n}=\frac{\Delta_{n}}{\Delta_{n-1}} \tag{3}\]  
where \(\Delta_{n}\) is the GCD of all \(n\times n\) minors of \(S(s)\). Obviously, this textbook method loses computational efficiency with the increasing number of sub-systems \(n\); e.g., for \(n=10\), the GCD among 100 different matrices needs to be computed. Hence, an alternative, efficient solution suitable for a general formulation, which exploits the particular structure of the transfer function matrix (1), is provided hereafter. For the case of \(n\) interacting sub-systems and with the polynomial \(h(s)=(sT_{1}+1)(sT_{2}+1)\cdots(sT_{3n-1}+1)(sT_{3n}+1)\), which collects all the distinct denominators from (1), hence the \(3n\) degree, the Smith matrix \(S(s)\) has the following entries:  
\[S_{1,1}(s) = c_{1}\gamma_{1}(T_{2}s+1)(T_{3}s+1)\cdots(T_{3n}s+1)\] \[S_{1,n}(s) = c_{1}(1-\gamma_{n})(T_{1}s+1)(T_{4}s+1)\cdots(T_{3n}s+1)\] \[\vdots\] \[S_{n,n}(s) = c_{n}\gamma_{n}(T_{1}s+1)\cdots(T_{3n-1}s+1)\]  
The above polynomials can be classified, with respect to the degree, into two disjoint sets. Hence, \(\mathcal{Q}\) contains all the polynomials of \(3n-1\) degree (i.e., \(\mathcal{Q}=\{S_{1,1},S_{2,2},\cdots,S_{n,n}\}\)), and \(\mathcal{R}\) contains all the polynomials of \(3n-2\) degree (i.e., \(\mathcal{R}=\{S_{1,n},S_{2,1},\cdots,S_{n,n-1}\}\)).