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\title{Zeros, poles and system equivalence of time-delay systems }
\author{ {\bf Rafikul Alam} and Jibrail Ali\\ email: rafik@iitg.ac.in, jibrail@iitg.ac.in \\
Department of Mathematics\\ Indian Institute of Technology
Guwahati \\Guwahati-781039, INDIA }
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\noindent
{\bf Extended abstract:} Consider the linear time-invariant time-delay system (TDS)
\begin{eqnarray} \frac{\mathrm{d}\mathbf{x}(t)}{\mathrm{d}t} &=& A_0 \mathbf{x}(t) + \sum^{N_1}{j=1} A_j \mathbf{x}(t - \tau_j) + \sum^{N_2}{j=1} B_j \mathbf{u}(t- t_j), \ \mathbf{y}(t) &=& \sum^{N_3}{j=1} C_j \mathbf{x}(t-s_j) + \sum^{N_4}{j=1} D_j \mathbf{u}(t-h_j), \end{eqnarray}
where $\mathbf{x}(t) \in \mathbb{C}^r$ and $ \mathbf{u}(t) \in \mathbb{C}^n$ are state and control vectors, respectively, at a time $t$, $ (A_i, B_i, C_i, D_i) \in \mathbb{C}^{r\times r} \times \mathbb{C}^{r\times n} \times \mathbb{C}^{m\times r} \times \mathbb{C}^{m\times n}$ and $ ( \tau_i, t_i, s_i, h_i)$ are time-delay parameters. A system matrix $\mathbf{S}(\lambda)$ and the transfer function $\mathbf{M}(\lambda)$ of the TDS are given by
$$ \mathbf{S}(\lambda) := \left[\begin{array}{c|c} A(\lambda) & B(\lambda) \\ \hline -C(\lambda) & D(\lambda)\end{array}\right] \text{ and } \mathbf{M}(\lambda) := [D(\lambda) + C(\lambda) A(\lambda)^{-1} B(\lambda)], $$where $ A(\lambda) := \lambda I_r - A_0 - \sum^{N_1}{j=1} A_j e^{-\lambda \tau_j},$ $ B(\lambda) := \sum^{N_2}{j=1} B_j e^{-\lambda t_j},$ $C(\lambda) := \sum^{N_3}{j=1} C_j e^{-\lambda s_j}$ and $ D(\lambda) := \sum^{N_4}{j=1} D_j e^{-\lambda h_j}$ are entire matrix-valued functions.
Our main aim is two-fold. First, to study the canonical forms of $\mathbf{S}(\lambda)$ and $\mathbf{M}(\lambda)$ so as to analyze zeros and poles of the TDS. Second, to investigate system equivalence, that is, if $\mathbf{S}1(\lambda)$ and $\mathbf{S}_2(\lambda)$ are system matrices of time-delay systems, then investigate Rosenbrock system equivalence (written as $\mathbf{S}_1(\lambda)\sim{rse} \mathbf{S}2(\lambda)$) as well as Fuhrmann system equivalence (written as $\mathbf{S}_1(\lambda)\sim{fse} \mathbf{S}2(\lambda)$).
We show that $\mathbf{M}(\lambda)$ admits a Smith-McMillan form $\Sigma{\mathbf{M}}(\lambda)$ given by
\begin{equation} \Sigma_{\mathbf{M}}(\lambda) = \left[\begin{array}{ccc|c}\phi_1(\lambda)/\psi_1(\lambda) & & & \\ & \ddots& & \\ & & \phi_p(\lambda)/\psi_p(\lambda) & \\ \hline & & & 0_{m-p\times n-p} \end{array}\right],\end{equation} where $ \phi_1, \ldots, \phi_p$ and $ \psi_1, \ldots, \psi_p$ are entire functions, $\phi_j$ and $\psi_j$ are relatively prime. Further, $ \phi_j$ divides $\phi_{j+1}$ and $ \psi_{j+1}$ divides $\psi_j$ for $ j=1:p-1.$ Furthermore, $\phi_1,\cdots, \phi_p$ and $ \psi_1, \ldots, \psi_p$ are given by $$ \phi_j(\lambda) = \prod^\infty_{\ell=1} (\lambda- \lambda_\ell)^{\partial_j(\lambda_\ell)}u_{j\ell}(\lambda) \text{ and } \psi_j(\lambda) = \prod^\infty_{\ell=1} (\lambda- \mu_\ell)^{\delta_j(\mu_\ell)}v_{j\ell}(\lambda),$$where $ u_{j\ell}, v_{j\ell} $ are entire functions with no zeros in $\mathbb{C}$, $\lambda_\ell$ and $\mu_\ell$ are zeros and poles of $\mathbf{M}(\lambda)$, $\partial_j({\lambda_\ell})$ are $\delta_j(\mu_\ell)$ are appropriate non-negative integers for $j=1:p$ and $\ell \in \mathbb{N}.$
We also show that $\mathbf{M}(\lambda)$ admits a right coprime matrix-fraction description (MFD), that is, there exist entire matrix-valued functions $N(\lambda)$ and $D(\lambda)$ such that $N(\lambda)$ and $D(\lambda)$ are right coprime, $D(\lambda)$ is regular, and $$ \mathbf{M}(\lambda) = N(\lambda) D(\lambda)^{-1}.$$ Further, $\sigma_{\mathbb{C}}(\mathbf{M}) = \sigma_{\mathbb{C}}(N)$ and $\wp_{\mathbb{C}}(\mathbf{M}) = \sigma_{\mathbb{C}}(D),$ where $\sigma_{\mathbb{C}}(X)$ is the spectrum (set of zeros) and $\wp_{\mathbb{C}}(X)$ is the set of poles of a meromorphic matrix-valued function $X(\lambda)$.
For system matrices, we show that $$\mathbf{S}_1(\lambda) \sim_{rse} \mathbf{S}_2(\lambda) \Longleftrightarrow \mathbf{S}_1(\lambda) \sim_{fse} \mathbf{S}_2(\lambda).$$ Further, if $\mathbf{S}1(\lambda)$ and $\mathbf{S}_2(\lambda)$ are {\em irreducible} then $\mathbf{S}_1(\lambda) \sim{rse} \mathbf{S}2(\lambda) \Longleftrightarrow \mathbf{S}_1(\lambda) \text{ and } \mathbf{S}_2(\lambda)$ have the same transfer function.
Finally, if $\mathbf{S}(\lambda)$ is irreducible and $\Sigma{\mathbf{M}}(\lambda)$ is the Smith-McMillan form of $\mathbf{M}(\lambda)$ as given above, then we show that
\begin{eqnarray*} S_A(\lambda) &=& I_{r-p} \oplus \mathrm{diag}(\psi_p(\lambda), \psi_{p-1}(\lambda), \psi_1(\lambda)) \\ S_{\mathbf{S}}(\lambda) &= & I_r \oplus \mathrm{diag}(\phi_1(\lambda), \ldots, \phi_p(\lambda)) \oplus 0_{m-p, n-p} \end{eqnarray*} are the Smith forms of $A(\lambda)$ and $\mathbf{S}(\lambda),$ respectively. Hence we show that $ \sigma_{\mathbb{C}}(\mathbf{M}) = \sigma_{\mathbb{C}}(\mathbf{S}) $ and $ \wp_{\mathbb{C}}(\mathbf{M}) = \sigma_{\mathbb{C}}(A).$
%
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% We characterize holomorphic system matrices having the same transfer function $\mathbb{M}(\lambda)$ and present, among other things, the Smith form of $\mathbf{S}(\lambda)$ and the Smith-McMillan form of $\mathbb{M}(\lambda).$ With a view to analyzing zeros and poles of the TDS, we establish relationship between canonical forms of the system matrix $\mathbf{S}(\lambda), $ the state matrix $A(\lambda)$ and the transfer function $ \mathbb{M}(\lambda).$
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\noindent {\bf References:}
\begin{itemize}
\item[1.] M. Frost and C. Storey, ``{A note on the controllability of linear constant delay differential systems}", {\em International J. Control,} Vol.~28, No.~5, pp.~673-679, 1978.
\item[2.] M. Frost and C. Storey, ``{Further remarks on the controllability of linear constant delay-differential systems}", {\em International J. Control,} Vol.~30, No.~5, pp.~863-870, 1979.
\item[3.] A. I. G. Vardulakis, {\em Linear Multivariable Control}, John Wiley $\&$ Sons Ltd., 1991.
%\item[4.] L. Weiss, ``{On the controllability of delay-differential systems}", {\em SIAM J. Control,} Vol.~5, No.~4, pp.~575-587, 1967.
\end{itemize}
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