Speaker
Description
Consider the linear time-invariant time-delay system (TDS)
$$ \begin{align*} \frac{\mathrm{d}\mathbf{x}(t)}{\mathrm{d}t} &= A_0 \mathbf{x}(t) + \sum_{j=1}^{N_1} A_j \mathbf{x}(t - \tau_j) + \sum_{j=1}^{N_2} B_j \mathbf{u}(t- t_j), \\\\ \mathbf{y}(t) &= \sum_{j=1}^{N_3} C_j \mathbf{x}(t-s_j) + \sum_{j=1}^{N_4} D_j \mathbf{u}(t-h_j), \end{align*} $$ 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 $$ \mathbf{S}(\lambda) := \begin{bmatrix} A(\lambda) & B(\lambda) \\\hline -C(\lambda) & D(\lambda) \end{bmatrix} \quad \text{and} \quad \mathbf{M}(\lambda) := D(\lambda) + C(\lambda) A(\lambda)^{-1} B(\lambda) $$ where - $A(\lambda) := \lambda I_r - A_0 - \sum_{j=1}^{N_1} A_j e^{-\lambda \tau_j}$ - $B(\lambda) := \sum_{j=1}^{N_2} B_j e^{-\lambda t_j}$ - $C(\lambda) := \sum_{j=1}^{N_3} C_j e^{-\lambda s_j}$ - $D(\lambda) := \sum_{j=1}^{N_4} D_j e^{-\lambda h_j}$ 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 $$ \Sigma_{\mathbf{M}}(\lambda) = \begin{bmatrix} \frac{\phi_1(\lambda)}{\psi_1(\lambda)} & & & \\\\ & \ddots & & \\\\ & & \frac{\phi_p(\lambda)}{\psi_p(\lambda)} & \\\\ \hline & & & 0_{m-p\times n-p} \end{bmatrix}, $$ 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 $j=1 \ldots p-1$. Furthermore, $$ \phi_j(\lambda) = \prod_{\ell=1}^\infty (\lambda- \lambda_\ell)^{\partial_j(\lambda_\ell)} u_{j\ell}(\lambda), $$ and $$ \psi_j(\lambda) = \prod_{\ell=1}^\infty (\lambda- \mu_\ell)^{\delta_j(\mu_\ell)} v_{j\ell}(\lambda), $$ where $u_{j\ell}$ and $v_{j\ell}$ are entire functions with no zeros in $\mathbb{C}$, $\lambda_\ell$ and $\mu_\ell$ are zeros and poles of $\mathbf{M}(\lambda)$, $\partial_j({\lambda_\ell})$ and $\delta_j(\mu_\ell)$ 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 *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 $$ \begin{align*} S_A(\lambda) &= I_{r-p} \oplus \mathrm{diag}(\psi_p(\lambda), \psi_{p-1}(\lambda), \psi_1(\lambda)) \\\\ S_{\mathbf{S}}(\lambda) &= I_r \oplus \mathrm{diag}(\phi_1(\lambda), \ldots, \phi_p(\lambda)) \oplus 0_{m-p, n-p} \end{align*} $$ 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 $\wp_{\mathbb{C}}(\mathbf{M}) = \sigma_{\mathbb{C}}(A)$.
