EMOSC 25: Energy-based modeling, simulation, and control of dynamical systems - Workshop in honor of Volker Mehrmann’s 70th birthday

Zeros, poles and system equivalence of time-delay systems

27 May 2025, 12:00

30m

H2036 (TU Berlin)

Talk Talks

Speaker

Rafikul Alam (IIT Guwahati)

Description

Consider the linear time-invariant time-delay system (TDS)

$$\begin{array}{rl}\frac{\mathrm{d}\mathbf{x}(t)}{\mathrm{d}t}& =A_0\mathbf{x}(t)+\sum _{j=1}^{N_1}{A}_j\mathbf{x}(t-{\tau }_j)+\sum _{j=1}^{N_2}{B}_j\mathbf{u}(t-t_j),\\ \\ \mathbf{y}(t)& =\sum _{j=1}^{N_3}{C}_j\mathbf{x}(t-s_j)+\sum _{j=1}^{N_4}{D}_j\mathbf{u}(t-h_j),\end{array}$$ 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 $$\mathbf{S}(\lambda ):=\left[\begin{array}{cc}A(\lambda )& B(\lambda )\\ -C(\lambda )& D(\lambda )\end{array}\right]\text{and}\mathbf{M}(\lambda ):=D(\lambda )+C(\lambda )A(\lambda {)}^{-1}B(\lambda )$$ where - $A(\lambda ):=\lambda I_r-A_0-\sum _{j=1}^{N_1}{A}_j{e}^{-\lambda {\tau }_j}$ - $B(\lambda ):=\sum _{j=1}^{N_2}{B}_j{e}^{-\lambda t_j}$ - $C(\lambda ):=\sum _{j=1}^{N_3}{C}_j{e}^{-\lambda s_j}$ - $D(\lambda ):=\sum _{j=1}^{N_4}{D}_j{e}^{-\lambda h_j}$ 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$\mathbf{S}_2(\lambda)$aresystemmatricesoftime-delaysystems,theninvestigateRosenbrocksystemequivalence(writtenas$\mathbf{S}_1(\lambda)\sim{rse} \mathbf{S}2(\lambda)$)aswellasFuhrmannsystemequivalence(writtenas$\mathbf{S}_1(\lambda)\sim{fse} \mathbf{S}2(\lambda)$).Weshowthat$\mathbf{M}(\lambda)$admitsaSmith-McMillanform$\Sigma{\mathbf{M}}(\lambda)$givenby$${\mathrm{\Sigma }}_{\mathbf{M}}(\lambda )=\left[\begin{array}{cccc}\frac{{\varphi }_1(\lambda )}{{\psi }_1(\lambda )}& & & \\ \\ & \ddots & & \\ \\ & & \frac{{\varphi }_p(\lambda )}{{\psi }_p(\lambda )}& \\ \\ & & & 0_{m-p\times n-p}\end{array}\right],$$where$\phi_1, \ldots, \phi_p$and$\psi_1, \ldots, \psi_p$areentirefunctions,$\phi_j$and$\psi_j$arerelativelyprime.Further,$\phi_j$divides$\phi_{j+1}$and$\psi_{j+1}$divides$\psi_j$for$j=1 \ldots p-1$.Furthermore,$${\varphi }_j(\lambda )=\prod _{\ell =1}^{\mathrm{\infty }}(\lambda -{\lambda }_{\ell }{)}^{{\partial }_j({\lambda }_{\ell })}{u}_{j\ell }(\lambda ),$$and$${\psi }_j(\lambda )=\prod _{\ell =1}^{\mathrm{\infty }}(\lambda -{\mu }_{\ell }{)}^{{\delta }_j({\mu }_{\ell })}{v}_{j\ell }(\lambda ),$$where$u_{j\ell}$and$v_{j\ell}$areentirefunctionswithnozerosin$\mathbb{C}$,$\lambda_\ell$and$\mu_\ell$arezerosandpolesof$\mathbf{M}(\lambda)$,$\partial_j({\lambda_\ell})$and$\delta_j(\mu_\ell)$areappropriatenon-negativeintegersfor$j=1:p$and$\ell \in \mathbb{N}$.Wealsoshowthat$\mathbf{M}(\lambda)$admitsarightcoprimematrix-fractiondescription(MFD),thatis,thereexistentirematrix-valuedfunctions$N(\lambda)$and$D(\lambda)$suchthat$N(\lambda)$and$D(\lambda)$arerightcoprime,$D(\lambda)$isregular,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$\sigma_{\mathbb{C}}(X)$isthespectrum(setofzeros)and$\wp_{\mathbb{C}}(X)$isthesetofpolesofameromorphicmatrix-valuedfunction$X(\lambda)$.Forsystemmatrices,weshowthat$${\mathbf{S}}_1(\lambda ){\sim }_{rse}{\mathbf{S}}_2(\lambda )⟺{\mathbf{S}}_1(\lambda ){\sim }_{fse}{\mathbf{S}}_2(\lambda ).$$Further,if$\mathbf{S}1(\lambda)$and$\mathbf{S}_2(\lambda)$are\ast irreducible\ast then$${\mathbf{S}}_1(\lambda ){\sim }_{rse}{\mathbf{S}}_2(\lambda )⟺{\mathbf{S}}_1(\lambda )\text{ and }{\mathbf{S}}_2(\lambda )$$havethesametransferfunction.Finally,if$\mathbf{S}(\lambda)$isirreducibleand$\Sigma{\mathbf{M}}(\lambda)$istheSmith-McMillanformof$\mathbf{M}(\lambda)$asgivenabove,thenweshowthat$$\begin{array}{rl}{S}_A(\lambda )& =I_{r-p}\oplus \mathrm{diag}({\psi }_p(\lambda ),{\psi }_{p-1}(\lambda ),{\psi }_1(\lambda ))\\ \\ S_{\mathbf{S}}(\lambda )& =I_r\oplus \mathrm{diag}({\varphi }_1(\lambda ),\dots ,{\varphi }_p(\lambda ))\oplus 0_{m-p,n-p}\end{array}$$aretheSmithformsof$A(\lambda)$and$\mathbf{S}(\lambda)$,respectively.Henceweshowthat$\sigma_{\mathbb{C}}(\mathbf{M}) = \sigma_{\mathbb{C}}(\mathbf{S})$and$\wp_{\mathbb{C}}(\mathbf{M}) = \sigma_{\mathbb{C}}(A)$.