On Extended Model Order Reduction for Linear Time Delay Systems

Abstract

This chapter presents a so-called extended model-reduction technique for linear delay differential equations. The presented technique preserves the infinite-dimensional nature of the system and facilitates the preservation of properties such as system parameterizations (uncertainties). It is proved in this chapter that the extended model-reduction technique also preserves stability properties and provides a guaranteed a-priori bound on the reduction error. The reduction technique relies on the solution of matrix inequalities that characterize controllability and observability properties for time delay systems. This work presents conditions on the feasibility of these inequalities, and studies the applicability of the extended model reduction to a spatio-temporal model of neuronal activity, known as delay neural fields. Lastly, it discusses the relevance of this technique in the scope of model reduction of uncertain time delay systems, which is supported by a numerical example.

Appendix A. Derivation of X(r)

We consider $w_{ij}(r,r')$, for $i,j=1,2$. This function can be written in the following general form

$$ \begin{aligned} W_{ij}(r,r')&={k_{ij}}\exp \left( {-\frac{{{{\left| {r - r' - {\mu _{ij}}} \right| }^2}}}{{2{\sigma _{ij} }}}} \right) \\&= {k_{ij}}\exp \left( {-\frac{{{{\left| r \right| }^2}}}{{2{\sigma _{ij} }}}} \right) \exp \left( {-\frac{{{{\left| {r' + {\mu _{ij}}} \right| }^2}}}{{2{\sigma _{ij}}}}} \right) \exp \left( {\frac{{r(r'+\mu _{ij})}}{{{\sigma _{ij}}}}} \right) \end{aligned} $$

for some constants $k_{ij}, \sigma _{ij}$ and $\mu _{ij}$. We wish to decompose $w_{ij}(r,r')$ into a multiplication of only-r and only-$r'$ dependent functions. However, the term $\exp (r(r'+\mu _{ij})/\sigma _{ij})$ cannot be directly decomposed into such a desirable form. To cope with this issue, we use the Taylor series approximation of order $\rho$ of this term to obtain

$$\exp \left( {\frac{{r(r'+\mu _{ij})}}{{{\sigma _{ij}}}}} \right) \approx \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 1&r&{{r^2}} \end{array}}&{...}&r^{\rho } \end{array}} \right] {\left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 1&{\frac{{(r'+\mu _{ij})}}{{{\sigma _{ij}}}}}&{\frac{{(r'+\mu _{ij}){^2}}}{{2\sigma _{ij}^2}}} \end{array}}&{...}&{\frac{{(r'+\mu _{ij}){^{\rho }}}}{{\rho !\sigma _{ij}^\rho }}} \end{array}} \right] ^T},$$

where $\rho$ is the order of approximation. With this approximation, we can now write

$$\begin{aligned} {w_{ij}}(r,r') \approx {f_{ij}}(r)g_{ij}^T(r') \end{aligned}$$

where

$$\begin{aligned} {f_{ij}}(r)&= \left[ {\begin{array}{*{20}{c}} {{f_{ij,0}}(r)}&\cdots&{{f_{ij,\rho }}(r)} \end{array}} \right] ,\\ {g_{ij}}(r)&= \left[ {\begin{array}{*{20}{c}} {{g_{ij,0}}(r)}&\cdots&{{g_{ij,\rho }}(r)} \end{array}} \right] \end{aligned}$$

with

$$\begin{aligned} {f_{ij,m}}(r)&= {r^{m }}\exp \left( -{\frac{{{{\left| r \right| }^2}}}{{2{\sigma _{ij}}}}} \right) ,\\{g_{ij,m}}(r')&={k_{ij}} \frac{{{{\left( {r' + {\mu _{ij}}} \right) }^{m }}}}{{m!\sigma _{ij}^{m }}}\exp \left( -{\frac{{{{\left| {r' + {\mu _{ij}}} \right| }^2}}}{{2{\sigma _{ij}}}}} \right) ,\quad m = 0,2,...,\rho . \end{aligned}$$

With this representation of $w(r,r')$, we may choose

$$\begin{aligned} {X_1}&= \left[ {\begin{array}{*{20}{c}} 0\\ {{f_{22}}} \end{array}} \right] ,\quad {X_2} = \left[ {\begin{array}{*{20}{c}} {{f_{12}}} &{}0\\ 0 &{}{{f_{21}}} \end{array}} \right] ,\\ {Y_1}&= \left[ {\begin{array}{*{20}{c}} 0\\ {{g_{22}}} \end{array}} \right] ,\quad {Y_2} = \left[ {\begin{array}{*{20}{c}} 0 &{}{{g_{21}}}\\ {{g_{12}}} &{}0 \end{array}} \right] . \end{aligned}$$

With this choice of $X_1$ and $X_2$, we obtain $N_1 = \rho$ and $N_2 = 2\rho$. We also note that this choice of $X_1$ and $X_2$ leads to $w_{11}=0$.