Synthetic parametric model

Introduction

On this page you will find a purely synthetic parametric model. The goal is to have a simple parametric model which one can use to experiment with different system orders, parameter values etc.

System description

The parameter $\theta$ scales the real part of the system poles, that is, $p_k=\theta a_k+j{b}_k$. If the system is in pole-residue form, then

$H(s)=\sum _{i=1}^n\frac{r_i}{s-p_i}=\sum _{i=1}^n\frac{r_i}{s-(\theta a_i+j{b}_i)},$

which has the state-space realisation

$\hat{A}=\theta \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}([a_1,\dots ,a_n])+\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}([j{b}_1,\dots ,j{b}_n]),$

$\hat{B}=[1,\dots ,1{]}^T,\hat{C}=[r_1,\dots ,r_n],D=0.$

Notice that the system matrices have complex entries.

For simplicity, assume that $n$ is even, $n=2k$, and that all system poles are complex and ordered in complex conjugate pairs, i.e.

$p_1=a_1+j{b}_1,p_2=a_1-j{b}_1,\dots ,p_{n-1}=a_k+j{b}_k,p_n=a_k-j{b}_k.$

Which also implies that the residues form complex conjugate pairs $r_1,{\overline{r}}_1,\dots ,r_k,{\overline{r}}_k.$

Then a realization with matrices having real entries is given by

$A=T\hat{A}{T}^{*},B=T\hat{B},C=\hat{C}{T}^{*},D=0,$

with the matrix $T$ defined using $2\times 2$ diagonal blocks.