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, $p_k=\theta a_k+jb_k$. If the system is in pole-residue form, then

$H(s) = \sum_{i=1}^{n}\frac{r_i}{s-p_i} = \sum_{i=1}^{n}\frac{r_i}{s-(\theta a_i+jb_i)} ,$

which has the state-space realisation

$\widehat{A} = \theta \mathrm{diag}~([a_1,\ldots,a_n])+\mathrm{diag}~([jb_1,\ldots,jb_n]) ,$

$\widehat{B} = [1,\ldots,1]^T,\quad \widehat{C} = [r_1,\ldots,r_n],\quad D = 0.$

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.

$p_1 = a_1+jb_1, p_2 = a_1-jb_1, \ldots, p_{n-1} = a_k+jb_k, p_n = a_k-jb_k.$

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

Then a realization with matrices having real entries is given by

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

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