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 $\epsilon$ scales the real part of the system poles, that is, $p_i=\epsilon a_i+j{b}_i$. For a system in pole-residue form

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

we can write down the state-space realisation $H(s,\epsilon )=\hat{C}(sI-\epsilon {\hat{A}}_{\epsilon }-{\hat{A}}_0{)}^{-1}\hat{B}+D$ with

$\epsilon {\hat{A}}_{\epsilon }+{\hat{A}}_0=\epsilon \left[\begin{array}{ccc}{a}_1& & \\ & \ddots & \\ & & a_n\end{array}\right]+\left[\begin{array}{ccc}j{b}_1& & \\ & \ddots & \\ & & j{b}_n\end{array}\right],$

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

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.

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

which, for real systems, 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

$A_{\epsilon }=T{\hat{A}}_{\epsilon }{T}^{*},A_0=T{\hat{A}}_0{T}^{*},B=T\hat{B},C=\hat{C}{T}^{*},D=0,$

with $T=\left[\begin{array}{ccc}{T}_0& & \\ & \ddots & \\ & & T_0\end{array}\right]$ and $T_0=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& -j\\ 1& j\end{array}\right]$.

Numerical values

We construct a system of order $n=100$. The numerical values for the different variables are

• $r_i=1$,

• $a_i$ equally spaced in $[10,10^3]$,

• $b_i$ equally spaced in $[10,10^3]$,

• $\epsilon$$\in [1/50,1]$.

In MATLAB the system matrices are easily formed as follows

n = 100;

a = -linspace(1e1,1e3,n/2);

b = linspace(1e1,1e3,n/2);