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.