Synthetic parametric model

Introduction

On this page you will find a synthetic parametric model for which one can easily experiment with different system orders $n$, values of the parameter $\epsilon$, as well as different poles and residues.

Also, the decay of the Hankel singular values can be changed indirectly through the parameter $\epsilon$.

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.

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

and the residues also form complex conjugate pairs

$r_1=c_1+j{d}_1,r_2=c_1-j{d}_1,\dots ,r_{n-1}=c_k+j{d}_k,r_n=c_k-j{d}_k.$

Then a realization with matrices having real entries is given by

$A_{\epsilon }=\left[\begin{array}{ccc}{A}_{\epsilon ,1}& & \\ & \ddots & \\ & & A_{\epsilon ,k}\end{array}\right],A_0=\left[\begin{array}{ccc}{A}_{0,1}& & \\ & \ddots & \\ & & A_{0,k}\end{array}\right],B=\left[\begin{array}{c}{B}_1\\ ⋮\\ B_k\end{array}\right],C=\left[\begin{array}{ccc}{C}_1& \cdots & C_k\end{array}\right],D=0,$

with $A_{\epsilon ,i}=\left[\begin{array}{cc}{a}_i& 0\\ 0& a_i\end{array}\right]$, $A_{0,i}=\left[\begin{array}{cc}0& b_i\\ -b_i& 0\end{array}\right]$, $B_i=\left[\begin{array}{c}2\\ 0\end{array}\right]$, $C_i=\left[\begin{array}{cc}{c}_i& d_i\end{array}\right]$.

Numerical values

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

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

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

• $c_i=1$,

• $d_i=0$,

• $\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).';
c = ones(n/2,1);                d = zeros(n/2,1);
aa(1:2:n-1,1) = a;              aa(2:2:n,1) = a;
bb(1:2:n-1,1) = b;              bb(2:2:n-2,1) = 0;
Ae = spdiags(aa,0,n,n);
A0 = spdiags([0;bb],1,n,n) + spdiags(-bb,-1,n,n);
B = 2*sparse(mod([1:n],2)).';
C(1:2:n-1) = c.';               C(2:2:n) = d.';   C = sparse(C);

Plots

We plot the frequency response and poles for a few different parameter values $\epsilon \in [1/50,1/20,1/10,1/5,1/2,1]$.

Frequency response of synthetic parametrized system, for parameter values 1/50 (blue), 1/20 (green), 1/10 (red), 1/5 (teal), 1/2 (purple), 1 (yellow).

Poles of synthetic parametrized system, for parameter values 1/50 (blue), 1/20 (green), 1/10 (red), 1/5 (teal), 1/2 (purple), 1 (yellow).

These plots can be obtained in MATLAB using the following commands

r(1:2:n-1,1) = c+1j*d;     r(2:2:n,1) = c-1j*d;
ep = [1/50; 1/20; 1/10; 1/5; 1/2; 1];                       % parameter epsilon
jw = 1j*linspace(0,1.2e3,5000).';                           % frequency grid
for j = 1:length(ep)
  p(:,j) = [ep(j)*a+1j*b; ep(j)*a-1j*b];                    % poles
  [jww,pp] = meshgrid(jw,p(:,j));
  Hjw(j,:) = (r.')*(1./(jww-pp));                           % freq. resp.
end
figure,  loglog(imag(jw),abs(Hjw),'LineWidth',2)
         axis tight,    xlim([6 1200])
         xlabel('frequency (rad/sec)')
         ylabel('magnitude')
         title('Frequency response for different \epsilon')
figure,  plot(real(p),imag(p),'.')
         title('Poles for different \epsilon')