Synthetic parametric model

Introduction

On this page you will find a synthetic parametric model for which one can easily change system order, parameter values, poles, residues. The decay of Hankel singular values can also be changed indirectly by changing the value of the parameter.

System description

The parameter $\varepsilon$ scales the real part of the system poles, that is, $p_i=\varepsilon a_i+jb_i$. For a system in pole-residue form

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

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

$\varepsilon \widehat{A}_\varepsilon + \widehat{A}_0 = \varepsilon \left[\begin{array}{ccc} a_1 & & \\ & \ddots & \\ & & a_n\end{array}\right] +\left[\begin{array}{ccc} jb_1 & & \\ & \ddots & \\ & & jb_n\end{array}\right] ,$

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

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.

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

and the residues also form complex conjugate pairs

$r_1 = c_1+jd_1, r_2 = c_1-jd_1, \ldots, r_{n-1} = c_k+jd_k, r_n = c_k-jd_k.$

Then a realization with matrices having real entries is given by

$A_\varepsilon = \left[\begin{array}{ccc} A_{\varepsilon,1} & & \\ & \ddots & \\ & & A_{\varepsilon,k}\end{array}\right], \quad A_0 = \left[\begin{array}{ccc} A_{0,1} & & \\ & \ddots & \\ & & A_{0,k}\end{array}\right], \quad B = \left[\begin{array}{c} B_1 \\ \vdots \\ B_k\end{array}\right], \quad C = \left[\begin{array}{ccc} C_1 & \cdots & C_k\end{array}\right], \quad D = 0,$

with $A_{\varepsilon,i} = \left[\begin{array}{cc} a_i& 0 \\ 0 & a_i \end{array}\right]$, $A_{0,i} = \left[\begin{array}{cc} 0& b_i \\ -b_i & 0 \end{array}\right]$, $B_{i} = \left[\begin{array}{c} 2 \\ 0 \end{array}\right]$, $C_{i} = \left[\begin{array}{cc} c_i& d_i\end{array}\right]$.

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 $[10, 10^3]$,

• $c_i = 1$,

• $d_i = 0$,

• $\varepsilon$$\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 $\varepsilon \in [1/50, 1/20, 1/10, 1/5, 1/2, 1]$.

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')