Description
On this page you will find a synthetic parametric model with one parameter for which one can easily experiment with different system orders, values of the parameter, as well as different poles and residues. Also, the decay of the Hankel singular values can be changed indirectly through the parameter.
Model
We consider a dynamical system in the frequency domain given by its pole-residue form of the transfer function
\[ \begin{align} H(s,\varepsilon) & = \sum_{k=1}^{n}\frac{r_{k}}{s-p_{k}}\\ & = \sum_{k=1}^{n}\frac{r_{k}}{s-(\varepsilon a_{k} + jb_{k})}, \end{align} \]
with \(p_{k} = \varepsilon a_{k} + jb_{k}\) the poles of the system, \(j\) the imaginary unit, and \(r_{k}\) the residues. The parameter \(\varepsilon\) is used to scale the real part of the system poles. We can write down the state-space realization of the system's transfer function as
\[ \begin{align} H(s,\varepsilon) = \widehat{C}(sI_{n} - (\varepsilon \widehat{A}_{\varepsilon} + \widehat{A}_{0}))^{-1}\widehat{B}, \end{align} \]
with the corresponding system matrices \(\widehat{A}_{\varepsilon} \in \mathbb{R}^{n \times n}\), \(\widehat{A}_{0} \in \mathbb{C}^{n \times n}\), \(\widehat{B} \in \mathbb{R}^{n}\), and \(\widehat{C}^{T} \in \mathbb{C}^{n}\) given by
\[ \begin{align} \varepsilon\widehat{A}_{\varepsilon} + \widehat{A}_{0} & = \varepsilon \begin{bmatrix} a_{1} & & \\ & \ddots & \\ & & a_{n} \end{bmatrix} + \begin{bmatrix} jb_{1} & & \\ & \ddots & \\ & & jb_{n} \end{bmatrix},\\ \widehat{B} & = \begin{bmatrix}1, & \ldots, & 1 \end{bmatrix}^{T},\\ \widehat{C} & = \begin{bmatrix}r_{1}, & \ldots, & r_{n} \end{bmatrix}. \end{align} \]
One notices that the system matrices \(\widehat{A}_{0}\) and \(\widehat{C}\) have complex entries. For rewriting the system with real matrices, we assume that \(n\) is even, \(n=2m\), and that all system poles are complex and ordered in complex conjugate pairs, i.e.,
\[ \begin{align} p_{1} & = \varepsilon a_{1} + jb_{1},\\ p_{2} & = \varepsilon a_{1} - jb_{1},\\ & \ldots\\ p_{n-1} & = \varepsilon a_{m} + jb_{m},\\ p_{n} & = \varepsilon a_{m} - jb_{m}. \end{align} \]
Corresponding to the system poles, also the residues are written in complex conjugate pairs
\[ \begin{align} r_{1} & = c_{1} + jd_{1},\\ r_{2} & = c_{1} - jd_{1},\\ & \ldots\\ r_{n-1} & = c_{m} + jd_{m},\\ r_n & = c_{m} - jd_{m}. \end{align} \]
Using this, the realization of the dynamical system can be written with matrices having real entries by
\[ \begin{align} A_{\varepsilon} & = \begin{bmatrix} A_{\varepsilon, 1} & & \\ & \ddots & \\ & & A_{\varepsilon, m} \end{bmatrix}, & A_{0} & = \begin{bmatrix} A_{0, 1} & & \\ & \ddots & \\ & & A_{0, m} \end{bmatrix}, & B & = \begin{bmatrix} B_{1} \\ \vdots \\ B_{m} \end{bmatrix}, & C & = \begin{bmatrix} C_{1}, & \cdots, & C_{m} \end{bmatrix}, \end{align} \]
with \(A_{\varepsilon, k} = \begin{bmatrix} a_{k} & 0 \\ 0 & a_{k} \end{bmatrix}\), \(A_{0, k} = \begin{bmatrix} 0 & b_{k} \\ -b_{k} & 0 \end{bmatrix}\), \(B_{k} = \begin{bmatrix} 2 \\ 0 \end{bmatrix}\), \(C_{k} = \begin{bmatrix} c_k, & d_k \end{bmatrix}\).
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);
The above system matrices \(A_\varepsilon, A_0, B, C\) are also available in MatrixMarket format Synth_matrices.tar.gz.
Plots
We plot the frequency response \(H(s,\varepsilon) = C\big(sI-\varepsilon A_\varepsilon - A_0\big)^{-1}B\) and poles for parameter values \(\varepsilon \in [1/50, 1/20, 1/10, 1/5, 1/2, 1] \).
- Error creating thumbnail: Unable to save thumbnail to destinationFrequency response of synthetic parametric system, for parameter values 1/50 (blue), 1/20 (green), 1/10 (red), 1/5 (teal), 1/2 (purple), 1 (yellow).
In MATLAB, the plots are generated 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')
Other interesting plots result for small values of the parameter. For example, for \(\varepsilon = 1/100, 1/1000 \), the peaks in the frequency response become more pronounced, since the poles move closer to the imaginary axis.
Next, for \(\varepsilon \in [1/50, 1/20, 1/10, 1/5, 1/2, 1] \), we also plot the decay of the Hankel singular values. Notice that for small values of the parameter, the decay of the Hankel singular values is very slow.