Synthetic parametric model

2011-11-29T11:47:42Z

Ionita: /* Numerical values */

== 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 <math>\varepsilon</math> scales the real part of the system poles, that is, <math>p_i=\varepsilon a_i+jb_i</math>.
For a system in pole-residue form

:<math> 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)} </math>

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

:<math>\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] ,</math>

:<math>\widehat{B} = [1,\ldots,1]^T,\quad \widehat{C} = [r_1,\ldots,r_n],\quad D = 0.</math>

Notice that the system matrices have complex entries.

For simplicity, assume that <math> n </math> is even, <math> n=2k </math>, and that all system poles are complex and ordered in complex conjugate pairs, i.e.

<math> 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, </math>

and the residues also form complex conjugate pairs

<math> 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. </math>

Then a realization with matrices having real entries is given by

:<math> 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,</math>

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

== Numerical values ==

We construct a system of order <math>n = 100</math>. The numerical values for the different variables are

* <math>a_i </math> equally spaced in <math> [-10^3, -10]</math>,

* <math>b_i </math> equally spaced in <math>[10, 10^3]</math>,

* <math> c_i = 1</math>,

* <math> d_i = 0</math>,

* <math>\varepsilon</math><math> \in [1/50,1]</math>.

In MATLAB the system matrices are easily formed as follows

<tt>
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);
</tt>

Synthetic parametric model

2011-11-29T11:39:40Z

Ionita: /* Numerical values */

== 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 <math>\varepsilon</math> scales the real part of the system poles, that is, <math>p_i=\varepsilon a_i+jb_i</math>.
For a system in pole-residue form

:<math> 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)} </math>

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

:<math>\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] ,</math>

:<math>\widehat{B} = [1,\ldots,1]^T,\quad \widehat{C} = [r_1,\ldots,r_n],\quad D = 0.</math>

Notice that the system matrices have complex entries.

For simplicity, assume that <math> n </math> is even, <math> n=2k </math>, and that all system poles are complex and ordered in complex conjugate pairs, i.e.

<math> 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, </math>

and the residues also form complex conjugate pairs

<math> 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. </math>

Then a realization with matrices having real entries is given by

:<math> 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,</math>

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

== Numerical values ==

We construct a system of order <math>n = 100</math>. The numerical values for the different variables are

* <math>a_i </math> equally spaced in <math> [-10^3, -10]</math>,

* <math>b_i </math> equally spaced in <math>[10, 10^3]</math>,

* <math> c_i = 1</math>,

* <math> d_i = 0</math>,

* <math>\varepsilon</math><math> \in [1/50,1]</math>.

In MATLAB the system matrices are easily formed as follows

:<tt>
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);

</tt>

Synthetic parametric model

2011-11-29T11:38:39Z

Ionita: /* Numerical values */

== 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 <math>\varepsilon</math> scales the real part of the system poles, that is, <math>p_i=\varepsilon a_i+jb_i</math>.
For a system in pole-residue form

:<math> 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)} </math>

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

:<math>\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] ,</math>

:<math>\widehat{B} = [1,\ldots,1]^T,\quad \widehat{C} = [r_1,\ldots,r_n],\quad D = 0.</math>

Notice that the system matrices have complex entries.

For simplicity, assume that <math> n </math> is even, <math> n=2k </math>, and that all system poles are complex and ordered in complex conjugate pairs, i.e.

<math> 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, </math>

and the residues also form complex conjugate pairs

<math> 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. </math>

Then a realization with matrices having real entries is given by

:<math> 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,</math>

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

== Numerical values ==

We construct a system of order <math>n = 100</math>. The numerical values for the different variables are

* <math>a_i </math> equally spaced in <math> [-10^3, -10]</math>,

* <math>b_i </math> equally spaced in <math>[10, 10^3]</math>,

* <math> c_i = 1</math>,

* <math> d_i = 0</math>,

* <math>\varepsilon</math><math> \in [1/50,1]</math>.

In MATLAB the system matrices are easily formed as follows

:<tt>
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);

</tt>

Synthetic parametric model

2011-11-29T11:34:18Z

Ionita: /* Numerical values */

== 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 <math>\varepsilon</math> scales the real part of the system poles, that is, <math>p_i=\varepsilon a_i+jb_i</math>.
For a system in pole-residue form

:<math> 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)} </math>

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

:<math>\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] ,</math>

:<math>\widehat{B} = [1,\ldots,1]^T,\quad \widehat{C} = [r_1,\ldots,r_n],\quad D = 0.</math>

Notice that the system matrices have complex entries.

For simplicity, assume that <math> n </math> is even, <math> n=2k </math>, and that all system poles are complex and ordered in complex conjugate pairs, i.e.

<math> 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, </math>

and the residues also form complex conjugate pairs

<math> 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. </math>

Then a realization with matrices having real entries is given by

:<math> 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,</math>

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

== Numerical values ==

We construct a system of order <math>n = 100</math>. The numerical values for the different variables are

* <math>a_i </math> equally spaced in <math> [10, 10^3]</math>,

* <math>b_i </math> equally spaced in <math>[10, 10^3]</math>,

* <math> c_i = 1</math>,

* <math> d_i = 0</math>,

* <math>\varepsilon</math><math> \in [1/50,1]</math>.

In MATLAB the system matrices are easily formed as follows

:<tt>
n = 100; % system order n
a = -linspace(1e1,1e3,n/2).'; % poles p = a+jb
b = linspace(1e1,1e3,n/2).';
c = ones(n/2,1); % residues p = c+jd
d = zeros(n/2,1);
aa(1:2:n-1,1) = a; aa(2:2:n,1) = a; % system matrices
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);
</tt>