(Synthetic parametric model, created for experimental purposes.) |
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== System description == |
== System description == |
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− | The parameter <math>\ |
+ | The parameter <math>\varepsilon</math> scales the real part of the system poles, that is, <math>p_k=\varepsilon a_k+jb_k</math>. |
If the system is in pole-residue form, then |
If the system is in pole-residue form, then |
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− | <math>H(s) = \sum_{i=1}^{n}\frac{r_i}{s-p_i} = \sum_{i=1}^{n}\frac{r_i}{s-(\ |
+ | <math> H(s) = \sum_{i=1}^{n}\frac{r_i}{s-p_i} = \sum_{i=1}^{n}\frac{r_i}{s-(\varepsilon a_i+jb_i)} ,</math> |
which has the state-space realisation |
which has the state-space realisation |
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− | <math>\widehat{A} = \ |
+ | <math>\widehat{A} = \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] = \varepsilon \widehat{A}_\varepsilon + \widehat{A}_0,</math> |
<math>\widehat{B} = [1,\ldots,1]^T,\quad \widehat{C} = [r_1,\ldots,r_n],\quad D = 0.</math> |
<math>\widehat{B} = [1,\ldots,1]^T,\quad \widehat{C} = [r_1,\ldots,r_n],\quad D = 0.</math> |
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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. |
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. |
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− | <math> 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. </math> |
+ | <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> |
Which also implies that the residues form complex conjugate pairs <math>r_1, \bar{r}_1,\ldots , r_k, \bar{r}_k.</math> |
Which also implies that the residues form complex conjugate pairs <math>r_1, \bar{r}_1,\ldots , r_k, \bar{r}_k.</math> |
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with the matrix <math> T </math> defined using <math> 2\times 2 </math> diagonal blocks. |
with the matrix <math> T </math> defined using <math> 2\times 2 </math> diagonal blocks. |
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+ | == Example == |
Revision as of 12:22, 28 November 2011
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 scales the real part of the system poles, that is,
.
If the system is in pole-residue form, then
which has the state-space realisation
Notice that the system matrices have complex entries.
For simplicity, assume that is even,
, and that all system poles are complex and ordered in complex conjugate pairs, i.e.
Which also implies that the residues form complex conjugate pairs
Then a realization with matrices having real entries is given by
with the matrix defined using
diagonal blocks.