Synthetic parametric model: Difference between revisions

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Revision as of 10:22, 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 $ε$ scales the real part of the system poles, that is, $p_{k} = ε a_{k} + j b_{k}$ . If the system is in pole-residue form, then

$H (s) = \sum_{i = 1}^{n} \frac{r_{i}}{s - p_{i}} = \sum_{i = 1}^{n} \frac{r_{i}}{s - (ε a_{i} + j b_{i})},$

which has the state-space realisation

$\hat{A} = ε [\begin{array}{ccc} a_{1} \\ ⋱ \\ a_{n} \end{array}] + [\begin{array}{ccc} j b_{1} \\ ⋱ \\ j b_{n} \end{array}] = ε {\hat{A}}_{ε} + {\hat{A}}_{0},$

$\hat{B} = [1, \dots, 1]^{T}, \hat{C} = [r_{1}, \dots, r_{n}], D = 0 .$

Notice that the system matrices have complex entries.

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

$p_{1} = ε a_{1} + j b_{1}, p_{2} = ε a_{1} - j b_{1}, \dots, p_{n - 1} = ε a_{k} + j b_{k}, p_{n} = ε a_{k} - j b_{k} .$

Which also implies that the residues form complex conjugate pairs $r_{1}, {\bar{r}}_{1}, \dots, r_{k}, {\bar{r}}_{k} .$

Then a realization with matrices having real entries is given by

$A = T \hat{A} T^{*}, B = T \hat{B}, C = \hat{C} T^{*}, D = 0,$

with the matrix $T$ defined using $2 \times 2$ diagonal blocks.

@@ Line 5: / Line 5: @@
 == System description ==
-The parameter <math>\theta</math> scales the real part of the system poles, that is, <math>p_k=\theta a_k+jb_k</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
-<math>H(s) =  \sum_{i=1}^{n}\frac{r_i}{s-p_i} =  \sum_{i=1}^{n}\frac{r_i}{s-(\theta a_i+jb_i)} ,</math>
+<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
-<math>\widehat{A} = \theta \mathrm{diag}~([a_1,\ldots,a_n])+\mathrm{diag}~([jb_1,\ldots,jb_n]) ,</math>
+<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>
@@ Line 20: / Line 20: @@
 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 = 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>
@@ Line 29: / Line 29: @@
 with the matrix <math> T </math> defined using <math> 2\times 2 </math> diagonal blocks.
+== Example ==