Synthetic parametric model: Difference between revisions

Help

Revision as of 11:22, 29 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, $p_{i} = ε a_{i} + j b_{i}$ . For a system in pole-residue form

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

we can write down the state-space realisation $H (s, ε) = \hat{C} (s I - ε {\hat{A}}_{ε} - {\hat{A}}_{0})^{- 1} \hat{B} + D$ with

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

\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.

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

and the residues also form complex conjugate pairs

$r_{1} = c_{1} + j d_{1}, r_{2} = c_{1} - j d_{1}, \dots, r_{n - 1} = c_{k} + j d_{k}, r_{n} = c_{k} - j d_{k} .$

Then a realization with matrices having real entries is given by

A_{ε} = [\begin{array}{ccc} A_{ε}^{1} \\ ⋱ \\ A_{ε}^{k} \end{array}], A_{0} = [\begin{array}{ccc} A_{0}^{1} \\ ⋱ \\ A_{0}^{k} \end{array}], B = T \hat{B}, C = \hat{C} T^{*}, D = 0,

with $T = [\begin{array}{ccc} T_{0} \\ ⋱ \\ T_{0} \end{array}]$ and $T_{0} = \frac{1}{\sqrt{2}} [\begin{array}{cc} 1 & - j \\ 1 & j \end{array}]$ .

Numerical values

We construct a system of order $n = 100$ . The numerical values for the different variables are

$r_{i} = 1$ ,

$a_{i}$ equally spaced in $[10, 1 0^{3}]$ ,

$b_{i}$ equally spaced in $[10, 1 0^{3}]$ ,

$ε$ $\in [1 / 50, 1]$ .

In MATLAB the system matrices are easily formed as follows

n = 100; a = -linspace(1e1,1e3,n/2);

b = linspace(1e1,1e3,n/2);

@@ Line 27: / Line 27: @@
 <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, for real systems, also implies that the residues form complex conjugate pairs <math>r_1, \bar{r}_1,\ldots , r_k, \bar{r}_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 = T\widehat{A}_\varepsilon T^*, \quad A_0 = T\widehat{A}_0 T^*, \quad B = T\widehat{B}, \quad C = \widehat{C}T^*, \quad D = 0,</math>
+:<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 = T\widehat{B}, \quad C = \widehat{C}T^*, \quad D = 0,</math>