fBasis – IQClab

The function $\psi=fBasis(l,p,n_r,type)$ creates minimum state space realizations for various basis functions that are used for the parametrization of IQC-multipliers. Here

$l$ is the length of the basis function, which equals the McMillan degree $m$ plus 1 (i.e., $l=m+1$ ).
$p$ is the pole location of the basis function.
$n_r$ is the number of diagonal repetitions of the basis function
$type$ is the type of basis function. The current version includes 4 versions as specified next:

\[\psi=\left(\begin{array}{c}1\\\frac{(i\omega+pl)}{(i\omega-pl)}\\ \frac{(i\omega+pl)^2}{ (i\omega-pl)^2}\\\vdots\\ \frac{(i\omega+pl)^l}{ (i\omega-pl)^l} \end{array}\right)\otimes I_{n_r}\]

Type	Description
$type=1$	$\psi=\left(\begin{array}{c}1\\\frac{(i\omega+pl)}{(i\omega-pl)}\\ \frac{(i\omega+pl)^2}{ (i\omega-pl)^2}\\\vdots\\ \frac{(i\omega+pl)^l}{ (i\omega-pl)^l} \end{array}\right)\otimes I_{n_r}$
$type=2$	$\psi=\left(\begin{array}{c}1\\ \frac{(i\omega)}{(i\omega-pl)^{l-1}}\\ \vdots\\\frac{(i\omega)^{l-1}}{(i\omega-pl)^{l-1}}\end{array}\right)\otimes I_{n_r}$
$type=3$	$\psi=\left(\begin{array}{c}1\\\frac{1}{(i\omega-pl)}\\ \frac{1}{ (i\omega-pl)^2}\\\vdots\\ \frac{1}{(i\omega-pl)^l}\end{array}\right)\otimes I_{n_r}$
$type=4$	This is the complex conjugate of $type=3$ (i.e. $\psi=-(type=3)^*$ ).
$type=5$	This is the discrete time version of $type=1$
$type=6$	This is the discrete time version of $type=2$ (not yet implemented)
$type=7$	This is the discrete time version of $type=3$