fHinfbalreal – IQClab

The function $[G_\mathrm{b},T_\mathrm{b} ,T_\mathrm{bi}]=fHinfbalreal(G)$ computes the $H_\infty$ balanced realization for the stable system $G=ss(A,B,C,D)$ .

As output you obtain the $H_\infty$ balanced realization

$G_\mathrm{b}=\left[\begin{array}{c|c}T_\mathrm{b}AT_\mathrm{bi}& T_\mathrm{b}B\\\hline CT_\mathrm{bi}&D\end{array}\right]$

with $T_\mathrm{bi}=T_\mathrm{b}^{-1}$ .

The balancing proceeds as follows:

First, the function computes the $H_\infty$ -norm, $\gamma$ of the stable system $G$ .
With
$\begin{array}{c}D_h^TD_h=-\left(\frac{1}{\gamma}D^TD-\gamma I\right)\prec0\\D_cD_c^T=-\left(\frac{1}{\gamma}DD^T-\gamma I\right)\prec0\end{array}$
compute the solutions of the algebraic Riccati (ARE) equations
$\begin{array}{c}XA+A^TX+\frac{1}{\gamma}C^TC+\left(\bullet}\right)^T\!\!\left(D_h^TD_h-\gamma I\right)^{-1}\!\!\left(XB+\frac{1}{\gamma}C^TD\right)\\AY+YA^T+\frac{1}{\gamma}BB^T+\left(\bullet\right)^T\!\!\left(D_cD_c^T-\gamma I\right)^{-1}\!\!\left(XC^T+\frac{1}{\gamma}BD^T\right)\end{array},$
which are guaranteed to have a solution $X$ and $Y$ respectively.
These solutions can now be used to perform a standard balancing.