fCoprime – IQClab

The function $[C_\mathrm{l},C_\mathrm{r}]=fCoprime(G)$ computes the coprime factors of the stabilizable and detectable state space realization $G$ .

For each proper real rational transfer matrix $G$ , there exist eight transfer matrices in $RH_\infty$ satisfying the equations:

$G=NM^{-1}=\tilde{M}^{-1}\tilde{N}$

and

$\left(\begin{array}{cc}\tilde{X}&-\tilde{Y}\\-\tilde{N}&\tilde{M} \end{array}\right) \left(\begin{array}{cc}M&Y\\N&X\end{array}\right)=I.$

The pairs $(N,M)$ and $(\tilde{N},\tilde{M})$ constitute the right and left coprime factors of the transfer matrix $G$ , respectively. They are said to be normalized when each of the transfer matrices $\left(\begin{array}{c}M\\N\end{array}\right)$ , $\left(\begin{array}{cc}\tilde{N}&\tilde{M}\end{array}\right)$ is norm preserving. The second equation is known as the Bezout identity. The state space formulae for normalized factors, and their certificates of coprimeness are given as follows.

Let $G=ss(A,B,C,D)$ be a stabilizable and detectable realization and choose $R$ , $S$ , and $\tilde{A}$ as:

$\begin{array}{c}R^TR=(I+D^TD)^{-1}\\ SS^T=(I+DD^T)^{-1}\\\tilde{A}=A-B (I+D^TD)^{-1}D^TC\end{array}$

Let $P$ and $Z$ be the unique stabilizing solutions for each of the following Algebraic Riccati equations (AREs), respectively:

$\begin{array}{c}\tilde{A}^TP+P\tilde{A}-PB(I+D^TD)^{-1}B^TP+C^T(I+DD^T)^{-1}C=0\\ \tilde{A} Z+ Z\tilde{A}^T-ZC^T(I+DD^T)^{-1}CZ+B(I+D^TD)^{-1}B^T=0\end{array}$

Define the state feedback and observer gains $F$ and $H$ as:

$\begin{array}{c}F=-(I+D^TD)^{-1}(B^TP+D^TC) \\H=-(BD^T+ZC^T)(I+DD^T)^{-1}\end{array}$

Then

$\begin{array}{c} T_\mathrm{l}=\left(\begin{array}{cc} \tilde{X}&-\tilde{Y}\\-\tilde{N}&\tilde{M} \end{array}\right)= \left[\begin{array}{c|cc}A+HC&-(B+HD)&H\\ \hline R^{-1}F& R^{-1} &0\\ SC&-SD&S\end{array}\right]\\T_\mathrm{r}=\left(\begin{array}{cc}M&Y\\N&X\end{array}\right)= \left[\begin{array}{c|cc}A+BF&BR&-HS^{-1}\\ \hline F&R&0\\ C+DF&DR&D^{-1}\end{array}\right]\end{array}$

satisfy the Bezout identity and the pairs $(N,M)$ and $(\tilde{N},\tilde{M})$ are normalized according to the definition above.

As output the function provides the structures $C_\mathrm{l}$ and $C_\mathrm{r}$ with the realizations of respectively:

$T_\mathrm{l}$ , $M$ , $N$ , $X$ , $Y$
$T_\mathrm{r}$ , $\tilde{M}$ , $\tilde{N}$ , $\tilde{X}$ , $\tilde{Y}$