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:



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

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:

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


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

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}

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