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Rosenbrock System Matrix
The Rosenbrock System Matrix (or Rosenbrock's system matrix) of an linear time invariant system is an useful representation of a state-space system.

Definition
Consider the dynamic systems
 * $$\dot{x}= Ax +Bu,$$
 * $$y= Cx +Du,$$

then the Rosenbrock system matrix is given by
 * $$P(s)=\begin{pmatrix}

sI-A & B\\ -C & D \end{pmatrix}.$$ In the original work by Rosenbrock, the constant matrix $$D$$ is allowed to be a polynomial in $$s$$.

The transfer function between the input $$i$$ and output $$j$$ is given by
 * $$g_{ij}=\frac{\begin{vmatrix}

sI-A & b_i\\ -c_j & d_{ij} \end{vmatrix}}{|sI-A|}$$ where $$b_i$$ is the column $$i$$ of $$B$$ and $$c_j$$ is the row $$j$$ of $$C$$.

Based in this representation, Rosenbrock developed a version of the PHB test.

Rosenbrock form
In page 79 in, the Rosembrock form of the system is proposed as follows
 * $$\begin{pmatrix}

A & B\\ C & D \end{pmatrix},$$ for computational purposes. The Rosenbrock form of a system has widely used in $$H_\infty$$, also referred to as packed form, see command pck in

The first application of the Rosenbrock form was the development of a computational method to Kalman Decomposition. It is based on pivot element method.