Rosenbrock system matrix

In applied mathematics, the Rosenbrock system matrix or Rosenbrock's system matrix of a linear time-invariant system is a useful representation bridging state-space representation and transfer function matrix form. It was proposed in 1967 by Howard H. Rosenbrock.
Definition.
Consider the dynamic system
The Rosenbrock system matrix is given by 
In the original work by Rosenbrock, the constant matrix formula_4 is allowed to be a polynomial in formula_5.
The transfer function between the input formula_6 and output formula_7 is given by 
where formula_9 is the column formula_6 of formula_11 and formula_12 is the row formula_7 of formula_14.
Based in this representation, Rosenbrock developed his version of the PHB test.
Short form.
For computational purposes, a short form of the Rosenbrock system matrix is more appropriate and given by
The short form of the Rosenbrock system matrix has been widely used in H-infinity methods in control theory, where it is also referred to as packed form; see command pck in MATLAB. An interpretation of the Rosenbrock System Matrix as a Linear Fractional Transformation can be found in.
One of the first applications of the Rosenbrock form was the development of an efficient computational method for Kalman decomposition, which is based on the pivot element method. A variant of Rosenbrock’s method is implemented in the minreal command of Matlab and
GNU Octave.