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.[1]

Definition

Consider the dynamic system

${\displaystyle {\dot {x}}=Ax+Bu,}$

${\displaystyle y=Cx+Du.}$

The Rosenbrock system matrix is given by

${\displaystyle P(s)={\begin{pmatrix}sI-A&-B\\C&D\end{pmatrix}}.}$

In the original work by Rosenbrock, the constant matrix ${\displaystyle D}$ is allowed to be a polynomial in ${\displaystyle s}$.

The transfer function between the input ${\displaystyle i}$ and output ${\displaystyle j}$ is given by

${\displaystyle g_{ij}={\frac {\begin{vmatrix}sI-A&-b_{i}\\c_{j}&d_{ij}\end{vmatrix}}{|sI-A|}}}$

where ${\displaystyle b_{i}}$ is the column ${\displaystyle i}$ of ${\displaystyle B}$ and ${\displaystyle c_{j}}$ is the row ${\displaystyle j}$ of ${\displaystyle C}$.

Based in this representation, Rosenbrock developed his version of the PBH test.

Short form

For computational purposes, a short form of the Rosenbrock system matrix is more appropriate[2] and given by

${\displaystyle P\sim {\begin{pmatrix}A&B\\C&D\end{pmatrix}}.}$

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.[3] An interpretation of the Rosenbrock System Matrix as a Linear Fractional Transformation can be found in.[4]

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[5] and GNU Octave.

References

1. Rosenbrock, H. H. (1967). "Transformation of linear constant system equations". Proc. IEE. 114: 541–544.
2. Rosenbrock, H. H. (1970). State-Space and Multivariable Theory. Nelson.
3. "Mu Analysis and Synthesis Toolbox". Retrieved 25 August 2014.
4. Zhou, Kemin; Doyle, John C.; Glover, Keith (1995). Robust and Optimal Control. Prentice Hall.
5. De Schutter, B. (2000). "Minimal state-space realization in linear system theory: an overview". Journal of Computational and Applied Mathematics. 121 (1–2): 331–354. Bibcode:2000JCoAM.121..331S. doi:10.1016/S0377-0427(00)00341-1.