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.{{cite journal|last1=Rosenbrock|first1=H. H.|title=Transformation of linear constant system equations|journal=Proc. IEE|date=1967|volume=114|pages=541–544}}

Definition

Consider the dynamic system

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

:: $y= Cx +Du.$

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 his version of the PBH test.

Short form

For computational purposes, a short form of the Rosenbrock system matrix is more appropriate{{cite book|last1=Rosenbrock|first1=H. H.|title=State-Space and Multivariable Theory|date=1970|publisher=Nelson}} and given by

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

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.{{cite web|title=Mu Analysis and Synthesis Toolbox|url=http://radio.feld.cvut.cz/matlab/toolbox/mutools/pck.html|accessdate=25 August 2014}} An interpretation of the Rosenbrock System Matrix as a Linear Fractional Transformation can be found in.{{cite book|last1=Zhou|first1=Kemin|last2=Doyle|first2=John C.|last3=Glover|first3=Keith|title=Robust and Optimal Control|date=1995|publisher=Prentice Hall}}

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{{cite journal|last1=De Schutter|first1=B.|title=Minimal state-space realization in linear system theory: an overview|journal=Journal of Computational and Applied Mathematics|date=2000|volume=121|issue=1–2|pages=331–354|doi=10.1016/S0377-0427(00)00341-1|bibcode=2000JCoAM.121..331S |doi-access=free}} and

GNU Octave.

References

Category:1967 introductions

Category:Control theory

Category:Matrices (mathematics)