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

{\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 PHB test.

Short form

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

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References

Rosenbrock, H. H. (1967). "Transformation of linear constant system equations". Proc. IEE. 114: 541–544.
Rosenbrock, H. H. (1970). State-Space and Multivariable Theory. Nelson.
"Mu Analysis and Synthesis Toolbox". Retrieved 25 August 2014.
Zhou, Kemin; Doyle, John C.; Glover, Keith (1995). Robust and Optimal Control. Prentice Hall.
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. doi:10.1016/S0377-0427(00)00341-1.

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[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. doi:10.1016/S0377-0427(00)00341-1.