Rosenbrock system matrix

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.^[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] as well as GNU Octave.

References

↑ Rosenbrock, H.H. (1967). "Transformation of linear constant system equations". Proc. I.E.E. 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.