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
- [math]\displaystyle{ \dot{x}= Ax +Bu, }[/math]
- [math]\displaystyle{ y= Cx +Du. }[/math]
The Rosenbrock system matrix is given by
- [math]\displaystyle{ P(s)=\begin{pmatrix} sI-A & -B\\ C & D \end{pmatrix}. }[/math]
In the original work by Rosenbrock, the constant matrix [math]\displaystyle{ D }[/math] is allowed to be a polynomial in [math]\displaystyle{ s }[/math].
The transfer function between the input [math]\displaystyle{ i }[/math] and output [math]\displaystyle{ j }[/math] is given by
- [math]\displaystyle{ g_{ij}=\frac{\begin{vmatrix} sI-A & -b_i\\ c_j & d_{ij} \end{vmatrix}}{|sI-A|} }[/math]
where [math]\displaystyle{ b_i }[/math] is the column [math]\displaystyle{ i }[/math] of [math]\displaystyle{ B }[/math] and [math]\displaystyle{ c_j }[/math] is the row [math]\displaystyle{ j }[/math] of [math]\displaystyle{ C }[/math].
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
- [math]\displaystyle{ P\sim\begin{pmatrix} A & B\\ C & D \end{pmatrix}. }[/math]
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
- ↑ 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". http://radio.feld.cvut.cz/matlab/toolbox/mutools/pck.html. 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. Bibcode: 2000JCoAM.121..331S.
 |  |
