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.

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