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 In control engineering, a state-space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations or difference equations. State variables are variables wh ...

and

transfer function matrix In control system theory, and various branches of engineering, a transfer function matrix, or just transfer matrix is a generalisation of the transfer functions of single-input single-output (SISO) systems to multiple-input and multiple-output (M ...

form. It was proposed in 1967 by Howard H. Rosenbrock.

Definition

Consider the dynamic system ::

\dot= Ax +Bu,

y= Cx +Du.

The Rosenbrock system matrix is given by ::

P(s)=\begin
sI-A & -B\\
C  & D
\end.

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_=\frac

where

b_i

is the column

i

B

and

c_j

is the row

j

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 and given by ::

P\sim\begin
A & B\\
C  & D
\end.

The short form of the Rosenbrock system matrix has been widely used in

H-infinity methods in control theory ''H''∞ (i.e. "''H''-infinity") methods are used in control theory to synthesize controllers to achieve stabilization with guaranteed performance. To use ''H''∞ methods, a control designer expresses the control problem as a mathematical optimiza ...

, where it is also referred to as packed form; see command pck in MATLAB. An interpretation of the Rosenbrock System Matrix as a Linear Fractional Transformation can be found in. One of the first applications of the Rosenbrock form was the development of an efficient computational method for

Kalman decomposition In control theory, a Kalman decomposition provides a mathematical means to convert a representation of any linear time-invariant (LTI) control system to a form in which the system can be decomposed into a standard form which makes clear the observa ...

, which is based on the pivot element method. A variant of Rosenbrock’s method is implemented in the minreal command of Matlab and

GNU Octave GNU Octave is a high-level programming language primarily intended for scientific computing and numerical computation. Octave helps in solving linear and nonlinear problems numerically, and for performing other numerical experiments using a langu ...

References

{{Reflist 1967 introductions Control theory Matrices