This information is part of the Modelica Standard Library maintained by the Modelica Association.

Syntax

              To = Matrices.Utilities.reorderRSF(T, Q, alphaReal, alphaImag);
(To, Qo, wr, wi) = Matrices.Utilities.reorderRSF(T, Q, alphaReal, alphaImag, iscontinuous);

Description

Function reorderRSF() reorders a real Schur form such that the stable eigenvalues of the system are in the 1-by-1 and 2-by-2 diagonal blocks of the block upper triangular matrix. If the Schur form is referenced to a continuous system the staple eigenvalues are in the left complex half plane. The stable eigenvalues of a discrete system are inside the complex unit circle.
This function is used for example to solve algebraic Riccati equations (continuousRiccati, discreteRiccati). In this context the Schur form as well as the corresponding eigenvalues and the transformation matrix Q are known, why the eigenvalues and the transformation matrix are inputs to reorderRSF().
The Schur vector matrix Qo is also reordered according to To. The vectors wr and wi contains the real and imaginary parts of the reordered eigenvalues respectively.

Example

T := [-1,2, 3,4;
       0,2, 6,5;
       0,0,-3,5;
       0,0, 0,6];
To := Matrices.Utilities.reorderRSF(T,identity(4),{-1, 2, -3, 6},{0, 0, 0, 0}, true);

// To = [-1.0, -0.384, 3.585, 4.0;
//        0.0, -3.0,   6.0,   0.64;
//        0.0,  0.0,   2.0,   7.04;
//        0.0,  0.0,   0.0,   6.0]

See also Matrices.realSchur

(To, Qo, wr, wi) = reorderRSF(T, Q, alphaReal, alphaImag, iscontinuous)

T	Type: Real[:,:] Description: Real Schur form
Q	Type: Real[:,size(T, 2)] Description: Schur vector Matrix
alphaReal	Type: Real[size(T, 1)] Description: Real part of eigenvalue=alphaReal+i*alphaImag
alphaImag	Type: Real[size(T, 1)] Description: Imaginary part of eigenvalue=alphaReal+i*alphaImag
iscontinuous	Default Value: true Type: Boolean Description: = true, if the according system is continuous. False for discrete systems

To	Type: Real[size(T, 1),size(T, 2)] Description: Reordered Schur form
Qo	Type: Real[size(T, 1),size(T, 2)] Description: Reordered Schur vector matrix
wr	Type: Real[size(T, 2)] Description: Reordered eigenvalues, real part
wi	Type: Real[size(T, 2)] Description: Reordered eigenvalues, imaginary part

• 2010/04/30 by Marcus Baur, DLR-RM