.Modelica.Math.Matrices.continuousLyapunov

Information

Syntax

X = Matrices.continuousLyapunov(A, C);
X = Matrices.continuousLyapunov(A, C, ATisSchur, eps);

Description

This function computes the solution X of the continuous-time Lyapunov equation

X*A + A'*X = C

using the Schur method for Lyapunov equations proposed by Bartels and Stewart [1].

In a nutshell, the problem is reduced to the corresponding problem

Y*R' + R*Y = D

with R=U'*A'*U is the real Schur form of A' and D=U'*C*U and Y=U'*X*U are the corresponding transformations of C and X. This problem is solved sequentially for the 1x1 or 2x2 Schur blocks by exploiting the block triangular form of R. Finally the solution of the original problem is recovered as X=U*Y*U'.
The Boolean input "ATisSchur" indicates to omit the transformation to Schur in the case that A' has already Schur form.

References

[1] Bartels, R.H. and Stewart G.W.
    Algorithm 432: Solution of the matrix equation AX + XB = C.
    Comm. ACM., Vol. 15, pp. 820-826, 1972.

Example

A = [1, 2,  3,  4;
     3, 4,  5, -2;
    -1, 2, -3, -5;
     0, 2,  0,  6];

C =  [-2, 3, 1, 0;
      -6, 8, 0, 1;
       2, 3, 4, 5;
      0, -2, 0, 0];

X = continuousLyapunov(A, C);

results in:

X = [1.633, -0.761,  0.575, -0.656;
    -1.158,  1.216,  0.047,  0.343;
    -1.066, -0.052, -0.916,  1.61;
    -2.473,  0.717, -0.986,  1.48]

See also

Matrices.continuousSylvester, Matrices.discreteLyapunov

Interface

function continuousLyapunov
  extends Modelica.Icons.Function;
  import Modelica.Math.Matrices;
  input Real A[:, size(A, 1)] "Square matrix A in X*A + A'*X = C";
  input Real C[size(A, 1), size(A, 2)] "Square matrix C in X*A + A'*X = C";
  input Boolean ATisSchur = false "= true, if transpose(A) has already real Schur form";
  input Real eps = Modelica.Math.Matrices.norm(A, 1) * 10 * 1e-15 "Tolerance eps";
  output Real X[size(A, 1), size(A, 2)] "Solution X of the Lyapunov equation X*A + A'*X = C";
end continuousLyapunov;

Revisions

• 2010/05/31 by Marcus Baur, DLR-RM