functioncontinuousLyapunov

Return solution X of the continuous-time Lyapunov equation X*A + A'*X = C

Extends from Modelica.Icons.Function (Icon for functions).

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]

Inputs

Type	Name	Default	Description
Real[:,size(A, 1)]	A		Square matrix A in XA + A'X = C
Real[size(A, 1),size(A, 2)]	C		Square matrix C in XA + A'X = C
Boolean	ATisSchur	false	= true, if transpose(A) has already real Schur form
Real	eps	Modelica.Math.Matrices.norm(A, 1)101e-15	Tolerance eps

Outputs

Type	Name	Default	Description
Real[size(A, 1),size(A, 2)]	X		Solution X of the Lyapunov equation XA + A'X = C

Revisions

2010/05/31 by Marcus Baur, DLR-RM