.Modelica.Math.Matrices.discreteLyapunov

Information

Syntax

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

Description

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

 A'*X*A + sgn*X = C

where sgn=1 or sgn =-1. For sgn = -1, the discrete Lyapunov equation is a special case of the Stein equation:

 A*X*B - X + Q = 0.

The algorithm uses the Schur method for Lyapunov equations proposed by Bartels and Stewart [1].

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

 R*Y*R' + sgn*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 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 = discreteLyapunov(A, C, sgn=-1);

  results in:

  X  = [7.5735,   -3.1426,  2.7205, -2.5958;
       -2.6105,    1.2384, -0.9232,  0.9632;
        6.6090,   -2.6775,  2.6415, -2.6928;
       -0.3572,    0.2298,  0.0533, -0.27410];

See also

Matrices.discreteSylvester, Matrices.continuousLyapunov

Interface

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

Revisions

• 2010/05/31 by Marcus Baur, DLR-RM