.Modelica.Math.Matrices.discreteRiccati

Information

Syntax

                        X = Matrices.discreteRiccati(A, B, R, Q);
(X, alphaReal, alphaImag) = Matrices.discreteRiccati(A, B, R, Q, true);

Description

Function discreteRiccati computes the solution X of the discrete-time algebraic Riccati equation

A'*X*A - X - A'*X*B*inv(R + B'*X*B)*B'*X*A + Q = 0

using the Schur vector approach proposed by Laub [1].

It is assumed that Q is symmetric and positive semidefinite and R is symmetric, nonsingular and positive definite, (A,B) is stabilizable and (A,Q) is detectable. Using this method, A has also to be invertible.

These assumptions are not checked in this function !!!

The assumptions guarantee that the Hamiltonian matrix.

H = [A + G*T*Q, -G*T; -T*Q, T]

with

     -T
T = A

and

       -1
G = B*R *B'

has no eigenvalue on the unit circle and can be put to an ordered real Schur form

U'*H*U = S = [S11, S12; 0, S22]

with orthogonal similarity transformation U. S is ordered in such a way, that S11 contains the n stable eigenvalues of the closed loop system with system matrix

                  -1
A - B*(R + B'*X*B)  *B'*X*A

If U is partitioned to

U = [U11, U12; U21, U22]

according to S, the solution X can be calculated by

X*U11 = U21.

References

[1] Laub, A.J.
    A Schur Method for Solving Algebraic Riccati equations.
    IEEE Trans. Auto. Contr., AC-24, pp. 913-921, 1979.

Example

A  = [4.0    3.0]
     -4.5,  -3.5];

B  = [ 1.0;
      -1.0];

R = [1.0];

Q = [9.0, 6.0;
     6.0, 4.0]

X = discreteRiccati(A, B, R, Q);

  results in:

X = [14.5623, 9.7082;
      9.7082, 6.4721];

See also

Matrices.continuousRiccati

Interface

function discreteRiccati
  extends Modelica.Icons.Function;
  import Modelica.Math.Matrices;
  input Real A[:, size(A, 1)] "Square matrix A in DARE";
  input Real B[size(A, 1), :] "Matrix B in DARE";
  input Real R[size(B, 2), size(B, 2)] = identity(size(B, 2)) "Matrix R in DARE";
  input Real Q[size(A, 1), size(A, 1)] = identity(size(A, 1)) "Matrix Q in DARE";
  input Boolean refine = false "True for subsequent refinement";
  output Real X[size(A, 1), size(A, 2)] "orthogonal matrix of the Schur vectors associated to ordered rsf";
  output Real alphaReal[2 * size(A, 1)] "Real part of eigenvalue=alphaReal+i*alphaImag";
  output Real alphaImag[2 * size(A, 1)] "Imaginary part of eigenvalue=alphaReal+i*alphaImag";
end discreteRiccati;

Revisions

• 2010/05/31 by Marcus Baur, DLR-RM