.Modelica.Math.Matrices.continuousRiccati

Information

Syntax

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

Description

Function continuousRiccati computes the solution X of the continuous-time algebraic Riccati equation

 A'*X + X*A - X*G*X + Q = 0

with G = B*inv(R)*B' 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.

These assumptions are not checked in this function !!

The assumptions guarantee that the Hamiltonian matrix

H = [A, -G; -Q, -A']

has no pure imaginary eigenvalue 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 A - B*inv(R)*B'*X. If U is partitioned to

U = [U11, U12; U21, U22]

with dimensions according to S, the solution X is calculated by

X*U11 = U21.

With optional input refinement=true a subsequent iterative refinement based on Newton's method with exact line search is applied. See continuousRiccatiIterative for more information.

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 = [0.0, 1.0;
       0.0, 0.0];

  B = [0.0;
       1.0];

  R = [1];

  Q = [1.0, 0.0;
       0.0, 2.0];

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

  results in:

X = [2.0, 1.0;
     1.0, 2.0];

See also

Matrices.Utilities.continuousRiccatiIterative, Matrices.discreteRiccati

Interface

function continuousRiccati
  extends Modelica.Icons.Function;
  import Modelica.Math.Matrices;
  input Real A[:, size(A, 1)] "Square matrix A in CARE";
  input Real B[size(A, 1), :] "Matrix B in CARE";
  input Real R[size(B, 2), size(B, 2)] = identity(size(B, 2)) "Matrix R in CARE";
  input Real Q[size(A, 1), size(A, 1)] = identity(size(A, 1)) "Matrix Q in CARE";
  input Boolean refine = false "True for subsequent refinement";
  output Real X[size(A, 1), size(A, 2)] "stabilizing solution of CARE";
  output Real alphaReal[2 * size(A, 1)] "Real parts of eigenvalue=alphaReal+i*alphaImag";
  output Real alphaImag[2 * size(A, 1)] "Imaginary parts of eigenvalue=alphaReal+i*alphaImag";
end continuousRiccati;

Revisions

• 2010/05/31 by Marcus Baur, DLR-RM