discreteRiccatiReturn solution of discrete-time algebraic Riccati equation A'*X*A - X - A'*X*B*inv(R + B'*X*B)*B'*X*A + Q = 0 (dare) |
This information is part of the Modelica Standard Library maintained by the Modelica Association.
X = Matrices.discreteRiccati(A, B, R, Q); (X, alphaReal, alphaImag) = Matrices.discreteRiccati(A, B, R, Q, true);
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.
[1] Laub, A.J. A Schur Method for Solving Algebraic Riccati equations. IEEE Trans. Auto. Contr., AC-24, pp. 913-921, 1979.
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];
A |
Type: Real[:,size(A, 1)] Description: Square matrix A in DARE |
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B |
Type: Real[size(A, 1),:] Description: Matrix B in DARE |
R |
Default Value: identity(size(B, 2)) Type: Real[size(B, 2),size(B, 2)] Description: Matrix R in DARE |
Q |
Default Value: identity(size(A, 1)) Type: Real[size(A, 1),size(A, 1)] Description: Matrix Q in DARE |
refine |
Default Value: false Type: Boolean Description: True for subsequent refinement |
X |
Type: Real[size(A, 1),size(A, 2)] Description: orthogonal matrix of the Schur vectors associated to ordered rsf |
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alphaReal |
Type: Real[2 * size(A, 1)] Description: Real part of eigenvalue=alphaReal+i*alphaImag |
alphaImag |
Type: Real[2 * size(A, 1)] Description: Imaginary part of eigenvalue=alphaReal+i*alphaImag |