functioncontinuousRiccati

Return solution X of the continuous-time algebraic Riccati equation A'*X + X*A - X*B*inv(R)*B'*X + Q = 0 (care)

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

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];

Inputs

Type	Name	Default	Description
Real[:,size(A, 1)]	A		Square matrix A in CARE
Real[size(A, 1),:]	B		Matrix B in CARE
Real[size(B, 2),size(B, 2)]	R	identity(size(B, 2))	Matrix R in CARE
Real[size(A, 1),size(A, 1)]	Q	identity(size(A, 1))	Matrix Q in CARE
Boolean	refine	false	True for subsequent refinement

Outputs

Type	Name	Description
Real[size(A, 1),size(A, 2)]	X	Stabilizing solution of CARE
Real[2*size(A, 1)]	alphaReal	Real parts of eigenvalue=alphaReal+i*alphaImag
Real[2*size(A, 1)]	alphaImag	Imaginary parts of eigenvalue=alphaReal+i*alphaImag

Revisions

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