Methods for dealing with discrete-time Sylvester equations. More...

#include <sylvester_discrete.h>

Static Public Member Functions
static bool	hasUniqueSolution (const Eigen::Ref< const Eigen::MatrixXd > &A, const Eigen::Ref< const Eigen::MatrixXd > &B)
	Determine if the Sylvester equation exhibits a unique solution. More...

static bool	solve (const Eigen::Ref< const Eigen::MatrixXd > &A, const Eigen::Ref< const Eigen::MatrixXd > &B, const Eigen::Ref< const Eigen::MatrixXd > &C, Eigen::MatrixXd &X)
	Solve discrete-time Sylvester equation. More...

Detailed Description

Methods for dealing with discrete-time Sylvester equations.

The discrete-time Sylvester equation is given by

$A X B - X + C = 0$

A, B and C are not required to be square, but must be compatible to each other. In particular if size(C) = m x n, then size(A) = n x n and size(B) = m x m.

Remarks: If you want to deal with a Lyapunov equation you might want to refer to our specialized implementation in order to improve efficency.

See also: SylvesterDiscrete LyapunovContinuous LyapunovDiscrete

Author: Christoph Rösmann (chris.nosp@m.toph.nosp@m..roes.nosp@m.mann.nosp@m.@tu-d.nosp@m.ortm.nosp@m.und.d.nosp@m.e)

Todo:

Add support for the generalized Sylvester equations $ A X A^T - E X E^T + C = 0 $ .

Allow the user to precompute the Schur decomposition for subsequent calls of solve() with varying $ C $ .

Definition at line 56 of file sylvester_discrete.h.

Member Function Documentation

◆ hasUniqueSolution()

bool corbo::SylvesterDiscrete::hasUniqueSolution	(	const Eigen::Ref< const Eigen::MatrixXd > &	A,
		const Eigen::Ref< const Eigen::MatrixXd > &	B
	)

static

Determine if the Sylvester equation exhibits a unique solution.

The solution is unique if for Eigenvalues of A and B it is: $\alpha_i \beta_j \neq 1, \forall i,j.$

Parameters

[in]	A	Matrix (might be non-square)
[in]	B	Matrix (might be non-square)

Returns: true if the solution is unique, false otherwise (also if A and B are not square)

Definition at line 95 of file sylvester_discrete.cpp.

◆ solve()

bool corbo::SylvesterDiscrete::solve	(	const Eigen::Ref< const Eigen::MatrixXd > &	A,
		const Eigen::Ref< const Eigen::MatrixXd > &	B,
		const Eigen::Ref< const Eigen::MatrixXd > &	C,
		Eigen::MatrixXd &	X
	)

static

Solve discrete-time Sylvester equation.

Solve $ A X B - X + C = 0 $ w.r.t. $ X $ . A, B and C are not required to be square, but must be compatible to each other. In particular if size(C) = m x n, then size(A) = n x n and size(B) = m x m. The solution is unique if for Eigenvalues of A and B it is: $\alpha_i \beta_j \neq 1, \forall i,j.$

The solution is obtained via Schur decomposition [1-3].

[1] A. Y. Barraud. "A numerical algorithm to solve AT XA − X = Q". IEEE Trans. Automat. Control, AC-22, pp. 883–885, 1977. [2] V. Simoncini. "Computational Methods for Linear Matrix Equations". SIAM Review 58 (3): 377-441, 2016. [3] G. Kitagawa. "An Algorithm for Solving the Matrix Equation X = F X F' + S". International Journal of Control. 25 (5): 745–753, 1977.

Parameters

[in]	A	Matrix (might be non-square)
[in]	B	Matrix (might be non-square)
[in]	C	Matrix (might be non-square)
[out]	X	Solution with same size as C (size(X) must not be preallocated a-priori)

Returns: true if a finite solution is found, false otherwise.

Definition at line 36 of file sylvester_discrete.cpp.

The documentation for this class was generated from the following files:

Static Public Member Functions

Detailed Description

Member Function Documentation

◆ hasUniqueSolution()

◆ solve()