Methods for dealing with continuous-time sylvester equations. More...

#include <sylvester_continuous.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 continuous-time Sylvester equation. More...

Detailed Description

Methods for dealing with continuous-time sylvester equations.

The Sylvester equation is given by

$A X + X B + C = 0$

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 E^T + E X B + 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_continuous.h.

Member Function Documentation

◆ hasUniqueSolution()

bool corbo::SylvesterContinuous::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: $\alpha_i \neq -\beta_j, \forall i,j.$

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 92 of file sylvester_continuous.cpp.

◆ solve()

bool corbo::SylvesterContinuous::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 continuous-time Sylvester equation.

Solve $ A X + X B + 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: $\alpha_i \neq -\beta_j, \forall i,j.$

The solution is obtained via Schur decomposition [1-4].

[1] R.H. Bartels, G. W. Stewart. "Algorithm 432: Solution of the matrix equation AX + XB = C". Comm. ACM. 15 (9): 820–826, 1972. [2] V. Simoncini. "Computational Methods for Linear Matrix Equations". SIAM Review 58 (3): 377-441, 2016. [3] https://people.kth.se/~eliasj/NLA/matrixeqs.pdf [4] https://github.com/ajt60gaibb/freeLYAP

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 37 of file sylvester_continuous.cpp.

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

Static Public Member Functions

Detailed Description

Member Function Documentation

◆ hasUniqueSolution()

◆ solve()