Methods for dealing with discrete-time algebraic Riccati equations. More...

#include <algebraic_riccati_discrete.h>

Static Public Member Functions
static bool	isClosedLoopStable (const Eigen::Ref< const Eigen::MatrixXd > &A, const Eigen::Ref< const Eigen::MatrixXd > &B, const Eigen::Ref< const Eigen::MatrixXd > &G)
	Determine if the closed-loop system is stable. More...

static bool	solve (const Eigen::Ref< const Eigen::MatrixXd > &A, const Eigen::Ref< const Eigen::MatrixXd > &B, const Eigen::Ref< const Eigen::MatrixXd > &Q, const Eigen::Ref< const Eigen::MatrixXd > &R, Eigen::MatrixXd &X, Eigen::MatrixXd *G=nullptr)
	Solve discrete-time algebraic Riccati equation. More...

Static Protected Member Functions
static bool	isInsideUnitCircle (const Eigen::Ref< const Eigen::MatrixXd > &block)

static bool	solveRiccatiHamiltonianSchur (const Eigen::MatrixXd &H, Eigen::MatrixXd &X)
	Solve Hamiltonian via Schur method. More...

Detailed Description

Methods for dealing with discrete-time algebraic Riccati equations.

The discrete-time algebraic Riccati equation is given by

$A^T X A - X - A^T X B ( B^T X B + R )^{-1} B^T X A + Q = 0$

$ A $ and $ B $ must be stabilizable.

The resulting gain matrix is given by

$G = (B^T X B R)^{-1} B^T X A$

See also: AlgebraicRiccatiContinuous LyapunovDiscrete LyapunovContinuous SylvesterContinuous SylvesterDiscrete

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 a method that determiens if sufficient conditions hold and that (A,B) is stabilizable.

Add support for the more generalized Riccati equation (augmented by matrix S and E) E.g. Hamiltonian for augmentation with S: http://web.dcsc.tudelft.nl/~cscherer/books/thesis.pdf

Definition at line 58 of file algebraic_riccati_discrete.h.

Member Function Documentation

◆ isClosedLoopStable()

bool corbo::AlgebraicRiccatiDiscrete::isClosedLoopStable	(	const Eigen::Ref< const Eigen::MatrixXd > &	A,
		const Eigen::Ref< const Eigen::MatrixXd > &	B,
		const Eigen::Ref< const Eigen::MatrixXd > &	G
	)

static

Determine if the closed-loop system is stable.

The closed-loop system is stable if the closed-loop eigenvalues are within the unit circle. This method returns true if $ A - B G $ has only eigenvalues within the unit circle. Hereby, G denotes the gain matrix obtained from solve().

Parameters

[in]	A	Open-loop system matrix [n x n]
[in]	B	Input matrix [n x m]
[in]	G	Feedback gain matrix according to the solution of the algebraic Riccati equation [m x n]

Returns: true if the closed-loop system is stable, false otherwise.

Definition at line 102 of file algebraic_riccati_discrete.cpp.

◆ isInsideUnitCircle()

static bool corbo::AlgebraicRiccatiDiscrete::isInsideUnitCircle ( const Eigen::Ref< const Eigen::MatrixXd > & block )

inlinestaticprotected

Definition at line 132 of file algebraic_riccati_discrete.h.

◆ solve()

bool corbo::AlgebraicRiccatiDiscrete::solve	(	const Eigen::Ref< const Eigen::MatrixXd > &	A,
		const Eigen::Ref< const Eigen::MatrixXd > &	B,
		const Eigen::Ref< const Eigen::MatrixXd > &	Q,
		const Eigen::Ref< const Eigen::MatrixXd > &	R,
		Eigen::MatrixXd &	X,
		Eigen::MatrixXd *	G = `nullptr`
	)

static

Solve discrete-time algebraic Riccati equation.

Solve $A^T X A - X - A^T X B ( B^T X B + R )^{-1} B^T X A + Q = 0$ w.r.t. $ X $ .

$ A $ and $ B $ must be stabilizable (all eigenvalues of A outside the unit circle must be controllable). In addition, the associated symplectic pencil must have no eigenvalue on the unit circle. Sufficient conditions: R > 0.

Also, $ A $ must be invertible for this implementation.

This method optionally returns the gain matrix $G = (B^T X B + R)^{-1} B^T X A$ .

Warning: Input matrix sizes are checked only in debug mode and occuring issues immediately break execution

Parameters

[in]	A	[n x n] matrix
[in]	B	[n x m] matrix
[in]	Q	[n x n] matrix
[in]	R	[m x m] matrix
[out]	X	Solution of the algebraic Riccati equation with size [n x n] (must not be preallocated).
[out]	G	Gain matrix with size [m x n] (optional)

Returns: true if a finite solution is found, false otherwise (but no NaN checking is performend on X, use X.allFinite()).

Definition at line 40 of file algebraic_riccati_discrete.cpp.

◆ solveRiccatiHamiltonianSchur()

bool corbo::AlgebraicRiccatiDiscrete::solveRiccatiHamiltonianSchur	(	const Eigen::MatrixXd &	H,
		Eigen::MatrixXd &	X
	)

staticprotected

Solve Hamiltonian via Schur method.

The Hamiltonian constructed in solve() is solved via the Schur decomposition. The Real Schur form of A is reordered such that blocks on the diagonal with eigenvalues contained in the unit circle. The corresponding column in the orthogonal transformation matrix U spans the invariant subspace: [U1 U2]^T. The solution of the Riccati equation is then obtained by

$X = U_2 U_1^{-1}$

The implementation is based on [1].

[1] A. J. Laub. "A Schur method for solving algebraic Riccati equations", IEEE Conference on Decision and Control, 1978.

Parameters

[in]	H	Hamiltonian matrix of the Riccati equation [square]
[out]	X	Resuling solution: symmetric matrix with size(H).

Returns: true if solving was successful (but no NaN checking is performend on X, use X.allFinite()).

Definition at line 71 of file algebraic_riccati_discrete.cpp.

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

Static Public Member Functions

Static Protected Member Functions

Detailed Description

Member Function Documentation

◆ isClosedLoopStable()

◆ isInsideUnitCircle()

◆ solve()

◆ solveRiccatiHamiltonianSchur()