Geometric_certificates.cpp File Reference

Numerical-range, small-gain, and sampled DW-shell certificates. More...

#include "Geometric_certificates.h"
#include <algorithm>
#include <cmath>
#include <limits>

Include dependency graph for Geometric_certificates.cpp:

Functions
double	excessPassivity (const Eigen::MatrixXcd &A)
	Hermitian / excess passivity ν = λ_min(Her A); shortage = max(0,-ν).
double	shortagePassivity (const Eigen::MatrixXcd &A)
double	spectralNorm (const Eigen::MatrixXcd &A)
	Operator (spectral) norm σ_max(A).
std::vector< std::complex< double > >	numericalRangeBoundary (const Eigen::MatrixXcd &A, int n_theta)
	Trace ∂W(A) via the support function: z(θ) = e^{iθ} λ_max(Her(e^{-iθ} A)).
NumericalRangeCertificate	certifyNumericalRange (const Eigen::MatrixXcd &A, double phase_limit_deg, int n_theta)
	Numerical-range small-phase certificate.
DwShellCertificate	certifyDwShell (const Eigen::MatrixXcd &A, double phase_limit_deg, double gain_limit, double excess_eps, int n_theta, int n_sphere)
	Sample DW(A) on a dense set of unit vectors (exact NR boundary + sphere grid).
Eigen::MatrixXcd	loopShiftedSector (const Eigen::MatrixXcd &A, double alpha, double beta)
	Classic loop-shifted (sector) matrix for SISO/MIMO sector [α, β]: T = (A - α I)(β I - A)^{-1} when invertible; empty matrix on failure.
bool	certifySector (const Eigen::MatrixXcd &A, double alpha, double beta, double eps)
	True if Her(T) ≻ eps for the loop-shifted sector map (when defined).

Detailed Description

Numerical-range, small-gain, and sampled DW-shell certificates.

Function Documentation

certifyDwShell()

DwShellCertificate certifyDwShell	(	const Eigen::MatrixXcd &	A,
		double	phase_limit_deg = 90.0,
		double	gain_limit = 1.0,
		double	excess_eps = 0.0,
		int	n_theta = 180,
		int	n_sphere = 24
	)

Sample DW(A) on a dense set of unit vectors (exact NR boundary + sphere grid).

Parameters

gain_limit	Require σ_max(A) ≤ gain_limit when used as small-gain gate.
excess_eps	Require λ_min(Her A) ≥ excess_eps.

References certifyNumericalRange(), DwShellCertificate::excess_passivity, excessPassivity(), DwShellPoint::gain2, j, DwShellCertificate::max_singular, NumericalRangeCertificate::min_real, DwShellCertificate::nr, DwShellCertificate::pass_excess, DwShellCertificate::pass_small_gain, DwShellPoint::re, DwShellCertificate::samples, DwShellCertificate::shortage_passivity, shortagePassivity(), spectralNorm(), DwShellCertificate::xz_max_gain2, and DwShellCertificate::xz_min_re.

Referenced by plot_certificate_dw_shell().

certifyNumericalRange()

NumericalRangeCertificate certifyNumericalRange	(	const Eigen::MatrixXcd &	A,
		double	phase_limit_deg = 90.0,
		int	n_theta = 180
	)

Numerical-range small-phase certificate.

Parameters

phase_limit_deg

Max allowed |arg| of W(A) (90° ≈ passivity for sectorial sets).

References NumericalRangeCertificate::boundary, NumericalRangeCertificate::contains_zero, NumericalRangeCertificate::max_phase_deg, NumericalRangeCertificate::min_real, numericalRangeBoundary(), NumericalRangeCertificate::pass_phase, NumericalRangeCertificate::pass_zero_outside, and NumericalRangeCertificate::zero_margin.

Referenced by certifyDwShell(), plot_certificate_dw_slice(), and plot_certificate_numerical_range().

certifySector()

bool certifySector	(	const Eigen::MatrixXcd &	A,
		double	alpha,
		double	beta,
		double	eps
	)

True if Her(T) ≻ eps for the loop-shifted sector map (when defined).

References loopShiftedSector().

excessPassivity()

double excessPassivity

(

const Eigen::MatrixXcd &

A

)

Hermitian / excess passivity ν = λ_min(Her A); shortage = max(0,-ν).

Referenced by certifyDwShell().

loopShiftedSector()

Eigen::MatrixXcd loopShiftedSector	(	const Eigen::MatrixXcd &	A,
		double	alpha,
		double	beta
	)

Classic loop-shifted (sector) matrix for SISO/MIMO sector [α, β]: T = (A - α I)(β I - A)^{-1} when invertible; empty matrix on failure.

Passivity of T is a sector certificate for A (when defined).

Referenced by certifySector().

numericalRangeBoundary()

std::vector< std::complex< double > > numericalRangeBoundary	(	const Eigen::MatrixXcd &	A,
		int	n_theta
	)

Trace ∂W(A) via the support function: z(θ) = e^{iθ} λ_max(Her(e^{-iθ} A)).

Referenced by certifyNumericalRange().

shortagePassivity()

double shortagePassivity

(

const Eigen::MatrixXcd &

A

)

Referenced by certifyDwShell().

spectralNorm()

double spectralNorm

(

const Eigen::MatrixXcd &

A

)

Operator (spectral) norm σ_max(A).

Referenced by certifyDwShell(), and sweepReturnRatioCertificate().