H2 State Observer. More...

#include <h2_state_observer.hpp>

Public Types
using	Model = linalg::remove_cvref_t< ObsModel >
	extracting alias and types

using	T = plant_type_t< Model >

using	QMat = linalg::Matrix< T, StateDim, StateDim >

using	WMat = linalg::Matrix< T, StateDim, MeasurementDim >

using	LMat = linalg::Matrix< T, StateDim, MeasurementDim >

using	ZMat = linalg::Matrix< T, ExogenousDim, ExogenousDim >

using	DecVarSpace = linalg::RowVector< T, DecVarDim >

Public Member Functions
template<convex::AtomLike DecVar>
constexpr auto	operator() (const ObsModel &model, const DecVar &Z_hat, T margin=1e-6) const
	operator overload to form H2 state observer

template<convex::AtomLike DecVar>
constexpr auto	apply (const ObsModel &model, const DecVar &Z_hat, T margin=1e-6) const
	apply

template<convex::AtomLike DecVar>
constexpr auto	stability_constraint (const ObsModel &model, const DecVar &Z_hat) const
	stability_constraint

template<convex::AtomLike DecVar>
constexpr auto	z_constraint (const ObsModel &model, const DecVar &Z_hat, T margin=1e-6) const
	z_constraint

template<convex::AtomLike DecVar>
constexpr auto	objective (const DecVar &Z_hat) const
	objective

constexpr auto	L (const QMat &Q, const WMat &W) const
	L.

Static Public Attributes
static constexpr std::size_t	StateDim = plant_traits<Model>::StateDim

static constexpr std::size_t	ExogenousDim = plant_traits<Model>::ExogenousDim

static constexpr std::size_t	MeasurementDim

static constexpr std::size_t	PerformanceDim

static constexpr std::size_t	QDim = StateDim * (StateDim + 1) / 2

static constexpr std::size_t	WDim = StateDim * MeasurementDim

static constexpr std::size_t	ZDim = ExogenousDim * (ExogenousDim + 1) / 2

static constexpr std::size_t	DecVarDim = QDim + WDim + ZDim

Detailed Description

template<ContinuousPlantModelLike ObsModel>
struct control::H2StateObserver< ObsModel >

H2 State Observer.

LDI governed by H2 norm

Given the LTI System with additive disturbance xdot = A x + B_u u + B_w w y = C_y x + D_yw w z_e = C_e e

Observer design hat{xdot} = A hat{x} + B_u u + L(y - C_y hat{x}) e = x - hat{x} edot = (A - L C_y) e + (B_w - L D_yw) w

A_cl = A - L C_y B_cl = B_w - L D_yw

The observability gramian (Q) requirement A_cl^T Q + Q A_cl + C_e^T C_e < 0

The H2 norm squared (total impulse response energy) ||T_{wz_e}||_2^2 = Tr(B_cl^T Q B_cl)

Dimensions: n = state dimension m_w = disturbance input dimension p_y = measurement output dimension p_e = performance output dimension on the estimation error

A : n x n B_w : n x m_w C_y : p_y x n D_yw : p_y x m_w C_e : p_e x n

Decision blocks: Q : n x n W : n x p_y L : n x p_y Z : m_w x m_w DecVar = [svec(Q); vec(W); svec(Z)]

We have a bilinear term in Q L, so let W = Q L where L = Q^{-1} W

H2 objective can be lifted via epigraph with Z Z >= (Q B_w - W D_yw)^T Q^{-1} (Q B_w - W D_yw)

Convex Problem is now min Tr(Z) s.t. A^T Q + Q A - C_y^T W^T - W C_y + C_e^T C_e < 0 [ Z (Q B_w - W D_yw)^T ] >= 0 [ Q B_w - W D_yw Q ]

Shape checks: A^T Q, Q A, C_y^T W^T, W C_y, C_e^T C_e : n x n Q B_w - W D_yw : n x m_w Epigraph LMI : (m_w + n) x (m_w + n)

Definition at line 82 of file h2_state_observer.hpp.

Member Typedef Documentation

◆ DecVarSpace

template<ContinuousPlantModelLike ObsModel>

using control::H2StateObserver< ObsModel >::DecVarSpace = linalg::RowVector<T, DecVarDim>

Definition at line 105 of file h2_state_observer.hpp.

◆ LMat

template<ContinuousPlantModelLike ObsModel>

using control::H2StateObserver< ObsModel >::LMat = linalg::Matrix<T, StateDim, MeasurementDim>

Definition at line 97 of file h2_state_observer.hpp.

◆ Model

template<ContinuousPlantModelLike ObsModel>

using control::H2StateObserver< ObsModel >::Model = linalg::remove_cvref_t<ObsModel>

extracting alias and types

Definition at line 84 of file h2_state_observer.hpp.

◆ QMat

template<ContinuousPlantModelLike ObsModel>

using control::H2StateObserver< ObsModel >::QMat = linalg::Matrix<T, StateDim, StateDim>

Definition at line 95 of file h2_state_observer.hpp.

◆ T

template<ContinuousPlantModelLike ObsModel>

using control::H2StateObserver< ObsModel >::T = plant_type_t<Model>

Definition at line 86 of file h2_state_observer.hpp.

◆ WMat

template<ContinuousPlantModelLike ObsModel>

using control::H2StateObserver< ObsModel >::WMat = linalg::Matrix<T, StateDim, MeasurementDim>

Definition at line 96 of file h2_state_observer.hpp.

◆ ZMat

template<ContinuousPlantModelLike ObsModel>

using control::H2StateObserver< ObsModel >::ZMat = linalg::Matrix<T, ExogenousDim, ExogenousDim>

Definition at line 98 of file h2_state_observer.hpp.

Member Function Documentation

◆ apply()

template<ContinuousPlantModelLike ObsModel>

template<convex::AtomLike DecVar>

constexpr auto control::H2StateObserver< ObsModel >::apply	(	const ObsModel &	model,
		const DecVar &	Z_hat,
		T	margin = `1e-6`
	)		const

inlineconstexpr

apply

Parameters

model	(PlantModel)
Z_hat	([svec(Q); vec(W); svec(Z)])

Returns: H2 LMI

Definition at line 130 of file h2_state_observer.hpp.

References control::is_continuous_v, control::H2StateObserver< ObsModel >::stability_constraint(), and control::H2StateObserver< ObsModel >::z_constraint().

Referenced by control::H2StateObserver< ObsModel >::operator()().

◆ L()

template<ContinuousPlantModelLike ObsModel>

constexpr auto control::H2StateObserver< ObsModel >::L	(	const QMat &	Q,
		const WMat &	W
	)		const

inlineconstexpr

L.

L = Q^{-1} W

Parameters

Q	observability gramian
W	lifted observer product (W = Q L)

Returns: L

Definition at line 275 of file h2_state_observer.hpp.

References control::is_continuous_v, and control::recover_state_observer_gain().

◆ objective()

template<ContinuousPlantModelLike ObsModel>

template<convex::AtomLike DecVar>

constexpr auto control::H2StateObserver< ObsModel >::objective ( const DecVar & Z_hat ) const

inlineconstexpr

objective

min Tr(Z)

Parameters

Z_hat (DecVar)

Returns: Tr(Z)

Definition at line 260 of file h2_state_observer.hpp.

References control::is_continuous_v.

◆ operator()()

template<ContinuousPlantModelLike ObsModel>

template<convex::AtomLike DecVar>

constexpr auto control::H2StateObserver< ObsModel >::operator()	(	const ObsModel &	model,
		const DecVar &	Z_hat,
		T	margin = `1e-6`
	)		const

inlineconstexpr

operator overload to form H2 state observer

Parameters

model	(PlantModel)
Z_hat	(DecVar)

Returns: H2 LMIs

Definition at line 116 of file h2_state_observer.hpp.

References control::H2StateObserver< ObsModel >::apply(), and control::is_continuous_v.

◆ stability_constraint()

template<ContinuousPlantModelLike ObsModel>

template<convex::AtomLike DecVar>

constexpr auto control::H2StateObserver< ObsModel >::stability_constraint	(	const ObsModel &	model,
		const DecVar &	Z_hat
	)		const

inlineconstexpr

stability_constraint

Stability constraint on LMI: A^T Q + Q A - C_y^T W^T - W C_y + C_e^T C_e < 0

Parameters

model	(PlantModel)
Z_hat	(Dec Var)

Returns: stability constraint as LMI

Definition at line 156 of file h2_state_observer.hpp.

References control::is_continuous_v, control::H2StateObserver< ObsModel >::QDim, and control::H2StateObserver< ObsModel >::WDim.

Referenced by control::H2StateObserver< ObsModel >::apply().

◆ z_constraint()

template<ContinuousPlantModelLike ObsModel>

template<convex::AtomLike DecVar>

constexpr auto control::H2StateObserver< ObsModel >::z_constraint	(	const ObsModel &	model,
		const DecVar &	Z_hat,
		T	margin = `1e-6`
	)		const

inlineconstexpr

z_constraint

PSD constraint on Z (epigraph) [ Z (Q B_w - W D_yw)^T ] >= 0 [ Q B_w - W D_yw Q ]

FMat Dim (m+n x m+n) [ mxm mxn ] [ nxm nxn ]

Parameters

model	(PlantModel)
Z_hat	(DecVar)
margin	for PD

Returns: Z constraint as LMI

Definition at line 204 of file h2_state_observer.hpp.

References control::EXOGENOUS, control::H2StateObserver< ObsModel >::ExogenousDim, control::is_continuous_v, control::MEASUREMENT, control::H2StateObserver< ObsModel >::QDim, control::H2StateObserver< ObsModel >::StateDim, control::H2StateObserver< ObsModel >::WDim, and control::H2StateObserver< ObsModel >::ZDim.

Referenced by control::H2StateObserver< ObsModel >::apply().

Member Data Documentation

◆ DecVarDim

template<ContinuousPlantModelLike ObsModel>

constexpr std::size_t control::H2StateObserver< ObsModel >::DecVarDim = QDim + WDim + ZDim

staticconstexpr

Definition at line 103 of file h2_state_observer.hpp.

◆ ExogenousDim

template<ContinuousPlantModelLike ObsModel>

constexpr std::size_t control::H2StateObserver< ObsModel >::ExogenousDim = plant_traits<Model>::ExogenousDim

staticconstexpr

Definition at line 89 of file h2_state_observer.hpp.

Referenced by control::H2StateObserver< ObsModel >::z_constraint().

◆ MeasurementDim

template<ContinuousPlantModelLike ObsModel>

constexpr std::size_t control::H2StateObserver< ObsModel >::MeasurementDim

staticconstexpr

Initial value:

=

plant_traits<Model>::MeasurementDim

Definition at line 90 of file h2_state_observer.hpp.

◆ PerformanceDim

template<ContinuousPlantModelLike ObsModel>

constexpr std::size_t control::H2StateObserver< ObsModel >::PerformanceDim

staticconstexpr

Initial value:

=

plant_traits<Model>::PerformanceDim

Definition at line 92 of file h2_state_observer.hpp.

◆ QDim

template<ContinuousPlantModelLike ObsModel>

constexpr std::size_t control::H2StateObserver< ObsModel >::QDim = StateDim * (StateDim + 1) / 2

staticconstexpr

Definition at line 100 of file h2_state_observer.hpp.

Referenced by control::H2StateObserver< ObsModel >::stability_constraint(), and control::H2StateObserver< ObsModel >::z_constraint().

◆ StateDim

template<ContinuousPlantModelLike ObsModel>

constexpr std::size_t control::H2StateObserver< ObsModel >::StateDim = plant_traits<Model>::StateDim

staticconstexpr

Definition at line 88 of file h2_state_observer.hpp.

Referenced by control::H2StateObserver< ObsModel >::z_constraint().

◆ WDim

template<ContinuousPlantModelLike ObsModel>

constexpr std::size_t control::H2StateObserver< ObsModel >::WDim = StateDim * MeasurementDim

staticconstexpr

Definition at line 101 of file h2_state_observer.hpp.

Referenced by control::H2StateObserver< ObsModel >::stability_constraint(), and control::H2StateObserver< ObsModel >::z_constraint().

◆ ZDim

template<ContinuousPlantModelLike ObsModel>

constexpr std::size_t control::H2StateObserver< ObsModel >::ZDim = ExogenousDim * (ExogenousDim + 1) / 2

staticconstexpr

Definition at line 102 of file h2_state_observer.hpp.

Referenced by control::H2StateObserver< ObsModel >::z_constraint().

The documentation for this struct was generated from the following file:

include/akreon/control/estimation/h2_state_observer.hpp

Public Types

Public Member Functions

Static Public Attributes

Detailed Description

Member Typedef Documentation

◆ DecVarSpace

◆ LMat

◆ Model

◆ QMat

◆ T

◆ WMat

◆ ZMat

Member Function Documentation

◆ apply()

◆ L()

◆ objective()

◆ operator()()

◆ stability_constraint()

◆ z_constraint()

Member Data Documentation

◆ DecVarDim

◆ ExogenousDim

◆ MeasurementDim

◆ PerformanceDim

◆ QDim

◆ StateDim

◆ WDim

◆ ZDim