Hinf State Feedback. More...

#include <hinf_state_feedback.hpp>

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

using	T = plant_type_t< Model >

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

using	YMat = linalg::Matrix< T, InputDim, StateDim >

using	KMat = linalg::Matrix< T, InputDim, StateDim >

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

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

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

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

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

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

constexpr auto	K (const QMat &Q, const YMat &Y) const
	K.

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

static constexpr PortRole	ExogenousPortRole = PortRole::EXOGENOUS

static constexpr std::size_t	InputDim = plant_traits<Model>::ControlDim

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

static constexpr std::size_t	OutputDim = plant_traits<Model>::PerformanceDim

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

static constexpr std::size_t	YDim = InputDim * StateDim

static constexpr std::size_t	GammaDim = 1

static constexpr std::size_t	DecVarDim = QDim + YDim + GammaDim

Detailed Description

template<ContinuousPlantModelLike SFModel>
struct control::HinfStateFeedback< SFModel >

Hinf State Feedback.

We seek a state-feedback gain K s.t. L2 gain sup_{||w||_2 = 1} = ||z||_2 = sup_{||w||_2!=0}frac{||z||_2}{||w||_2}

of the CL system is less than an arbitrary gamma

L2 gain for LTI is equal to H_infinity norm of the corresponding TF. exists K s.t. L2 gain of an LTI system is less than gamma if the inverse BRL Lyapunov variable Q > 0. xdot = A x + B_xw w + B_xu u z = C_zx x + D_zw w + D_zu u y = C_yx x + D_yw w + D_yu u

Plant = [ A B_xw B_xu ] [ C_zx D_zw D_zu ] [ C_yx D_yw D_yu ]

let u = K x

xdot = (A + B_xu K) x + B_xw w z = (C_zx + D_zu K) x + D_zw w

A_cl = A + B_xu K B_cl = B_xw C_cl = C_zx + D_zu K D_cl = D_zw

Certifies ||T_zw||_infinity < gamma where T_zw is the complementary sensitivity TF from w to z

General BRL form with Lyapunov variable S > 0: [ (A_cl^T S + S A_cl) (S B_xw) C_cl^T ] < 0 [ (B_xw^T S) (-gamma I) D_zw^T ] [ C_cl D_zw -gamma I ]

Hinf synthesis decision variables: Q : n x n, inverse BRL Lyapunov variable Q = S^{-1} Y : m_u x n, lifted feedback product Y = K Q K : m_u x n gamma : scalar induced L2-gain bound DecVar = [svec(Q); vec(Y); gamma]

Objective: min gamma

We have a bilinear term in K Q, so let Y = K Q where K = Y Q^-1 [ A Q + Q A^T + B_xu Y + Y^T B_xu^T B_xw (C_zx Q + D_zu Y)^T ] < 0 [ B_xw^T (-gamma I) D_zw^T ] [ C_zx Q + D_zu Y D_zw (-gamma I) ]

Convex problem: min gamma s.t. Q > 0 BRL(Q, Y, gamma) < 0

Full decision space: QDim = n(n + 1) / 2 YDim = m_u n GammaDim = 1 DecVarDim = QDim + YDim + GammaDim DecVar in R^DecVarDim

Shape checks: A Q, Q A^T, B_xu Y, Y^T B_xu^T : n x n B_xw : n x m_w C_zx Q + D_zu Y : p_z x n D_zw : p_z x m_w gamma I_w : m_w x m_w gamma I_z : p_z x p_z BRL LMI : (n + m_w + p_z) x (n + m_w + p_z)

Definition at line 96 of file hinf_state_feedback.hpp.