Control Systems

Collaboration diagram for Control Systems:

Classes
class	InverseDynamicsController< T >
	A state feedback controller that uses a PidController to generate desired accelerations, which are then converted into torques using InverseDynamics. More...
class	LinearModelPredictiveController< T >
	Implements a basic Model Predictive Controller that linearizes the system about an equilibrium condition and regulates to the same point by solving an optimal control problem over a finite time horizon. More...
class	PidControlledSystem< T >
	A system that encapsulates a PidController and a controlled System (a.k.a the "plant"). More...
class	PidController< T >
	Implements the PID controller. More...

Functions
LinearQuadraticRegulatorResult	LinearQuadraticRegulator (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, const Eigen::Ref< const Eigen::MatrixXd > &N=Eigen::Matrix< double, 0, 0 >::Zero())
	Computes the optimal feedback controller, u=-Kx, and the optimal cost-to-go J = x'Sx for the problem: More...
LinearQuadraticRegulatorResult	DiscreteTimeLinearQuadraticRegulator (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)
	Computes the optimal feedback controller, u=-Kx, and the optimal cost-to-go J = x'Sx for the problem: More...
std::unique_ptr< systems::LinearSystem< double > >	LinearQuadraticRegulator (const LinearSystem< double > &system, const Eigen::Ref< const Eigen::MatrixXd > &Q, const Eigen::Ref< const Eigen::MatrixXd > &R, const Eigen::Ref< const Eigen::MatrixXd > &N=Eigen::Matrix< double, 0, 0 >::Zero())
	Creates a system that implements the optimal time-invariant linear quadratic regulator (LQR). More...
std::unique_ptr< systems::AffineSystem< double > >	LinearQuadraticRegulator (const System< double > &system, const Context< double > &context, const Eigen::Ref< const Eigen::MatrixXd > &Q, const Eigen::Ref< const Eigen::MatrixXd > &R, const Eigen::Ref< const Eigen::MatrixXd > &N=Eigen::Matrix< double, 0, 0 >::Zero())
	Linearizes the System around the specified Context, computes the optimal time-invariant linear quadratic regulator (LQR), and returns a System which implements that regulator in the original System's coordinates. More...
Eigen::MatrixXd	ControllabilityMatrix (const LinearSystem< double > &sys)
	Returns the controllability matrix: R = [B, AB, ..., A^{n-1}B]. More...
bool	IsControllable (const LinearSystem< double > &sys, optional< double > threshold)
	Returns true iff the controllability matrix is full row rank. More...

Detailed Description

Function Documentation

Eigen::MatrixXd ControllabilityMatrix

(

const LinearSystem< double > &

sys

)

Returns the controllability matrix: R = [B, AB, ..., A^{n-1}B].

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LinearQuadraticRegulatorResult DiscreteTimeLinearQuadraticRegulator	(	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
	)

Computes the optimal feedback controller, u=-Kx, and the optimal cost-to-go J = x'Sx for the problem:

\[ x[n+1] = Ax[n] + Bu[n] \]

\[ \min_u \sum_0^\infty x'Qx + u'Ru \]

Parameters

A	The state-space dynamics matrix of size num_states x num_states.
B	The state-space input matrix of size num_states x num_inputs.
Q	A symmetric positive semi-definite cost matrix of size num_states x num_states.
R	A symmetric positive definite cost matrix of size num_inputs x num_inputs.

Returns

A structure that contains the optimal feedback gain K and the quadratic cost term S. The optimal feedback control is u = -Kx;

Exceptions

std::runtime_error

if R is not positive definite.

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bool IsControllable	(	const LinearSystem< double > &	sys,
		optional< double >	threshold
	)

Returns true iff the controllability matrix is full row rank.

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LinearQuadraticRegulatorResult LinearQuadraticRegulator	(	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,
		const Eigen::Ref< const Eigen::MatrixXd > &	N = Eigen::Matrix< double, 0, 0 >::Zero()
	)

Computes the optimal feedback controller, u=-Kx, and the optimal cost-to-go J = x'Sx for the problem:

\[ \dot{x} = Ax + Bu \]

\[ \min_u \int_0^\infty x'Qx + u'Ru + 2x'Nu dt \]

Parameters

A	The state-space dynamics matrix of size num_states x num_states.
B	The state-space input matrix of size num_states x num_inputs.
Q	A symmetric positive semi-definite cost matrix of size num_states x num_states.
R	A symmetric positive definite cost matrix of size num_inputs x num_inputs.
N	A cost matrix of size num_states x num_inputs.

Returns

A structure that contains the optimal feedback gain K and the quadratic cost term S. The optimal feedback control is u = -Kx;

Exceptions

std::runtime_error

if R is not positive definite.

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std::unique_ptr< LinearSystem< double > > LinearQuadraticRegulator	(	const LinearSystem< double > &	system,
		const Eigen::Ref< const Eigen::MatrixXd > &	Q,
		const Eigen::Ref< const Eigen::MatrixXd > &	R,
		const Eigen::Ref< const Eigen::MatrixXd > &	N = Eigen::Matrix< double, 0, 0 >::Zero()
	)

Creates a system that implements the optimal time-invariant linear quadratic regulator (LQR).

If system is a continuous-time system, then solves the continuous-time LQR problem:

\[ \min_u \int_0^\infty x^T(t)Qx(t) + u^T(t)Ru(t) dt. \]

If system is a discrete-time system, then solves the discrete-time LQR problem:

\[ \min_u \sum_0^\infty x^T[n]Qx[n] + u^T[n]Ru[n]. \]

Parameters

system	The System to be controlled.
Q	A symmetric positive semi-definite cost matrix of size num_states x num_states.
R	A symmetric positive definite cost matrix of size num_inputs x num_inputs.
N	A cost matrix of size num_states x num_inputs.

Returns

A system implementing the optimal controller in the original system coordinates.

Exceptions

std::runtime_error

if R is not positive definite.

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std::unique_ptr< AffineSystem< double > > LinearQuadraticRegulator	(	const System< double > &	system,
		const Context< double > &	context,
		const Eigen::Ref< const Eigen::MatrixXd > &	Q,
		const Eigen::Ref< const Eigen::MatrixXd > &	R,
		const Eigen::Ref< const Eigen::MatrixXd > &	N = Eigen::Matrix< double, 0, 0 >::Zero()
	)

Linearizes the System around the specified Context, computes the optimal time-invariant linear quadratic regulator (LQR), and returns a System which implements that regulator in the original System's coordinates.

If system is a continuous-time system, then solves the continuous-time LQR problem:

\[ \min_u \int_0^\infty (x-x_0)^TQ(x-x_0) + (u-u_0)^TR(u-u_0) dt. \]

If system is a discrete-time system, then solves the discrete-time LQR problem:

\[ \min_u \sum_0^\infty (x-x_0)^TQ(x-x_0) + (u-u_0)^TR(u-u_0), \]

where \( x_0 \) is the nominal state and \( u_0 \) is the nominal input. The system is considered discrete if it has a single discrete state vector and a single unique periodic update event declared.

Parameters

system	The System to be controlled.
context	Defines the desired state and control input to regulate the system to. Note that this state/input must be an equilibrium point of the system. See drake::systems::Linearize for more details.
Q	A symmetric positive semi-definite cost matrix of size num_states x num_states.
R	A symmetric positive definite cost matrix of size num_inputs x num_inputs.
N	A cost matrix of size num_states x num_inputs.

Returns

A system implementing the optimal controller in the original system coordinates.

Exceptions

std::runtime_error

if R is not positive definite.

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