Drake
ImplicitEulerIntegrator< T > Class Template Referencefinal

Detailed Description

template<class T> class drake::systems::ImplicitEulerIntegrator< T >

A first-order, fully implicit integrator with second order error estimation.

Template Parameters
 T The vector element type, which must be a valid Eigen scalar.

This class uses Drake's -inl.h pattern. When seeing linker errors from this class, please refer to https://drake.mit.edu/cxx_inl.html.

Instantiated templates for the following kinds of T's are provided:

• double
• AutoDiffXd

This integrator uses the following update rule:

x(t+h) = x(t) + h f(t+h,x(t+h))

where x are the state variables, h is the integration step size, and f() returns the time derivatives of the state variables. Contrast this update rule to that of an explicit first-order integrator:

x(t+h) = x(t) + h f(t, x(t))

Thus implicit first-order integration must solve a nonlinear system of equations to determine both the state at t+h and the time derivatives of that state at that time. Cast as a nonlinear system of equations, we seek the solution to:

x(t+h) - x(t) - h f(t+h,x(t+h)) = 0

given unknowns x(t+h).

This "implicit Euler" method is known to be L-Stable, meaning both that applying it at a fixed integration step to the "test" equation y(t) = eᵏᵗ yields zero (for k < 0 and t → ∞) and that it is also A-Stable. A-Stability, in turn, means that the method can integrate the linear constant coefficient system dx/dt = Ax at any step size without the solution becoming unstable (growing without bound). The practical effect of L-Stability is that the integrator tends to be stable for any given step size on an arbitrary system of ordinary differential equations. See [Lambert, 1991], Ch. 6 for an approachable discussion on stiff differential equations and L- and A-Stability.

This implementation uses Newton-Raphson (NR) and relies upon the obvious convergence to a solution for g = 0 where g(x(t+h)) ≡ x(t+h) - x(t) - h f(t+h,x(t+h)) as h becomes sufficiently small. It also uses the implicit trapezoid method- fed the result from implicit Euler for (hopefully) faster convergence- to compute the error estimate. General implementational details were gleaned from [Hairer, 1996].

• [Hairer, 1996] E. Hairer and G. Wanner. Solving Ordinary Differential Equations II (Stiff and Differential-Algebraic Problems). Springer, 1996.
• [Lambert, 1991] J. D. Lambert. Numerical Methods for Ordinary Differential Equations. John Wiley & Sons, 1991.
Note
This integrator uses the integrator accuracy setting, even when run in fixed-step mode, to limit the error in the underlying Newton-Raphson process. See IntegratorBase::set_target_accuracy() for more info.
ImplicitIntegrator class documentation for information about implicit integration methods in general.

#include <drake/systems/analysis/implicit_euler_integrator.h>

Public Member Functions

~ImplicitEulerIntegrator () override=default

ImplicitEulerIntegrator (const System< T > &system, Context< T > *context=nullptr)

bool supports_error_estimation () const final
The integrator supports error estimation. More...

int get_error_estimate_order () const final
This first-order integrator uses embedded second order methods to compute estimates of the local truncation error. More...

Does not allow copy, move, or assignment
ImplicitEulerIntegrator (const ImplicitEulerIntegrator &)=delete

ImplicitEulerIntegratoroperator= (const ImplicitEulerIntegrator &)=delete

ImplicitEulerIntegrator (ImplicitEulerIntegrator &&)=delete

ImplicitEulerIntegratoroperator= (ImplicitEulerIntegrator &&)=delete Public Member Functions inherited from ImplicitIntegrator< T >
virtual ~ImplicitIntegrator ()

ImplicitIntegrator (const System< T > &system, Context< T > *context=nullptr)

int max_newton_raphson_iterations () const
The maximum number of Newton-Raphson iterations to take before the Newton-Raphson process decides that convergence will not be attained. More...

void set_reuse (bool reuse)
Sets whether the integrator attempts to reuse Jacobian matrices and iteration matrix factorizations (default is true). More...

bool get_reuse () const
Gets whether the integrator attempts to reuse Jacobian matrices and iteration matrix factorizations. More...

void set_jacobian_computation_scheme (JacobianComputationScheme scheme)
Sets the Jacobian computation scheme. More...

JacobianComputationScheme get_jacobian_computation_scheme () const

int64_t get_num_derivative_evaluations_for_jacobian () const
Gets the number of ODE function evaluations (calls to EvalTimeDerivatives()) used only for computing the Jacobian matrices since the last call to ResetStatistics(). More...

int64_t get_num_jacobian_evaluations () const
Gets the number of Jacobian computations (i.e., the number of times that the Jacobian matrix was reformed) since the last call to ResetStatistics(). More...

int64_t get_num_newton_raphson_iterations () const
Gets the number of iterations used in the Newton-Raphson nonlinear systems of equation solving process since the last call to ResetStatistics(). More...

int64_t get_num_iteration_matrix_factorizations () const
Gets the number of factorizations of the iteration matrix since the last call to ResetStatistics(). More...

int64_t get_num_error_estimator_derivative_evaluations () const
Gets the number of ODE function evaluations (calls to EvalTimeDerivatives()) used only for the error estimation process since the last call to ResetStatistics(). More...

int64_t get_num_error_estimator_derivative_evaluations_for_jacobian () const

int64_t get_num_error_estimator_newton_raphson_iterations () const
Gets the number of iterations used in the Newton-Raphson nonlinear systems of equation solving process for the error estimation process since the last call to ResetStatistics(). More...

int64_t get_num_error_estimator_jacobian_evaluations () const
Gets the number of Jacobian matrix computations used only during the error estimation process since the last call to ResetStatistics(). More...

int64_t get_num_error_estimator_iteration_matrix_factorizations () const
Gets the number of factorizations of the iteration matrix used only during the error estimation process since the last call to ResetStatistics(). More... Public Member Functions inherited from IntegratorBase< T >
IntegratorBase (const System< T > &system, Context< T > *context=nullptr)
Maintains references to the system being integrated and the context used to specify the initial conditions for that system (if any). More...

virtual ~IntegratorBase ()=default
Destructor. More...

void set_fixed_step_mode (bool flag)
Sets an integrator with error control to fixed step mode. More...

bool get_fixed_step_mode () const
Gets whether an integrator is running in fixed step mode. More...

void set_target_accuracy (double accuracy)
Request that the integrator attempt to achieve a particular accuracy for the continuous portions of the simulation. More...

double get_target_accuracy () const
Gets the target accuracy. More...

double get_accuracy_in_use () const
Gets the accuracy in use by the integrator. More...

void set_maximum_step_size (const T &max_step_size)
Sets the maximum step size that may be taken by this integrator. More...

const T & get_maximum_step_size () const
Gets the maximum step size that may be taken by this integrator. More...

void Reset ()
Resets the integrator to initial values, i.e., default construction values. More...

void Initialize ()
An integrator must be initialized before being used. More...

void request_initial_step_size_target (const T &step_size)
Request that the first attempted integration step have a particular size. More...

const T & get_initial_step_size_target () const
Gets the target size of the first integration step. More...

StepResult IntegrateNoFurtherThanTime (const T &publish_time, const T &update_time, const T &boundary_time)
(Internal use only) Integrates the system forward in time by a single step with step size subject to integration error tolerances (assuming that the integrator supports error estimation). More...

double get_stretch_factor () const
Gets the stretch factor (> 1), which is multiplied by the maximum (typically user-designated) integration step size to obtain the amount that the integrator is able to stretch the maximum time step toward hitting an upcoming publish or update event in IntegrateNoFurtherThanTime(). More...

void IntegrateWithMultipleStepsToTime (const T &t_final)
Stepping function for integrators operating outside of Simulator that advances the continuous state exactly to t_final. More...

DRAKE_NODISCARD bool IntegrateWithSingleFixedStepToTime (const T &t_target)
Stepping function for integrators operating outside of Simulator that advances the continuous state using a single step to t_target. More...

const T & get_ideal_next_step_size () const
Return the step size the integrator would like to take next, based primarily on the integrator's accuracy prediction. More...

const Context< T > & get_context () const
Returns a const reference to the internally-maintained Context holding the most recent state in the trajectory. More...

Context< T > * get_mutable_context ()
Returns a mutable pointer to the internally-maintained Context holding the most recent state in the trajectory. More...

void reset_context (Context< T > *context)
Replace the pointer to the internally-maintained Context with a different one. More...

const System< T > & get_system () const
Gets a constant reference to the system that is being integrated (and was provided to the constructor of the integrator). More...

bool is_initialized () const
Indicates whether the integrator has been initialized. More...

const T & get_previous_integration_step_size () const
Gets the size of the last (previous) integration step. More...

const ContinuousState< T > * get_error_estimate () const
Gets the error estimate (used only for integrators that support error estimation). More...

IntegratorBase (const IntegratorBase &)=delete

IntegratorBaseoperator= (const IntegratorBase &)=delete

IntegratorBase (IntegratorBase &&)=delete

IntegratorBaseoperator= (IntegratorBase &&)=delete

void set_requested_minimum_step_size (const T &min_step_size)
Sets the requested minimum step size h_min that may be taken by this integrator. More...

const T & get_requested_minimum_step_size () const
Gets the requested minimum step size h_min for this integrator. More...

void set_throw_on_minimum_step_size_violation (bool throws)
Sets whether the integrator should throw a std::runtime_error exception when the integrator's step size selection algorithm determines that it must take a step smaller than the minimum step size (for, e.g., purposes of error control). More...

bool get_throw_on_minimum_step_size_violation () const
Reports the current setting of the throw_on_minimum_step_size_violation flag. More...

get_working_minimum_step_size () const
Gets the current value of the working minimum step size h_work(t) for this integrator, which may vary with the current time t as stored in the integrator's context. More...

void ResetStatistics ()
Forget accumulated statistics. More...

int64_t get_num_substep_failures () const
Gets the number of failed sub-steps (implying one or more step size reductions was required to permit solving the necessary nonlinear system of equations). More...

int64_t get_num_step_shrinkages_from_substep_failures () const
Gets the number of step size shrinkages due to sub-step failures (e.g., integrator convergence failures) since the last call to ResetStatistics() or Initialize(). More...

int64_t get_num_step_shrinkages_from_error_control () const
Gets the number of step size shrinkages due to failure to meet targeted error tolerances, since the last call to ResetStatistics or Initialize(). More...

int64_t get_num_derivative_evaluations () const
Returns the number of ODE function evaluations (calls to CalcTimeDerivatives()) since the last call to ResetStatistics() or Initialize(). More...

const T & get_actual_initial_step_size_taken () const
The actual size of the successful first step. More...

const T & get_smallest_adapted_step_size_taken () const
The size of the smallest step taken as the result of a controlled integration step adjustment since the last Initialize() or ResetStatistics() call. More...

const T & get_largest_step_size_taken () const
The size of the largest step taken since the last Initialize() or ResetStatistics() call. More...

int64_t get_num_steps_taken () const
The number of integration steps taken since the last Initialize() or ResetStatistics() call. More...

void add_derivative_evaluations (double evals)
Manually increments the statistic for the number of ODE evaluations. More...

void StartDenseIntegration ()
Starts dense integration, allocating a new dense output for this integrator to use. More...

const DenseOutput< T > * get_dense_output () const
Returns a const pointer to the integrator's current DenseOutput instance, holding a representation of the continuous state trajectory since the last StartDenseIntegration() call. More...

std::unique_ptr< DenseOutput< T > > StopDenseIntegration ()
Stops dense integration, yielding ownership of the current dense output to the caller. More...

const Eigen::VectorXd & get_generalized_state_weight_vector () const
Gets the weighting vector (equivalent to a diagonal matrix) applied to weighting both generalized coordinate and velocity state variable errors, as described in the group documentation. More...

Eigen::VectorBlock< Eigen::VectorXd > get_mutable_generalized_state_weight_vector ()
Gets a mutable weighting vector (equivalent to a diagonal matrix) applied to weighting both generalized coordinate and velocity state variable errors, as described in the group documentation. More...

const Eigen::VectorXd & get_misc_state_weight_vector () const
Gets the weighting vector (equivalent to a diagonal matrix) for weighting errors in miscellaneous continuous state variables z. More...

Eigen::VectorBlock< Eigen::VectorXd > get_mutable_misc_state_weight_vector ()
Gets a mutable weighting vector (equivalent to a diagonal matrix) for weighting errors in miscellaneous continuous state variables z. More... Public Types inherited from ImplicitIntegrator< T >
enum  JacobianComputationScheme { kForwardDifference, kCentralDifference, kAutomatic } Public Types inherited from IntegratorBase< T >
enum  StepResult {
kReachedPublishTime = 1, kReachedZeroCrossing = 2, kReachedUpdateTime = 3, kTimeHasAdvanced = 4,
kReachedBoundaryTime = 5, kReachedStepLimit = 6
}
Status returned by StepOnceAtMost(). More... Protected Member Functions inherited from ImplicitIntegrator< T >
virtual int do_max_newton_raphson_iterations () const
Derived classes can override this method to change the number of Newton-Raphson iterations (10 by default) to take before the Newton-Raphson process decides that convergence will not be attained. More...

bool MaybeFreshenMatrices (const T &t, const VectorX< T > &xt, const T &h, int trial, const std::function< void(const MatrixX< T > &J, const T &h, typename ImplicitIntegrator< T >::IterationMatrix *)> &compute_and_factor_iteration_matrix, typename ImplicitIntegrator< T >::IterationMatrix *iteration_matrix)
Computes necessary matrices (Jacobian and iteration matrix) for Newton-Raphson (NR) iterations, as necessary. More...

bool IsUpdateZero (const VectorX< T > &xc, const VectorX< T > &dxc, double eps=-1.0) const
Checks whether a proposed update is effectively zero, indicating that the Newton-Raphson process converged. More...

bool IsBadJacobian (const MatrixX< T > &J) const
Checks to see whether a Jacobian matrix is "bad" (has any NaN or Inf values) and needs to be recomputed. More...

MatrixX< T > & get_mutable_jacobian ()

void DoResetStatistics () override
Resets any statistics particular to a specific integrator. More...

const MatrixX< T > & CalcJacobian (const T &tf, const VectorX< T > &xtplus)

void ComputeForwardDiffJacobian (const System< T > &, const T &t, const VectorX< T > &xc, Context< T > *, MatrixX< T > *J)

void ComputeCentralDiffJacobian (const System< T > &, const T &t, const VectorX< T > &xc, Context< T > *, MatrixX< T > *J)

void ComputeAutoDiffJacobian (const System< T > &system, const T &t, const VectorX< T > &xc, const Context< T > &context, MatrixX< T > *J)

template<>
void ComputeAutoDiffJacobian (const System< AutoDiffXd > &, const AutoDiffXd &, const VectorX< AutoDiffXd > &, const Context< AutoDiffXd > &, MatrixX< AutoDiffXd > *) Protected Member Functions inherited from IntegratorBase< T >
const ContinuousState< T > & EvalTimeDerivatives (const Context< T > &context)
Evaluates the derivative function and updates call statistics. More...

template<typename U >
const ContinuousState< U > & EvalTimeDerivatives (const System< U > &system, const Context< U > &context)
Evaluates the derivative function (and updates call statistics). More...

void set_accuracy_in_use (double accuracy)
Sets the working ("in use") accuracy for this integrator. More...

void InitializeAccuracy (double default_accuracy, double loosest_accuracy, double max_step_fraction)
Generic code for validating (and resetting, if need be) the integrator working accuracy for error controlled integrators. More...

bool StepOnceErrorControlledAtMost (const T &dt_max)
Default code for advancing the continuous state of the system by a single step of dt_max (or smaller, depending on error control). More...

CalcStateChangeNorm (const ContinuousState< T > &dx_state) const
Computes the infinity norm of a change in continuous state. More...

std::pair< bool, T > CalcAdjustedStepSize (const T &err, const T &attempted_step_size, bool *at_minimum_step_size) const
Calculates adjusted integrator step sizes toward keeping state variables within error bounds on the next integration step. More...

virtual void DoReset ()
Derived classes can override this method to perform routines when Reset() is called. More...

virtual std::unique_ptr< StepwiseDenseOutput< T > > DoStartDenseIntegration ()
Derived classes can override this method to provide a continuous extension of their own when StartDenseIntegration() is called. More...

StepwiseDenseOutput< T > * get_mutable_dense_output ()
Returns a mutable pointer to the internally-maintained StepwiseDenseOutput instance, holding a representation of the continuous state trajectory since the last time StartDenseIntegration() was called. More...

virtual bool DoDenseStep (const T &dt)
Derived classes may implement this method to (1) integrate the continuous portion of this system forward by a single step of size dt, (2) set the error estimate (via get_mutable_error_estimate()) and (3) update their own dense output implementation (via get_mutable_dense_output()). More...

ContinuousState< T > * get_mutable_error_estimate ()
Gets an error estimate of the state variables recorded by the last call to StepOnceFixedSize(). More...

void set_actual_initial_step_size_taken (const T &dt)

void set_smallest_adapted_step_size_taken (const T &dt)
Sets the size of the smallest-step-taken statistic as the result of a controlled integration step adjustment. More...

void set_largest_step_size_taken (const T &dt)

void set_ideal_next_step_size (const T &dt)

◆ ImplicitEulerIntegrator() [1/3]

 ImplicitEulerIntegrator ( const ImplicitEulerIntegrator< T > & )
delete

◆ ImplicitEulerIntegrator() [2/3]

 ImplicitEulerIntegrator ( ImplicitEulerIntegrator< T > && )
delete

◆ ~ImplicitEulerIntegrator()

 ~ImplicitEulerIntegrator ( )
overridedefault

◆ ImplicitEulerIntegrator() [3/3]

 ImplicitEulerIntegrator ( const System< T > & system, Context< T > * context = nullptr )
explicit

◆ get_error_estimate_order()

 int get_error_estimate_order ( ) const
finalvirtual

This first-order integrator uses embedded second order methods to compute estimates of the local truncation error.

The order of the asymptotic difference between these two methods (from which the error estimate is computed) is O(h²).

Implements IntegratorBase< T >.

◆ operator=() [1/2]

 ImplicitEulerIntegrator& operator= ( ImplicitEulerIntegrator< T > && )
delete

◆ operator=() [2/2]

 ImplicitEulerIntegrator& operator= ( const ImplicitEulerIntegrator< T > & )
delete

◆ supports_error_estimation()

 bool supports_error_estimation ( ) const
finalvirtual

The integrator supports error estimation.

Implements IntegratorBase< T >.

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