:class:`DSystem` - Discrete System wrapper for Midpoint Variational Integrators
.. currentmodule:: trep.discopt
:class:`DSystem` objects represent a :class:`MidpointVI` variational integrators as first order discrete systems of the form X(k+1) = f(X(k), U(k), k). This representation is used by :class:`DOptimizer` for discrete trajectory optimization. :class:`DSystem` also provides methods for automatically calculating linearizations and feedback controllers along trajectories.
The discrete state consists of the variational integrator's full configuration, the dynamic momentum, and the kinematic velocity:
X(k) = \begin{bmatrix} q_d(k) \\ q_k(k) \\ p(k) \\ v_k(k) \end{bmatrix}
The configuration and momentum are the same as in :class:`MidpointVI`. The kinematic velocity is calculated as:
v_k(k) = \frac{q_k(k) - q_k(k-1)}{t(k) - t(k-1)}
The discrete input consists of the variational integrator's force inputs and the future state of the kinematic configurations:
U(k) = \begin{bmatrix} u(k) \\ \rho(k) \end{bmatrix}
where the kinematic inputs are denoted by \rho thoughout this code (i.e, q_k(k+1) = \rho(k)). Additionally, the state input U is always capitalized to distinguish it from the force input u which is always lower case.
:class:`DSystem` provides methods for converting between trajectories for the discrete system and trajectories for the variational integrator.
Examples
pend-on-cart-optimization.py
| param varint: | The variational integrator being represented. |
|---|---|
| type varint: | :class:`MidpointVI` instance |
| param t: | An array of times |
| type t: | numpy array of floats, shape (N) |
Create a discrete system wrapper for the variational integrator varint and the time t. The time t is the array t(k) that maps a discrete time index to a time. It should have the same length N as trajectories used with the system.
.. attribute:: DSystem.nX Number of states to the discrete system. *int*
.. attribute:: DSystem.nU Number of inputs to the discrete system. *int*
.. attribute:: DSystem.varint The variational integrator wrapped by this instance. :class:`MidpointVI`
.. attribute:: DSystem.system The mechanical system modeled by the variational integrator. :class:`System`
.. attribute:: DSystem.time The time of the discrete steps. numpy array, shape (N)
.. attribute:: xk Current state of the system. numpy array, shape (nX)
.. attribute:: uk Current input of the system. numpy array, shape (nU)
.. attribute:: k Current discrete time of the system. int
.. method:: DSystem.kf() :rtype: int Return the last available state that the system can be set to. This is one less than ``len(self.time)``.
.. method:: DSystem.build_state([Q=None, p=None, v=None]) :param Q: A configuration vector :type Q: numpy array, shape (nQ) :param p: A momentum vector :type p: numpy array, shape (nd) :param v: A kinematic velocity vector :type v: numpy array, shape (nk) Build a state vector *Xk* from components. Unspecified components are set to zero.
.. method:: DSystem.build_input([u=None, rho=None]) :param u: An input force vector :type u: numpy array, shape (nu) :param rho: A kinematic input vector :type rho: numpy array, shape (nk) :type: numpy array, shape (nU) Build an input vector *Uk* from components. Unspecified components are set to zero.
.. method:: DSystem.build_trajectory([Q=None, p=None, v=None, u=None, rho=None])
:param Q: A configuration trajectory
:type Q: numpy array, shape (N, nQ)
:param p: A momentum trajectory
:type p: numpy array, shape (N, nd)
:param v: A velocity trajectory
:type v: numpy array, shape (N, nk)
:param u: An input force trajectory
:type u: numpy array, shape (N-1, nu)
:param rho: A kinematic input trajectory
:type rho: numpy array, shape (N-1, nk)
:rtype: named tuple of (*X*, *U*)
Combine component trajectories into a state and input trajectories.
The state length is the same as the time base, the input length is
one less than the time base. Unspecified components are set to
zero::
>>> dsys.build_trajectory() # Create a zero state and input trajectory
.. method:: DSystem.split_state([X=None]) :param X: A state vector for the system :type X: numpy array, shape (nX) :rtype: named tuple of (*Q*, *p*, *v*) Split a state vector into its configuration, moementum, and kinematic velocity parts. If *X* is :data:`None`, returns zero arrays for each component.
.. method:: DSystem.split_input([U=None]) :param U: An input vector for the system :type U: numpy array, shape (nU) :rtype: named tuple of (*u*, *rho*) Split a state input vector *U* into its force and kinematic input parts, (*u*, *rho*). If *U* is :data:`None`, returns zero arrays of the appropriate size.
.. method:: DSystem.split_trajectory([X=None, U=None]) :param X: A state trajectory :type X: numpy array, shape (N, nX) :param U: An input trajectory :type U: numpy array, shape (N-1, nU) :rtype: named tuple of (*Q*, *p*, *v*, *u*, *rho*) Split the state trajectory (*X*, *U*) into its *Q*, *p*, *v*, *u*, *rho* components. If *X* or *U* are :data:`None`, the corresponding components are arrays of zero.
.. method:: DSystem.convert_trajectory(dsys_a, X, U)
:param dsys_a: Another discrete system
:type dsys_a: :class:`DSystem`
:param X: A state trajectory for *dsys_a*
:type X: numpy array, shape (N, nX)
:param U: An input trajectory for *dsys_a*
:type U: numpy array, shape (N, nU)
:rtype: trajectory for this system, named tuple (*X*, *U*)
Convert the trajectory (*X*, *U*) for *dsys_a* into a trajectory
for this system. This reorders the trajectory components according
to the configuration and input variable names and drops components
that aren't in this system. Variables in this system that are not
in *dsys_a* are replaced with zero.
.. note::
The returned path may not be a valid trajectory for this system
in the sense that :math:`x(k+1) = f(x(k), u(k), k)`. This
function only reorders the information.
.. method:: DSystem.save_state_trajectory(filename, [X=None, U=None]) :param filename: Location to save the trajectory :type filename: string :param X: A state trajectory :type X: numpy array, shape (N, nX) :param U: An input trajectory :type U: numpy array, shape (N-1, nU) Save a trajectory to a file. This splits the trajectory with :meth:`split_trajectory` and saves the results with :func:`trep.save_traejctory`. If *X* or *U* are not specified, they are replaced with zero arrays.
.. method:: DSystem.load_state_trajectory(filename) :param filename: Location of saved trajectory :type filename: string :rtype: named tuple of (X, U) Load a trajectory from a file that was stored with :meth:`save_state_trajectory` or :func:`trep.save_trajectory`. If the file does not contain complete information for the system (e.g, it was saved for a different system with different states, or the inputs were not saved), the missing components will be filled with zeros.
.. method:: DSystem.set(self, xk, uk, k[, \
xk_hint=None, lambda_hint=None])
Set the current state, input, and time of the discrete system.
If *xk_hint* and *lambda_hint* are provided, these are used to
provide hints to hints to :meth:`MidpointVI.step`. If the solution
is known (for example, if you are calculating the linearization
about a known trajectory) this can result in faster performance by
reducing the number of root solver iterations in the variational
integrator.
.. method:: DSystem.step(self, uk[, xk_hint=None, lambda_hint=None]) Advance the system to the next discrete time using the given input *uk*. This is equivalent to calling ``self.set(self.f(), uk, self.k+1)``. If *xk_hint* and *lambda_hint* are provided, these are used to provide hints to hints to :meth:`MidpointVI.step`. If the solution is known (for example, if you are calculating the linearization about a known trajectory) this can result in faster performance by reducing the number of root solver iterations in the variational integrator.
.. method:: DSystem.f() :rtype: numpy array, shape (nX) Get the next state of the system, :math:`x(k+1)`.
.. method:: DSystem.fdx() :rtype: numpy array, shape (nX, nX)
.. method:: DSystem.fdu() :rtype: numpy array, shape (nX, nU) These functions return first derivatives of the system dynamics :math:`f()` as numpy arrays with the derivatives across the rows.
.. method:: DSystem.fdxdx(z) :param z: adjoint vector :type z: numpy array, shape (nX) :rtype: numpy array, shape (nU, nU)
.. method:: DSystem.fdxdu(z) :param z: adjoint vector :type z: numpy array, shape (nX) :rtype: numpy array, shape (nX, nU)
.. method:: DSystem.fdudu(z)
:param z: adjoint vector
:type z: numpy array, shape (nX)
:rtype: numpy array, shape (nU, nU)
These functions return the product of the 1D array *z* and the
second derivative of f. For example:
.. math::
z^T \derivII[f]{u}{u}
.. method:: DSystem.linearize_trajectory(X, U) :rtype: named tuple (A, B) Calculate the linearization of the system dynamics about a trajectory. *X* and *U* do not have to be an exact trajectory of the system. Returns the linearization in a named tuple ``(A, B)``\ .
.. method:: DSystem.project(bX, bU[, Kproj=None])
:rtyple: named tuple (X, U)
Project *bX* and *bU* into a nearby trajectory for the system using
a linear feedback law::
X[0] = bX[0]
U[k] = bU[k] - Kproj * (X[k] - bU[k])
X[k+1] = f(X[k], U[k], k)
If no feedback law is specified, one will be created by
:meth:`calc_feedback_controller` along *bX* and *bU*. This is
typically a **bad idea** if *bX* and *bU* are not very close to an
actual trajectory for the system.
Returns the projected trajectory in a named tuple ``(X, U)``\ .
.. method:: DSystem.dproject(A, B, bdX, bdU, K) :rtyple: named tuple (dX, dU) Project *bdX* and *bdU* into the tangent trajectory space of the system. *A* and *B* are the linearization of the system about the trajectory. *K* is a stabilizing feedback controller. Returns the projected tangent trajectory ``(dX, dU)``\ .
.. method:: DSystem.calc_feedback_controller(X, U[, Q=None, R=None, return_linearization=False]) :rtype: K or named tuple (K, A, B) Calculate a stabilizing feedback controller for the system about a trajectory *X* and *U*. The feedback law is calculated by solving the discrete LQR problem for the linearization of the system about *X* and *U*. *X* and *U* do not have to be an exact trajectory of the system, but if they are not close, the controller is unlikely to be effective. If the LQR weights *Q* and *R* are not specified, identity matrices are used. If *return_linearization* is :data:`False`, the return value is the feedback control law, *K*. If *return_linearization* is :data:`True`, the method returns the linearization as well in a named tuple: ``(K, A, B)``.
.. method:: DSystem.check_fdx(xk, uk, k[, delta=1e-5])
DSystem.check_fdu(xk, uk, k[, delta=1e-5])
DSystem.check_fdxdx(xk, uk, k[, delta=1e-5])
DSystem.check_fdxdu(xk, uk, k[, delta=1e-5])
DSystem.check_fdudu(xk, uk, k[, delta=1e-5])
:param xk: A valid state of the system
:type xk: numpy array, shape (nX)
:param uk: A valid input to the system
:type uk: numpy array, shape (nU)
:param k: A valid discrete time index
:type k: int
:param delta: The perturbation for approximating the derivative.
These functions check derivatives of the discrete state dynamics
against numeric approximations generated from lower derivatives
(e.g, fdx() from f(), and fdudu() from fdu()). A three point
approximation is used::
approx_deriv = (f(x + delta) - f(x - delta)) / (2 * delta)
Each function returns a named tuple ``(error, exact_norm, approx_norm)``
where ``error`` is the norm of the difference between the exact and
approximate derivative, ``exact_norm`` is the norm of the exact
derivative, ``approx_norm`` is the norm of the approximate derivative.