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Commit522a8d7

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add documentation for discrete time optimal control problems
1 parent33fafa1 commit522a8d7

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2 files changed

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‎control/optimal.py

Lines changed: 45 additions & 14 deletions
Original file line numberDiff line numberDiff line change
@@ -66,7 +66,7 @@ class OptimalControlProblem():
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`(fun, lb, ub)`. The constraints will be applied at each time point
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along the trajectory.
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terminal_cost : callable, optional
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Function that returns the terminal cost given thecurrent state
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Function that returns the terminal cost given thefinal state
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and input. Called as terminal_cost(x, u).
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trajectory_method : string, optional
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Method to use for carrying out the optimization. Currently supported
@@ -287,12 +287,16 @@ def __init__(
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# time point and we use a trapezoidal approximation to compute the
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# integral cost, then add on the terminal cost.
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#
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# For shooting methods, given the input U = [u[0], ... u[N]] we need to
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# For shooting methods, given the input U = [u[t_0], ... u[t_N]] we need to
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# compute the cost of the trajectory generated by that input. This
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# means we have to simulate the system to get the state trajectory X =
293-
# [x[0], ..., x[N]] and then compute the cost at each point:
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# [x[t_0], ..., x[t_N]] and then compute the cost at each point:
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#
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# cost = sum_k integral_cost(x[k], u[k]) + terminal_cost(x[N], u[N])
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# cost = sum_k integral_cost(x[t_k], u[t_k])
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# + terminal_cost(x[t_N], u[t_N])
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#
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# The actual calculation is a bit more complex: for continuous time
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# systems, we use a trapezoidal approximation for the integral cost.
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#
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# The initial state used for generating the simulation is stored in the
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# class parameter `x` prior to calling the optimization algorithm.
@@ -325,8 +329,8 @@ def _cost_function(self, coeffs):
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# Sum the integral cost over the time (second) indices
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# cost += self.integral_cost(states[:,i], inputs[:,i])
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cost=sum(map(
328-
self.integral_cost,np.transpose(states[:, :-1]),
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np.transpose(inputs[:, :-1])))
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self.integral_cost,states[:, :-1].transpose(),
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inputs[:, :-1].transpose()))
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# Terminal cost
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ifself.terminal_costisnotNone:
@@ -954,7 +958,22 @@ def solve_ocp(
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transpose=None,return_states=True,print_summary=True,log=False,
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**kwargs):
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"""Compute the solution to an optimal control problem
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"""Compute the solution to an optimal control problem.
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The optimal trajectory (states and inputs) is computed so as to
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approximately mimimize a cost function of the following form (for
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continuous time systems):
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J(x(.), u(.)) = \int_0^T L(x(t), u(t)) dt + V(x(T)),
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where T is the time horizon.
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Discrete time systems use a similar formulation, with the integral
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replaced by a sum:
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J(x[.], u[.]) = \sum_0^{N-1} L(x_k, u_k) + V(x_N),
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where N is the time horizon (corresponding to timepts[-1]).
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Parameters
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----------
@@ -968,7 +987,7 @@ def solve_ocp(
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Initial condition (default = 0).
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cost : callable
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Function that returns the integral cost given the current state
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Function that returns the integral cost(L)given the current state
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and input. Called as `cost(x, u)`.
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trajectory_constraints : list of tuples, optional
@@ -990,8 +1009,10 @@ def solve_ocp(
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The constraints are applied at each time point along the trajectory.
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terminal_cost : callable, optional
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Function that returns the terminal cost given the current state
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and input. Called as terminal_cost(x, u).
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Function that returns the terminal cost (V) given the final state
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and input. Called as terminal_cost(x, u). (For compatibility with
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the form of the cost function, u is passed even though it is often
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not part of the terminal cost.)
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terminal_constraints : list of tuples, optional
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List of constraints that should hold at the end of the trajectory.
@@ -1044,9 +1065,19 @@ def solve_ocp(
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Notes
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-----
1047-
Additional keyword parameters can be used to fine-tune the behavior of
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the underlying optimization and integration functions. See
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:func:`OptimalControlProblem` for more information.
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1. For discrete time systems, the final value of the timepts vector
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specifies the final time t_N, and the trajectory cost is computed
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from time t_0 to t_{N-1}. Note that the input u_N does not affect
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the state x_N and so it should always be returned as 0. Further, if
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neither a terminal cost nor a terminal constraint is given, then the
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input at time point t_{N-1} does not affect the cost function and
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hence u_{N-1} will also be returned as zero. If you want the
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trajectory cost to include state costs at time t_{N}, then you can
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set `terminal_cost` to be the same function as `cost`.
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2. Additional keyword parameters can be used to fine-tune the behavior
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of the underlying optimization and integration functions. See
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:func:`OptimalControlProblem` for more information.
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"""
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# Process keyword arguments
@@ -1116,7 +1147,7 @@ def create_mpc_iosystem(
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See :func:`~control.optimal.solve_ocp` for more details.
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terminal_cost : callable, optional
1119-
Function that returns the terminal cost given thecurrent state
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Function that returns the terminal cost given thefinal state
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and input. Called as terminal_cost(x, u).
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terminal_constraints : list of tuples, optional

‎doc/optimal.rst

Lines changed: 7 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -65,6 +65,13 @@ can be on the input, the state, or combinations of input and state,
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depending on the form of:math:`g_i`. Furthermore, these constraints are
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intended to hold at all instants in time along the trajectory.
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For a discrete time system, the same basic formulation applies except
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that the cost function is given by
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..math::
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J(x, u) =\sum_{k=0}^{N-1} L(x_k, u_k)\, dt + V\bigl( x_N\bigr).
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A common use of optimization-based control techniques is the implementation
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of model predictive control (also called receding horizon control). In
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model predictive control, a finite horizon optimal control problem is solved,

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