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# optimal.py - optimization based control module
#
# RMM, 11 Feb 2021
#
"""Optimization-based control module.
This module provides support for optimization-based controllers for
nonlinear systems with state and input constraints.
The docstring examples assume the following import commands::
>>> import numpy as np
>>> import control as ct
>>> import control.optimal as opt
"""
import logging
import time
import warnings
import numpy as np
import scipy as sp
import scipy.optimize as opt
import control as ct
from . import config
from .iosys import _process_control_disturbance_indices, _process_labels
# Define module default parameter values
_optimal_trajectory_methods = {'shooting', 'collocation'}
_optimal_defaults = {
'optimal.minimize_method': None,
'optimal.minimize_options': {},
'optimal.minimize_kwargs': {},
'optimal.solve_ivp_method': None,
'optimal.solve_ivp_options': {},
}
class OptimalControlProblem():
"""Description of a finite horizon, optimal control problem.
The `OptimalControlProblem` class holds all of the information required to
specify an optimal control problem: the system dynamics, cost function,
and constraints. As much as possible, the information used to specify an
optimal control problem matches the notation and terminology of the SciPy
`optimize.minimize` module, with the hope that this makes it easier to
remember how to describe a problem.
Parameters
----------
sys : InputOutputSystem
I/O system for which the optimal input will be computed.
timepts : 1D array_like
List of times at which the optimal input should be computed.
integral_cost : callable
Function that returns the integral cost given the current state
and input. Called as integral_cost(x, u).
trajectory_constraints : list of constraints, optional
List of constraints that should hold at each point in the time
vector. Each element of the list should be an object of type
`~scipy.optimize.LinearConstraint` with arguments `(A, lb,
ub)` or `~scipy.optimize.NonlinearConstraint` with arguments
`(fun, lb, ub)`. The constraints will be applied at each time point
along the trajectory.
terminal_cost : callable, optional
Function that returns the terminal cost given the final state
and input. Called as terminal_cost(x, u).
trajectory_method : string, optional
Method to use for carrying out the optimization. Currently supported
methods are 'shooting' and 'collocation' (continuous time only). The
default value is 'shooting' for discrete time systems and
'collocation' for continuous time systems.
initial_guess : (tuple of) 1D or 2D array_like
Initial states and/or inputs to use as a guess for the optimal
trajectory. For shooting methods, an array of inputs for each time
point should be specified. For collocation methods, the initial
guess is either the input vector or a tuple consisting guesses for
the state and the input. Guess should either be a 2D vector of
shape (ninputs, ntimepts) or a 1D input of shape (ninputs,) that
will be broadcast by extension of the time axis.
log : bool, optional
If `True`, turn on logging messages (using Python logging module).
Use :py`logging.basicConfig` to enable logging output
(e.g., to a file).
Returns
-------
ocp : OptimalControlProblem
Optimal control problem object, to be used in computing optimal
controllers.
Other Parameters
----------------
basis : BasisFamily, optional
Use the given set of basis functions for the inputs instead of
setting the value of the input at each point in the timepts vector.
terminal_constraints : list of constraints, optional
List of constraints that should hold at the terminal point in time,
in the same form as `trajectory_constraints`.
solve_ivp_method : str, optional
Set the method used by `scipy.integrate.solve_ivp`.
solve_ivp_kwargs : str, optional
Pass additional keywords to `scipy.integrate.solve_ivp`.
minimize_method : str, optional
Set the method used by `scipy.optimize.minimize`.
minimize_options : str, optional
Set the options keyword used by `scipy.optimize.minimize`.
minimize_kwargs : str, optional
Pass additional keywords to `scipy.optimize.minimize`.
Notes
-----
To describe an optimal control problem we need an input/output system,
a set of time points over a a fixed horizon, a cost function, and
(optionally) a set of constraints on the state and/or input, either
along the trajectory and at the terminal time. This class sets up an
optimization over the inputs at each point in time, using the integral
and terminal costs as well as the trajectory and terminal constraints.
The `compute_trajectory` method sets up an optimization problem that
can be solved using `scipy.optimize.minimize`.
The `_cost_function` method takes the information computes the cost of the
trajectory generated by the proposed input. It does this by calling a
user-defined function for the integral_cost given the current states and
inputs at each point along the trajectory and then adding the value of a
user-defined terminal cost at the final point in the trajectory.
The `_constraint_function` method evaluates the constraint functions along
the trajectory generated by the proposed input. As in the case of the
cost function, the constraints are evaluated at the state and input along
each point on the trajectory. This information is compared against the
constraint upper and lower bounds. The constraint function is processed
in the class initializer, so that it only needs to be computed once.
If `basis` is specified, then the optimization is done over coefficients
of the basis elements. Otherwise, the optimization is performed over the
values of the input at the specified times (using linear interpolation for
continuous systems).
The default values for `minimize_method`, `minimize_options`,
`minimize_kwargs`, `solve_ivp_method`, and `solve_ivp_options` can
be set using config.defaults['optimal.<keyword>'].
"""
def __init__(
self, sys, timepts, integral_cost, trajectory_constraints=None,
terminal_cost=None, terminal_constraints=None, initial_guess=None,
trajectory_method=None, basis=None, log=False, kwargs_check=True,
**kwargs):
"""Set up an optimal control problem."""
# Save the basic information for use later
self.system = sys
self.timepts = timepts
self.integral_cost = integral_cost
self.terminal_cost = terminal_cost
self.terminal_constraints = terminal_constraints
self.basis = basis
# Keep track of what type of trajectory method we are using
if trajectory_method is None:
trajectory_method = 'collocation' if sys.isctime() else 'shooting'
elif trajectory_method not in _optimal_trajectory_methods:
raise NotImplementedError(f"Unknown method {trajectory_method}")
self.shooting = trajectory_method in {'shooting'}
self.collocation = trajectory_method in {'collocation'}
# Process keyword arguments
self.solve_ivp_kwargs = {}
self.solve_ivp_kwargs['method'] = kwargs.pop(
'solve_ivp_method', config.defaults['optimal.solve_ivp_method'])
self.solve_ivp_kwargs.update(kwargs.pop(
'solve_ivp_kwargs', config.defaults['optimal.solve_ivp_options']))
self.minimize_kwargs = {}
self.minimize_kwargs['method'] = kwargs.pop(
'minimize_method', config.defaults['optimal.minimize_method'])
self.minimize_kwargs['options'] = kwargs.pop(
'minimize_options', config.defaults['optimal.minimize_options'])
self.minimize_kwargs.update(kwargs.pop(
'minimize_kwargs', config.defaults['optimal.minimize_kwargs']))
# Check to make sure arguments for discrete-time systems are OK
if sys.isdtime(strict=True):
if self.solve_ivp_kwargs['method'] is not None or \
len(self.solve_ivp_kwargs) > 1:
raise TypeError(
"solve_ivp method, kwargs not allowed for"
" discrete time systems")
# Make sure there were no extraneous keywords
if kwargs_check and kwargs:
raise TypeError("unrecognized keyword(s): ", str(kwargs))
self.trajectory_constraints = _process_constraints(
trajectory_constraints, "trajectory")
self.terminal_constraints = _process_constraints(
terminal_constraints, "terminal")
#
# Compute and store constraints
#
# While the constraints are evaluated during the execution of the
# SciPy optimization method itself, we go ahead and pre-compute the
# `scipy.optimize.NonlinearConstraint` function that will be passed to
# the optimizer on initialization, since it doesn't change. This is
# mainly a matter of computing the lower and upper bound vectors,
# which we need to "stack" to account for the evaluation at each
# trajectory time point plus any terminal constraints (in a way that
# is consistent with the `_constraint_function` that is used at
# evaluation time.
#
# TODO: when using the collocation method, linear constraints on the
# states and inputs can potentially maintain their linear structure
# rather than being converted to nonlinear constraints.
#
constraint_lb, constraint_ub, eqconst_value = [], [], []
# Go through each time point and stack the bounds
for t in self.timepts:
for type, fun, lb, ub in self.trajectory_constraints:
if np.all(lb == ub):
# Equality constraint
eqconst_value.append(lb)
else:
# Inequality constraint
constraint_lb.append(lb)
constraint_ub.append(ub)
# Add on the terminal constraints
for type, fun, lb, ub in self.terminal_constraints:
if np.all(lb == ub):
# Equality constraint
eqconst_value.append(lb)
else:
# Inequality constraint
constraint_lb.append(lb)
constraint_ub.append(ub)
# Turn constraint vectors into 1D arrays
self.constraint_lb = np.hstack(constraint_lb) if constraint_lb else []
self.constraint_ub = np.hstack(constraint_ub) if constraint_ub else []
self.eqconst_value = np.hstack(eqconst_value) if eqconst_value else []
# Create the constraints (inequality and equality)
# TODO: for collocation method, keep track of linear vs nonlinear
self.constraints = []
if len(self.constraint_lb) != 0:
self.constraints.append(sp.optimize.NonlinearConstraint(
self._constraint_function, self.constraint_lb,
self.constraint_ub))
if len(self.eqconst_value) != 0:
self.constraints.append(sp.optimize.NonlinearConstraint(
self._eqconst_function, self.eqconst_value,
self.eqconst_value))
if self.collocation:
# Add the collocation constraints
colloc_zeros = np.zeros(sys.nstates * self.timepts.size)
self.colloc_vals = np.zeros((sys.nstates, self.timepts.size))
self.constraints.append(sp.optimize.NonlinearConstraint(
self._collocation_constraint, colloc_zeros, colloc_zeros))
# Initialize run-time statistics
self._reset_statistics(log)
# Process the initial guess
self.initial_guess = self._process_initial_guess(initial_guess)
# Store states, input (used later to minimize re-computation)
self.last_x = np.full(self.system.nstates, np.nan)
self.last_coeffs = np.full(self.initial_guess.shape, np.nan)
# Log information
if log:
logging.info("New optimal control problem initailized")
#
# Cost function
#
# For collocation methods we are given the states and inputs at each
# time point and we use a trapezoidal approximation to compute the
# integral cost, then add on the terminal cost.
#
# For shooting methods, given the input U = [u[t_0], ... u[t_N]] we need to
# compute the cost of the trajectory generated by that input. This
# means we have to simulate the system to get the state trajectory X =
# [x[t_0], ..., x[t_N]] and then compute the cost at each point:
#
# cost = sum_k integral_cost(x[t_k], u[t_k])
# + terminal_cost(x[t_N], u[t_N])
#
# The actual calculation is a bit more complex: for continuous time
# systems, we use a trapezoidal approximation for the integral cost.
#
# The initial state used for generating the simulation is stored in the
# class parameter `x` prior to calling the optimization algorithm.
#
def _cost_function(self, coeffs):
if self.log:
start_time = time.process_time()
logging.info("_cost_function called at: %g", start_time)
# Compute the states and inputs
states, inputs = self._compute_states_inputs(coeffs)
# Trajectory cost
if ct.isctime(self.system):
# Evaluate the costs
costs = [self.integral_cost(states[:, i], inputs[:, i]) for
i in range(self.timepts.size)]
# Compute the time intervals
dt = np.diff(self.timepts)
# Integrate the cost
costs = np.array(costs)
# Approximate the integral using trapezoidal rule
cost = np.sum(0.5 * (costs[:-1] + costs[1:]) * dt)
else:
# Sum the integral cost over the time (second) indices
# cost += self.integral_cost(states[:,i], inputs[:,i])
cost = sum(map(
self.integral_cost, states[:, :-1].transpose(),
inputs[:, :-1].transpose()))
# Terminal cost
if self.terminal_cost is not None:
cost += self.terminal_cost(states[:, -1], inputs[:, -1])
# Update statistics
self.cost_evaluations += 1
if self.log:
stop_time = time.process_time()
self.cost_process_time += stop_time - start_time
logging.info(
"_cost_function returning %g; elapsed time: %g",
cost, stop_time - start_time)
# Return the total cost for this input sequence
return cost
#
# Constraints
#
# We are given the constraints along the trajectory and the terminal
# constraints, which each take inputs [x, u] and evaluate the
# constraint. How we handle these depends on the type of constraint:
#
# * For linear constraints (LinearConstraint), a combined (hstack'd)
# vector of the state and input is multiplied by the polytope A matrix
# for comparison against the upper and lower bounds.
#
# * For nonlinear constraints (NonlinearConstraint), a user-specific
# constraint function having the form
#
# constraint_fun(x, u)
#
# is called at each point along the trajectory and compared against the
# upper and lower bounds.
#
# * If the upper and lower bound for the constraint are identical, then
# we separate out the evaluation into two different constraints, which
# allows the SciPy optimizers to be more efficient (and stops them
# from generating a warning about mixed constraints). This is handled
# through the use of the `_eqconst_function` and `eqconst_value`
# members.
#
# In both cases, the constraint is specified at a single point, but we
# extend this to apply to each point in the trajectory. This means
# that for N time points with m trajectory constraints and p terminal
# constraints we need to compute N*m + p constraints, each of which
# holds at a specific point in time, and implements the original
# constraint.
#
# For collocation methods, we can directly evaluate the constraints at
# the collocation points.
#
# For shooting methods, we do this by creating a function that simulates
# the system dynamics and returns a vector of values corresponding to
# the value of the function at each time. The class initialization
# methods takes care of replicating the upper and lower bounds for each
# point in time so that the SciPy optimization algorithm can do the
# proper evaluation.
#
def _constraint_function(self, coeffs):
if self.log:
start_time = time.process_time()
logging.info("_constraint_function called at: %g", start_time)
# Compute the states and inputs
states, inputs = self._compute_states_inputs(coeffs)
#
# Evaluate the constraint function along the trajectory
#
value = []
for i, t in enumerate(self.timepts):
for ctype, fun, lb, ub in self.trajectory_constraints:
if np.all(lb == ub):
# Skip equality constraints
continue
elif ctype == opt.LinearConstraint:
# `fun` is the A matrix associated with the polytope...
value.append(fun @ np.hstack([states[:, i], inputs[:, i]]))
elif ctype == opt.NonlinearConstraint:
value.append(fun(states[:, i], inputs[:, i]))
else: # pragma: no cover
# Checked above => we should never get here
raise TypeError(f"unknown constraint type {ctype}")
# Evaluate the terminal constraint functions
for ctype, fun, lb, ub in self.terminal_constraints:
if np.all(lb == ub):
# Skip equality constraints
continue
elif ctype == opt.LinearConstraint:
value.append(fun @ np.hstack([states[:, -1], inputs[:, -1]]))
elif ctype == opt.NonlinearConstraint:
value.append(fun(states[:, -1], inputs[:, -1]))
else: # pragma: no cover
# Checked above => we should never get here
raise TypeError(f"unknown constraint type {ctype}")
# Update statistics
self.constraint_evaluations += 1
if self.log:
stop_time = time.process_time()
self.constraint_process_time += stop_time - start_time
logging.info(
"_constraint_function elapsed time: %g",
stop_time - start_time)
# Debugging information
logging.debug(
"constraint values\n" + str(value) + "\n" +
"lb, ub =\n" + str(self.constraint_lb) + "\n" +
str(self.constraint_ub))
# Return the value of the constraint function
return np.hstack(value)
def _eqconst_function(self, coeffs):
if self.log:
start_time = time.process_time()
logging.info("_eqconst_function called at: %g", start_time)
# Compute the states and inputs
states, inputs = self._compute_states_inputs(coeffs)
# Evaluate the constraint function along the trajectory
value = []
for i, t in enumerate(self.timepts):
for ctype, fun, lb, ub in self.trajectory_constraints:
if np.any(lb != ub):
# Skip inequality constraints
continue
elif ctype == opt.LinearConstraint:
# `fun` is the A matrix associated with the polytope...
value.append(fun @ np.hstack([states[:, i], inputs[:, i]]))
elif ctype == opt.NonlinearConstraint:
value.append(fun(states[:, i], inputs[:, i]))
else: # pragma: no cover
# Checked above => we should never get here
raise TypeError(f"unknown constraint type {ctype}")
# Evaluate the terminal constraint functions
for ctype, fun, lb, ub in self.terminal_constraints:
if np.any(lb != ub):
# Skip inequality constraints
continue
elif ctype == opt.LinearConstraint:
value.append(fun @ np.hstack([states[:, -1], inputs[:, -1]]))
elif ctype == opt.NonlinearConstraint:
value.append(fun(states[:, -1], inputs[:, -1]))
else: # pragma: no cover
# Checked above => we should never get here
raise TypeError("unknown constraint type {ctype}")
# Update statistics
self.eqconst_evaluations += 1
if self.log:
stop_time = time.process_time()
self.eqconst_process_time += stop_time - start_time
logging.info(
"_eqconst_function elapsed time: %g", stop_time - start_time)
# Debugging information
logging.debug(
"eqconst values\n" + str(value) + "\n" +
"desired =\n" + str(self.eqconst_value))
# Return the value of the constraint function
return np.hstack(value)
def _collocation_constraint(self, coeffs):
# Compute the states and inputs
states, inputs = self._compute_states_inputs(coeffs)
if self.system.isctime():
# Compute the collocation constraints
for i, t in enumerate(self.timepts):
if i == 0:
# Initial condition constraint (self.x = initial point)
self.colloc_vals[:, 0] = states[:, 0] - self.x
fk = self.system._rhs(
self.timepts[0], states[:, 0], inputs[:, 0])
continue
# From M. Kelly, SIAM Review (2017), equation (3.2), i = k+1
# x[k+1] - x[k] = 0.5 hk (f(x[k+1], u[k+1] + f(x[k], u[k]))
fkp1 = self.system._rhs(t, states[:, i], inputs[:, i])
self.colloc_vals[:, i] = states[:, i] - states[:, i-1] - \
0.5 * (self.timepts[i] - self.timepts[i-1]) * (fkp1 + fk)
fk = fkp1
else:
raise NotImplementedError(
"collocation not yet implemented for discrete time systems")
# Return the value of the constraint function
return self.colloc_vals.reshape(-1)
#
# Initial guess processing
#
# We store an initial guess in case it is not specified later. Note
# that create_mpc_iosystem() will reset the initial guess based on
# the current state of the MPC controller.
#
# The functions below are used to process the initial guess, which can
# either consist of an input only (for shooting methods) or an input
# and/or state trajectory (for collocaiton methods).
#
# Note: The initial input guess is passed as the inputs at the given time
# vector. If a basis is specified, this is converted to coefficient
# values (which are generally of smaller dimension).
#
def _process_initial_guess(self, initial_guess):
# Sort out the input guess and the state guess
if self.collocation and initial_guess is not None and \
isinstance(initial_guess, tuple):
state_guess, input_guess = initial_guess
else:
state_guess, input_guess = None, initial_guess
# Process the input guess
if input_guess is not None:
input_guess = self._broadcast_initial_guess(
input_guess, (self.system.ninputs, self.timepts.size))
# If we were given a basis, project onto the basis elements
if self.basis is not None:
input_guess = self._inputs_to_coeffs(input_guess)
else:
input_guess = np.zeros(
self.system.ninputs *
(self.timepts.size if self.basis is None else self.basis.N))
# Process the state guess
if self.collocation:
if state_guess is None:
# Run a simulation to get the initial guess
if self.basis:
inputs = self._coeffs_to_inputs(input_guess)
else:
inputs = input_guess.reshape(self.system.ninputs, -1)
state_guess = self._simulate_states(
np.zeros(self.system.nstates), inputs)
else:
state_guess = self._broadcast_initial_guess(
state_guess, (self.system.nstates, self.timepts.size))
# Reshape for use by scipy.optimize.minimize()
return np.hstack([
input_guess.reshape(-1), state_guess.reshape(-1)])
else:
# Reshape for use by scipy.optimize.minimize()
return input_guess.reshape(-1)
# Utility function to broadcast an initial guess to the right shape
def _broadcast_initial_guess(self, initial_guess, shape):
# Convert to a 1D array (or higher)
initial_guess = np.atleast_1d(initial_guess)
# See whether we got entire guess or just first time point
if initial_guess.ndim == 1:
# Broadcast inputs to entire time vector
try:
initial_guess = np.broadcast_to(
initial_guess.reshape(-1, 1), shape)
except ValueError:
raise ValueError("initial guess is the wrong shape")
elif initial_guess.shape != shape:
raise ValueError("initial guess is the wrong shape")
return initial_guess
#
# Utility function to convert input vector to coefficient vector
#
# Initial guesses from the user are passed as input vectors as a
# function of time, but internally we store the guess in terms of the
# basis coefficients. We do this by solving a least squares problem to
# find coefficients that match the input functions at the time points
# (as much as possible, if the problem is under-determined).
#
def _inputs_to_coeffs(self, inputs):
# If there is no basis function, just return inputs as coeffs
if self.basis is None:
return inputs
# Solve least squares problems (M x = b) for coeffs on each input
coeffs = []
for i in range(self.system.ninputs):
# Set up the matrices to get inputs
M = np.zeros((self.timepts.size, self.basis.var_ncoefs(i)))
b = np.zeros(self.timepts.size)
# Evaluate at each time point and for each basis function
# TODO: vectorize
for j, t in enumerate(self.timepts):
for k in range(self.basis.var_ncoefs(i)):
M[j, k] = self.basis(k, t)
b[j] = inputs[i, j]
# Solve a least squares problem for the coefficients
alpha, residuals, rank, s = np.linalg.lstsq(M, b, rcond=None)
coeffs.append(alpha)
return np.hstack(coeffs)
# Utility function to convert coefficient vector to input vector
def _coeffs_to_inputs(self, coeffs):
# TODO: vectorize
# TODO: use BasisFamily eval() method (if more efficient)?
inputs = np.zeros((self.system.ninputs, self.timepts.size))
offset = 0
for i in range(self.system.ninputs):
length = self.basis.var_ncoefs(i)
for j, t in enumerate(self.timepts):
for k in range(length):
inputs[i, j] += coeffs[offset + k] * self.basis(k, t)
offset += length
return inputs
#
# Log and statistics
#
# To allow some insight into where time is being spent, we keep track of
# the number of times that various functions are called and (optionally)
# how long we spent inside each function.
#
def _reset_statistics(self, log=False):
"""Reset counters for keeping track of statistics"""
self.log = log
self.cost_evaluations, self.cost_process_time = 0, 0
self.constraint_evaluations, self.constraint_process_time = 0, 0
self.eqconst_evaluations, self.eqconst_process_time = 0, 0
self.system_simulations = 0
def _print_statistics(self, reset=True):
"""Print out summary statistics from last run"""
print("Summary statistics:")
print("* Cost function calls:", self.cost_evaluations)
if self.log:
print("* Cost function process time:", self.cost_process_time)
if self.constraint_evaluations:
print("* Constraint calls:", self.constraint_evaluations)
if self.log:
print(
"* Constraint process time:", self.constraint_process_time)
if self.eqconst_evaluations:
print("* Eqconst calls:", self.eqconst_evaluations)
if self.log:
print(
"* Eqconst process time:", self.eqconst_process_time)
print("* System simulations:", self.system_simulations)
if reset:
self._reset_statistics(self.log)
#
# Compute the states and inputs from the coefficient vector
#
# These internal functions return the states and inputs at the
# collocation points given the ceofficient (optimizer state) vector.
# They keep track of whether a shooting method is being used or not and
# simulate the dynamics if needed.
#
# Compute the states and inputs from the coefficients
def _compute_states_inputs(self, coeffs):
#
# Compute out the states and inputs
#
if self.collocation:
# States are appended to end of (input) coefficients
states = coeffs[-self.system.nstates * self.timepts.size:].reshape(
self.system.nstates, -1)
coeffs = coeffs[:-self.system.nstates * self.timepts.size]
# Compute input at time points
if self.basis:
inputs = self._coeffs_to_inputs(coeffs)
else:
inputs = coeffs.reshape((self.system.ninputs, -1))
if self.shooting:
# See if we already have a simulation for this condition
if np.array_equal(coeffs, self.last_coeffs) \
and np.array_equal(self.x, self.last_x):
states = self.last_states
else:
states = self._simulate_states(self.x, inputs)
self.last_x = self.x
self.last_states = states
self.last_coeffs = coeffs
return states, inputs
# Simulate the system dynamis to retrieve the state
def _simulate_states(self, x0, inputs):
if self.log:
logging.debug(
"calling input_output_response from state\n" + str(x0))
logging.debug("input =\n" + str(inputs))
# Simulate the system to get the state
# TODO: update to use response object; remove return_x
_, _, states = ct.input_output_response(
self.system, self.timepts, inputs, x0, return_x=True,
solve_ivp_kwargs=self.solve_ivp_kwargs, t_eval=self.timepts)
self.system_simulations += 1
if self.log:
logging.debug(
"input_output_response returned states\n" + str(states))
return states
#
# Optimal control computations
#
# Compute the optimal trajectory from the current state
# TODO: update docstring to refer to primary function (?)
def compute_trajectory(
self, x, squeeze=None, transpose=None, return_states=True,
initial_guess=None, print_summary=True, **kwargs):
"""Compute the optimal input at state x.
Parameters
----------
x : array-like or number, optional
Initial state for the system.
initial_guess : (tuple of) 1D or 2D array_like
Initial states and/or inputs to use as a guess for the optimal
trajectory. For shooting methods, an array of inputs for each
time point should be specified. For collocation methods, the
initial guess is either the input vector or a tuple consisting
guesses for the state and the input. Guess should either be a
2D vector of shape (ninputs, ntimepts) or a 1D input of shape
(ninputs,) that will be broadcast by extension of the time axis.
return_states : bool, optional
If True (default), return the values of the state at each time.
squeeze : bool, optional
If True and if the system has a single output, return the system
output as a 1D array rather than a 2D array. If False, return the
system output as a 2D array even if the system is SISO. Default
value set by config.defaults['control.squeeze_time_response'].
transpose : bool, optional
If True, assume that 2D input arrays are transposed from the
standard format. Used to convert MATLAB-style inputs to our
format.
print_summary : bool, optional
If `True` (default), print a short summary of the computation.
Returns
-------
res : OptimalControlResult
Bundle object with the results of the optimal control problem.
res.success : bool
Boolean flag indicating whether the optimization was successful.
res.time : array
Time values of the input.
res.inputs : array
Optimal inputs for the system. If the system is SISO and squeeze
is not True, the array is 1D (indexed by time). If the system is
not SISO or squeeze is False, the array is 2D (indexed by the
output number and time).
res.states : array
Time evolution of the state vector (if return_states=True).
"""
# Allow 'return_x` as a synonym for 'return_states'
return_states = ct.config._get_param(
'optimal', 'return_x', kwargs, return_states, pop=True, last=True)
# Store the initial state (for use in _constraint_function)
self.x = x
# Allow the initial guess to be overriden
if initial_guess is None:
initial_guess = self.initial_guess
else:
initial_guess = self._process_initial_guess(initial_guess)
# Call ScipPy optimizer
res = sp.optimize.minimize(
self._cost_function, initial_guess,
constraints=self.constraints, **self.minimize_kwargs)
# Process and return the results
return OptimalControlResult(
self, res, transpose=transpose, return_states=return_states,
squeeze=squeeze, print_summary=print_summary)
# Compute the current input to apply from the current state (MPC style)
def compute_mpc(self, x, squeeze=None):
"""Compute the optimal input at state x.
This function calls the :meth:`compute_trajectory` method and returns
the input at the first time point.
Parameters
----------
x : array-like or number, optional
Initial state for the system.
squeeze : bool, optional
If True and if the system has a single output, return the system
output as a 1D array rather than a 2D array. If False, return the
system output as a 2D array even if the system is SISO. Default
value set by config.defaults['control.squeeze_time_response'].
Returns
-------
input : array
Optimal input for the system at the current time. If the system
is SISO and squeeze is not True, the array is 1D (indexed by
time). If the system is not SISO or squeeze is False, the array
is 2D (indexed by the output number and time). Set to `None`
if the optimization failed.
"""
res = self.compute_trajectory(x, squeeze=squeeze)
return res.inputs[:, 0]
# Create an input/output system implementing an MPC controller
def create_mpc_iosystem(self, **kwargs):
"""Create an I/O system implementing an MPC controller"""
# Check to make sure we are in discrete time
if self.system.dt == 0:
raise ct.ControlNotImplemented(
"MPC for continuous time systems not implemented")
def _update(t, x, u, params={}):
coeffs = x.reshape((self.system.ninputs, -1))
if self.basis:
# Keep the coeffecients unchanged
# TODO: could compute input vector, shift, and re-project (?)
self.initial_guess = coeffs
else:
# Shift the basis elements by one time step
self.initial_guess = np.hstack(
[coeffs[:, 1:], coeffs[:, -1:]]).reshape(-1)
res = self.compute_trajectory(u, print_summary=False)
# New state is the new input vector
return res.inputs.reshape(-1)
def _output(t, x, u, params={}):
# Start with initial guess and recompute based on input state (u)
self.initial_guess = x
res = self.compute_trajectory(u, print_summary=False)
return res.inputs[:, 0]
# Define signal names, if they are not already given
if kwargs.get('inputs') is None:
kwargs['inputs'] = self.system.state_labels
if kwargs.get('outputs') is None:
kwargs['outputs'] = self.system.input_labels
if kwargs.get('states') is None:
kwargs['states'] = self.system.ninputs * \
(self.timepts.size if self.basis is None else self.basis.N)
return ct.NonlinearIOSystem(
_update, _output, dt=self.system.dt, **kwargs)
# Optimal control result
class OptimalControlResult(sp.optimize.OptimizeResult):
"""Result from solving an optimal control problem.
This class is a subclass of `scipy.optimize.OptimizeResult` with
additional attributes associated with solving optimal control problems.
Attributes
----------
inputs : ndarray
The optimal inputs associated with the optimal control problem.
states : ndarray
If `return_states` was set to true, stores the state trajectory
associated with the optimal input.
success : bool
Whether or not the optimizer exited successful.
problem : OptimalControlProblem
Optimal control problem that generated this solution.
cost : float
Final cost of the return solution.
system_simulations, {cost, constraint, eqconst}_evaluations : int
Number of system simulations and evaluations of the cost function,
(inequality) constraint function, and equality constraint function
performed during the optimzation.
{cost, constraint, eqconst}_process_time : float
If logging was enabled, the amount of time spent evaluating the cost
and constraint functions.
"""
def __init__(
self, ocp, res, return_states=True, print_summary=False,
transpose=None, squeeze=None):
"""Create a OptimalControlResult object"""
# Copy all of the fields we were sent by sp.optimize.minimize()
for key, val in res.items():
setattr(self, key, val)
# Remember the optimal control problem that we solved
self.problem = ocp
# Parse the optimization variables into states and inputs
states, inputs = ocp._compute_states_inputs(res.x)
# See if we got an answer
if not res.success:
warnings.warn(
"unable to solve optimal control problem\n"
"scipy.optimize.minimize returned " + res.message, UserWarning)
# Save the final cost
self.cost = res.fun
# Optionally print summary information
if print_summary:
ocp._print_statistics()
print("* Final cost:", self.cost)
# Process data as a time response (with "outputs" = inputs)
response = ct.TimeResponseData(
ocp.timepts, inputs, states, issiso=ocp.system.issiso(),
transpose=transpose, return_x=return_states, squeeze=squeeze)
self.time = response.time
self.inputs = response.outputs
self.states = response.states
# Compute the input for a nonlinear, (constrained) optimal control problem
def solve_ocp(
sys, timepts, X0, cost, trajectory_constraints=None, terminal_cost=None,
terminal_constraints=None, initial_guess=None, basis=None, squeeze=None,
transpose=None, return_states=True, print_summary=True, log=False,
**kwargs):
r"""Compute the solution to an optimal control problem.
The optimal trajectory (states and inputs) is computed so as to
approximately mimimize a cost function of the following form (for
continuous time systems):
J(x(.), u(.)) = \int_0^T L(x(t), u(t)) dt + V(x(T)),
where T is the time horizon.
Discrete time systems use a similar formulation, with the integral
replaced by a sum:
J(x[.], u[.]) = \sum_0^{N-1} L(x_k, u_k) + V(x_N),
where N is the time horizon (corresponding to timepts[-1]).
Parameters
----------
sys : InputOutputSystem
I/O system for which the optimal input will be computed.
timepts : 1D array_like
List of times at which the optimal input should be computed.
X0 : array-like or number, optional
Initial condition (default = 0).