APMonitor
commented
4 years ago You set initial conditions when you define your position and velocity.
import numpy as np
import matplotlib.pyplot as plt
from gekko import GEKKO
m = GEKKO()
ntime = 51
m.time = np.linspace(0, 2, ntime)
# set initial conditions [10,5]
x = m.Var(value=10)
v = m.Var(value=5)
U = m.MV()
U.STATUS = 1
m.Equation(x.dt() == v)
m.Equation(v.dt() == U)
# fix endpoint
m.fix(x, pos=len(m.time)-1, val=0)
m.fix(v, pos=len(m.time)-1, val=0)
m.Obj(U**2)
m.options.IMODE = 6
m.open_folder()
m.solve()
plt.figure(figsize=(4, 3))
plt.subplot(2, 1, 1)
plt.plot(m.time, x.value, label=r"$x$")
plt.plot(m.time, v.value, label=r"$v$")
plt.legend()
plt.subplot(2, 1, 2)
plt.plot(m.time, U.value, label=r"$U$")
plt.legend()
plt.xlabel("Time")
plt.show()Alternatively, you can set x.value = 10 and v.value = 5 before the solve to define the initial conditions. The initial conditions also give a guess for the remainder of the time horizon so that the solver starts from those values for all time points. You can also feed in a vector of values if you'd like to provide a better starting guess. You are correct that value is used for input to the model and to store the results after the solve is complete. You'll also see that x.value is an object with two additional x.valve.value that stores the values and x.value.change that is set to True when the value is updated. This is so that the prior values are used in applications like MPC where the prior solution is time-shifted to initialize the next solution.
The reason that your m.fix() doesn't work for initial conditions is because of 2 reasons:
- The underlying APM is 1-based instead of 0-based so the first point is p(1).n(1).x in the collocation horizon. You can see the .apm text file that Gekko writes by using
m.open_folder(). - The issue that you mentioned about fixing the state value also fixes the derivative. Here is an alternative way to use soft constraints. There is also a way to use hard constraints (without specifying the derivative) to fix any point in the horizon but I'd recommend just using
x.value=10instead for initial conditions.
There are a couple similar applications (pendulum cart and inverted pendulum) on the Machine Learning and Dynamic Optimization Course to yours.
whoburg
loganbeal
This is possibly related to #59, but possibly a separate issue. Here is a GEKKO implementation of a simple optimal control problem -- moving a double integrator (i.e. a frictionless brick) to the origin in a fixed amount of time.
In the code below,
fixis used to set both the initial condition (right of the origin and moving right) and the desired final condition of x=0, v=0.The code runs and generates the expected plots, but also prints four unexpected
Warning: connection failed,unable to locate variablemessages.A related question: what is the recommended way to set an initial condition in GEKKO when IMODE=6? Most examples in the documentation seem to set initial conditions using the
valuekwarg ofGEKKO.Var. As a new user I find this somewhat confusing -valueseems to be used both to set an initial guess (to then later be manipulated by the optimizer) and to fix initial conditions. Does the behavior really change from a) fixing, for the 0-index of the variable, to b) setting an initial guess, for all other indices of the variable?