我如何选择时间跨度内的特定时间段将我的目标 function 整合到 Gekko 中？

Question

I want to choose a specific time period over the time span to define my objective function in Gekko.

For example, I want to minimize the integral of u^2 from t0=5 to tf=8. However, there seems to be no such way in Gekko.

#%%Import packages
import numpy as np
from gekko import GEKKO
import matplotlib.pyplot as plt

#%% Build model

#initialize GEKKO model
m = GEKKO()

#time
m.time = np.linspace(0,10,101)

#Parameters
mass1 = m.Param(value=10)
mass2 = m.Param(value=1)
final = np.zeros(np.size(m.time))
for i in range(np.size(m.time)):
    if m.time[i] >= 6.2:
        final[i] = 1
    else:   
        final[i] = 0
final = m.Param(value=final)

#Manipulated variable
u = m.Var(value=0)

#Variables
theta = m.Var(value=0)
q = m.Var(value=0)
#Controlled Variable
y = m.Var(value=-1)
v = m.Var(value=0)

#Equations
m.Equations([y.dt() == v,
             v.dt() == mass2/(mass1+mass2) * theta + u,
             theta.dt() == q,
             q.dt() == -theta - u])

#Objective
m.Obj(final * (y**2 + v**2 + theta**2 + q**2))
m.Obj(0.001 * u**2)
#%% Tuning
#global
m.options.IMODE = 6 #control

#%% Solve
m.solve()

#%% Plot solution
plt.figure()
plt.subplot(4,1,1)
plt.plot(m.time,u.value,'r-',lw=2)
plt.ylabel('Force')
plt.legend(['u'],loc='best')
plt.subplot(4,1,2)
plt.plot(m.time,v.value,'b--',lw=2)
plt.legend(['v'],loc='best')
plt.ylabel('Velocity')
plt.subplot(4,1,3)
plt.plot(m.time,y.value,'g:',lw=2)
plt.legend(['y'],loc='best')
plt.ylabel('Position')
plt.subplot(4,1,4)
plt.plot(m.time,theta.value,'m-',lw=2)
plt.plot(m.time,q.value,'k.-',lw=2)
plt.legend([r'$\theta$','q'],loc='best')
plt.ylabel('Angle')
plt.xlabel('Time')
plt.show()

The best I can do now is

m.Obj(m.integral(u**2))

But this will generate the integral over the entire time span instead of the period I want.

I want to do something like,

m.Obj(m.integral(u**2, t0 = 5, tf = 8))

Is there any way to achieve my goal in Gekko?

Answer 1

Use a new parameter interval that is 1 in the range t>=5 and t<=8 .

p = np.zeros(101)
p[np.where((m.time>=5)&(m.time<=8))]=1
interval = m.Param(p)

This can then be applied in the objective function.

m.Minimize(0.001 * interval* u**2)

Here are the results:

#%%Import packages
import numpy as np
from gekko import GEKKO
import matplotlib.pyplot as plt

#%% Build model

#initialize GEKKO model
m = GEKKO(remote=False)

#time
m.time = np.linspace(0,10,101)

p = np.zeros(101)
p[np.where((m.time>=5)&(m.time<=8))]=1
interval = m.Param(p)

#Parameters
mass1 = m.Param(value=10)
mass2 = m.Param(value=1)
final = np.zeros(np.size(m.time))
for i in range(np.size(m.time)):
    if m.time[i] >= 6.2:
        final[i] = 1
    else:   
        final[i] = 0
final = m.Param(value=final)

#Manipulated variable
u = m.Var(value=0)

#Variables
theta = m.Var(value=0)
q = m.Var(value=0)
#Controlled Variable
y = m.Var(value=-1)
v = m.Var(value=0)

#Equations
m.Equations([y.dt() == v,
             v.dt() == mass2/(mass1+mass2) * theta + u,
             theta.dt() == q,
             q.dt() == -theta - u])

#Objective
m.Minimize(final * (y**2 + v**2 + theta**2 + q**2))
m.Minimize(0.001 * interval* u**2)
#%% Tuning
#global
m.options.IMODE = 6 #control

#%% Solve
m.solve()

#%% Plot solution
plt.figure()
plt.subplot(4,1,1)
plt.plot(m.time,u.value,'r-',lw=2)
plt.ylabel('Force')
plt.legend(['u'],loc='best')
plt.subplot(4,1,2)
plt.plot(m.time,v.value,'b--',lw=2)
plt.legend(['v'],loc='best')
plt.ylabel('Velocity')
plt.subplot(4,1,3)
plt.plot(m.time,y.value,'g:',lw=2)
plt.legend(['y'],loc='best')
plt.ylabel('Position')
plt.subplot(4,1,4)
plt.plot(m.time,theta.value,'m-',lw=2)
plt.plot(m.time,q.value,'k.-',lw=2)
plt.legend([r'$\theta$','q'],loc='best')
plt.ylabel('Angle')
plt.xlabel('Time')
plt.savefig('results.png',dpi=300)
plt.show()