[英]Python GEKKO: Objective function not showing correct results
I am trying to optimize the trajectory of a thrust propelled system.我正在尝试优化推力推进系统的轨迹。 The control variable is the mass flow rate, and the final objective is to maximize the mass of the robot, minimizing the amount of propelled used.
控制变量是质量流率,最终目标是最大化机器人的质量,最小化所使用的推进量。 The trajectory resembles a ballistic one, with an initial ascent phase and a final descent phase.
轨迹类似于弹道,具有初始上升阶段和最终下降阶段。
I think i managed to get a good initial guess, however the algorithm does not converge.我想我设法得到了一个很好的初始猜测,但是算法没有收敛。 I checked the output in the console and it seems that the objective function is not working correctly, and I think this is why it is not converging.
我检查了控制台中的 output,似乎目标 function 没有正常工作,我认为这就是它没有收敛的原因。
Here is my code这是我的代码
~# -*- coding: utf-8 -*-
from gekko import GEKKO
import numpy as np
import matplotlib.pyplot as plt
# create GEKKO model
m = GEKKO(remote=False)
# m = GEKKO()
print(m.path)
g_0 = 9.81
Isp = 46.28
m_fb=1
m_prop=0.1*m_fb
m_dry=value=m_fb-m_prop
c = 0.8
a = 0.4
g = 1.62
R_moon = 1737.4e3
J_y = 0.06666
m_dot_up=2.5*m_fb*g/(Isp*g_0)
m_dot_low=0*m_fb*g/(Isp*g_0)
theta_0=70*np.pi/180
v0= 3
x_f=np.sin(2*theta_0)*v0*v0/g
v0_x= v0*np.cos(theta_0)
y_max=v0*v0/(2*g)
m0=1
m1=(np.e**(-v0/(Isp*g_0)))
time_burn= m_fb*(m0-m1)/m_dot_up
tf=2*v0*np.sin(theta_0)/g + 2*time_burn
t0=0
nr_intervals=30
step=tf/nr_intervals
t=np.linspace(t0, tf, nr_intervals)
time_burn_node= min(range(len(t)), key=lambda i: abs(t[i]-time_burn))
m.time=t
m0=1
m1=(np.e**(-v0/(Isp*g_0)))
time_burn= m_fb*(m0-m1)/m_dot_up
time_burn_node= min(range(len(t)), key=lambda i: abs(t[i]-time_burn))
y_f=1
v_f_x = 1
v_f_y=3
gamma_f = -90*np.pi/180
alpha_f= 180*np.pi/180
alpha_dot_f=5*np.pi/180
parabola_profile= lambda x: -4*y_max/(x_f*x_f)*x*x+4*y_max*x/x_f
velocity_profile_y=lambda t: (v0*np.sin(theta_0)-g*t)
velocity_profile_x= lambda t: v0*np.cos(theta_0)
velocity_profile=lambda t: np.sqrt(velocity_profile_x(t)*velocity_profile_x(t)+velocity_profile_y(t)*velocity_profile_y(t))
x2 = m.Var(value=[np.linspace(0,v0,time_burn_node+1)[int(i)] for i in np.linspace(0,time_burn_node,time_burn_node+1)]+[velocity_profile(i) for i in t[1:int(nr_intervals)-time_burn_node]],lb=0,ub=1e3)
x3 = m.Var(value=[np.linspace(np.pi/2,theta_0,time_burn_node+1)[int(i)] for i in np.linspace(0,time_burn_node,time_burn_node+1)]+[np.arctan2(velocity_profile_y(i),velocity_profile_x(i)) for i in t[1:int(nr_intervals)-2*time_burn_node-1]]+[np.linspace(np.arctan2(velocity_profile_y(t[int(nr_intervals)-2*time_burn_node]),velocity_profile_x(t[int(nr_intervals)-2*time_burn_node])),-np.pi/2,time_burn_node+1)[int(i)] for i in np.linspace(0,time_burn_node,time_burn_node+1)],lb=-np.pi*2,ub=np.pi*2)
x0 = m.Var(value=[np.trapz(np.multiply(x2.value[0:i],np.cos(x3.value[0:i])),x=t[0:i]) for i in range(0,int(nr_intervals))],lb=0,ub=x_f+1)
x1 = m.Var(value=[np.trapz(np.multiply(x2.value[0:i],np.sin(x3.value[0:i])),x=t[0:i]) for i in range(0,int(nr_intervals))],lb=0,ub=x_f+15)
x4 = m.Var(value=np.concatenate((np.zeros(int(nr_intervals/2)),np.pi*np.ones(int(nr_intervals)-int(nr_intervals/2)))),lb=-np.pi*2,ub=np.pi*2)
cc= [x*180/3.1415 for x in x4.value]
x5 = m.Var(value=[(x4.value[i+1]-x4.value[i])/step for i in range(0,len(x4.value)-1)]+[(x4[-1]-x4[-2])/step],lb=-step/J_y,ub=step/J_y)
x6 = m.Var(value=np.concatenate((np.linspace(m0,m1,time_burn_node),m1*np.ones(len(t)-time_burn_node))),lb=0,ub=m0)
x2 = m.Var(value=[np.linspace(0,v0,time_burn_node+1)[int(i)] for i in np.linspace(0,time_burn_node,time_burn_node+1)]+[velocity_profile(i) for i in t[1:int(nr_intervals)-time_burn_node]],lb=0,ub=1e3)
x3 = m.Var(value=[np.linspace(np.pi/2,theta_0,time_burn_node+1)[int(i)] for i in np.linspace(0,time_burn_node,time_burn_node+1)]+[np.arctan2(velocity_profile_y(i),velocity_profile_x(i)) for i in t[1:int(nr_intervals)-2*time_burn_node-1]]+[np.linspace(np.arctan2(velocity_profile_y(t[int(nr_intervals)-2*time_burn_node]),velocity_profile_x(t[int(nr_intervals)-2*time_burn_node])),-np.pi/2,time_burn_node+1)[int(i)] for i in np.linspace(0,time_burn_node,time_burn_node+1)],lb=-np.pi*2,ub=np.pi*2)
x0 = m.Var(value=[np.trapz(np.multiply(x2.value[0:i],np.cos(x3.value[0:i])),x=t[0:i]) for i in range(0,int(nr_intervals))],lb=0,ub=x_f+1)
x1 = m.Var(value=[np.trapz(np.multiply(x2.value[0:i],np.sin(x3.value[0:i])),x=t[0:i]) for i in range(0,int(nr_intervals))],lb=0,ub=x_f+15)
x4 = m.Var(value=np.concatenate((np.zeros(int(nr_intervals/2)),np.pi*np.ones(int(nr_intervals)-int(nr_intervals/2)))),lb=-np.pi*2,ub=np.pi*2)
x5 = m.Var(value=[(x4.value[i+1]-x4.value[i])/step for i in range(0,len(x4.value)-1)]+[(x4[-1]-x4[-2])/step],lb=-step/J_y,ub=step/J_y)
x6 = m.Var(value=np.concatenate((np.linspace(m0,m1,time_burn_node),m1*np.ones(len(t)-time_burn_node))),lb=0,ub=m0)
m_dot= m.MV(value=np.concatenate((m_dot_up*np.ones(time_burn_node),np.zeros(len(t)-time_burn_node))),lb=m_dot_low,ub=m_dot_up)
m_dot.value=m_dot.value[0:len(x1.value)]
p = np.zeros(len(x1.value))
p[-1]=1
final = m.Param(value=p,lb=0,ub=1)
m.Equation(x0.dt()==x2*m.cos(x3))
m.Equation(x1.dt()==x2*m.sin(x3))
m.Equation(x6*x2.dt()==(Isp*m_dot*g_0)*m.cos(x4)-x6*g*m.sin(x3))
m.Equation(x6*x2*x3.dt()==x2**2*m.cos(x3)*x6/R_moon+(Isp*m_dot*g_0)*m.sin(x4)-x6*g*m.cos(x3))
m.Equation(x6.dt()==-m_dot)
m.fix_final(x0,x0.value[-1])
m.Equation(x1*final<=1)
m.Minimize((-x6*final))
m.options.MAX_ITER = 1000 # adjust maximum iterations
m.options.SOLVER = 3
m.options.IMODE = 6
m.options.NODES = 3
m.solve()
print(final.value)
print(f"Final Mass: {x6.value[-1]:.3f} s")
Here is the output of the console, showing the objective function going from negative to positive, which makes no sense as the final mass is always positive这是控制台的 output,显示目标 function 从负到正,这没有意义,因为最终质量始终为正
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
90r-8.8685190e-001 2.32e+000 9.25e+003 0.9 3.41e+001 - 1.18e-001 2.17e-002f 1
91 -8.8659478e-001 3.03e+000 1.79e+005 0.3 5.68e+001 - 2.78e-001 1.27e-001f 2
92 -8.8283316e-001 6.47e+000 2.86e+007 1.6 1.16e+002 - 9.91e-001 4.76e-002f 1
93 1.8507003e+000 1.87e+001 1.55e+007 1.6 5.71e+003 - 2.41e-001 3.44e-001f 1
94 1.8506996e+000 1.86e+001 1.16e+011 1.6 1.25e+001 10.6 6.17e-003 2.28e-003h 1
95 1.8506996e+000 1.86e+001 1.16e+011 1.6 1.25e+001 10.1 5.75e-002 2.28e-005h 1
I have also outputted the final values for the final vector and final mass, and they are showing the correct results:我还输出了最终矢量和最终质量的最终值,它们显示了正确的结果:
[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0]
Final Mass: 0.993 s
Thank you very much for any suggestion非常感谢您的任何建议
Examine the model file with m.open_folder()
and view the gk0_model.apm
file in a text editor.使用
m.open_folder()
检查 model 文件并在文本编辑器中查看gk0_model.apm
文件。
Model
Parameters
p1<= 0.008920571233734825, >= 0.0
p2<= 1, >= 0
End Parameters
Variables
v1<= 1000.0, >= 0
v2<= 6.283185307179586, >= -6.283185307179586
v3<= 4.57104227603633, >= 0
v4<= 18.57104227603633, >= 0
v5<= 6.283185307179586, >= -6.283185307179586
v6<= 2.478718169538892, >= -2.478718169538892
v7<= 1, >= 0
v8<= 1000.0, >= 0
v9<= 6.283185307179586, >= -6.283185307179586
v10<= 4.57104227603633, >= 0
v11<= 18.57104227603633, >= 0
v12<= 6.283185307179586, >= -6.283185307179586
v13<= 2.478718169538892, >= -2.478718169538892
v14<= 1, >= 0
End Variables
Equations
$v10=((v8)*(cos(v9)))
$v11=((v8)*(sin(v9)))
((v14)*($v8))=(((((((46.28)*(p1)))*(9.81)))*(cos(v12)))-((((v14)*(1.62)))*(sin(v9))))
((((v14)*(v8)))*($v9))=((((((((((v8)^(2)))*(cos(v9))))*(v14)))/(1737400.0))+((((((46.28)*(p1)))*(9.81)))*(sin(v12))))-((((v14)*(1.62)))*(cos(v9))))
$v14=(-p1)
((v11)*(p2))<=1
minimize (((-v14))*(p2))
End Equations
Connections
p(end).n(end).v10=4.25390890270255
p(end).n(end).v10=fixed
End Connections
End Model
The objective is minimize (((-v14))*(p2))
and the solver is not constrained to take a feasible search path to the optimum.目标是
minimize (((-v14))*(p2))
并且求解器不受限于采用可行的搜索路径达到最优。 The value of v14
is likely oscillating between positive and negative values. v14
的值可能在正值和负值之间振荡。
Initialization and softening the constraints can help find a solution.初始化和软化约束可以帮助找到解决方案。 Try replacing the hard terminal constraints with soft constraints such as:
尝试用软约束替换硬终端约束,例如:
#m.fix_final(x0,x0.value[-1])
#m.Equation(x1*final<=1)
m.Minimize(final*(x0-x0.value[-1])**2)
m.Minimize(final*(x1-1)**2)
Another strategy is to first solve for a feasible solution where the decision variables are fixed at reasonable values.另一种策略是首先求解一个可行的解决方案,其中决策变量固定在合理的值。 Setting
m.options.COLDSTART=1
turns off the MVs to find a feasible solution.设置
m.options.COLDSTART=1
关闭 MV 以找到可行的解决方案。
# -*- coding: utf-8 -*-
from gekko import GEKKO
import numpy as np
import matplotlib.pyplot as plt
# create GEKKO model
m = GEKKO(remote=False)
# m = GEKKO()
print(m.path)
g_0 = 9.81
Isp = 46.28
m_fb=1
m_prop=0.1*m_fb
m_dry=value=m_fb-m_prop
c = 0.8
a = 0.4
g = 1.62
R_moon = 1737.4e3
J_y = 0.06666
m_dot_up=2.5*m_fb*g/(Isp*g_0)
m_dot_low=0*m_fb*g/(Isp*g_0)
theta_0=70*np.pi/180
v0= 3
x_f=np.sin(2*theta_0)*v0*v0/g
v0_x= v0*np.cos(theta_0)
y_max=v0*v0/(2*g)
m0=1
m1=(np.e**(-v0/(Isp*g_0)))
time_burn= m_fb*(m0-m1)/m_dot_up
tf=2*v0*np.sin(theta_0)/g + 2*time_burn
t0=0
nr_intervals=30
step=tf/nr_intervals
t=np.linspace(t0, tf, nr_intervals)
time_burn_node= min(range(len(t)), key=lambda i: abs(t[i]-time_burn))
m.time=t
m0=1
m1=(np.e**(-v0/(Isp*g_0)))
time_burn= m_fb*(m0-m1)/m_dot_up
time_burn_node= min(range(len(t)), key=lambda i: abs(t[i]-time_burn))
y_f=1
v_f_x = 1
v_f_y=3
gamma_f = -90*np.pi/180
alpha_f= 180*np.pi/180
alpha_dot_f=5*np.pi/180
parabola_profile= lambda x: -4*y_max/(x_f*x_f)*x*x+4*y_max*x/x_f
velocity_profile_y=lambda t: (v0*np.sin(theta_0)-g*t)
velocity_profile_x= lambda t: v0*np.cos(theta_0)
velocity_profile=lambda t: np.sqrt(velocity_profile_x(t)*velocity_profile_x(t)+velocity_profile_y(t)*velocity_profile_y(t))
x2 = m.Var(value=[np.linspace(0,v0,time_burn_node+1)[int(i)] for i in np.linspace(0,time_burn_node,time_burn_node+1)]+[velocity_profile(i) for i in t[1:int(nr_intervals)-time_burn_node]],lb=0,ub=1e3)
x3 = m.Var(value=[np.linspace(np.pi/2,theta_0,time_burn_node+1)[int(i)] for i in np.linspace(0,time_burn_node,time_burn_node+1)]+[np.arctan2(velocity_profile_y(i),velocity_profile_x(i)) for i in t[1:int(nr_intervals)-2*time_burn_node-1]]+[np.linspace(np.arctan2(velocity_profile_y(t[int(nr_intervals)-2*time_burn_node]),velocity_profile_x(t[int(nr_intervals)-2*time_burn_node])),-np.pi/2,time_burn_node+1)[int(i)] for i in np.linspace(0,time_burn_node,time_burn_node+1)],lb=-np.pi*2,ub=np.pi*2)
x0 = m.Var(value=[np.trapz(np.multiply(x2.value[0:i],np.cos(x3.value[0:i])),x=t[0:i]) for i in range(0,int(nr_intervals))],lb=0,ub=x_f+1)
x1 = m.Var(value=[np.trapz(np.multiply(x2.value[0:i],np.sin(x3.value[0:i])),x=t[0:i]) for i in range(0,int(nr_intervals))],lb=0,ub=x_f+15)
x4 = m.Var(value=np.concatenate((np.zeros(int(nr_intervals/2)),np.pi*np.ones(int(nr_intervals)-int(nr_intervals/2)))),lb=-np.pi*2,ub=np.pi*2)
cc= [x*180/3.1415 for x in x4.value]
x5 = m.Var(value=[(x4.value[i+1]-x4.value[i])/step for i in range(0,len(x4.value)-1)]+[(x4[-1]-x4[-2])/step],lb=-step/J_y,ub=step/J_y)
x6 = m.Var(value=np.concatenate((np.linspace(m0,m1,time_burn_node),m1*np.ones(len(t)-time_burn_node))),lb=0,ub=m0)
x2 = m.Var(value=[np.linspace(0,v0,time_burn_node+1)[int(i)] for i in np.linspace(0,time_burn_node,time_burn_node+1)]+[velocity_profile(i) for i in t[1:int(nr_intervals)-time_burn_node]],lb=0,ub=1e3)
x3 = m.Var(value=[np.linspace(np.pi/2,theta_0,time_burn_node+1)[int(i)] for i in np.linspace(0,time_burn_node,time_burn_node+1)]+[np.arctan2(velocity_profile_y(i),velocity_profile_x(i)) for i in t[1:int(nr_intervals)-2*time_burn_node-1]]+[np.linspace(np.arctan2(velocity_profile_y(t[int(nr_intervals)-2*time_burn_node]),velocity_profile_x(t[int(nr_intervals)-2*time_burn_node])),-np.pi/2,time_burn_node+1)[int(i)] for i in np.linspace(0,time_burn_node,time_burn_node+1)],lb=-np.pi*2,ub=np.pi*2)
x0 = m.Var(value=[np.trapz(np.multiply(x2.value[0:i],np.cos(x3.value[0:i])),x=t[0:i]) for i in range(0,int(nr_intervals))],lb=0,ub=x_f+1)
x1 = m.Var(value=[np.trapz(np.multiply(x2.value[0:i],np.sin(x3.value[0:i])),x=t[0:i]) for i in range(0,int(nr_intervals))],lb=0,ub=x_f+15)
x4 = m.Var(value=np.concatenate((np.zeros(int(nr_intervals/2)),np.pi*np.ones(int(nr_intervals)-int(nr_intervals/2)))),lb=-np.pi*2,ub=np.pi*2)
x5 = m.Var(value=[(x4.value[i+1]-x4.value[i])/step for i in range(0,len(x4.value)-1)]+[(x4[-1]-x4[-2])/step],lb=-step/J_y,ub=step/J_y)
x6 = m.Var(value=np.concatenate((np.linspace(m0,m1,time_burn_node),m1*np.ones(len(t)-time_burn_node))),lb=0,ub=m0)
m_dot= m.MV(value=np.concatenate((m_dot_up*np.ones(time_burn_node),np.zeros(len(t)-time_burn_node))),lb=m_dot_low,ub=m_dot_up)
m_dot.value=m_dot.value[0:len(x1.value)]
p = np.zeros(len(x1.value))
p[-1]=1
final = m.Param(value=p,lb=0,ub=1)
m.Equation(x0.dt()==x2*m.cos(x3))
m.Equation(x1.dt()==x2*m.sin(x3))
m.Equation(x6*x2.dt()==(Isp*m_dot*g_0)*m.cos(x4)-x6*g*m.sin(x3))
m.Equation(x6*x2*x3.dt()==x2**2*m.cos(x3)*x6/R_moon+(Isp*m_dot*g_0)*m.sin(x4)-x6*g*m.cos(x3))
m.Equation(x6.dt()==-m_dot)
#m.fix_final(x0,x0.value[-1])
#m.Equation(x1*final<=1)
m.Minimize(final*(x0-x0.value[-1])**2)
m.Minimize(final*(x1-1)**2)
m.Minimize((-x6*final))
m.options.MAX_ITER = 2000 # adjust maximum iterations
m.options.SOLVER = 2
m.options.IMODE = 6
m.options.NODES = 3
m.open_folder()
m.options.COLDSTART = 1
m.solve()
print(final.value)
print(f"Final Mass: {x6.value[-1]:.3f} s")
This gives an error:这给出了一个错误:
Number of state variables: 1247
Number of total equations: - 725
Number of slack variables: - 0
---------------------------------------
Degrees of freedom : 522
@error: Degrees of Freedom
* Error: DOF must be zero for this mode
STOPPING...
Traceback (most recent call last):
File "C:\Users\johnh\Desktop\test.py", line 111, in <module>
m.solve()
File "C:\Users\johnh\Python39\lib\site-packages\gekko\gekko.py", line 2185, in solve
raise Exception(response)
Exception: @error: Degrees of Freedom
* Error: DOF must be zero for this mode
STOPPING...
Try removing any constraints and identify the pairing of each variable with a constraint.尝试移除任何约束并确定每个变量与约束的配对。 If the degrees of freedom are zero (same number of equations and variables), then the solver can sometimes find an initial solution that can subsequently be optimized.
如果自由度为零(相同数量的方程和变量),则求解器有时可以找到随后可以优化的初始解。
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