不可行性，未找到解决方案 - Gekko 错误
[英]Infeasibilities, Solution Not Found - Gekko Error
问题描述
I'm Arthur.
我是亚瑟。 I'm very new to gekko, thus I might be missing something very obvious here.我对gekko很陌生，因此我可能在这里遗漏了一些非常明显的东西。
I have been trying to find the solution to an optimal control problem, namely trajectory optimization of a spacecraft rendezvous problem (Chaser -> Target), under certain final constraints to the relative final distance and velocity and certain distance along their trip.我一直在努力寻找最优控制问题的解决方案，即航天器交会问题的轨迹优化（追赶者 - >目标），在相对最终距离和速度以及他们行程的特定距离的某些最终约束下。 In order to do this, I tried using soft constraints to the final path.为此，我尝试对最终路径使用软约束。 As a objective function, I use a norm 1 of the force generated from Thrusters.作为目标 function，我使用推进器产生的力的范数 1。
The computation keeps going until the maximum no.计算一直进行到最大数。 of iterations (2000) is reached and cancels.迭代次数 (2000) 已达到并取消。 Is there maybe a way to discretize the search space of this problem to make it solvable in acceptable time trading off computation time for accuracy?有没有办法离散化这个问题的搜索空间，使其在可接受的时间内解决，以牺牲计算时间来换取准确性？
import matplotlib.pyplot as plt
import numpy as np
from gekko import GEKKO
# create GEKKO model
m = GEKKO()
print(m.path)
nt = 501
m.time = np.linspace(0,500,nt)
# Variables
# Initial Position and Velocity - Chaser
r1_c = -1182.339411 # [km]
r2_c = 6816.939420 # [km]
r3_c = 904.891745 # [km]
v1_c = 0.1175776 # [km/s]
v2_c = -0.963776 # [km/s]
v3_c = 7.494102 # [km/s]
# Initial Position and Velocity - Target
r1_t = -1182.959348 # [km]
r2_t = 6817.396210 # [km]
r3_t = 904.495486 # [km]
v1_t = 0.1175776 # [km/s]
v2_t = -0.963776 # [km/s]
v3_t = 7.494102 # [km/s]
# initial values
x = m.Var(value = r1_c)
y = m.Var(value = r2_c)
z = m.Var(value = r3_c)
vx = m.Var(value = v1_c)
vy = m.Var(value = v2_c)
vz = m.Var(value = v3_c)
# initial values
x2 = m.Var(value = r1_t)
y2 = m.Var(value = r2_t)
z2 = m.Var(value = r3_t)
vx2 = m.Var(value = v1_t)
vy2 = m.Var(value = v2_t)
vz2 = m.Var(value = v3_t)
# Cost Initial - Integrated thrust (fuel usage)
J = m.Var(value = 0)
# Manipulated Variable
U = m.MV(value=0, lb=-0.001, ub=0.001)
U.STATUS = 1
# Mask time
p = np.zeros(nt) # mask final time point
p[-1] = 500
final = m.Param(value=p)
# Parameters
RT = m.Const(6378.139); # [km]
mu_t = m.Const(3.9860064*10**(5)) # [km^3/s^2]
# Equations
# Define intermediate quantities - Chaser
Rc = m.Intermediate((x**2 + y**2 + z**2)**0.5)
v = m.Intermediate((vx**2 + vy**2 + vz**2)**0.5)
ax = m.Intermediate(x * -mu_t / Rc**3 + U *vx / v )
ay = m.Intermediate(y * -mu_t / Rc**3 + U *vy / v )
az = m.Intermediate(z * -mu_t / Rc**3 + U *vz / v )
# Define intermediate quantities - Target
Rt = m.Intermediate((x2**2 + y2**2 + z2**2)**0.5)
v2 = m.Intermediate((vx2**2 + vy2**2 + vz2**2)**0.5)
ax2 = m.Intermediate(x2 * -mu_t / Rt**3 )
ay2 = m.Intermediate(y2 * -mu_t / Rt**3 )
az2 = m.Intermediate(z2 * -mu_t / Rt**3 )
# Governing equations
m.Equations((vx.dt() == ax, vy.dt() == ay, vz.dt() == az))
m.Equations((x.dt() == vx, y.dt() == vy, z.dt() == vz))
m.Equations((vx2.dt() == ax2, vy2.dt() == ay2, vz2.dt() == az2))
m.Equations((x2.dt() == vx2, y2.dt() == vy2, z2.dt() == vz2))
# Equation relating thrust to fuel usage
m.Equation(J.dt() == m.abs2(U))
# Path Constraints
# specify endpoint conditions
m.fix(x, pos=len(m.time)-1, val=x2)
m.fix(y, pos=len(m.time)-1, val=y2)
m.fix(z, pos=len(m.time)-1, val=z2)
m.fix(vx, pos=len(m.time)-1, val=vx2)
m.fix(vy, pos=len(m.time)-1, val=vy2)
m.fix(vz, pos=len(m.time)-1, val=vz2)
# Constraints
m.Equations((m.abs2(Rc - RT) > 200, m.abs2(Rc - RT) < 1000, m.abs2(Rc*final - Rt*final) == 0, m.abs2(v*final - v2*final) == 0))
# Objective, minimizing fuel usage
m.Obj(J * final)
# Set solver mode - Optimal Control
m.options.IMODE = 6
# Increase maximum number of allowed iterations
m.options.MAX_ITER = 2000
# Set number of nodes per time segment
m.options.NODES = 3
# Run solver and display progress
m.solve(disp=True)
---------------------------------------------------
Solver : IPOPT (v3.12)
Solution time : 3663.92820000000 sec
Objective : 185346862893.104
Unsuccessful with error code 0
---------------------------------------------------
Creating file: infeasibilities.txt
Use command apm_get(server,app,'infeasibilities.txt') to retrieve file
@error: Solution Not Found
---------------------------------------------------------------------------
It does not find the solution and I do not find the file infeasibilities.txt .它没有找到解决方案，我也没有找到文件infeasibilities.txt 。
Can you help me please?你能帮我吗？
I would like to find the reason from results (Solution Not Found).我想从结果中找到原因（未找到解决方案）。 However someone can help me with structure of the problem becoming it feasible.但是，有人可以帮助我解决问题的结构，使其变得可行。
1 个解决方案
解决方案1
0 2023-02-01 15:58:03
One infeasible set of constraints is that Rc-RT is set to >200 and <1000 everywhere, but it must also equal zero at the final time.一组不可行的约束是Rc-RT在所有地方都设置为>200和<1000 ，但它在最后时刻也必须等于 0。 The solver can't satisfy these constraints.求解器无法满足这些约束。
m.abs2(Rc - RT) > 200
m.abs2(Rc - RT) < 1000
m.abs2(Rc*final - Rt*final) == 0
I recommend replacing these constraints with something that is numerically simple with the definition of a new variable with upper and lower bounds and the **2 to avoid sqrt() and abs() .我建议将这些约束替换为在数值上简单的东西，即定义一个具有上限和下限的新变量以及**2以避免sqrt()和abs() 。
RcRT = m.Var(lb=200,ub=1000)
m.Equation(RcRT**2 == (Rc-RT)**2)
The MPCC m.abs2() does not work very well for most problems. MPCC m.abs2()不能很好地解决大多数问题。 m.abs3() works better, but can take more time to solve as a mixed integer form. m.abs3()效果更好，但可能需要更多时间来解决混合 integer 形式。 Try to avoid the use of abs() in the model, when there is an alternative.当有替代方法时，尽量避免在 model 中使用abs() 。 Equations such as m.abs2(v*final - v2*final) == 0 will have no solution if the Chaser is not able to match the velocity of the Target.如果 Chaser 无法匹配 Target 的速度，诸如m.abs2(v*final - v2*final) == 0等方程式将无解。 Try a different soft constraint (objective-based) that allows the problem to remain feasible and minimize the difference.尝试不同的软约束（基于目标），使问题保持可行并最小化差异。
m.Minimize(final*(v-v2)**2)
Likewise, these hard constraints may be infeasible:同样，这些硬约束可能是不可行的：
# hard constraints
m.fix(x, pos=len(m.time)-1, val=x2)
m.fix(y, pos=len(m.time)-1, val=y2)
m.fix(z, pos=len(m.time)-1, val=z2)
m.fix(vx, pos=len(m.time)-1, val=vx2)
m.fix(vy, pos=len(m.time)-1, val=vy2)
m.fix(vz, pos=len(m.time)-1, val=vz2)
Try soft constraints:尝试软约束：
# soft constraints
m.Minimize(final*(v-v2)**2)
m.Minimize(final*(x-x2)**2)
m.Minimize(final*(y-y2)**2)
m.Minimize(final*(z-z2)**2)
m.Minimize(final*(vx-vx2)**2)
m.Minimize(final*(vy-vy2)**2)
m.Minimize(final*(vz-vz2)**2)
Here is the complete problem:这是完整的问题：
import matplotlib.pyplot as plt
import numpy as np
from gekko import GEKKO
# create GEKKO model
m = GEKKO()
print(m.path)
nt = 101
m.time = np.linspace(0,500,nt)
# Variables
# Initial Position and Velocity - Chaser
r1_c = -1182.339411 # [km]
r2_c = 6816.939420 # [km]
r3_c = 904.891745 # [km]
v1_c = 0.1175776 # [km/s]
v2_c = -0.963776 # [km/s]
v3_c = 7.494102 # [km/s]
# Initial Position and Velocity - Target
r1_t = -1182.959348 # [km]
r2_t = 6817.396210 # [km]
r3_t = 904.495486 # [km]
v1_t = 0.1175776 # [km/s]
v2_t = -0.963776 # [km/s]
v3_t = 7.494102 # [km/s]
# initial values
x = m.Var(value = r1_c)
y = m.Var(value = r2_c)
z = m.Var(value = r3_c)
vx = m.Var(value = v1_c)
vy = m.Var(value = v2_c)
vz = m.Var(value = v3_c)
# initial values
x2 = m.Var(value = r1_t)
y2 = m.Var(value = r2_t)
z2 = m.Var(value = r3_t)
vx2 = m.Var(value = v1_t)
vy2 = m.Var(value = v2_t)
vz2 = m.Var(value = v3_t)
# Cost Initial - Integrated thrust (fuel usage)
J = m.Var(value = 0)
# Manipulated Variable
U = m.MV(value=0, lb=-0.001, ub=0.001)
U.STATUS = 1
# Mask time
p = np.zeros(nt) # mask final time point
p[-1] = 500
final = m.Param(value=p)
# Parameters
RT = m.Const(6378.139); # [km]
mu_t = m.Const(3.9860064*10**(5)) # [km^3/s^2]
# Equations
# Define intermediate quantities - Chaser
Rc = m.Intermediate((x**2 + y**2 + z**2)**0.5)
v = m.Intermediate((vx**2 + vy**2 + vz**2)**0.5)
ax = m.Intermediate(x * -mu_t / Rc**3 + U *vx / v )
ay = m.Intermediate(y * -mu_t / Rc**3 + U *vy / v )
az = m.Intermediate(z * -mu_t / Rc**3 + U *vz / v )
# Define intermediate quantities - Target
Rt = m.Intermediate((x2**2 + y2**2 + z2**2)**0.5)
v2 = m.Intermediate((vx2**2 + vy2**2 + vz2**2)**0.5)
ax2 = m.Intermediate(x2 * -mu_t / Rt**3 )
ay2 = m.Intermediate(y2 * -mu_t / Rt**3 )
az2 = m.Intermediate(z2 * -mu_t / Rt**3 )
# Governing equations
m.Equations((vx.dt() == ax, vy.dt() == ay, vz.dt() == az))
m.Equations((x.dt() == vx, y.dt() == vy, z.dt() == vz))
m.Equations((vx2.dt() == ax2, vy2.dt() == ay2, vz2.dt() == az2))
m.Equations((x2.dt() == vx2, y2.dt() == vy2, z2.dt() == vz2))
# Equation relating thrust to fuel usage
#m.Equation(J.dt() == m.abs2(U))
m.Equation(J.dt()**2 == U**2)
# Path Constraints
# specify endpoint conditions
# soft constraints
m.Minimize(final*(v-v2)**2)
m.Minimize(final*(x-x2)**2)
m.Minimize(final*(y-y2)**2)
m.Minimize(final*(z-z2)**2)
m.Minimize(final*(vx-vx2)**2)
m.Minimize(final*(vy-vy2)**2)
m.Minimize(final*(vz-vz2)**2)
# Constraints
# m.abs2(Rc - RT) > 200
# m.abs2(Rc - RT) < 1000
# m.abs2(Rc*final - Rt*final) == 0
RcRT = m.Var(lb=200,ub=1000)
m.Equation(RcRT**2 == (Rc-RT)**2)
# Objective, minimizing fuel usage
m.Minimize(J * final)
# Set solver mode - Optimal Control
m.options.IMODE = 6
# Increase maximum number of allowed iterations
m.options.MAX_ITER = 2000
# Set number of nodes per time segment
m.options.NODES = 3
# Run solver and display progress
m.solve(disp=True)
IPOPT finds an optimal solution after 1375 iterations with a reduced the number of time points to 101 for testing. IPOPT 在 1375 次迭代后找到了一个最优解，测试的时间点数量减少到 101 个。 You can try to increase the number of time points and resolve.您可以尝试增加时间点的数量并解决。
1375 -1.3996401e+02 2.04e-09 8.64e-07 -10.5 1.61e-06 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 1375
(scaled) (unscaled)
Objective...............: -4.4944986703206206e-03 -1.3996401074391898e+02
Dual infeasibility......: 8.6432001202008113e-07 2.6915948656834957e-02
Constraint violation....: 7.7449958484773991e-10 2.0372681319713593e-09
Complementarity.........: 3.3628245954861400e-11 1.0472233998435268e-06
Overall NLP error.......: 8.6432001202008113e-07 2.6915948656834957e-02
Number of objective function evaluations = 1974
Number of objective gradient evaluations = 1149
Number of equality constraint evaluations = 1982
Number of inequality constraint evaluations = 1982
Number of equality constraint Jacobian evaluations = 1389
Number of inequality constraint Jacobian evaluations = 1389
Number of Lagrangian Hessian evaluations = 1375
Total CPU secs in IPOPT (w/o function evaluations) = 83.311
Total CPU secs in NLP function evaluations = 112.375
EXIT: Optimal Solution Found.
The solution was found.
The final value of the objective function is -139.964010743919
---------------------------------------------------
Solver : IPOPT (v3.12)
Solution time : 199.320599999999 sec
Objective : -139.964010743919
Successful solution
---------------------------------------------------
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