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Atsushi Sakai
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"""
Nonlinear MPC simulation with CGMRES
author Atsushi Sakai (@Atsushi_twi)
Ref:
- Shunichi09/nonlinear_control: Implementing the nonlinear model predictive control, sliding mode control https://github.com/Shunichi09/nonlinear_control
"""
import numpy as np
import matplotlib.pyplot as plt
import math
class TwoWheeledSystem():
def __init__(self, init_x_1=0., init_x_2=0., init_x_3=0.):
self.x_1 = init_x_1
self.x_2 = init_x_2
self.x_3 = init_x_3
self.history_x_1 = [init_x_1]
self.history_x_2 = [init_x_2]
self.history_x_3 = [init_x_3]
def update_state(self, u_1, u_2, dt=0.01):
self.x_1 += dt * math.cos(self.x_3) * u_1
self.x_2 += dt * math.sin(self.x_3) * u_1
self.x_3 += dt * u_2
# save
self.history_x_1.append(self.x_1)
self.history_x_2.append(self.x_2)
self.history_x_3.append(self.x_3)
class NMPCSimulatorSystem():
def calc_predict_and_adjoint_state(self, x_1, x_2, x_3, u_1s, u_2s, N, dt):
"""main
Parameters
------------
x_1 : float
current state
x_2 : float
current state
x_3 : float
current state
u_1s : list of float
estimated optimal input Us for N steps
u_2s : list of float
estimated optimal input Us for N steps
N : int
predict step
dt : float
sampling time
Returns
--------
x_1s : list of float
predicted x_1s for N steps
x_2s : list of float
predicted x_2s for N steps
x_3s : list of float
predicted x_3s for N steps
lam_1s : list of float
adjoint state of x_1s, lam_1s for N steps
lam_2s : list of float
adjoint state of x_2s, lam_2s for N steps
lam_3s : list of float
adjoint state of x_3s, lam_3s for N steps
"""
x_1s, x_2s, x_3s = self._calc_predict_states(
x_1, x_2, x_3, u_1s, u_2s, N, dt) # by usin state equation
lam_1s, lam_2s, lam_3s = self._calc_adjoint_states(
x_1s, x_2s, x_3s, u_1s, u_2s, N, dt) # by using adjoint equation
return x_1s, x_2s, x_3s, lam_1s, lam_2s, lam_3s
def _calc_predict_states(self, x_1, x_2, x_3, u_1s, u_2s, N, dt):
"""
Parameters
------------
x_1 : float
current state
x_2 : float
current state
x_3 : float
current state
u_1s : list of float
estimated optimal input Us for N steps
u_2s : list of float
estimated optimal input Us for N steps
N : int
predict step
dt : float
sampling time
Returns
--------
x_1s : list of float
predicted x_1s for N steps
x_2s : list of float
predicted x_2s for N steps
x_3s : list of float
predicted x_3s for N steps
"""
# initial state
x_1s = [x_1]
x_2s = [x_2]
x_3s = [x_3]
for i in range(N):
temp_x_1, temp_x_2, temp_x_3 = self._predict_state_with_oylar(
x_1s[i], x_2s[i], x_3s[i], u_1s[i], u_2s[i], dt)
x_1s.append(temp_x_1)
x_2s.append(temp_x_2)
x_3s.append(temp_x_3)
return x_1s, x_2s, x_3s
def _calc_adjoint_states(self, x_1s, x_2s, x_3s, u_1s, u_2s, N, dt):
"""
Parameters
------------
x_1s : list of float
predicted x_1s for N steps
x_2s : list of float
predicted x_2s for N steps
x_3s : list of float
predicted x_3s for N steps
u_1s : list of float
estimated optimal input Us for N steps
u_2s : list of float
estimated optimal input Us for N steps
N : int
predict step
dt : float
sampling time
Returns
--------
lam_1s : list of float
adjoint state of x_1s, lam_1s for N steps
lam_2s : list of float
adjoint state of x_2s, lam_2s for N steps
lam_3s : list of float
adjoint state of x_2s, lam_2s for N steps
"""
# final state
# final_state_func
lam_1s = [x_1s[-1]]
lam_2s = [x_2s[-1]]
lam_3s = [x_3s[-1]]
for i in range(N - 1, 0, -1):
temp_lam_1, temp_lam_2, temp_lam_3 = self._adjoint_state_with_oylar(
x_1s[i], x_2s[i], x_3s[i], lam_1s[0], lam_2s[0], lam_3s[0], u_1s[i], u_2s[i], dt)
lam_1s.insert(0, temp_lam_1)
lam_2s.insert(0, temp_lam_2)
lam_3s.insert(0, temp_lam_3)
return lam_1s, lam_2s, lam_3s
def final_state_func(self):
"""this func usually need
"""
pass
def _predict_state_with_oylar(self, x_1, x_2, x_3, u_1, u_2, dt):
"""in this case this function is the same as simulator
Parameters
------------
x_1 : float
system state
x_2 : float
system state
x_3 : float
system state
u_1 : float
system input
u_2 : float
system input
dt : float in seconds
sampling time
Returns
--------
next_x_1 : float
next state, x_1 calculated by using state equation
next_x_2 : float
next state, x_2 calculated by using state equation
next_x_3 : float
next state, x_3 calculated by using state equation
"""
k0 = [0. for _ in range(3)]
functions = [self.func_x_1, self.func_x_2, self.func_x_3]
for i, func in enumerate(functions):
k0[i] = dt * func(x_1, x_2, x_3, u_1, u_2)
next_x_1 = x_1 + k0[0]
next_x_2 = x_2 + k0[1]
next_x_3 = x_3 + k0[2]
return next_x_1, next_x_2, next_x_3
def func_x_1(self, y_1, y_2, y_3, u_1, u_2):
"""
Parameters
------------
y_1 : float
y_2 : float
y_3 : float
u_1 : float
system input
u_2 : float
system input
"""
y_dot = math.cos(y_3) * u_1
return y_dot
def func_x_2(self, y_1, y_2, y_3, u_1, u_2):
"""
Parameters
------------
y_1 : float
y_2 : float
y_3 : float
u_1 : float
system input
u_2 : float
system input
"""
y_dot = math.sin(y_3) * u_1
return y_dot
def func_x_3(self, y_1, y_2, y_3, u_1, u_2):
"""
Parameters
------------
y_1 : float
y_2 : float
y_3 : float
u_1 : float
system input
u_2 : float
system input
"""
y_dot = u_2
return y_dot
def _adjoint_state_with_oylar(self, x_1, x_2, x_3, lam_1, lam_2, lam_3, u_1, u_2, dt):
"""
Parameters
------------
x_1 : float
system state
x_2 : float
system state
x_3 : float
system state
lam_1 : float
adjoint state
lam_2 : float
adjoint state
lam_3 : float
adjoint state
u_1 : float
system input
u_2 : float
system input
dt : float in seconds
sampling time
Returns
--------
pre_lam_1 : float
pre, 1 step before lam_1 calculated by using adjoint equation
pre_lam_2 : float
pre, 1 step before lam_2 calculated by using adjoint equation
pre_lam_3 : float
pre, 1 step before lam_3 calculated by using adjoint equation
"""
k0 = [0. for _ in range(3)]
functions = [self._func_lam_1, self._func_lam_2, self._func_lam_3]
for i, func in enumerate(functions):
k0[i] = dt * func(x_1, x_2, x_3, lam_1, lam_2, lam_3, u_1, u_2)
pre_lam_1 = lam_1 + k0[0]
pre_lam_2 = lam_2 + k0[1]
pre_lam_3 = lam_3 + k0[2]
return pre_lam_1, pre_lam_2, pre_lam_3
def _func_lam_1(self, y_1, y_2, y_3, y_4, y_5, y_6, u_1, u_2):
"""calculating -\dot{lam_1}
Parameters
------------
y_1 : float
means x_1
y_2 : float
means x_2
y_3 : float
means x_3
y_4 : float
means lam_1
y_5 : float
means lam_2
y_6 : float
means lam_3
u_1 : float
means system input
u_2 : float
means system input
Returns
---------
y_dot : float
means -\dot{lam_1}
"""
y_dot = 0.
return y_dot
def _func_lam_2(self, y_1, y_2, y_3, y_4, y_5, y_6, u_1, u_2):
"""calculating -\dot{lam_2}
Parameters
------------
y_1 : float
means x_1
y_2 : float
means x_2
y_3 : float
means x_3
y_4 : float
means lam_1
y_5 : float
means lam_2
y_6 : float
means lam_3
u_1 : float
means system input
u_2 : float
means system input
Returns
---------
y_dot : float
means -\dot{lam_2}
"""
y_dot = 0.
return y_dot
def _func_lam_3(self, y_1, y_2, y_3, y_4, y_5, y_6, u_1, u_2):
"""calculating -\dot{lam_3}
Parameters
------------
y_1 : float
means x_1
y_2 : float
means x_2
y_3 : float
means x_3
y_4 : float
means lam_1
y_5 : float
means lam_2
y_6 : float
means lam_3
u_1 : float
means system input
u_2 : float
means system input
Returns
---------
y_dot : float
means -\dot{lam_3}
"""
y_dot = - y_4 * math.sin(y_3) * u_1 + y_5 * math.cos(y_3) * u_1
return y_dot
class NMPCController_with_CGMRES():
"""
Attributes
------------
zeta : float
gain of optimal answer stability
ht : float
update value of NMPC this should be decided by zeta
tf : float
predict time
alpha : float
gain of predict time
N : int
predicte step, discritize value
threshold : float
cgmres's threshold value
input_num : int
system input length, this should include dummy u and constraint variables
max_iteration : int
decide by the solved matrix size
simulator : NMPCSimulatorSystem class
u_1s : list of float
estimated optimal system input
u_2s : list of float
estimated optimal system input
dummy_u_1s : list of float
estimated dummy input
dummy_u_2s : list of float
estimated dummy input
raw_1s : list of float
estimated constraint variable
raw_2s : list of float
estimated constraint variable
history_u_1 : list of float
time history of actual system input
history_u_2 : list of float
time history of actual system input
history_dummy_u_1 : list of float
time history of actual dummy u_1
history_dummy_u_2 : list of float
time history of actual dummy u_2
history_raw_1 : list of float
time history of actual raw_1
history_raw_2 : list of float
time history of actual raw_2
history_f : list of float
time history of error of optimal
"""
def __init__(self):
"""
Parameters
-----------
None
"""
# parameters
self.zeta = 100. # 安定化ゲイン
self.ht = 0.01 # 差分近似の幅
self.tf = 1. # 最終時間
self.alpha = 0.5 # 時間の上昇ゲイン
self.N = 10 # 分割数
self.threshold = 0.001 # break値
self.input_num = 6 # dummy, 制約条件に対するuにも合わせた入力の数
self.max_iteration = self.input_num * self.N
# simulator
self.simulator = NMPCSimulatorSystem()
# initial
self.u_1s = np.ones(self.N) * 1.
self.u_2s = np.ones(self.N) * 0.1
self.dummy_u_1s = np.ones(self.N) * 0.1
self.dummy_u_2s = np.ones(self.N) * 2.5
self.raw_1s = np.ones(self.N) * 0.8
self.raw_2s = np.ones(self.N) * 0.8
# for fig
self.history_u_1 = []
self.history_u_2 = []
self.history_dummy_u_1 = []
self.history_dummy_u_2 = []
self.history_raw_1 = []
self.history_raw_2 = []
self.history_f = []
def calc_input(self, x_1, x_2, x_3, time):
"""
Parameters
------------
x_1 : float
current state
x_2 : float
current state
x_3 : float
current state
time : float in seconds
now time
Returns
--------
u_1s : list of float
estimated optimal system input
u_2s : list of float
estimated optimal system input
"""
# calculating sampling time
dt = self.tf * (1. - np.exp(-self.alpha * time)) / float(self.N)
# x_dot
x_1_dot = self.simulator.func_x_1(
x_1, x_2, x_3, self.u_1s[0], self.u_2s[0])
x_2_dot = self.simulator.func_x_2(
x_1, x_2, x_3, self.u_1s[0], self.u_2s[0])
x_3_dot = self.simulator.func_x_3(
x_1, x_2, x_3, self.u_1s[0], self.u_2s[0])
dx_1 = x_1_dot * self.ht
dx_2 = x_2_dot * self.ht
dx_3 = x_3_dot * self.ht
x_1s, x_2s, x_3s, lam_1s, lam_2s, lam_3s = self.simulator.calc_predict_and_adjoint_state(
x_1 + dx_1, x_2 + dx_2, x_3 + dx_3, self.u_1s, self.u_2s, self.N, dt)
# Fxt
Fxt = self._calc_f(x_1s, x_2s, x_3s, lam_1s, lam_2s, lam_3s, self.u_1s, self.u_2s, self.dummy_u_1s, self.dummy_u_2s,
self.raw_1s, self.raw_2s, self.N, dt)
# F
x_1s, x_2s, x_3s, lam_1s, lam_2s, lam_3s = self.simulator.calc_predict_and_adjoint_state(
x_1, x_2, x_3, self.u_1s, self.u_2s, self.N, dt)
F = self._calc_f(x_1s, x_2s, x_3s, lam_1s, lam_2s, lam_3s, self.u_1s, self.u_2s, self.dummy_u_1s, self.dummy_u_2s,
self.raw_1s, self.raw_2s, self.N, dt)
right = -self.zeta * F - ((Fxt - F) / self.ht)
du_1 = self.u_1s * self.ht
du_2 = self.u_2s * self.ht
ddummy_u_1 = self.dummy_u_1s * self.ht
ddummy_u_2 = self.dummy_u_2s * self.ht
draw_1 = self.raw_1s * self.ht
draw_2 = self.raw_2s * self.ht
x_1s, x_2s, x_3s, lam_1s, lam_2s, lam_3s = self.simulator.calc_predict_and_adjoint_state(
x_1 + dx_1, x_2 + dx_2, x_3 + dx_3, self.u_1s + du_1, self.u_2s + du_2, self.N, dt)
Fuxt = self._calc_f(x_1s, x_2s, x_3s, lam_1s, lam_2s, lam_3s, self.u_1s + du_1, self.u_2s + du_2, self.dummy_u_1s + ddummy_u_1, self.dummy_u_2s + ddummy_u_2,
self.raw_1s + draw_1, self.raw_2s + draw_2, self.N, dt)
left = ((Fuxt - Fxt) / self.ht)
# calculationg cgmres
r0 = right - left
r0_norm = np.linalg.norm(r0)
# 数×iterarion回数
vs = np.zeros((self.max_iteration, self.max_iteration + 1))
vs[:, 0] = r0 / r0_norm # 最初の基底を算出
hs = np.zeros((self.max_iteration + 1, self.max_iteration + 1))
# in this case the state is 3(u and dummy_u)
e = np.zeros((self.max_iteration + 1, 1))
e[0] = 1.
for i in range(self.max_iteration):
du_1 = vs[::self.input_num, i] * self.ht
du_2 = vs[1::self.input_num, i] * self.ht
ddummy_u_1 = vs[2::self.input_num, i] * self.ht
ddummy_u_2 = vs[3::self.input_num, i] * self.ht
draw_1 = vs[4::self.input_num, i] * self.ht
draw_2 = vs[5::self.input_num, i] * self.ht
x_1s, x_2s, x_3s, lam_1s, lam_2s, lam_3s = self.simulator.calc_predict_and_adjoint_state(
x_1 + dx_1, x_2 + dx_2, x_3 + dx_3, self.u_1s + du_1, self.u_2s + du_2, self.N, dt)
Fuxt = self._calc_f(x_1s, x_2s, x_3s, lam_1s, lam_2s, lam_3s, self.u_1s + du_1, self.u_2s + du_2, self.dummy_u_1s + ddummy_u_1, self.dummy_u_2s + ddummy_u_2,
self.raw_1s + draw_1, self.raw_2s + draw_2, self.N, dt)
Av = ((Fuxt - Fxt) / self.ht)
sum_Av = np.zeros(self.max_iteration)
for j in range(i + 1): # グラムシュミットの直交化法です、和を取って差分を取って算出します
hs[j, i] = np.dot(Av, vs[:, j])
sum_Av = sum_Av + hs[j, i] * vs[:, j]
v_est = Av - sum_Av
hs[i + 1, i] = np.linalg.norm(v_est)
vs[:, i + 1] = v_est / hs[i + 1, i]
inv_hs = np.linalg.pinv(hs[:i + 1, :i]) # この辺は教科書（実時間の方）にのっています
ys = np.dot(inv_hs, r0_norm * e[:i + 1])
judge_value = r0_norm * e[:i + 1] - np.dot(hs[:i + 1, :i], ys[:i])
if np.linalg.norm(judge_value) < self.threshold or i == self.max_iteration - 1:
update_value = np.dot(vs[:, :i - 1], ys_pre[:i - 1]).flatten()
du_1_new = du_1 + update_value[::self.input_num]
du_2_new = du_2 + update_value[1::self.input_num]
ddummy_u_1_new = ddummy_u_1 + update_value[2::self.input_num]
ddummy_u_2_new = ddummy_u_2 + update_value[3::self.input_num]
draw_1_new = draw_1 + update_value[4::self.input_num]
draw_2_new = draw_2 + update_value[5::self.input_num]
break
ys_pre = ys
# update
self.u_1s += du_1_new * self.ht
self.u_2s += du_2_new * self.ht
self.dummy_u_1s += ddummy_u_1_new * self.ht
self.dummy_u_2s += ddummy_u_2_new * self.ht
self.raw_1s += draw_1_new * self.ht
self.raw_2s += draw_2_new * self.ht
x_1s, x_2s, x_3s, lam_1s, lam_2s, lam_3s = self.simulator.calc_predict_and_adjoint_state(
x_1, x_2, x_3, self.u_1s, self.u_2s, self.N, dt)
F = self._calc_f(x_1s, x_2s, x_3s, lam_1s, lam_2s, lam_3s, self.u_1s, self.u_2s, self.dummy_u_1s, self.dummy_u_2s,
self.raw_1s, self.raw_2s, self.N, dt)
print("check F = {0}".format(np.linalg.norm(F)))
# for save
self.history_f.append(np.linalg.norm(F))
self.history_u_1.append(self.u_1s[0])
self.history_u_2.append(self.u_2s[0])
self.history_dummy_u_1.append(self.dummy_u_1s[0])
self.history_dummy_u_2.append(self.dummy_u_2s[0])
self.history_raw_1.append(self.raw_1s[0])
self.history_raw_2.append(self.raw_2s[0])
return self.u_1s, self.u_2s
def _calc_f(self, x_1s, x_2s, x_3s, lam_1s, lam_2s, lam_3s, u_1s, u_2s, dummy_u_1s, dummy_u_2s, raw_1s, raw_2s, N, dt):
"""
Parameters
------------
x_1s : list of float
predicted x_1s for N steps
x_2s : list of float
predicted x_2s for N steps
x_3s : list of float
predicted x_3s for N steps
lam_1s : list of float
adjoint state of x_1s, lam_1s for N steps
lam_2s : list of float
adjoint state of x_2s, lam_2s for N steps
lam_3s : list of float
adjoint state of x_2s, lam_3s for N steps
u_1s : list of float
estimated optimal system input
u_2s : list of float
estimated optimal system input
dummy_u_1s : list of float
estimated dummy input
dummy_u_2s : list of float
estimated dummy input
raw_1s : list of float
estimated constraint variable
raw_2s : list of float
estimated constraint variable
N : int
predict time step
dt : float
sampling time of system
"""
F = []
for i in range(N):
F.append(u_1s[i] + lam_1s[i] * math.cos(x_3s[i]) +
lam_2s[i] * math.sin(x_3s[i]) + 2 * raw_1s[i] * u_1s[i])
F.append(u_2s[i] + lam_3s[i] + 2 * raw_2s[i] * u_2s[i])
F.append(-0.01 + 2. * raw_1s[i] * dummy_u_1s[i])
F.append(-0.01 + 2. * raw_2s[i] * dummy_u_2s[i])
F.append(u_1s[i]**2 + dummy_u_1s[i]**2 - 1.**2)
F.append(u_2s[i]**2 + dummy_u_2s[i]**2 - 1.5**2)
return np.array(F)
def plot_figures(plant_system, controller, iteration_num, dt):
# figure
# time history
fig_p = plt.figure()
fig_u = plt.figure()
fig_f = plt.figure()
# traj
fig_t = plt.figure()
fig_traj = fig_t.add_subplot(111)
fig_traj.set_aspect('equal')
x_1_fig = fig_p.add_subplot(311)
x_2_fig = fig_p.add_subplot(312)
x_3_fig = fig_p.add_subplot(313)
u_1_fig = fig_u.add_subplot(411)
u_2_fig = fig_u.add_subplot(412)
dummy_1_fig = fig_u.add_subplot(413)
dummy_2_fig = fig_u.add_subplot(414)
raw_1_fig = fig_f.add_subplot(311)
raw_2_fig = fig_f.add_subplot(312)
f_fig = fig_f.add_subplot(313)
x_1_fig.plot(np.arange(iteration_num) * dt, plant_system.history_x_1)
x_1_fig.set_xlabel("time [s]")
x_1_fig.set_ylabel("x_1")
x_2_fig.plot(np.arange(iteration_num) * dt, plant_system.history_x_2)
x_2_fig.set_xlabel("time [s]")
x_2_fig.set_ylabel("x_2")
x_3_fig.plot(np.arange(iteration_num) * dt, plant_system.history_x_3)
x_3_fig.set_xlabel("time [s]")
x_3_fig.set_ylabel("x_3")
u_1_fig.plot(np.arange(iteration_num - 1) * dt, controller.history_u_1)
u_1_fig.set_xlabel("time [s]")
u_1_fig.set_ylabel("u_v")
u_2_fig.plot(np.arange(iteration_num - 1) * dt, controller.history_u_2)
u_2_fig.set_xlabel("time [s]")
u_2_fig.set_ylabel("u_omega")
dummy_1_fig.plot(np.arange(iteration_num - 1) *
dt, controller.history_dummy_u_1)
dummy_1_fig.set_xlabel("time [s]")
dummy_1_fig.set_ylabel("dummy u_1")
dummy_2_fig.plot(np.arange(iteration_num - 1) *
dt, controller.history_dummy_u_2)
dummy_2_fig.set_xlabel("time [s]")
dummy_2_fig.set_ylabel("dummy u_2")
raw_1_fig.plot(np.arange(iteration_num - 1) * dt, controller.history_raw_1)
raw_1_fig.set_xlabel("time [s]")
raw_1_fig.set_ylabel("raw_1")
raw_2_fig.plot(np.arange(iteration_num - 1) * dt, controller.history_raw_2)
raw_2_fig.set_xlabel("time [s]")
raw_2_fig.set_ylabel("raw_2")
f_fig.plot(np.arange(iteration_num - 1) * dt, controller.history_f)
f_fig.set_xlabel("time [s]")
f_fig.set_ylabel("optimal error")
fig_traj.plot(plant_system.history_x_1,
plant_system.history_x_2, color="b", linestyle="dashed")
fig_traj.set_xlabel("x [m]")
fig_traj.set_ylabel("y [m]")
plt.show()
def main():
# simulation time
dt = 0.01
iteration_time = 15.0 # [s]
# plant
plant_system = TwoWheeledSystem(
init_x_1=-4.5, init_x_2=-2.5, init_x_3=0.25)
# controller
controller = NMPCController_with_CGMRES()
iteration_num = int(iteration_time / dt)
for i in range(1, iteration_num):
time = float(i) * dt
x_1 = plant_system.x_1
x_2 = plant_system.x_2
x_3 = plant_system.x_3
# make input
u_1s, u_2s = controller.calc_input(x_1, x_2, x_3, time)
# update state
plant_system.update_state(u_1s[0], u_2s[0])
plot_figures(plant_system, controller, iteration_num, dt)
if __name__ == "__main__":
main()