648 lines
18 KiB
Python
648 lines
18 KiB
Python
import numpy as np
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import matplotlib.pyplot as plt
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import math
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class SampleSystem():
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"""SampleSystem, this is the simulator
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Attributes
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-----------
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x_1 : float
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system state 1
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x_2 : float
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system state 2
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history_x_1 : list
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time history of system state 1 (x_1)
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history_x_2 : list
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time history of system state 2 (x_2)
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"""
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def __init__(self, init_x_1=0., init_x_2=0.):
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"""
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Palameters
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-----------
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init_x_1 : float, optional
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initial value of x_1, default is 0.
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init_x_2 : float, optional
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initial value of x_2, default is 0.
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"""
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self.x_1 = init_x_1
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self.x_2 = init_x_2
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self.history_x_1 = [init_x_1]
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self.history_x_2 = [init_x_2]
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def update_state(self, u, dt=0.01):
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"""
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Palameters
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------------
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u : float
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input of system in some cases this means the reference
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dt : float in seconds, optional
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sampling time of simulation, default is 0.01 [s]
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"""
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# for theta 1, theta 1 dot, theta 2, theta 2 dot
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k0 = [0.0 for _ in range(2)]
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k1 = [0.0 for _ in range(2)]
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k2 = [0.0 for _ in range(2)]
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k3 = [0.0 for _ in range(2)]
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functions = [self._func_x_1, self._func_x_2]
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# solve Runge-Kutta
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for i, func in enumerate(functions):
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k0[i] = dt * func(self.x_1, self.x_2, u)
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for i, func in enumerate(functions):
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k1[i] = dt * func(self.x_1 + k0[0]/2., self.x_2 + k0[1]/2., u)
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for i, func in enumerate(functions):
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k2[i] = dt * func(self.x_1 + k1[0]/2., self.x_2 + k1[1]/2., u)
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for i, func in enumerate(functions):
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k3[i] = dt * func(self.x_1 + k2[0], self.x_2 + k2[1], u)
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self.x_1 += (k0[0] + 2. * k1[0] + 2. * k2[0] + k3[0]) / 6.
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self.x_2 += (k0[1] + 2. * k1[1] + 2. * k2[1] + k3[1]) / 6.
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# save
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self.history_x_1.append(self.x_1)
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self.history_x_2.append(self.x_2)
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def _func_x_1(self, y_1, y_2, u):
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"""
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Parameters
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------------
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y_1 : float
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y_2 : float
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u : float
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system input
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"""
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y_dot = y_2
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return y_dot
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def _func_x_2(self, y_1, y_2, u):
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"""
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Parameters
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------------
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y_1 : float
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y_2 : float
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u : float
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system input
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"""
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y_dot = (1. - y_1**2 - y_2**2) * y_2 - y_1 + u
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return y_dot
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class NMPCSimulatorSystem():
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"""SimulatorSystem for nmpc, this is the simulator of nmpc
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the reason why I seperate the real simulator and nmpc's simulator is sometimes the modeling error, disturbance can include in real simulator
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Attributes
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-----------
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"""
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def __init__(self):
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"""
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Parameters
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-----------
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None
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"""
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pass
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def calc_predict_and_adjoint_state(self, x_1, x_2, us, N, dt):
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"""main
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Parameters
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------------
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x_1 : float
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current state
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x_2 : float
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current state
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us : list of float
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estimated optimal input Us for N steps
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N : int
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predict step
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dt : float
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sampling time
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Returns
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--------
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x_1s : list of float
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predicted x_1s for N steps
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x_2s : list of float
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predicted x_2s for N steps
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lam_1s : list of float
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adjoint state of x_1s, lam_1s for N steps
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lam_2s : list of float
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adjoint state of x_2s, lam_2s for N steps
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"""
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x_1s, x_2s = self._calc_predict_states(x_1, x_2, us, N, dt) # by usin state equation
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lam_1s, lam_2s = self._calc_adjoint_states(x_1s, x_2s, us, N, dt) # by using adjoint equation
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return x_1s, x_2s, lam_1s, lam_2s
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def _calc_predict_states(self, x_1, x_2, us, N, dt):
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"""
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Parameters
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------------
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x_1 : float
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current state
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x_2 : float
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current state
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us : list of float
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estimated optimal input Us for N steps
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N : int
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predict step
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dt : float
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sampling time
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Returns
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--------
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x_1s : list of float
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predicted x_1s for N steps
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x_2s : list of float
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predicted x_2s for N steps
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"""
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# initial state
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x_1s = [x_1]
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x_2s = [x_2]
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for i in range(N):
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temp_x_1, temp_x_2 = self._predict_state_with_oylar(x_1s[i], x_2s[i], us[i], dt)
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x_1s.append(temp_x_1)
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x_2s.append(temp_x_2)
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return x_1s, x_2s
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def _calc_adjoint_states(self, x_1s, x_2s, us, N, dt):
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"""
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Parameters
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------------
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x_1s : list of float
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predicted x_1s for N steps
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x_2s : list of float
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predicted x_2s for N steps
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us : list of float
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estimated optimal input Us for N steps
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N : int
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predict step
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dt : float
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sampling time
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Returns
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--------
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lam_1s : list of float
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adjoint state of x_1s, lam_1s for N steps
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lam_2s : list of float
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adjoint state of x_2s, lam_2s for N steps
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"""
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# final state
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# final_state_func
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lam_1s = [x_1s[-1]]
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lam_2s = [x_2s[-1]]
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for i in range(N-1, 0, -1):
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temp_lam_1, temp_lam_2 = self._adjoint_state_with_oylar(x_1s[i], x_2s[i], lam_1s[0] ,lam_2s[0], us[i], dt)
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lam_1s.insert(0, temp_lam_1)
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lam_2s.insert(0, temp_lam_2)
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return lam_1s, lam_2s
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def final_state_func(self):
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"""this func usually need
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"""
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pass
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def _predict_state_with_oylar(self, x_1, x_2, u, dt):
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"""in this case this function is the same as simulator
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Parameters
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------------
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x_1 : float
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system state
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x_2 : float
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system state
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u : float
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system input
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dt : float in seconds
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sampling time
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Returns
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--------
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next_x_1 : float
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next state, x_1 calculated by using state equation
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next_x_2 : float
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next state, x_2 calculated by using state equation
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"""
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k0 = [0. for _ in range(2)]
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functions = [self.func_x_1, self.func_x_2]
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for i, func in enumerate(functions):
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k0[i] = dt * func(x_1, x_2, u)
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next_x_1 = x_1 + k0[0]
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next_x_2 = x_2 + k0[1]
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return next_x_1, next_x_2
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def func_x_1(self, y_1, y_2, u):
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"""calculating \dot{x_1}
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Parameters
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------------
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y_1 : float
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means x_1
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y_2 : float
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means x_2
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u : float
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means system input
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Returns
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---------
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y_dot : float
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means \dot{x_1}
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"""
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y_dot = y_2
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return y_dot
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def func_x_2(self, y_1, y_2, u):
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"""calculating \dot{x_2}
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Parameters
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------------
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y_1 : float
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means x_1
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y_2 : float
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means x_2
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u : float
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means system input
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Returns
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---------
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y_dot : float
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means \dot{x_2}
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"""
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y_dot = (1. - y_1**2 - y_2**2) * y_2 - y_1 + u
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return y_dot
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def _adjoint_state_with_oylar(self, x_1, x_2, lam_1, lam_2, u, dt):
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"""
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Parameters
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------------
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x_1 : float
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system state
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x_2 : float
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system state
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lam_1 : float
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adjoint state
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lam_2 : float
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adjoint state
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u : float
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system input
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dt : float in seconds
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sampling time
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Returns
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--------
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pre_lam_1 : float
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pre, 1 step before lam_1 calculated by using adjoint equation
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pre_lam_2 : float
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pre, 1 step before lam_2 calculated by using adjoint equation
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"""
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k0 = [0. for _ in range(2)]
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functions = [self._func_lam_1, self._func_lam_2]
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for i, func in enumerate(functions):
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k0[i] = dt * func(x_1, x_2, lam_1, lam_2, u)
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pre_lam_1 = lam_1 + k0[0]
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pre_lam_2 = lam_2 + k0[1]
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return pre_lam_1, pre_lam_2
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def _func_lam_1(self, y_1, y_2, y_3, y_4, u):
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"""calculating -\dot{lam_1}
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Parameters
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------------
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y_1 : float
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means x_1
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y_2 : float
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means x_2
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y_3 : float
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means lam_1
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y_4 : float
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means lam_2
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u : float
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means system input
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Returns
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---------
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y_dot : float
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means -\dot{lam_1}
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"""
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y_dot = y_1 - (2. * y_1 * y_2 + 1.) * y_4
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return y_dot
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def _func_lam_2(self, y_1, y_2, y_3, y_4, u):
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"""calculating -\dot{lam_2}
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Parameters
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------------
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y_1 : float
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means x_1
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y_2 : float
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means x_2
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y_3 : float
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means lam_1
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y_4 : float
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means lam_2
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u : float
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means system input
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Returns
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---------
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y_dot : float
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means -\dot{lam_2}
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"""
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y_dot = y_2 + y_3 + (-3. * (y_2**2) - y_1**2 + 1. ) * y_4
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return y_dot
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def calc_numerical_gradient(f, x, shape):
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"""
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Parameters
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------------
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f : function
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forward function of NN
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x : numpy.ndarray
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input
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shape : tuple
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jacobian shape
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Returns
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---------
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grad : numpy.ndarray, shape is the same as shape
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results of numercial gradient of the input
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References
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-----------
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- oreilly japan 0 から作るdeeplearning
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https://github.com/oreilly-japan/deep-learning-from-scratch/blob/master/common/gradient.py
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"""
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# check condition
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if not callable(f):
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raise TypeError("f should be callable")
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if not (isinstance(x, list) or isinstance(x, np.ndarray)):
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raise TypeError("x should be array-like")
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h = 1e-3 # 0.01
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grad = np.zeros(shape)
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for idx in range(x.size):
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# save
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tmp_val = x[idx]
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x[idx] = float(tmp_val) + h
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fxh1 = f(x) # f(x+h)
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x[idx] = float(tmp_val) - h
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fxh2 = f(x) # f(x-h)
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grad[:, idx] = (fxh1 - fxh2) / (2*h)
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x[idx] = tmp_val
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return np.array(grad)
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class NMPCControllerWithNewton():
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"""
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Attributes
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------------
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zeta : float
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gain of optimal answer stability
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tf : float
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predict time
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alpha : float
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gain of predict time
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N : int
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predicte step, discritize value
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threshold : float
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cgmres's threshold value
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input_num : int
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system input length, this should include dummy u and constraint variables
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max_iteration : int
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decide by the solved matrix size
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simulator : NMPCSimulatorSystem class
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us : list of float
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estimated optimal system input
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dummy_us : list of float
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estimated dummy input
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raws : list of float
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estimated constraint variable
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history_u : list of float
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time history of actual system input
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history_dummy_u : list of float
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time history of actual dummy u
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history_raw : list of float
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time history of actual raw
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history_f : list of float
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time history of error of optimal
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"""
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def __init__(self):
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"""
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Parameters
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-----------
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None
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"""
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# parameters
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self.tf = 1. # 最終時間
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self.N = 10 # 分割数
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self.threshold = 0.0001 # break値
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self.NUM_INPUT = 3 # dummy, 制約条件に対するrawにも合わせた入力の数
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self.Jacobian_size = self.NUM_INPUT * self.N
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# newton parameters
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self.MAX_ITERATION = 100
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# simulator
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self.simulator = NMPCSimulatorSystem()
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# initial
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self.us = np.zeros(self.N)
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self.dummy_us = np.ones(self.N) * 0.25
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self.raws = np.ones(self.N) * 0.01
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# for fig
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self.history_u = []
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self.history_dummy_u = []
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self.history_raw = []
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self.history_f = []
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def calc_input(self, x_1, x_2, time):
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"""
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Parameters
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------------
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x_1 : float
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current state
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x_2 : float
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current state
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time : float in seconds
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now time
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Returns
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--------
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us : list of float
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estimated optimal system input
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"""
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# calculating sampling time
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dt = 0.01
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# concat all us, shape (NUM_INPUT, N)
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all_us = np.stack((self.us, self.dummy_us, self.raws))
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all_us = all_us.T.flatten()
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# Newton method
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for i in range(self.MAX_ITERATION):
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# check
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# print("all_us = {}".format(all_us))
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# print("newton iteration in {}".format(i))
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# input()
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# calc all state
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x_1s, x_2s, lam_1s, lam_2s = self.simulator.calc_predict_and_adjoint_state(x_1, x_2, self.us, self.N, dt)
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# F
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F_hat = self._calc_f(x_1s, x_2s, lam_1s, lam_2s, all_us, self.N, dt)
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# judge
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# print("F_hat = {}".format(F_hat))
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# print(np.linalg.norm(F_hat))
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if np.linalg.norm(F_hat) < self.threshold:
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# print("break!!")
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break
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grad_f = lambda all_us : self._calc_f(x_1s, x_2s, lam_1s, lam_2s, all_us, self.N, dt)
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grads = calc_numerical_gradient(grad_f, all_us, (self.Jacobian_size, self.Jacobian_size))
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# make jacobian and calc inverse of it
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# all us
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all_us = all_us - np.dot(np.linalg.inv(grads), F_hat)
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# update
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self.us = all_us[::self.NUM_INPUT]
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self.dummy_us = all_us[1::self.NUM_INPUT]
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self.raws = all_us[2::self.NUM_INPUT]
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# final insert
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self.us = all_us[::self.NUM_INPUT]
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self.dummy_us = all_us[1::self.NUM_INPUT]
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self.raws = all_us[2::self.NUM_INPUT]
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x_1s, x_2s, lam_1s, lam_2s = self.simulator.calc_predict_and_adjoint_state(x_1, x_2, self.us, self.N, dt)
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F = self._calc_f(x_1s, x_2s, lam_1s, lam_2s, all_us, self.N, dt)
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# print("check val of F = {0}".format(np.linalg.norm(F)))
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# input()
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# for save
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self.history_f.append(np.linalg.norm(F))
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self.history_u.append(self.us[0])
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self.history_dummy_u.append(self.dummy_us[0])
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self.history_raw.append(self.raws[0])
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return self.us
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def _calc_f(self, x_1s, x_2s, lam_1s, lam_2s, all_us, N, dt):
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"""
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Parameters
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------------
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x_1s : list of float
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predicted x_1s for N steps
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x_2s : list of float
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predicted x_2s for N steps
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lam_1s : list of float
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adjoint state of x_1s, lam_1s for N steps
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lam_2s : list of float
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adjoint state of x_2s, lam_2s for N steps
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us : list of float
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estimated optimal system input
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dummy_us : list of float
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estimated dummy input
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raws : list of float
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estimated constraint variable
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N : int
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predict time step
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dt : float
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sampling time of system
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"""
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F = []
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us = all_us[::self.NUM_INPUT]
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dummy_us = all_us[1::self.NUM_INPUT]
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raws = all_us[2::self.NUM_INPUT]
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for i in range(N):
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F.append(0.5 * us[i] + lam_2s[i] + 2. * raws[i] * us[i])
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F.append(-0.01 + 2. * raws[i] * dummy_us[i])
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F.append(us[i]**2 + dummy_us[i]**2 - 0.5**2)
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return np.array(F)
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def main():
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# simulation time
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dt = 0.01
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iteration_time = 20.
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iteration_num = int(iteration_time/dt)
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# plant
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plant_system = SampleSystem(init_x_1=2., init_x_2=0.)
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# controller
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controller = NMPCControllerWithNewton()
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# for i in range(iteration_num)
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for i in range(1, iteration_num):
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print("iteration = {}".format(i))
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time = float(i) * dt
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x_1 = plant_system.x_1
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x_2 = plant_system.x_2
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# make input
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us = controller.calc_input(x_1, x_2, time)
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# update state
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plant_system.update_state(us[0])
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# figure
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fig = plt.figure()
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x_1_fig = fig.add_subplot(321)
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x_2_fig = fig.add_subplot(322)
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u_fig = fig.add_subplot(323)
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dummy_fig = fig.add_subplot(324)
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raw_fig = fig.add_subplot(325)
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f_fig = fig.add_subplot(326)
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x_1_fig.plot(np.arange(iteration_num)*dt, plant_system.history_x_1)
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x_1_fig.set_xlabel("time [s]")
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x_1_fig.set_ylabel("x_1")
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x_2_fig.plot(np.arange(iteration_num)*dt, plant_system.history_x_2)
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x_2_fig.set_xlabel("time [s]")
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x_2_fig.set_ylabel("x_2")
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u_fig.plot(np.arange(iteration_num - 1)*dt, controller.history_u)
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u_fig.set_xlabel("time [s]")
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u_fig.set_ylabel("u")
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dummy_fig.plot(np.arange(iteration_num - 1)*dt, controller.history_dummy_u)
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dummy_fig.set_xlabel("time [s]")
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dummy_fig.set_ylabel("dummy u")
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raw_fig.plot(np.arange(iteration_num - 1)*dt, controller.history_raw)
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raw_fig.set_xlabel("time [s]")
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raw_fig.set_ylabel("raw")
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f_fig.plot(np.arange(iteration_num - 1)*dt, controller.history_f)
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f_fig.set_xlabel("time [s]")
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f_fig.set_ylabel("optimal error")
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fig.tight_layout()
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plt.show()
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if __name__ == "__main__":
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main()
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