646 lines
18 KiB
Python
646 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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None
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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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class NMPCController_with_CGMRES():
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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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ht : float
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update value of NMPC this should be decided by zeta
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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.zeta = 100. # 安定化ゲイン
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self.ht = 0.01 # 差分近似の幅
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self.tf = 1. # 最終時間
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self.alpha = 0.5 # 時間の上昇ゲイン
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self.N = 10 # 分割数
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self.threshold = 0.001 # break値
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self.input_num = 3 # dummy, 制約条件に対するuにも合わせた入力の数
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self.max_iteration = self.input_num * self.N
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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) * 0.49
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self.raws = np.ones(self.N) * 0.011
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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 = self.tf * (1. - np.exp(-self.alpha * time)) / float(self.N)
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# x_dot
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x_1_dot = self.simulator.func_x_1(x_1, x_2, self.us[0])
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x_2_dot = self.simulator.func_x_2(x_1, x_2, self.us[0])
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dx_1 = x_1_dot * self.ht
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dx_2 = x_2_dot * self.ht
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x_1s, x_2s, lam_1s, lam_2s = self.simulator.calc_predict_and_adjoint_state(x_1 + dx_1, x_2 + dx_2, self.us, self.N, dt)
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# Fxt
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Fxt = self._calc_f(x_1s, x_2s, lam_1s, lam_2s, self.us, self.dummy_us,
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self.raws, self.N, dt)
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# F
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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, self.us, self.dummy_us,
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self.raws, self.N, dt)
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right = -self.zeta * F - ((Fxt - F) / self.ht)
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du = self.us * self.ht
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ddummy_u = self.dummy_us * self.ht
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draw = self.raws * self.ht
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x_1s, x_2s, lam_1s, lam_2s = self.simulator.calc_predict_and_adjoint_state(x_1 + dx_1, x_2 + dx_2, self.us + du, self.N, dt)
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Fuxt = self._calc_f(x_1s, x_2s, lam_1s, lam_2s, self.us + du, self.dummy_us + ddummy_u,
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self.raws + draw, self.N, dt)
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left = ((Fuxt - Fxt) / self.ht)
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# calculationg cgmres
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r0 = right - left
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r0_norm = np.linalg.norm(r0)
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vs = np.zeros((self.max_iteration, self.max_iteration + 1)) # 数×iterarion回数
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vs[:, 0] = r0 / r0_norm # 最初の基底を算出
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hs = np.zeros((self.max_iteration + 1, self.max_iteration + 1))
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e = np.zeros((self.max_iteration + 1, 1)) # in this case the state is 3(u and dummy_u)
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e[0] = 1.
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for i in range(self.max_iteration):
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du = vs[::3, i] * self.ht
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ddummy_u = vs[1::3, i] * self.ht
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draw = vs[2::3, i] * self.ht
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x_1s, x_2s, lam_1s, lam_2s = self.simulator.calc_predict_and_adjoint_state(x_1 + dx_1, x_2 + dx_2, self.us + du, self.N, dt)
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Fuxt = self._calc_f(x_1s, x_2s, lam_1s, lam_2s, self.us + du, self.dummy_us + ddummy_u,
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self.raws + draw, self.N, dt)
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Av = (( Fuxt - Fxt) / self.ht)
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sum_Av = np.zeros(self.max_iteration)
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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
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hs[j, i] = np.dot(Av, vs[:, j])
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sum_Av = sum_Av + hs[j, i] * vs[:, j]
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v_est = Av - sum_Av
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hs[i+1, i] = np.linalg.norm(v_est)
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vs[:, i+1] = v_est / hs[i+1, i]
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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
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ys = np.dot(inv_hs, r0_norm * e[:i+1])
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judge_value = r0_norm * e[:i+1] - np.dot(hs[:i+1, :i], ys[:i])
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if np.linalg.norm(judge_value) < self.threshold or i == self.max_iteration-1:
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update_value = np.dot(vs[:, :i-1], ys_pre[:i-1]).flatten()
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du_new = du + update_value[::3]
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ddummy_u_new = ddummy_u + update_value[1::3]
|
||
draw_new = draw + update_value[2::3]
|
||
break
|
||
|
||
ys_pre = ys
|
||
|
||
# update
|
||
self.us += du_new * self.ht
|
||
self.dummy_us += ddummy_u_new * self.ht
|
||
self.raws += draw_new * self.ht
|
||
|
||
x_1s, x_2s, lam_1s, lam_2s = self.simulator.calc_predict_and_adjoint_state(x_1, x_2, self.us, self.N, dt)
|
||
|
||
F = self._calc_f(x_1s, x_2s, lam_1s, lam_2s, self.us, self.dummy_us,
|
||
self.raws, 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.append(self.us[0])
|
||
self.history_dummy_u.append(self.dummy_us[0])
|
||
self.history_raw.append(self.raws[0])
|
||
|
||
return self.us
|
||
|
||
def _calc_f(self, x_1s, x_2s, lam_1s, lam_2s, us, dummy_us, raws, 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
|
||
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
|
||
us : list of float
|
||
estimated optimal system input
|
||
dummy_us : list of float
|
||
estimated dummy input
|
||
raws : 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(us[i] + lam_2s[i] + 2. * raws[i] * us[i])
|
||
F.append(-0.01 + 2. * raws[i] * dummy_us[i])
|
||
F.append(us[i]**2 + dummy_us[i]**2 - 0.5**2)
|
||
|
||
return np.array(F)
|
||
|
||
def main():
|
||
# simulation time
|
||
dt = 0.01
|
||
iteration_time = 20.
|
||
iteration_num = int(iteration_time/dt)
|
||
|
||
# plant
|
||
plant_system = SampleSystem(init_x_1=2., init_x_2=0.)
|
||
|
||
# controller
|
||
controller = NMPCController_with_CGMRES()
|
||
|
||
# for i in range(iteration_num)
|
||
for i in range(1, iteration_num):
|
||
time = float(i) * dt
|
||
x_1 = plant_system.x_1
|
||
x_2 = plant_system.x_2
|
||
# make input
|
||
us = controller.calc_input(x_1, x_2, time)
|
||
# update state
|
||
plant_system.update_state(us[0])
|
||
|
||
# figure
|
||
fig = plt.figure()
|
||
|
||
x_1_fig = fig.add_subplot(321)
|
||
x_2_fig = fig.add_subplot(322)
|
||
u_fig = fig.add_subplot(323)
|
||
dummy_fig = fig.add_subplot(324)
|
||
raw_fig = fig.add_subplot(325)
|
||
f_fig = fig.add_subplot(326)
|
||
|
||
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")
|
||
|
||
u_fig.plot(np.arange(iteration_num - 1)*dt, controller.history_u)
|
||
u_fig.set_xlabel("time [s]")
|
||
u_fig.set_ylabel("u")
|
||
|
||
dummy_fig.plot(np.arange(iteration_num - 1)*dt, controller.history_dummy_u)
|
||
dummy_fig.set_xlabel("time [s]")
|
||
dummy_fig.set_ylabel("dummy u")
|
||
|
||
raw_fig.plot(np.arange(iteration_num - 1)*dt, controller.history_raw)
|
||
raw_fig.set_xlabel("time [s]")
|
||
raw_fig.set_ylabel("raw")
|
||
|
||
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.tight_layout()
|
||
|
||
plt.show()
|
||
|
||
|
||
if __name__ == "__main__":
|
||
main()
|
||
|
||
|
||
|