add memo and IOC

mpc/basic_IOC/README.md

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+# Model Predictive Control Basic Tool
+This program is about template, generic function of linear model predictive control
+
+# Documentation of the MPC function
+Linear model predicitive control should have state equation.
+So if you want to use this function, you should model the plant as state equation.
+Therefore, the parameters of this class are as following
+
+**class MpcController()**
+
+Attributes :
+
+- A : numpy.ndarray
+     - system matrix
+- B : numpy.ndarray
+    - input matrix
+- Q : numpy.ndarray
+    - evaluation function weight for states
+- Qs : numpy.ndarray
+    - concatenated evaluation function weight for states
+- R : numpy.ndarray
+    - evaluation function weight for inputs
+- Rs : numpy.ndarray
+    - concatenated evaluation function weight for inputs
+- pre_step : int
+    - prediction step
+- state_size : int
+    - state size of the plant
+- input_size : int
+    - input size of the plant
+- dt_input_upper : numpy.ndarray, shape(input_size, ), optional
+    - constraints of input dt, default is None
+- dt_input_lower : numpy.ndarray, shape(input_size, ), optional
+    - constraints of input dt, default is None
+- input_upper : numpy.ndarray, shape(input_size, ), optional
+    - constraints of input, default is None
+- input_lower : numpy.ndarray, shape(input_size, ), optional
+    - constraints of input, default is None
+
+Methods:
+
+- initialize_controller() initialize the controller
+- calc_input(states, references) calculating optimal input
+
+More details, please look the **mpc_func_with_scipy.py** and **mpc_func_with_cvxopt.py**
+
+We have two function, mpc_func_with_cvxopt.py and mpc_func_with_scipy.py
+Both functions have same variable and member function. However the solver is different. 
+Plese choose the right method for your environment.
+
+- example of import
+
+```py
+from mpc_func_with_scipy import MpcController as MpcController_scipy
+from mpc_func_with_cvxopt import MpcController as MpcController_cvxopt
+```
+
+# Examples
+## Problem Formulation
+
+- **first order system**
+
+<a href="https://www.codecogs.com/eqnedit.php?latex=\frac{d}{dt}&space;\boldsymbol{X}&space;=&space;\begin{bmatrix}&space;-1/&space;\tau&space;&&space;0&space;&&space;0&space;&&space;0\\&space;0&space;&&space;-1/&space;\tau&space;&&space;0&space;&&space;0\\&space;1&space;&&space;0&space;&&space;0&space;&&space;0\\&space;0&space;&&space;1&space;&&space;0&space;&&space;0\\&space;\end{bmatrix}&space;\begin{bmatrix}&space;v_x&space;\\&space;v_y&space;\\&space;x&space;\\&space;y&space;\end{bmatrix}&space;&plus;&space;\begin{bmatrix}&space;1/&space;\tau&space;&&space;0&space;\\&space;0&space;&&space;1/&space;\tau&space;\\&space;0&space;&&space;0&space;\\&space;0&space;&&space;0&space;\\&space;\end{bmatrix}&space;\begin{bmatrix}&space;u_x&space;\\&space;u_y&space;\\&space;\end{bmatrix}&space;=&space;\boldsymbol{A}\boldsymbol{X}&space;&plus;&space;\boldsymbol{B}\boldsymbol{U}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\frac{d}{dt}&space;\boldsymbol{X}&space;=&space;\begin{bmatrix}&space;-1/&space;\tau&space;&&space;0&space;&&space;0&space;&&space;0\\&space;0&space;&&space;-1/&space;\tau&space;&&space;0&space;&&space;0\\&space;1&space;&&space;0&space;&&space;0&space;&&space;0\\&space;0&space;&&space;1&space;&&space;0&space;&&space;0\\&space;\end{bmatrix}&space;\begin{bmatrix}&space;v_x&space;\\&space;v_y&space;\\&space;x&space;\\&space;y&space;\end{bmatrix}&space;&plus;&space;\begin{bmatrix}&space;1/&space;\tau&space;&&space;0&space;\\&space;0&space;&&space;1/&space;\tau&space;\\&space;0&space;&&space;0&space;\\&space;0&space;&&space;0&space;\\&space;\end{bmatrix}&space;\begin{bmatrix}&space;u_x&space;\\&space;u_y&space;\\&space;\end{bmatrix}&space;=&space;\boldsymbol{A}\boldsymbol{X}&space;&plus;&space;\boldsymbol{B}\boldsymbol{U}" title="\frac{d}{dt} \boldsymbol{X} = \begin{bmatrix} -1/ \tau & 0 & 0 & 0\\ 0 & -1/ \tau & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ \end{bmatrix} \begin{bmatrix} v_x \\ v_y \\ x \\ y \end{bmatrix} + \begin{bmatrix} 1/ \tau & 0 \\ 0 & 1/ \tau \\ 0 & 0 \\ 0 & 0 \\ \end{bmatrix} \begin{bmatrix} u_x \\ u_y \\ \end{bmatrix} = \boldsymbol{A}\boldsymbol{X} + \boldsymbol{B}\boldsymbol{U}" /></a>
+
+- **ACC (Adaptive cruise control)**
+
+The two wheeled model are expressed the following equation.
+
+<a href="https://www.codecogs.com/eqnedit.php?latex=\frac{d}{dt}&space;\boldsymbol{X}=&space;\frac{d}{dt}&space;\begin{bmatrix}&space;x&space;\\&space;y&space;\\&space;\theta&space;\end{bmatrix}&space;=&space;\begin{bmatrix}&space;\cos(\theta)&space;&&space;0&space;\\&space;\sin(\theta)&space;&&space;0&space;\\&space;0&space;&&space;1&space;\\&space;\end{bmatrix}&space;\begin{bmatrix}&space;u_v&space;\\&space;u_\omega&space;\\&space;\end{bmatrix}&space;=&space;\boldsymbol{B}\boldsymbol{U}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\frac{d}{dt}&space;\boldsymbol{X}=&space;\frac{d}{dt}&space;\begin{bmatrix}&space;x&space;\\&space;y&space;\\&space;\theta&space;\end{bmatrix}&space;=&space;\begin{bmatrix}&space;\cos(\theta)&space;&&space;0&space;\\&space;\sin(\theta)&space;&&space;0&space;\\&space;0&space;&&space;1&space;\\&space;\end{bmatrix}&space;\begin{bmatrix}&space;u_v&space;\\&space;u_\omega&space;\\&space;\end{bmatrix}&space;=&space;\boldsymbol{B}\boldsymbol{U}" title="\frac{d}{dt} \boldsymbol{X}= \frac{d}{dt} \begin{bmatrix} x \\ y \\ \theta \end{bmatrix} = \begin{bmatrix} \cos(\theta) & 0 \\ \sin(\theta) & 0 \\ 0 & 1 \\ \end{bmatrix} \begin{bmatrix} u_v \\ u_\omega \\ \end{bmatrix} = \boldsymbol{B}\boldsymbol{U}" /></a>
+
+However, if we assume the velocity are constant, we can approximate the equation as following, 
+
+<a href="https://www.codecogs.com/eqnedit.php?latex=\frac{d}{dt}&space;\boldsymbol{X}=&space;\frac{d}{dt}&space;\begin{bmatrix}&space;y&space;\\&space;\theta&space;\end{bmatrix}&space;=&space;\begin{bmatrix}&space;0&space;&&space;V&space;\\&space;0&space;&&space;0&space;\\&space;\end{bmatrix}&space;\begin{bmatrix}&space;y&space;\\&space;\theta&space;\end{bmatrix}&space;&plus;&space;\begin{bmatrix}&space;0&space;\\&space;1&space;\end{bmatrix}&space;\begin{bmatrix}&space;u_\omega&space;\\&space;\end{bmatrix}&space;=&space;\boldsymbol{A}\boldsymbol{X}&space;&plus;&space;\boldsymbol{B}\boldsymbol{U}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\frac{d}{dt}&space;\boldsymbol{X}=&space;\frac{d}{dt}&space;\begin{bmatrix}&space;y&space;\\&space;\theta&space;\end{bmatrix}&space;=&space;\begin{bmatrix}&space;0&space;&&space;V&space;\\&space;0&space;&&space;0&space;\\&space;\end{bmatrix}&space;\begin{bmatrix}&space;y&space;\\&space;\theta&space;\end{bmatrix}&space;&plus;&space;\begin{bmatrix}&space;0&space;\\&space;1&space;\end{bmatrix}&space;\begin{bmatrix}&space;u_\omega&space;\\&space;\end{bmatrix}&space;=&space;\boldsymbol{A}\boldsymbol{X}&space;&plus;&space;\boldsymbol{B}\boldsymbol{U}" title="\frac{d}{dt} \boldsymbol{X}= \frac{d}{dt} \begin{bmatrix} y \\ \theta \end{bmatrix} = \begin{bmatrix} 0 & V \\ 0 & 0 \\ \end{bmatrix} \begin{bmatrix} y \\ \theta \end{bmatrix} + \begin{bmatrix} 0 \\ 1 \end{bmatrix} \begin{bmatrix} u_\omega \\ \end{bmatrix} = \boldsymbol{A}\boldsymbol{X} + \boldsymbol{B}\boldsymbol{U}" /></a>
+
+then we can apply this model to linear mpc, we should give the model reference V although.
+
+- **evaluation function**
+
+the both examples have same evaluation function form as following equation.
+
+<a href="https://www.codecogs.com/eqnedit.php?latex=J&space;=&space;\sum_{i&space;=&space;0}^{prestep}||\boldsymbol{\hat{X}}(k&plus;i|k)-\boldsymbol{r}(k&plus;i|k)&space;||^2_{{\boldsymbol{Q}}(i)}&space;&plus;&space;||\boldsymbol{\Delta&space;{U}}(k&plus;i|k)||^2_{{\boldsymbol{R}}(i)}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?J&space;=&space;\sum_{i&space;=&space;0}^{prestep}||\boldsymbol{\hat{X}}(k&plus;i|k)-\boldsymbol{r}(k&plus;i|k)&space;||^2_{{\boldsymbol{Q}}(i)}&space;&plus;&space;||\boldsymbol{\Delta&space;{U}}(k&plus;i|k)||^2_{{\boldsymbol{R}}(i)}" title="J = \sum_{i = 0}^{prestep}||\boldsymbol{\hat{X}}(k+i|k)-\boldsymbol{r}(k+i|k) ||^2_{{\boldsymbol{Q}}(i)} + ||\boldsymbol{\Delta {U}}(k+i|k)||^2_{{\boldsymbol{R}}(i)}" /></a>
+
+- <a href="https://www.codecogs.com/eqnedit.php?latex=\boldsymbol{\hat{X}}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\boldsymbol{\hat{X}}" title="\boldsymbol{\hat{X}}" /></a> is predicit state by using predict input
+
+- <a href="https://www.codecogs.com/eqnedit.php?latex=\boldsymbol{{r}}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\boldsymbol{{r}}" title="\boldsymbol{{r}}" /></a> is reference state
+
+- <a href="https://www.codecogs.com/eqnedit.php?latex=\boldsymbol{\Delta&space;\boldsymbol{U}}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\boldsymbol{\Delta&space;\boldsymbol{U}}" title="\boldsymbol{\Delta \boldsymbol{U}}" /></a> is predict amount of change of input
+
+- <a href="https://www.codecogs.com/eqnedit.php?latex=\boldsymbol{\boldsymbol{R}},&space;\boldsymbol{\boldsymbol{Q}}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\boldsymbol{\boldsymbol{R}},&space;\boldsymbol{\boldsymbol{Q}}" title="\boldsymbol{\boldsymbol{R}}, \boldsymbol{\boldsymbol{Q}}" /></a> are evaluation function weights
+
+## Expected Results
+
+- first order system
+
+- time history
+
+<img src = https://github.com/Shunichi09/linear_nonlinear_control/blob/demo_gif/mpc/basic/first_order_states.png width = 70%>
+
+- input
+
+<img src = https://github.com/Shunichi09/linear_nonlinear_control/blob/demo_gif/mpc/basic/first_order_input.png width = 70%>
+
+- ACC (Adaptive cruise control)
+
+- time history of states
+
+<img src = https://github.com/Shunichi09/linear_nonlinear_control/blob/demo_gif/mpc/basic/ACC_states.png width = 70%>
+
+- animation
+
+<img src = https://github.com/Shunichi09/linear_nonlinear_control/blob/demo_gif/mpc/basic/ACC.gif width = 70%>
+
+
+# Usage
+
+- for example(first order system)
+
+```
+$ python main_example.py
+```
+
+- for example(ACC (Adaptive cruise control))
+
+```
+$ python main_ACC.py
+```
+
+- for comparing two methods of optimization solvers
+
+```
+$ python test_compare_methods.py
+```
+
+# Requirement
+
+- python3.5 or more
+- numpy
+- matplotlib
+- cvxopt
+- scipy1.2.0 or more
+- python-control
+
+# Reference
+I`m sorry that main references are written in Japanese
+
+- モデル予測制御―制約のもとでの最適制御 著：Jan M. Maciejowski　訳：足立修一　東京電機大学出版局

mpc/basic_IOC/animation.py

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+import numpy as np
+import matplotlib.pyplot as plt
+import matplotlib.animation as ani
+import matplotlib.font_manager as fon
+import sys
+import math
+
+# default setting of figures
+plt.rcParams["mathtext.fontset"] = 'stix' # math fonts
+plt.rcParams['xtick.direction'] = 'in' # x axis in
+plt.rcParams['ytick.direction'] = 'in' # y axis in 
+plt.rcParams["font.size"] = 10
+plt.rcParams['axes.linewidth'] = 1.0 # axis line width
+plt.rcParams['axes.grid'] = True # make grid
+
+def coordinate_transformation_in_angle(positions, base_angle):
+    '''
+    Transformation the coordinate in the angle
+
+    Parameters
+    -------
+    positions : numpy.ndarray
+        this parameter is composed of xs, ys 
+        should have (2, N) shape 
+    base_angle : float [rad]
+    
+    Returns
+    -------
+    traslated_positions : numpy.ndarray
+        the shape is (2, N)
+    
+    '''
+    if positions.shape[0] != 2:
+        raise ValueError('the input data should have (2, N)')
+
+    positions = np.array(positions)
+    positions = positions.reshape(2, -1)
+
+    rot_matrix = [[np.cos(base_angle), np.sin(base_angle)],
+                  [-1*np.sin(base_angle), np.cos(base_angle)]]
+
+    rot_matrix = np.array(rot_matrix)
+    
+    translated_positions = np.dot(rot_matrix, positions)
+
+    return translated_positions
+
+def square_make_with_angles(center_x, center_y, size, angle):
+    '''
+    Create square matrix with angle line matrix(2D)
+    
+    Parameters
+    -------
+    center_x : float in meters
+        the center x position of the square
+    center_y : float in meters
+        the center y position of the square
+    size : float in meters
+        the square's half-size
+    angle : float in radians
+
+    Returns
+    -------
+    square xs : numpy.ndarray
+        lenght is 5 (counterclockwise from right-up)
+    square ys : numpy.ndarray
+        length is 5 (counterclockwise from right-up)
+    angle line xs : numpy.ndarray
+    angle line ys : numpy.ndarray
+    '''
+
+    # start with the up right points
+    # create point in counterclockwise
+    square_xys = np.array([[size, 0.5 * size], [-size, 0.5 * size], [-size, -0.5 * size], [size, -0.5 * size], [size, 0.5 * size]])
+    trans_points = coordinate_transformation_in_angle(square_xys.T, -angle) # this is inverse type
+    trans_points += np.array([[center_x], [center_y]])
+
+    square_xs = trans_points[0, :]
+    square_ys = trans_points[1, :]
+
+    angle_line_xs = [center_x, center_x + math.cos(angle) * size]
+    angle_line_ys = [center_y, center_y + math.sin(angle) * size]
+
+    return square_xs, square_ys, np.array(angle_line_xs), np.array(angle_line_ys)
+
+
+class AnimDrawer():
+    """create animation of path and robot
+    
+    Attributes
+    ------------
+    cars : 
+    anim_fig : figure of matplotlib
+    axis : axis of matplotlib
+
+    """
+    def __init__(self, objects):
+        """
+        Parameters
+        ------------
+        objects : list of objects
+        """
+        self.lead_car_history_state = objects[0]
+        self.follow_car_history_state = objects[1]
+        
+        self.history_xs = [self.lead_car_history_state[:, 0], self.follow_car_history_state[:, 0]]
+        self.history_ys = [self.lead_car_history_state[:, 1], self.follow_car_history_state[:, 1]]
+        self.history_ths = [self.lead_car_history_state[:, 2], self.follow_car_history_state[:, 2]]
+
+        # setting up figure
+        self.anim_fig = plt.figure(dpi=150)
+        self.axis = self.anim_fig.add_subplot(111)
+
+        # imgs
+        self.object_imgs = []
+        self.traj_imgs = []
+
+    def draw_anim(self, interval=50):
+        """draw the animation and save
+
+        Parameteres
+        -------------
+        interval : int, optional
+            animation's interval time, you should link the sampling time of systems
+            default is 50 [ms]
+        """
+        self._set_axis()
+        self._set_img()
+
+        self.skip_num = 1
+        frame_num = int((len(self.history_xs[0])-1) / self.skip_num)
+
+        animation = ani.FuncAnimation(self.anim_fig, self._update_anim, interval=interval, frames=frame_num)
+
+        # self.axis.legend()
+        print('save_animation?')
+        shuold_save_animation = int(input())
+
+        if shuold_save_animation: 
+            print('animation_number?')
+            num = int(input())
+            animation.save('animation_{0}.mp4'.format(num), writer='ffmpeg')
+            # animation.save("Sample.gif", writer = 'imagemagick') # gif保存
+
+        plt.show()
+
+    def _set_axis(self):
+        """ initialize the animation axies
+        """
+        # (1) set the axis name
+        self.axis.set_xlabel(r'$\it{x}$ [m]')
+        self.axis.set_ylabel(r'$\it{y}$ [m]')
+        self.axis.set_aspect('equal', adjustable='box')
+
+        # (2) set the xlim and ylim        
+        self.axis.set_xlim(-5, 50)
+        self.axis.set_ylim(-2, 5)         
+
+    def _set_img(self):
+        """ initialize the imgs of animation
+            this private function execute the make initial imgs for animation
+        """
+        # object imgs
+        obj_color_list = ["k", "k", "m", "m"]
+        obj_styles = ["solid", "solid", "solid", "solid"]
+
+        for i in range(len(obj_color_list)):
+            temp_img, = self.axis.plot([], [], color=obj_color_list[i], linestyle=obj_styles[i])
+            self.object_imgs.append(temp_img)
+        
+        traj_color_list = ["k", "m"]
+
+        for i in range(len(traj_color_list)):
+            temp_img, = self.axis.plot([],[], color=traj_color_list[i], linestyle="dashed")
+            self.traj_imgs.append(temp_img)
+    
+    def _update_anim(self, i):
+        """the update animation
+        this function should be used in the animation functions
+
+        Parameters
+        ------------
+        i : int
+            time step of the animation
+            the sampling time should be related to the sampling time of system
+
+        Returns
+        -----------
+        object_imgs : list of img
+        traj_imgs : list of img
+        """
+        i = int(i * self.skip_num)
+
+        self._draw_objects(i)
+        self._draw_traj(i)
+
+        return self.object_imgs, self.traj_imgs,
+        
+    def _draw_objects(self, i):
+        """
+        This private function is just divided thing of
+        the _update_anim to see the code more clear
+
+        Parameters
+        ------------
+        i : int
+            time step of the animation
+            the sampling time should be related to the sampling time of system
+        """
+        # cars
+        for j in range(2):
+            fix_j = j * 2
+            object_x, object_y, angle_x, angle_y = square_make_with_angles(self.history_xs[j][i],
+                                                                           self.history_ys[j][i], 
+                                                                           1.0,
+                                                                           self.history_ths[j][i])
+
+            self.object_imgs[fix_j].set_data([object_x, object_y])
+            self.object_imgs[fix_j + 1].set_data([angle_x, angle_y])
+    
+    def _draw_traj(self, i):
+        """
+        This private function is just divided thing of
+        the _update_anim to see the code more clear
+
+        Parameters
+        ------------
+        i : int
+            time step of the animation
+            the sampling time should be related to the sampling time of system
+        """
+        for j in range(2):
+            self.traj_imgs[j].set_data(self.history_xs[j][:i], self.history_ys[j][:i])

mpc/basic_IOC/main_ACC.py

@@ -0,0 +1,246 @@
+import numpy as np
+import matplotlib.pyplot as plt
+import math
+import copy
+
+from mpc_func_with_cvxopt import MpcController as MpcController_cvxopt
+from animation import AnimDrawer
+from control import matlab
+
+class TwoWheeledSystem():
+    """SampleSystem, this is the simulator
+    Attributes
+    -----------
+    xs : numpy.ndarray
+        system states, [x, y, theta]
+    history_xs : list
+        time history of state
+    """
+    def __init__(self, init_states=None):
+        """
+        Palameters
+        -----------
+        init_state : float, optional, shape(3, )
+            initial state of system default is None
+        """
+        self.xs = np.zeros(3)
+
+        if init_states is not None:
+            self.xs = copy.deepcopy(init_states)
+
+        self.history_xs = [init_states]
+
+    def update_state(self, us, dt=0.01):
+        """
+        Palameters
+        ------------
+        u : numpy.ndarray
+            inputs of system in some cases this means the reference
+        dt : float in seconds, optional
+            sampling time of simulation, default is 0.01 [s]
+        """
+        # for theta 1, theta 1 dot, theta 2, theta 2 dot
+        k0 = [0.0 for _ in range(3)]
+        k1 = [0.0 for _ in range(3)]
+        k2 = [0.0 for _ in range(3)]
+        k3 = [0.0 for _ in range(3)]
+
+        functions = [self._func_x_1, self._func_x_2, self._func_x_3]
+
+        # solve Runge-Kutta
+        for i, func in enumerate(functions):
+            k0[i] = dt * func(self.xs[0], self.xs[1], self.xs[2], us[0], us[1])
+
+        for i, func in enumerate(functions):
+            k1[i] = dt * func(self.xs[0] + k0[0]/2., self.xs[1] + k0[1]/2., self.xs[2] + k0[2]/2., us[0], us[1])
+        
+        for i, func in enumerate(functions):
+            k2[i] = dt * func(self.xs[0] + k0[0]/2., self.xs[1] + k0[1]/2., self.xs[2] + k0[2]/2., us[0], us[1])
+        
+        for i, func in enumerate(functions):
+            k3[i] =  dt * func(self.xs[0] + k2[0], self.xs[1] + k2[1], self.xs[2] + k2[2], us[0], us[1])
+        
+        self.xs[0] += (k0[0] + 2. * k1[0] + 2. * k2[0] + k3[0]) / 6.
+        self.xs[1] += (k0[1] + 2. * k1[1] + 2. * k2[1] + k3[1]) / 6.
+        self.xs[2] += (k0[2] + 2. * k1[2] + 2. * k2[2] + k3[2]) / 6.
+    
+        # save
+        save_states = copy.deepcopy(self.xs)
+        self.history_xs.append(save_states)
+        print(self.xs)
+
+    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 main():
+    dt = 0.05
+    simulation_time = 10 # in seconds
+    iteration_num = int(simulation_time / dt)
+
+    # you must be care about this matrix
+    # these A and B are for continuos system if you want to use discret system matrix please skip this step
+    # lineared car system
+    V = 5.0
+    A = np.array([[0., V], [0., 0.]])
+    B = np.array([[0.], [1.]])
+
+    C = np.eye(2)
+    D = np.zeros((2, 1))
+
+    # make simulator with coninuous matrix
+    init_xs_lead = np.array([5., 0., 0.])
+    init_xs_follow = np.array([0., 0., 0.])
+    lead_car = TwoWheeledSystem(init_states=init_xs_lead)
+    follow_car = TwoWheeledSystem(init_states=init_xs_follow)
+
+    # create system
+    sysc = matlab.ss(A, B, C, D)
+    # discrete system
+    sysd = matlab.c2d(sysc, dt)
+
+    Ad = sysd.A
+    Bd = sysd.B
+
+    # evaluation function weight
+    Q = np.diag([1., 1.])
+    R = np.diag([5.])
+    pre_step = 15
+
+    # make controller with discreted matrix
+    # please check the solver, if you want to use the scipy, set the MpcController_scipy
+    lead_controller = MpcController_cvxopt(Ad, Bd, Q, R, pre_step,
+                               dt_input_upper=np.array([30 * dt]), dt_input_lower=np.array([-30 * dt]),
+                               input_upper=np.array([30.]), input_lower=np.array([-30.]))
+
+    follow_controller = MpcController_cvxopt(Ad, Bd, Q, R, pre_step,
+                               dt_input_upper=np.array([30 * dt]), dt_input_lower=np.array([-30 * dt]),
+                               input_upper=np.array([30.]), input_lower=np.array([-30.]))
+
+    lead_controller.initialize_controller()
+    follow_controller.initialize_controller()
+
+    # reference
+    lead_reference = np.array([[0., 0.] for _ in range(pre_step)]).flatten()
+
+    for i in range(iteration_num):
+        print("simulation time = {0}".format(i))
+        # make lead car's move
+        if i > int(iteration_num / 3):
+            lead_reference = np.array([[4., 0.] for _ in range(pre_step)]).flatten()
+        
+        lead_states = lead_car.xs
+        lead_opt_u = lead_controller.calc_input(lead_states[1:], lead_reference)
+        lead_opt_u = np.hstack((np.array([V]), lead_opt_u))
+
+        # make follow car
+        follow_reference = np.array([lead_states[1:] for _ in range(pre_step)]).flatten()
+        follow_states = follow_car.xs
+       
+        follow_opt_u = follow_controller.calc_input(follow_states[1:], follow_reference)
+        follow_opt_u = np.hstack((np.array([V]), follow_opt_u))
+        
+        lead_car.update_state(lead_opt_u, dt=dt)
+        follow_car.update_state(follow_opt_u, dt=dt)
+
+    # figures and animation
+    lead_history_states = np.array(lead_car.history_xs)
+    follow_history_states = np.array(follow_car.history_xs)
+
+    time_history_fig = plt.figure()
+    x_fig = time_history_fig.add_subplot(311)
+    y_fig = time_history_fig.add_subplot(312)
+    theta_fig = time_history_fig.add_subplot(313)
+
+    car_traj_fig = plt.figure()
+    traj_fig = car_traj_fig.add_subplot(111)
+    traj_fig.set_aspect('equal')
+
+    x_fig.plot(np.arange(0, simulation_time+0.01, dt), lead_history_states[:, 0], label="lead")
+    x_fig.plot(np.arange(0, simulation_time+0.01, dt), follow_history_states[:, 0], label="follow")
+    x_fig.set_xlabel("time [s]")
+    x_fig.set_ylabel("x")
+    x_fig.legend()
+
+    y_fig.plot(np.arange(0, simulation_time+0.01, dt), lead_history_states[:, 1], label="lead")
+    y_fig.plot(np.arange(0, simulation_time+0.01, dt), follow_history_states[:, 1], label="follow")
+    y_fig.plot(np.arange(0, simulation_time+0.01, dt), [4. for _ in range(iteration_num+1)], linestyle="dashed")
+    y_fig.set_xlabel("time [s]")
+    y_fig.set_ylabel("y")
+    y_fig.legend()
+
+    theta_fig.plot(np.arange(0, simulation_time+0.01, dt), lead_history_states[:, 2], label="lead")
+    theta_fig.plot(np.arange(0, simulation_time+0.01, dt), follow_history_states[:, 2], label="follow")
+    theta_fig.plot(np.arange(0, simulation_time+0.01, dt), [0. for _ in range(iteration_num+1)], linestyle="dashed")
+    theta_fig.set_xlabel("time [s]")
+    theta_fig.set_ylabel("theta")
+    theta_fig.legend()
+
+    time_history_fig.tight_layout()
+
+    traj_fig.plot(lead_history_states[:, 0], lead_history_states[:, 1], label="lead")
+    traj_fig.plot(follow_history_states[:, 0], follow_history_states[:, 1], label="follow")
+    traj_fig.set_xlabel("x")
+    traj_fig.set_ylabel("y")
+    traj_fig.legend()
+    plt.show()
+
+    lead_history_us = np.array(lead_controller.history_us)
+    follow_history_us = np.array(follow_controller.history_us)
+    input_history_fig = plt.figure()
+    u_1_fig = input_history_fig.add_subplot(111)
+
+    u_1_fig.plot(np.arange(0, simulation_time+0.01, dt), lead_history_us[:, 0], label="lead")
+    u_1_fig.plot(np.arange(0, simulation_time+0.01, dt), follow_history_us[:, 0], label="follow")
+    u_1_fig.set_xlabel("time [s]")
+    u_1_fig.set_ylabel("u_omega")
+    
+    input_history_fig.tight_layout()
+    plt.show()
+
+    animdrawer = AnimDrawer([lead_history_states, follow_history_states])
+    animdrawer.draw_anim()
+
+if __name__ == "__main__":
+    main()

mpc/basic_IOC/main_ACC_TEMP.py

@@ -0,0 +1,243 @@
+import numpy as np
+import matplotlib.pyplot as plt
+import math
+import copy
+
+from mpc_func_with_cvxopt import MpcController as MpcController_cvxopt
+from animation import AnimDrawer
+# from control import matlab
+
+class TwoWheeledSystem():
+    """SampleSystem, this is the simulator
+    Attributes
+    -----------
+    xs : numpy.ndarray
+        system states, [x, y, theta]
+    history_xs : list
+        time history of state
+    """
+    def __init__(self, init_states=None):
+        """
+        Palameters
+        -----------
+        init_state : float, optional, shape(3, )
+            initial state of system default is None
+        """
+        self.xs = np.zeros(3)
+
+        if init_states is not None:
+            self.xs = copy.deepcopy(init_states)
+
+        self.history_xs = [init_states]
+
+    def update_state(self, us, dt=0.01):
+        """
+        Palameters
+        ------------
+        u : numpy.ndarray
+            inputs of system in some cases this means the reference
+        dt : float in seconds, optional
+            sampling time of simulation, default is 0.01 [s]
+        """
+        # for theta 1, theta 1 dot, theta 2, theta 2 dot
+        k0 = [0.0 for _ in range(3)]
+        k1 = [0.0 for _ in range(3)]
+        k2 = [0.0 for _ in range(3)]
+        k3 = [0.0 for _ in range(3)]
+
+        functions = [self._func_x_1, self._func_x_2, self._func_x_3]
+
+        # solve Runge-Kutta
+        for i, func in enumerate(functions):
+            k0[i] = dt * func(self.xs[0], self.xs[1], self.xs[2], us[0], us[1])
+
+        for i, func in enumerate(functions):
+            k1[i] = dt * func(self.xs[0] + k0[0]/2., self.xs[1] + k0[1]/2., self.xs[2] + k0[2]/2., us[0], us[1])
+        
+        for i, func in enumerate(functions):
+            k2[i] = dt * func(self.xs[0] + k0[0]/2., self.xs[1] + k0[1]/2., self.xs[2] + k0[2]/2., us[0], us[1])
+        
+        for i, func in enumerate(functions):
+            k3[i] =  dt * func(self.xs[0] + k2[0], self.xs[1] + k2[1], self.xs[2] + k2[2], us[0], us[1])
+        
+        self.xs[0] += (k0[0] + 2. * k1[0] + 2. * k2[0] + k3[0]) / 6.
+        self.xs[1] += (k0[1] + 2. * k1[1] + 2. * k2[1] + k3[1]) / 6.
+        self.xs[2] += (k0[2] + 2. * k1[2] + 2. * k2[2] + k3[2]) / 6.
+    
+        # save
+        save_states = copy.deepcopy(self.xs)
+        self.history_xs.append(save_states)
+        print(self.xs)
+
+    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 main():
+    dt = 0.05
+    simulation_time = 10 # in seconds
+    iteration_num = int(simulation_time / dt)
+
+    # you must be care about this matrix
+    # these A and B are for continuos system if you want to use discret system matrix please skip this step
+    # lineared car system
+    V = 5.0
+    Ad = np.array([[1., V * dt], [0., 1.]])
+    Bd = np.array([[0.], [1. * dt]])
+
+    C = np.eye(2)
+    D = np.zeros((2, 1))
+
+    # make simulator with coninuous matrix
+    init_xs_lead = np.array([5., 0., 0.])
+    init_xs_follow = np.array([0., 0., 0.])
+    lead_car = TwoWheeledSystem(init_states=init_xs_lead)
+    follow_car = TwoWheeledSystem(init_states=init_xs_follow)
+
+    # create system
+    # sysc = matlab.ss(A, B, C, D)
+    # discrete system
+    # sysd = matlab.c2d(sysc, dt)
+
+    # evaluation function weight
+    Q = np.diag([1., 1.])
+    R = np.diag([5.])
+    pre_step = 15
+
+    # make controller with discreted matrix
+    # please check the solver, if you want to use the scipy, set the MpcController_scipy
+    lead_controller = MpcController_cvxopt(Ad, Bd, Q, R, pre_step,
+                               dt_input_upper=np.array([30 * dt]), dt_input_lower=np.array([-30 * dt]),
+                               input_upper=np.array([30.]), input_lower=np.array([-30.]))
+
+    follow_controller = MpcController_cvxopt(Ad, Bd, Q, R, pre_step,
+                               dt_input_upper=np.array([30 * dt]), dt_input_lower=np.array([-30 * dt]),
+                               input_upper=np.array([30.]), input_lower=np.array([-30.]))
+
+    lead_controller.initialize_controller()
+    follow_controller.initialize_controller()
+
+    # reference
+    lead_reference = np.array([[0., 0.] for _ in range(pre_step)]).flatten()
+
+    for i in range(iteration_num):
+        print("simulation time = {0}".format(i))
+        # make lead car's move
+        if i > int(iteration_num / 3):
+            lead_reference = np.array([[4., 0.] for _ in range(pre_step)]).flatten()
+        
+        lead_states = lead_car.xs
+        lead_opt_u = lead_controller.calc_input(lead_states[1:], lead_reference)
+        lead_opt_u = np.hstack((np.array([V]), lead_opt_u))
+
+        # make follow car
+        follow_reference = np.array([lead_states[1:] for _ in range(pre_step)]).flatten()
+        follow_states = follow_car.xs
+       
+        follow_opt_u = follow_controller.calc_input(follow_states[1:], follow_reference)
+        follow_opt_u = np.hstack((np.array([V]), follow_opt_u))
+        
+        lead_car.update_state(lead_opt_u, dt=dt)
+        follow_car.update_state(follow_opt_u, dt=dt)
+
+    # figures and animation
+    lead_history_states = np.array(lead_car.history_xs)
+    follow_history_states = np.array(follow_car.history_xs)
+
+    time_history_fig = plt.figure()
+    x_fig = time_history_fig.add_subplot(311)
+    y_fig = time_history_fig.add_subplot(312)
+    theta_fig = time_history_fig.add_subplot(313)
+
+    car_traj_fig = plt.figure()
+    traj_fig = car_traj_fig.add_subplot(111)
+    traj_fig.set_aspect('equal')
+
+    x_fig.plot(np.arange(0, simulation_time+0.01, dt), lead_history_states[:, 0], label="lead")
+    x_fig.plot(np.arange(0, simulation_time+0.01, dt), follow_history_states[:, 0], label="follow")
+    x_fig.set_xlabel("time [s]")
+    x_fig.set_ylabel("x")
+    x_fig.legend()
+
+    y_fig.plot(np.arange(0, simulation_time+0.01, dt), lead_history_states[:, 1], label="lead")
+    y_fig.plot(np.arange(0, simulation_time+0.01, dt), follow_history_states[:, 1], label="follow")
+    y_fig.plot(np.arange(0, simulation_time+0.01, dt), [4. for _ in range(iteration_num+1)], linestyle="dashed")
+    y_fig.set_xlabel("time [s]")
+    y_fig.set_ylabel("y")
+    y_fig.legend()
+
+    theta_fig.plot(np.arange(0, simulation_time+0.01, dt), lead_history_states[:, 2], label="lead")
+    theta_fig.plot(np.arange(0, simulation_time+0.01, dt), follow_history_states[:, 2], label="follow")
+    theta_fig.plot(np.arange(0, simulation_time+0.01, dt), [0. for _ in range(iteration_num+1)], linestyle="dashed")
+    theta_fig.set_xlabel("time [s]")
+    theta_fig.set_ylabel("theta")
+    theta_fig.legend()
+
+    time_history_fig.tight_layout()
+
+    traj_fig.plot(lead_history_states[:, 0], lead_history_states[:, 1], label="lead")
+    traj_fig.plot(follow_history_states[:, 0], follow_history_states[:, 1], label="follow")
+    traj_fig.set_xlabel("x")
+    traj_fig.set_ylabel("y")
+    traj_fig.legend()
+    plt.show()
+
+    lead_history_us = np.array(lead_controller.history_us)
+    follow_history_us = np.array(follow_controller.history_us)
+    input_history_fig = plt.figure()
+    u_1_fig = input_history_fig.add_subplot(111)
+
+    u_1_fig.plot(np.arange(0, simulation_time+0.01, dt), lead_history_us[:, 0], label="lead")
+    u_1_fig.plot(np.arange(0, simulation_time+0.01, dt), follow_history_us[:, 0], label="follow")
+    u_1_fig.set_xlabel("time [s]")
+    u_1_fig.set_ylabel("u_omega")
+    
+    input_history_fig.tight_layout()
+    plt.show()
+
+    animdrawer = AnimDrawer([lead_history_states, follow_history_states])
+    animdrawer.draw_anim()
+
+if __name__ == "__main__":
+    main()

mpc/basic_IOC/main_example.py

@@ -0,0 +1,184 @@
+import numpy as np
+import matplotlib.pyplot as plt
+import math
+import copy
+
+from mpc_func_with_scipy import MpcController as MpcController_scipy
+from mpc_func_with_cvxopt import MpcController as MpcController_cvxopt
+from control import matlab
+
+class FirstOrderSystem():
+    """FirstOrderSystemWithStates
+
+    Attributes
+    -----------
+    xs : numpy.ndarray
+        system states
+    A : numpy.ndarray
+        system matrix
+    B : numpy.ndarray
+        control matrix
+    C : numpy.ndarray
+        observation matrix
+    history_xs : list
+        time history of state
+    """
+    def __init__(self, A, B, C, D=None, init_states=None):
+        """
+        Parameters
+        -----------
+        A : numpy.ndarray
+            system matrix
+        B : numpy.ndarray
+            control matrix
+        C : numpy.ndarray
+            observation matrix
+        D : numpy.ndarray
+            directly matrix
+        init_state : float, optional
+            initial state of system default is None
+        history_xs : list
+            time history of system states
+        """
+
+        self.A = A
+        self.B = B
+        self.C = C
+
+        if D is not None:
+            self.D = D
+
+        self.xs = np.zeros(self.A.shape[0])
+
+        if init_states is not None:
+            self.xs = copy.deepcopy(init_states)
+
+        self.history_xs = [init_states]
+
+    def update_state(self, u, dt=0.01):
+        """calculating input
+        Parameters
+        ------------
+        u : numpy.ndarray
+            inputs of system in some cases this means the reference
+        dt : float in seconds, optional
+            sampling time of simulation, default is 0.01 [s]
+        """
+        temp_x = self.xs.reshape(-1, 1)
+        temp_u = u.reshape(-1, 1)
+
+        # solve Runge-Kutta
+        k0 = dt * (np.dot(self.A, temp_x) + np.dot(self.B, temp_u)) 
+        k1 = dt * (np.dot(self.A, temp_x + k0/2.) + np.dot(self.B, temp_u))
+        k2 = dt * (np.dot(self.A, temp_x + k1/2.) + np.dot(self.B, temp_u))
+        k3 = dt * (np.dot(self.A, temp_x + k2) + np.dot(self.B, temp_u))
+
+        self.xs +=  ((k0 + 2 * k1 + 2 * k2 + k3) / 6.).flatten()
+
+        # for oylar
+        # self.xs += k0.flatten()
+        # print("xs = {0}".format(self.xs))
+
+        # save
+        save_states = copy.deepcopy(self.xs)
+        self.history_xs.append(save_states)
+
+def main():
+    dt = 0.05
+    simulation_time = 30 # in seconds
+    iteration_num = int(simulation_time / dt)
+
+    # you must be care about this matrix
+    # these A and B are for continuos system if you want to use discret system matrix please skip this step
+    tau = 0.63
+    A = np.array([[-1./tau, 0., 0., 0.],
+                  [0., -1./tau, 0., 0.],
+                  [1., 0., 0., 0.], 
+                  [0., 1., 0., 0.]])
+    B = np.array([[1./tau, 0.],
+                  [0., 1./tau],
+                  [0., 0.],
+                  [0., 0.]])
+
+    C = np.eye(4)
+    D = np.zeros((4, 2))
+
+    # make simulator with coninuous matrix
+    init_xs = np.array([0., 0., 0., 0.])
+    plant = FirstOrderSystem(A, B, C, init_states=init_xs)
+
+    # create system
+    sysc = matlab.ss(A, B, C, D)
+    # discrete system
+    sysd = matlab.c2d(sysc, dt)
+
+    Ad = sysd.A
+    Bd = sysd.B
+
+    # evaluation function weight
+    Q = np.diag([1., 1., 1., 1.])
+    R = np.diag([1., 1.])
+    pre_step = 10
+
+    # make controller with discreted matrix
+    # please check the solver, if you want to use the scipy, set the MpcController_scipy
+    controller = MpcController_cvxopt(Ad, Bd, Q, R, pre_step,
+                               dt_input_upper=np.array([0.25 * dt, 0.25 * dt]), dt_input_lower=np.array([-0.5 * dt, -0.5 * dt]),
+                               input_upper=np.array([1. ,3.]), input_lower=np.array([-1., -3.]))
+
+    controller.initialize_controller()
+
+    for i in range(iteration_num):
+        print("simulation time = {0}".format(i))
+        reference = np.array([[0., 0., -5., 7.5] for _ in range(pre_step)]).flatten()   
+        states = plant.xs
+        opt_u = controller.calc_input(states, reference)
+        plant.update_state(opt_u, dt=dt)
+
+    history_states = np.array(plant.history_xs)
+
+    time_history_fig = plt.figure()
+    x_fig = time_history_fig.add_subplot(411)
+    y_fig = time_history_fig.add_subplot(412)
+    v_x_fig = time_history_fig.add_subplot(413)
+    v_y_fig = time_history_fig.add_subplot(414)
+
+    v_x_fig.plot(np.arange(0, simulation_time+0.01, dt), history_states[:, 0])
+    v_x_fig.plot(np.arange(0, simulation_time+0.01, dt), [0. for _ in range(iteration_num+1)], linestyle="dashed")
+    v_x_fig.set_xlabel("time [s]")
+    v_x_fig.set_ylabel("v_x")
+
+    v_y_fig.plot(np.arange(0, simulation_time+0.01, dt), history_states[:, 1])
+    v_y_fig.plot(np.arange(0, simulation_time+0.01, dt), [0. for _ in range(iteration_num+1)], linestyle="dashed")
+    v_y_fig.set_xlabel("time [s]")
+    v_y_fig.set_ylabel("v_y")
+
+    x_fig.plot(np.arange(0, simulation_time+0.01, dt), history_states[:, 2])
+    x_fig.plot(np.arange(0, simulation_time+0.01, dt), [-5. for _ in range(iteration_num+1)], linestyle="dashed")
+    x_fig.set_xlabel("time [s]")
+    x_fig.set_ylabel("x")
+
+    y_fig.plot(np.arange(0, simulation_time+0.01, dt), history_states[:, 3])
+    y_fig.plot(np.arange(0, simulation_time+0.01, dt), [7.5 for _ in range(iteration_num+1)], linestyle="dashed")
+    y_fig.set_xlabel("time [s]")
+    y_fig.set_ylabel("y")
+    time_history_fig.tight_layout()
+    plt.show()
+
+    history_us = np.array(controller.history_us)
+    input_history_fig = plt.figure()
+    u_1_fig = input_history_fig.add_subplot(211)
+    u_2_fig = input_history_fig.add_subplot(212)
+
+    u_1_fig.plot(np.arange(0, simulation_time+0.01, dt), history_us[:, 0])
+    u_1_fig.set_xlabel("time [s]")
+    u_1_fig.set_ylabel("u_x")
+    
+    u_2_fig.plot(np.arange(0, simulation_time+0.01, dt), history_us[:, 1])
+    u_2_fig.set_xlabel("time [s]")
+    u_2_fig.set_ylabel("u_y")
+    input_history_fig.tight_layout()
+    plt.show()
+
+if __name__ == "__main__":
+    main()

mpc/basic_IOC/memo.md

@@ -0,0 +1,5 @@
+ # ラプラス近似
+ 参考
+ - http://triadsou.hatenablog.com/entry/20090217/1234865055
+
+ # 

mpc/basic_IOC/mpc_func_with_cvxopt.py

@@ -0,0 +1,256 @@
+import numpy as np
+import matplotlib.pyplot as plt
+import math
+import copy
+
+from cvxopt import matrix, solvers
+
+class MpcController():
+    """
+    Attributes
+    ------------
+    A : numpy.ndarray
+        system matrix
+    B : numpy.ndarray
+        input matrix
+    Q : numpy.ndarray
+        evaluation function weight for states
+    Qs : numpy.ndarray
+        concatenated evaluation function weight for states
+    R : numpy.ndarray
+        evaluation function weight for inputs
+    Rs : numpy.ndarray
+        concatenated evaluation function weight for inputs
+    pre_step : int
+        prediction step
+    state_size : int
+        state size of the plant
+    input_size : int
+        input size of the plant
+    dt_input_upper : numpy.ndarray, shape(input_size, ), optional
+        constraints of input dt, default is None
+    dt_input_lower : numpy.ndarray, shape(input_size, ), optional
+        constraints of input dt, default is None
+    input_upper : numpy.ndarray, shape(input_size, ), optional
+        constraints of input, default is None
+    input_lower : numpy.ndarray, shape(input_size, ), optional
+        constraints of input, default is None
+    """
+    def __init__(self, A, B, Q, R, pre_step, initial_input=None, dt_input_upper=None, dt_input_lower=None, input_upper=None, input_lower=None):
+        """
+        Parameters
+        ------------
+        A : numpy.ndarray
+            system matrix
+        B : numpy.ndarray
+            input matrix
+        Q : numpy.ndarray
+            evaluation function weight for states
+        R : numpy.ndarray
+            evaluation function weight for inputs
+        pre_step : int
+            prediction step
+        dt_input_upper : numpy.ndarray, shape(input_size, ), optional
+            constraints of input dt, default is None
+        dt_input_lower : numpy.ndarray, shape(input_size, ), optional
+            constraints of input dt, default is None
+        input_upper : numpy.ndarray, shape(input_size, ), optional
+            constraints of input, default is None
+        input_lower : numpy.ndarray, shape(input_size, ), optional
+            constraints of input, default is None
+        history_us : list
+            time history of optimal input us(numpy.ndarray)
+        """
+        self.A = np.array(A)
+        self.B = np.array(B)
+        self.Q = np.array(Q)
+        self.R = np.array(R)
+        self.pre_step = pre_step
+
+        self.Qs = None
+        self.Rs = None
+
+        self.state_size = self.A.shape[0]
+        self.input_size = self.B.shape[1]
+
+        self.history_us = [np.zeros(self.input_size)]
+
+        # initial state
+        if initial_input is not None:
+            self.history_us = [initial_input]
+
+        # constraints
+        self.dt_input_lower = dt_input_lower
+        self.dt_input_upper = dt_input_upper
+        self.input_upper = input_upper
+        self.input_lower = input_lower
+        
+        # about mpc matrix
+        self.W = None
+        self.omega = None
+        self.F = None
+        self.f = None
+        
+    def initialize_controller(self):
+        """
+        make matrix to calculate optimal control input
+        
+        """
+        A_factorials = [self.A]
+
+        self.phi_mat = copy.deepcopy(self.A)
+
+        for _ in range(self.pre_step - 1):
+            temp_mat = np.dot(A_factorials[-1], self.A)
+            self.phi_mat = np.vstack((self.phi_mat, temp_mat))
+
+            A_factorials.append(temp_mat) # after we use this factorials
+            
+        print("phi_mat = \n{0}".format(self.phi_mat))
+
+        self.gamma_mat = copy.deepcopy(self.B)
+        gammma_mat_temp = copy.deepcopy(self.B)
+        
+        for i in range(self.pre_step - 1):
+            temp_1_mat = np.dot(A_factorials[i], self.B)
+            gammma_mat_temp = temp_1_mat + gammma_mat_temp
+            self.gamma_mat = np.vstack((self.gamma_mat, gammma_mat_temp))
+
+        print("gamma_mat = \n{0}".format(self.gamma_mat))
+
+        self.theta_mat = copy.deepcopy(self.gamma_mat)
+
+        for i in range(self.pre_step - 1):
+            temp_mat = np.zeros_like(self.gamma_mat)
+            temp_mat[int((i + 1)*self.state_size): , :] = self.gamma_mat[:-int((i + 1)*self.state_size) , :]
+
+            self.theta_mat = np.hstack((self.theta_mat, temp_mat))
+
+        print("theta_mat = \n{0}".format(self.theta_mat))
+
+        # evaluation function weight
+        diag_Qs = np.array([np.diag(self.Q) for _ in range(self.pre_step)])
+        diag_Rs = np.array([np.diag(self.R) for _ in range(self.pre_step)])
+        
+        self.Qs = np.diag(diag_Qs.flatten())
+        self.Rs = np.diag(diag_Rs.flatten())
+
+        print("Qs = \n{0}".format(self.Qs))
+        print("Rs = \n{0}".format(self.Rs))
+
+        # constraints
+        # about dt U
+        if self.input_lower is not None:
+            # initialize
+            self.F = np.zeros((self.input_size * 2, self.pre_step * self.input_size))
+            for i in range(self.input_size):
+                self.F[i * 2: (i + 1) * 2, i] = np.array([1.,  -1.])
+                temp_F = copy.deepcopy(self.F)
+
+            print("F = \n{0}".format(self.F))
+
+            for i in range(self.pre_step - 1):
+                temp_F = copy.deepcopy(temp_F)
+
+                for j in range(self.input_size):
+                    temp_F[j * 2: (j + 1) * 2, ((i+1) * self.input_size) + j] = np.array([1.,  -1.])
+                
+                self.F = np.vstack((self.F, temp_F))
+
+            self.F1 = self.F[:, :self.input_size]
+            
+            temp_f = []
+
+            for i in range(self.input_size):
+                temp_f.append(-1 * self.input_upper[i])
+                temp_f.append(self.input_lower[i])
+
+            self.f = np.array([temp_f for _ in range(self.pre_step)]).flatten()
+
+            print("F = \n{0}".format(self.F))
+            print("F1 = \n{0}".format(self.F1))
+            print("f = \n{0}".format(self.f))
+
+        # about dt_u
+        if self.dt_input_lower is not None:
+            self.W = np.zeros((2, self.pre_step * self.input_size))
+            self.W[:, 0] = np.array([1.,  -1.])
+
+            for i in range(self.pre_step * self.input_size - 1):
+                temp_W = np.zeros((2, self.pre_step * self.input_size))
+                temp_W[:, i+1] = np.array([1.,  -1.])
+                self.W = np.vstack((self.W, temp_W))
+
+            temp_omega = []
+
+            for i in range(self.input_size):
+                temp_omega.append(self.dt_input_upper[i])
+                temp_omega.append(-1. * self.dt_input_lower[i])
+
+            self.omega = np.array([temp_omega for _ in range(self.pre_step)]).flatten()
+
+            print("W = \n{0}".format(self.W))
+            print("omega = \n{0}".format(self.omega))
+
+        # about state
+        print("check the matrix!! if you think rite, plese push enter")
+        input()
+
+    def calc_input(self, states, references):
+        """calculate optimal input
+        Parameters
+        -----------
+        states : numpy.ndarray, shape(state length, )
+            current state of system
+        references : numpy.ndarray, shape(state length * pre_step, )
+            reference of the system, you should set this value as reachable goal
+
+        References
+        ------------
+        opt_input : numpy.ndarray, shape(input_length, )
+            optimal input
+        """
+        temp_1 = np.dot(self.phi_mat, states.reshape(-1, 1))
+        temp_2 = np.dot(self.gamma_mat, self.history_us[-1].reshape(-1, 1))
+
+        error = references.reshape(-1, 1) - temp_1 - temp_2
+
+        G = 2. * np.dot(self.theta_mat.T, np.dot(self.Qs, error))
+
+        H = np.dot(self.theta_mat.T, np.dot(self.Qs, self.theta_mat)) + self.Rs
+
+        # constraints
+        A = [] 
+        b = []
+
+        if self.W is not None:
+            A.append(self.W)
+            b.append(self.omega.reshape(-1, 1))
+
+        if self.F is not None:
+            b_F = - np.dot(self.F1, self.history_us[-1].reshape(-1, 1)) - self.f.reshape(-1, 1)
+            A.append(self.F)
+            b.append(b_F)
+
+        A = np.array(A).reshape(-1, self.input_size * self.pre_step)
+
+        ub = np.array(b).flatten()
+
+        # make cvxpy problem formulation
+        P = 2*matrix(H)
+        q = matrix(-1 * G)
+        A = matrix(A)
+        b = matrix(ub)
+
+        # constraint
+        if self.W is not None or self.F is not None :
+            opt_result = solvers.qp(P, q, G=A, h=b)
+        
+        opt_dt_us = list(opt_result['x'])
+        
+        opt_u = opt_dt_us[:self.input_size] + self.history_us[-1]
+        
+        # save
+        self.history_us.append(opt_u)
+
+        return opt_u

mpc/basic_IOC/mpc_func_with_scipy.py

@@ -0,0 +1,262 @@
+import numpy as np
+import matplotlib.pyplot as plt
+import math
+import copy
+
+from scipy.optimize import minimize
+from scipy.optimize import LinearConstraint
+
+class MpcController():
+    """
+    Attributes
+    ------------
+    A : numpy.ndarray
+        system matrix
+    B : numpy.ndarray
+        input matrix
+    Q : numpy.ndarray
+        evaluation function weight for states
+    Qs : numpy.ndarray
+        concatenated evaluation function weight for states
+    R : numpy.ndarray
+        evaluation function weight for inputs
+    Rs : numpy.ndarray
+        concatenated evaluation function weight for inputs
+    pre_step : int
+        prediction step
+    state_size : int
+        state size of the plant
+    input_size : int
+        input size of the plant
+    dt_input_upper : numpy.ndarray, shape(input_size, ), optional
+        constraints of input dt, default is None
+    dt_input_lower : numpy.ndarray, shape(input_size, ), optional
+        constraints of input dt, default is None
+    input_upper : numpy.ndarray, shape(input_size, ), optional
+        constraints of input, default is None
+    input_lower : numpy.ndarray, shape(input_size, ), optional
+        constraints of input, default is None
+    """
+    def __init__(self, A, B, Q, R, pre_step, initial_input=None, dt_input_upper=None, dt_input_lower=None, input_upper=None, input_lower=None):
+        """
+        Parameters
+        ------------
+        A : numpy.ndarray
+            system matrix
+        B : numpy.ndarray
+            input matrix
+        Q : numpy.ndarray
+            evaluation function weight for states
+        R : numpy.ndarray
+            evaluation function weight for inputs
+        pre_step : int
+            prediction step
+        dt_input_upper : numpy.ndarray, shape(input_size, ), optional
+            constraints of input dt, default is None
+        dt_input_lower : numpy.ndarray, shape(input_size, ), optional
+            constraints of input dt, default is None
+        input_upper : numpy.ndarray, shape(input_size, ), optional
+            constraints of input, default is None
+        input_lower : numpy.ndarray, shape(input_size, ), optional
+            constraints of input, default is None
+        history_us : list
+            time history of optimal input us(numpy.ndarray)
+        """
+        self.A = np.array(A)
+        self.B = np.array(B)
+        self.Q = np.array(Q)
+        self.R = np.array(R)
+        self.pre_step = pre_step
+
+        self.Qs = None
+        self.Rs = None
+
+        self.state_size = self.A.shape[0]
+        self.input_size = self.B.shape[1]
+
+        self.history_us = [np.zeros(self.input_size)]
+
+        # initial state
+        if initial_input is not None:
+            self.history_us = [initial_input]
+
+        # constraints
+        self.dt_input_lower = dt_input_lower
+        self.dt_input_upper = dt_input_upper
+        self.input_upper = input_upper
+        self.input_lower = input_lower
+
+        self.W = None
+        self.omega = None
+        self.F = None
+        self.f = None
+        
+    def initialize_controller(self):
+        """
+        make matrix to calculate optimal control input
+        """
+        A_factorials = [self.A]
+
+        self.phi_mat = copy.deepcopy(self.A)
+
+        for _ in range(self.pre_step - 1):
+            temp_mat = np.dot(A_factorials[-1], self.A)
+            self.phi_mat = np.vstack((self.phi_mat, temp_mat))
+
+            A_factorials.append(temp_mat) # after we use this factorials
+            
+        print("phi_mat = \n{0}".format(self.phi_mat))
+
+        self.gamma_mat = copy.deepcopy(self.B)
+        gammma_mat_temp = copy.deepcopy(self.B)
+        
+        for i in range(self.pre_step - 1):
+            temp_1_mat = np.dot(A_factorials[i], self.B)
+            gammma_mat_temp = temp_1_mat + gammma_mat_temp
+            self.gamma_mat = np.vstack((self.gamma_mat, gammma_mat_temp))
+
+        print("gamma_mat = \n{0}".format(self.gamma_mat))
+
+        self.theta_mat = copy.deepcopy(self.gamma_mat)
+
+        for i in range(self.pre_step - 1):
+            temp_mat = np.zeros_like(self.gamma_mat)
+            temp_mat[int((i + 1)*self.state_size): , :] = self.gamma_mat[:-int((i + 1)*self.state_size) , :]
+
+            self.theta_mat = np.hstack((self.theta_mat, temp_mat))
+
+        print("theta_mat = \n{0}".format(self.theta_mat))
+
+        # evaluation function weight
+        diag_Qs = np.array([np.diag(self.Q) for _ in range(self.pre_step)])
+        diag_Rs = np.array([np.diag(self.R) for _ in range(self.pre_step)])
+        
+        self.Qs = np.diag(diag_Qs.flatten())
+        self.Rs = np.diag(diag_Rs.flatten())
+
+        print("Qs = \n{0}".format(self.Qs))
+        print("Rs = \n{0}".format(self.Rs))
+
+        # constraints
+        # about dt U
+        if self.input_lower is not None:
+            # initialize
+            self.F = np.zeros((self.input_size * 2, self.pre_step * self.input_size))
+            for i in range(self.input_size):
+                self.F[i * 2: (i + 1) * 2, i] = np.array([1.,  -1.])
+                temp_F = copy.deepcopy(self.F)
+
+            print("F = \n{0}".format(self.F))
+
+            for i in range(self.pre_step - 1):
+                temp_F = copy.deepcopy(temp_F)
+
+                for j in range(self.input_size):
+                    temp_F[j * 2: (j + 1) * 2, ((i+1) * self.input_size) + j] = np.array([1.,  -1.])
+                
+                self.F = np.vstack((self.F, temp_F))
+
+            self.F1 = self.F[:, :self.input_size]
+            
+            temp_f = []
+
+            for i in range(self.input_size):
+                temp_f.append(-1 * self.input_upper[i])
+                temp_f.append(self.input_lower[i])
+
+            self.f = np.array([temp_f for _ in range(self.pre_step)]).flatten()
+
+            print("F = \n{0}".format(self.F))
+            print("F1 = \n{0}".format(self.F1))
+            print("f = \n{0}".format(self.f))
+
+        # about dt_u
+        if self.dt_input_lower is not None:
+            self.W = np.zeros((2, self.pre_step * self.input_size))
+            self.W[:, 0] = np.array([1.,  -1.])
+
+            for i in range(self.pre_step * self.input_size - 1):
+                temp_W = np.zeros((2, self.pre_step * self.input_size))
+                temp_W[:, i+1] = np.array([1.,  -1.])
+                self.W = np.vstack((self.W, temp_W))
+
+            temp_omega = []
+
+            for i in range(self.input_size):
+                temp_omega.append(self.dt_input_upper[i])
+                temp_omega.append(-1. * self.dt_input_lower[i])
+
+            self.omega = np.array([temp_omega for _ in range(self.pre_step)]).flatten()
+
+            print("W = \n{0}".format(self.W))
+            print("omega = \n{0}".format(self.omega))
+
+        # about state
+        print("check the matrix!! if you think rite, plese push enter")
+        input()
+
+    def calc_input(self, states, references):
+        """calculate optimal input
+        Parameters
+        -----------
+        states : numpy.ndarray, shape(state length, )
+            current state of system
+        references : numpy.ndarray, shape(state length * pre_step, )
+            reference of the system, you should set this value as reachable goal
+
+        References
+        ------------
+        opt_input : numpy.ndarray, shape(input_length, )
+            optimal input
+        """
+        temp_1 = np.dot(self.phi_mat, states.reshape(-1, 1))
+        temp_2 = np.dot(self.gamma_mat, self.history_us[-1].reshape(-1, 1))
+
+        error = references.reshape(-1, 1) - temp_1 - temp_2
+
+        G = 2. * np.dot(self.theta_mat.T, np.dot(self.Qs, error) )
+
+        H = np.dot(self.theta_mat.T, np.dot(self.Qs, self.theta_mat)) + self.Rs
+
+        # constraints
+        A = [] 
+        b = []
+
+        if self.W is not None:
+            A.append(self.W)
+            b.append(self.omega.reshape(-1, 1))
+
+        if self.F is not None:
+            b_F = - np.dot(self.F1, self.history_us[-1].reshape(-1, 1)) - self.f.reshape(-1, 1)
+            A.append(self.F)
+            b.append(b_F)
+
+        A = np.array(A).reshape(-1, self.input_size * self.pre_step)
+
+        ub = np.array(b).flatten()
+
+        def optimized_func(dt_us):
+            """
+            """
+            temp_dt_us = np.array([dt_us[i] for i in range(self.input_size * self.pre_step)])
+
+            return (np.dot(temp_dt_us, np.dot(H, temp_dt_us.reshape(-1, 1))) - np.dot(G.T, temp_dt_us.reshape(-1, 1)))[0]
+
+        # constraint
+        lb = np.array([-np.inf for _ in range(len(ub))])
+        linear_cons = LinearConstraint(A, lb, ub)
+
+        init_dt_us = np.zeros(self.input_size * self.pre_step)
+
+        # constraint
+        if self.W is not None or self.F is not None :
+            opt_result = minimize(optimized_func, init_dt_us, constraints=[linear_cons])
+        
+        opt_dt_us = opt_result.x
+
+        opt_u = opt_dt_us[:self.input_size] + self.history_us[-1]
+
+        # save
+        self.history_us.append(opt_u)
+        
+        return opt_u

mpc/basic_IOC/test_compare_methods.py

@@ -0,0 +1,211 @@
+import numpy as np
+import matplotlib.pyplot as plt
+import math
+import copy
+
+from mpc_func_with_scipy import MpcController as MpcController_scipy
+from mpc_func_with_cvxopt import MpcController as MpcController_cvxopt
+from control import matlab
+
+class FirstOrderSystem():
+    """FirstOrderSystemWithStates
+
+    Attributes
+    -----------
+    states : float
+        system states
+    A : numpy.ndarray
+        system matrix
+    B : numpy.ndarray
+        control matrix
+    C : numpy.ndarray
+        observation matrix
+    history_state : list
+        time history of state
+    """
+    def __init__(self, A, B, C, D=None, init_states=None):
+        """
+        Parameters
+        -----------
+        A : numpy.ndarray
+            system matrix
+        B : numpy.ndarray
+            control matrix
+        C : numpy.ndarray
+            observation matrix
+        C : numpy.ndarray
+            directly matrix
+        init_state : float, optional
+            initial state of system default is None
+        history_xs : list
+            time history of system states
+        """
+
+        self.A = A
+        self.B = B
+        self.C = C
+
+        if D is not None:
+            self.D = D
+
+        self.xs = np.zeros(self.A.shape[0])
+
+        if init_states is not None:
+            self.xs = copy.deepcopy(init_states)
+
+        self.history_xs = [init_states]
+
+    def update_state(self, u, dt=0.01):
+        """calculating input
+        Parameters
+        ------------
+        u : float
+            input of system in some cases this means the reference
+        dt : float in seconds, optional
+            sampling time of simulation, default is 0.01 [s]
+        """
+        temp_x = self.xs.reshape(-1, 1)
+        temp_u = u.reshape(-1, 1)
+
+        # solve Runge-Kutta
+        k0 = dt * (np.dot(self.A, temp_x) + np.dot(self.B, temp_u)) 
+        k1 = dt * (np.dot(self.A, temp_x + k0/2.) + np.dot(self.B, temp_u))
+        k2 = dt * (np.dot(self.A, temp_x + k1/2.) + np.dot(self.B, temp_u))
+        k3 = dt * (np.dot(self.A, temp_x + k2) + np.dot(self.B, temp_u))
+
+        self.xs +=  ((k0 + 2 * k1 + 2 * k2 + k3) / 6.).flatten()
+
+        # for oylar
+        # self.xs += k0.flatten()
+
+        # print("xs = {0}".format(self.xs))
+        # a = input()
+        # save
+        save_states = copy.deepcopy(self.xs)
+        self.history_xs.append(save_states)
+        # print(self.history_xs)
+
+def main():
+    dt = 0.05
+    simulation_time = 50 # in seconds
+    iteration_num = int(simulation_time / dt)
+
+    # you must be care about this matrix
+    # these A and B are for continuos system if you want to use discret system matrix please skip this step
+    tau = 0.63
+    A = np.array([[-1./tau, 0., 0., 0.],
+                  [0., -1./tau, 0., 0.],
+                  [1., 0., 0., 0.], 
+                  [0., 1., 0., 0.]])
+    B = np.array([[1./tau, 0.],
+                  [0., 1./tau],
+                  [0., 0.],
+                  [0., 0.]])
+
+    C = np.eye(4)
+    D = np.zeros((4, 2))
+
+    # make simulator with coninuous matrix
+    init_xs = np.array([0., 0., 0., 0.])
+    plant_cvxopt = FirstOrderSystem(A, B, C, init_states=init_xs)
+    plant_scipy = FirstOrderSystem(A, B, C, init_states=init_xs)
+
+    # create system
+    sysc = matlab.ss(A, B, C, D)
+    # discrete system
+    sysd = matlab.c2d(sysc, dt)
+
+    Ad = sysd.A
+    Bd = sysd.B
+
+    # evaluation function weight
+    Q = np.diag([1., 1., 10., 10.])
+    R = np.diag([0.01, 0.01])
+    pre_step = 5
+
+    # make controller with discreted matrix
+    # please check the solver, if you want to use the scipy, set the MpcController_scipy
+    controller_cvxopt = MpcController_cvxopt(Ad, Bd, Q, R, pre_step,
+                               dt_input_upper=np.array([0.25 * dt, 0.25 * dt]), dt_input_lower=np.array([-0.5 * dt, -0.5 * dt]),
+                               input_upper=np.array([1. ,3.]), input_lower=np.array([-1., -3.]))
+    
+    controller_scipy = MpcController_scipy(Ad, Bd, Q, R, pre_step,
+                               dt_input_upper=np.array([0.25 * dt, 0.25 * dt]), dt_input_lower=np.array([-0.5 * dt, -0.5 * dt]),
+                               input_upper=np.array([1. ,3.]), input_lower=np.array([-1., -3.]))
+
+    controller_cvxopt.initialize_controller()
+    controller_scipy.initialize_controller()
+
+    for i in range(iteration_num):
+        print("simulation time = {0}".format(i))
+        reference = np.array([[0., 0., -5., 7.5] for _ in range(pre_step)]).flatten()   
+
+        states_cvxopt = plant_cvxopt.xs
+        states_scipy = plant_scipy.xs
+
+        opt_u_cvxopt = controller_cvxopt.calc_input(states_cvxopt, reference)
+        opt_u_scipy = controller_scipy.calc_input(states_scipy, reference)
+
+        plant_cvxopt.update_state(opt_u_cvxopt)
+        plant_scipy.update_state(opt_u_scipy)
+
+    history_states_cvxopt = np.array(plant_cvxopt.history_xs)
+    history_states_scipy = np.array(plant_scipy.history_xs)
+
+    time_history_fig = plt.figure(dpi=75)
+    x_fig = time_history_fig.add_subplot(411)
+    y_fig = time_history_fig.add_subplot(412)
+    v_x_fig = time_history_fig.add_subplot(413)
+    v_y_fig = time_history_fig.add_subplot(414)
+
+    v_x_fig.plot(np.arange(0, simulation_time+0.01, dt), history_states_cvxopt[:, 0], label="cvxopt")
+    v_x_fig.plot(np.arange(0, simulation_time+0.01, dt), history_states_scipy[:, 0], label="scipy", linestyle="dashdot")
+    v_x_fig.plot(np.arange(0, simulation_time+0.01, dt), [0. for _ in range(iteration_num+1)], linestyle="dashed")
+    v_x_fig.set_xlabel("time [s]")
+    v_x_fig.set_ylabel("v_x")
+    v_x_fig.legend()
+
+    v_y_fig.plot(np.arange(0, simulation_time+0.01, dt), history_states_cvxopt[:, 1], label="cvxopt")
+    v_y_fig.plot(np.arange(0, simulation_time+0.01, dt), history_states_scipy[:, 1], label="scipy", linestyle="dashdot")
+    v_y_fig.plot(np.arange(0, simulation_time+0.01, dt), [0. for _ in range(iteration_num+1)], linestyle="dashed")
+    v_y_fig.set_xlabel("time [s]")
+    v_y_fig.set_ylabel("v_y")
+    v_y_fig.legend()
+
+    x_fig.plot(np.arange(0, simulation_time+0.01, dt), history_states_cvxopt[:, 2], label="cvxopt")
+    x_fig.plot(np.arange(0, simulation_time+0.01, dt), history_states_scipy[:, 2], label="scipy", linestyle="dashdot")
+    x_fig.plot(np.arange(0, simulation_time+0.01, dt), [-5. for _ in range(iteration_num+1)], linestyle="dashed")
+    x_fig.set_xlabel("time [s]")
+    x_fig.set_ylabel("x")
+
+    y_fig.plot(np.arange(0, simulation_time+0.01, dt),  history_states_cvxopt[:, 3], label ="cvxopt")
+    y_fig.plot(np.arange(0, simulation_time+0.01, dt),  history_states_scipy[:, 3], label="scipy", linestyle="dashdot")
+    y_fig.plot(np.arange(0, simulation_time+0.01, dt), [7.5 for _ in range(iteration_num+1)], linestyle="dashed")
+    y_fig.set_xlabel("time [s]")
+    y_fig.set_ylabel("y")
+    time_history_fig.tight_layout()
+    plt.show()
+
+    history_us_cvxopt = np.array(controller_cvxopt.history_us)
+    history_us_scipy = np.array(controller_scipy.history_us)
+
+    input_history_fig = plt.figure(dpi=75)
+    u_1_fig = input_history_fig.add_subplot(211)
+    u_2_fig = input_history_fig.add_subplot(212)
+
+    u_1_fig.plot(np.arange(0, simulation_time+0.01, dt), history_us_cvxopt[:, 0], label="cvxopt")
+    u_1_fig.plot(np.arange(0, simulation_time+0.01, dt), history_us_scipy[:, 0], label="scipy", linestyle="dashdot")
+    u_1_fig.set_xlabel("time [s]")
+    u_1_fig.set_ylabel("u_x")
+    u_1_fig.legend()
+    
+    u_2_fig.plot(np.arange(0, simulation_time+0.01, dt), history_us_cvxopt[:, 1], label="cvxopt")
+    u_2_fig.plot(np.arange(0, simulation_time+0.01, dt), history_us_scipy[:, 1], label="scipy", linestyle="dashdot")
+    u_2_fig.set_xlabel("time [s]")
+    u_2_fig.set_ylabel("u_y")
+    u_2_fig.legend()
+    input_history_fig.tight_layout()
+    plt.show()
+
+if __name__ == "__main__":
+    main()

mpc/extend/main_track.py

@@ -317,7 +317,7 @@ def main():
     # input()
 
     # evaluation function weight
-    Q = np.diag([100., 1000., 1.])
+    Q = np.diag([1000000., 10., 1.])
     R = np.diag([0.1])
 
     # System model update