import numpy as np
import matplotlib.pyplot as plt
import math

class SampleSystem():
    """SampleSystem, this is the simulator
    Attributes
    -----------
    x_1 : float
        system state 1
    x_2 : float
        system state 2
    history_x_1 : list
        time history of system state 1 (x_1)
    history_x_2 : list
        time history of system state 2 (x_2)
    """
    def __init__(self, init_x_1=0., init_x_2=0.):
        """
        Palameters
        -----------
        init_x_1 : float, optional
            initial value of x_1, default is 0.
        init_x_2 : float, optional
            initial value of x_2, default is 0.
        """
        self.x_1 = init_x_1
        self.x_2 = init_x_2
        self.history_x_1 = [init_x_1]
        self.history_x_2 = [init_x_2]

    def update_state(self, u, dt=0.01):
        """
        Palameters
        ------------
        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]
        """
        # for theta 1, theta 1 dot, theta 2, theta 2 dot
        k0 = [0.0 for _ in range(2)]
        k1 = [0.0 for _ in range(2)]
        k2 = [0.0 for _ in range(2)]
        k3 = [0.0 for _ in range(2)]

        functions = [self._func_x_1, self._func_x_2]

        # solve Runge-Kutta
        for i, func in enumerate(functions):
            k0[i] = dt * func(self.x_1, self.x_2, u)
        
        for i, func in enumerate(functions):
            k1[i] = dt * func(self.x_1 + k0[0]/2., self.x_2 + k0[1]/2., u)
        
        for i, func in enumerate(functions):
            k2[i] = dt * func(self.x_1 + k1[0]/2., self.x_2 + k1[1]/2., u)
        
        for i, func in enumerate(functions):
            k3[i] =  dt * func(self.x_1 + k2[0], self.x_2 + k2[1], u)
        
        self.x_1 += (k0[0] + 2. * k1[0] + 2. * k2[0] + k3[0]) / 6.
        self.x_2 += (k0[1] + 2. * k1[1] + 2. * k2[1] + k3[1]) / 6.
    
        # save
        self.history_x_1.append(self.x_1)
        self.history_x_2.append(self.x_2)      

    def _func_x_1(self, y_1, y_2, u):
        """
        Parameters
        ------------
        y_1 : float
        y_2 : float
        u : float
            system input
        """
        y_dot = y_2
        return y_dot
    
    def _func_x_2(self, y_1, y_2, u):
        """
        Parameters
        ------------
        y_1 : float
        y_2 : float
        u : float
            system input
        """
        y_dot = (1. - y_1**2 - y_2**2) * y_2 - y_1 + u
        return y_dot


class NMPCSimulatorSystem():
    """SimulatorSystem for nmpc, this is the simulator of nmpc
    the reason why I seperate the real simulator and nmpc's simulator is sometimes the modeling error, disturbance can include in real simulator
    Attributes
    -----------
    None

    """
    def __init__(self):
        """
        Parameters
        -----------
        None
        """
        pass

    def calc_predict_and_adjoint_state(self, x_1, x_2, us, N, dt):
        """main
        Parameters
        ------------
        x_1 : float
            current state
        x_2 : float
            current state
        us : list of float
            estimated optimal input Us for N steps
        N : int
            predict step
        dt : float
            sampling time

        Returns
        --------
        x_1s : list of float
            predicted x_1s for N steps
        x_2s : list of float
            predicted x_2s for N steps
        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
        """

        x_1s, x_2s = self._calc_predict_states(x_1, x_2, us, N, dt) # by usin state equation
        lam_1s, lam_2s = self._calc_adjoint_states(x_1s, x_2s, us, N, dt) # by using adjoint equation

        return x_1s, x_2s, lam_1s, lam_2s

    def _calc_predict_states(self, x_1, x_2, us, N, dt):
        """ 
        Parameters
        ------------
        x_1 : float
            current state
        x_2 : float
            current state
        us : list of float
            estimated optimal input Us for N steps
        N : int
            predict step
        dt : float
            sampling time
            
        Returns
        --------
        x_1s : list of float
            predicted x_1s for N steps
        x_2s : list of float
            predicted x_2s for N steps
        """
        # initial state
        x_1s = [x_1]
        x_2s = [x_2]

        for i in range(N):
            temp_x_1, temp_x_2 = self._predict_state_with_oylar(x_1s[i], x_2s[i], us[i], dt)
            x_1s.append(temp_x_1)
            x_2s.append(temp_x_2)

        return x_1s, x_2s

    def _calc_adjoint_states(self, x_1s, x_2s, us, 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
        us : list of float
            estimated optimal input Us for N steps
        N : int
            predict step
        dt : float
            sampling time
            
        Returns
        --------
        lam_1s : list of float
            adjoint state of x_1s, lam_1s for N steps
        lam_2s : list of float
            adjoint state of x_2s, lam_2s for N steps
        """
        # final state
        # final_state_func
        lam_1s = [x_1s[-1]]
        lam_2s = [x_2s[-1]]

        for i in range(N-1, 0, -1): 
            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)
            lam_1s.insert(0, temp_lam_1)
            lam_2s.insert(0, temp_lam_2)

        return lam_1s, lam_2s

    def final_state_func(self):
        """this func usually need
        """
        pass

    def _predict_state_with_oylar(self, x_1, x_2, u, dt):
        """in this case this function is the same as simulator
        Parameters
        ------------
        x_1 : float
            system state
        x_2 : float
            system state
        u : float
            system input
        dt : float in seconds
            sampling time
        Returns
        --------
        next_x_1 : float
            next state, x_1 calculated by using state equation
        next_x_2 : float
            next state, x_2 calculated by using state equation
        """
        k0 = [0. for _ in range(2)]

        functions = [self.func_x_1, self.func_x_2]

        for i, func in enumerate(functions):
            k0[i] = dt * func(x_1, x_2, u)
                
        next_x_1 = x_1 + k0[0]
        next_x_2 = x_2 + k0[1]

        return next_x_1, next_x_2

    def func_x_1(self, y_1, y_2, u):
        """calculating \dot{x_1}
        Parameters
        ------------
        y_1 : float
            means x_1
        y_2 : float
            means x_2
        u : float
            means system input
        Returns
        ---------
        y_dot : float
            means \dot{x_1}
        """
        y_dot = y_2
        return y_dot
    
    def func_x_2(self, y_1, y_2, u):
        """calculating \dot{x_2}
        Parameters
        ------------
        y_1 : float
            means x_1
        y_2 : float
            means x_2
        u : float
            means system input
        Returns
        ---------
        y_dot : float
            means \dot{x_2}
        """
        y_dot = (1. - y_1**2 - y_2**2) * y_2 - y_1 + u
        return y_dot

    def _adjoint_state_with_oylar(self, x_1, x_2, lam_1, lam_2, u, dt):
        """
        Parameters
        ------------
        x_1 : float
            system state
        x_2 : float
            system state
        lam_1 : float
            adjoint state
        lam_2 : float
            adjoint state
        u : float
            system input
        dt : float in seconds
            sampling time
        Returns
        --------
        pre_lam_1 : float
            pre, 1 step before lam_1 calculated by using adjoint equation
        pre_lam_2 : float
            pre, 1 step before lam_2 calculated by using adjoint equation
        """
        k0 = [0. for _ in range(2)]

        functions = [self._func_lam_1, self._func_lam_2]

        for i, func in enumerate(functions):
            k0[i] = dt * func(x_1, x_2, lam_1, lam_2, u)
                
        pre_lam_1 = lam_1 + k0[0]
        pre_lam_2 = lam_2 + k0[1]

        return pre_lam_1, pre_lam_2

    def _func_lam_1(self, y_1, y_2, y_3, y_4, u):
        """calculating -\dot{lam_1}
        Parameters
        ------------
        y_1 : float
            means x_1
        y_2 : float
            means x_2
        y_3 : float
            means lam_1
        y_4 : float
            means lam_2
        u : float
            means system input
        Returns
        ---------
        y_dot : float
            means -\dot{lam_1}
        """
        y_dot = y_1 - (2. * y_1 * y_2 + 1.) * y_4
        return y_dot

    def _func_lam_2(self, y_1, y_2, y_3, y_4, u):
        """calculating -\dot{lam_2}
        Parameters
        ------------
        y_1 : float
            means x_1
        y_2 : float
            means x_2
        y_3 : float
            means lam_1
        y_4 : float
            means lam_2
        u : float
            means system input
        Returns
        ---------
        y_dot : float
            means -\dot{lam_2}
        """
        y_dot = y_2 + y_3 + (-3. * (y_2**2) - y_1**2 + 1. ) * y_4
        return y_dot

class NMPCController_with_CGMRES():
    """
    Attributes
    ------------
    zeta : float
        gain of optimal answer stability
    ht : float
        update value of NMPC this should be decided by zeta
    tf : float
        predict time
    alpha : float
        gain of predict time
    N : int
        predicte step, discritize value
    threshold : float
        cgmres's threshold value
    input_num : int
        system input length, this should include dummy u and constraint variables
    max_iteration : int
        decide by the solved matrix size
    simulator : NMPCSimulatorSystem class
    us : list of float
        estimated optimal system input
    dummy_us : list of float
        estimated dummy input
    raws : list of float
        estimated constraint variable
    history_u : list of float
        time history of actual system input
    history_dummy_u : list of float
        time history of actual dummy u
    history_raw : list of float
        time history of actual raw
    history_f : list of float
        time history of error of optimal
    """
    def __init__(self):
        """
        Parameters
        -----------
        None
        """
        # parameters
        self.zeta = 100. # 安定化ゲイン
        self.ht = 0.01 # 差分近似の幅
        self.tf = 1. # 最終時間
        self.alpha = 0.5 # 時間の上昇ゲイン
        self.N = 10 # 分割数
        self.threshold = 0.001 # break値

        self.input_num = 3 # dummy, 制約条件に対するuにも合わせた入力の数
        self.max_iteration = self.input_num * self.N

        # simulator
        self.simulator = NMPCSimulatorSystem()

        # initial
        self.us = np.zeros(self.N)
        self.dummy_us = np.ones(self.N) * 0.49
        self.raws = np.ones(self.N) * 0.011

        # for fig
        self.history_u = []
        self.history_dummy_u = []
        self.history_raw = []
        self.history_f = []

    def calc_input(self, x_1, x_2, time):
        """
        Parameters
        ------------
        x_1 : float
            current state
        x_2 : float
            current state
        time : float in seconds
            now time
        Returns
        --------
        us : list of float
            estimated optimal system input
        """
        # calculating sampling time
        dt = self.tf * (1. - np.exp(-self.alpha * time)) / float(self.N)

        # x_dot
        x_1_dot = self.simulator.func_x_1(x_1, x_2, self.us[0])
        x_2_dot = self.simulator.func_x_2(x_1, x_2, self.us[0])

        dx_1 = x_1_dot * self.ht
        dx_2 = x_2_dot * self.ht

        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)
        
        # Fxt
        Fxt = self._calc_f(x_1s, x_2s, lam_1s, lam_2s, self.us, self.dummy_us,
                            self.raws, self.N, dt)

        # F
        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)

        right = -self.zeta * F - ((Fxt - F) / self.ht)

        du = self.us * self.ht
        ddummy_u = self.dummy_us * self.ht
        draw = self.raws * self.ht

        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)

        Fuxt = self._calc_f(x_1s, x_2s, lam_1s, lam_2s, self.us + du, self.dummy_us + ddummy_u,
                           self.raws + draw, self.N, dt)

        left = ((Fuxt - Fxt) / self.ht)

        # calculationg cgmres
        r0 = right - left
        r0_norm = np.linalg.norm(r0)
        
        vs = np.zeros((self.max_iteration, self.max_iteration + 1)) # 数×iterarion回数
        
        vs[:, 0] = r0 / r0_norm # 最初の基底を算出

        hs = np.zeros((self.max_iteration + 1, self.max_iteration + 1))

        e = np.zeros((self.max_iteration + 1, 1)) # in this case the state is 3(u and dummy_u)
        e[0] = 1.

        for i in range(self.max_iteration):
            du = vs[::3, i] * self.ht
            ddummy_u = vs[1::3, i] * self.ht
            draw = vs[2::3, i] * self.ht

            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)

            Fuxt = self._calc_f(x_1s, x_2s, lam_1s, lam_2s, self.us + du, self.dummy_us + ddummy_u,
                           self.raws + draw, self.N, dt)

            Av = (( Fuxt - Fxt) / self.ht)

            sum_Av = np.zeros(self.max_iteration)

            for j in range(i + 1): # グラムシュミットの直交化法です、和を取って差分を取って算出します
                hs[j, i] = np.dot(Av, vs[:, j])
                sum_Av = sum_Av + hs[j, i] * vs[:, j]

            v_est = Av - sum_Av

            hs[i+1, i] = np.linalg.norm(v_est)

            vs[:, i+1] = v_est / hs[i+1, i]

            inv_hs = np.linalg.pinv(hs[:i+1, :i]) # この辺は教科書（実時間の方）にのっています
            ys = np.dot(inv_hs, r0_norm * e[:i+1])

            judge_value = r0_norm * e[:i+1] - np.dot(hs[:i+1, :i], ys[:i])

            if np.linalg.norm(judge_value) < self.threshold or i == self.max_iteration-1:
                update_value = np.dot(vs[:, :i-1], ys_pre[:i-1]).flatten()
                du_new = du + update_value[::3]
                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()