pysimenv 0.0.20

A framework for performing numerical simulation of dynamic systems

python-sim-env

python-sim-env is a framework for performing numerical simulation of dynamic systems based on python. python-sim-env provides several essential components including systems, controllers, and simulators. The motivation of python-sim-env can be summarized as

• To make easier to develop code for defining a new dynamic system, controller and simulation model.
• To make it easier to perform a simulation.
• To enhance re-usability of the already developed code.
• To require less computation time for performing a simulation.
• To provide a easy way to utilize simulation data.

python-sim-env achieves the goal by:

• Clarifying the role of each component.
• Making sure that each component only contains its essential parts.
• Supporting various types of systems and various ways of writing code.

Basic Usage

Example 1: Step response of a linear system

In this example, we will learn the following things.

• Defining a dynamic system.
• Performing a numerical simulation.
• Retrieving the simulation data.

First, let's import modules required for the simulation.

import numpy as np
import matplotlib.pyplot as plt
from pysimenv.core.base import DynSystem
from pysimenv.core.simulator import Simulator

We need a DynSystem object for defining the linear system and a Simulator for performing the simulation.

Consider the standard second-order system defined by a transfer function of $$ G(s)=\frac{\omega_{n}^{2}}{s^{2} + 2\zeta\omega_{n} + \omega_{n}^{2}} $$ where $\omega_{n}$ is the natural frequency and $\zeta$ is the damping ratio. The system can be expressed in state-space equation as $$ \dot{x} = \begin{bmatrix} 0 & 1 \ -\omega_{n}^{2} & -2\zeta\omega_{n} \end{bmatrix} x + \begin{bmatrix} 0 \ \omega_{n}^{2} \end{bmatrix}u $$ where $x=[x_{1},x_{2}]^{T}$ is the state and $u$ is the control input.

Inside the main function, we define a function named as deriv_fun representing the right-hand side of the equation. We use values of $\omega_{n}=1$, $\zeta=0.8$.

    def deriv_fun(x, u):
        omega = 1.
        zeta = 0.8
        A = np.array([[0, 1.], [-omega**2, -2*zeta*omega]])
        B = np.array([[0.], [omega**2]])
        x_dot = A.dot(x) + B.dot(u)
        return {'x': x_dot}

We define a dynamic system object using DynSystem class where the initial state is set as $x(0)=[0,0]^{T}$.

    sys = DynSystem(
        initial_states={'x': np.zeros(2)},
        deriv_fun=deriv_fun
    )

Then, we define a simulator and perform a numerical simulation with the integration interval $dt=0.01$, final time $t_{f}=10$, and unit step input for $u$.

    simulator = Simulator(sys)
    simulator.propagate(dt=0.01, time=10., save_history=True, u=np.array([1.]))

After the simulation, the time history, state history and input history can be retrieved as

    t = sys.history('t')
    x = sys.history('x')
    u = sys.history('u')

where the output of history method is returned in numpy.ndarray. The first axis of each array corresponds to the time index, and the second axis of each array corresponds to the state index. Therefore, we can visualize the simulation data as

    fig, ax = plt.subplots()
    for i in range(2):
        ax.plot(t, x[:, i], label="x_" + str(i + 1))
    ax.set_xlabel("Time (s)")
    ax.set_ylabel("x")
    ax.grid()
    ax.legend()

    fig, ax = plt.subplots()
    ax.plot(t, u)
    ax.set_xlabel("Time (s)")
    ax.set_ylabel("u")
    ax.grid()

    plt.show()

The entire code is written as

import numpy as np
import matplotlib.pyplot as plt
from pysimenv.core.base import DynSystem
from pysimenv.core.simulator import Simulator


def main():
    def deriv_fun(x, u):
        omega = 1.
        zeta = 0.8
        A = np.array([[0, 1.], [-omega**2, -2*zeta*omega]])
        B = np.array([[0.], [omega**2]])
        x_dot = A.dot(x) + B.dot(u)
        return {'x': x_dot}

    sys = DynSystem(
        initial_states={'x': np.zeros(2)},
        deriv_fun=deriv_fun
    )
    simulator = Simulator(sys)
    simulator.propagate(dt=0.01, time=10., save_history=True, u=np.array([1.]))

    t = sys.history('t')
    x = sys.history('x')
    u = sys.history('u')

    fig, ax = plt.subplots()
    for i in range(2):
        ax.plot(t, x[:, i], label="x_" + str(i + 1))
    ax.set_xlabel("Time (s)")
    ax.set_ylabel("x")
    ax.grid()
    ax.legend()

    fig, ax = plt.subplots()
    ax.plot(t, u)
    ax.set_xlabel("Time (s)")
    ax.set_ylabel("u")
    ax.grid()

    plt.show()


if __name__ == "__main__":
    main()

The following figures show the simulation result.

Example 2: LQR(Linear Quadratic Regulator) control of a linear system-Part 1

In this example, we will learn the following things.

• Defining a closed-loop dynamic system.

We begin with importing modules required for the simulation.

import numpy as np
import scipy.linalg as lin
from pysimenv.core.base import DynSystem, SimObject
from pysimenv.core.simulator import Simulator

DynSystem is used for defining a linear system, and SimObject is used for defining the closed-loop system.

Let us define the structure of the closed-loop system as a class inheriting from SimObject.

class ClosedLoopSys(SimObject):
    def __init__(self):
        super(ClosedLoopSys, self).__init__()

The open-loop system is defined as

        # open-loop system
        zeta = 0.1
        omega = 1.
        A = np.array([[0., 1.], [-omega**2, -2*zeta*omega]])
        B = np.array([[0.], [omega**2]])

        self.linear_sys = DynSystem(
            initial_states={'x': [0., 1.]},
            deriv_fun=lambda x, u: {'x': A.dot(x) + B.dot(u)},
            name="linear_sys"
        )

The control gain is calculated as

        # control gain
        Q = np.identity(2)
        R = np.identity(1)
        P = lin.solve_continuous_are(A, B, Q, R)
        self.K = np.linalg.inv(R).dot(B.transpose().dot(P))

Any instance of any subclass of SimObject defined inside any subclass of SimObject must be added to the list of simulation objects at the initialization phase. (The list of subclasses inheriting SimObject can be found in Overview section.) DynSystem class inherits SimObject, which means that self.linear_sys is also an instance of SimObject. Therefore, add self.linear_sys as

        self._add_sim_objs([self.linear_sys])

We define the feedback structure by implementing _forward method.

    def _forward(self):
        x = self.linear_sys.state('x')
        u_lqr = -self.K.dot(x)
        self.linear_sys.forward(u=u_lqr)

Now, we are ready to perform a simulation. The main function looks like:

def main():
    model = ClosedLoopSys()
    simulator = Simulator(model)
    simulator.propagate(dt=0.01, time=10., save_history=True)
    model.linear_sys.default_plot(show=True)

We used default_plot method defined in DynSystem to simply visualize the simulation result.

The entire code is written as

import numpy as np
import scipy.linalg as lin
from pysimenv.core.base import DynSystem, SimObject
from pysimenv.core.simulator import Simulator


class ClosedLoopSys(SimObject):
    def __init__(self):
        super(ClosedLoopSys, self).__init__()

        # open-loop system
        zeta = 0.1
        omega = 1.
        A = np.array([[0., 1.], [-omega ** 2, -2 * zeta * omega]])
        B = np.array([[0.], [omega ** 2]])

        self.linear_sys = DynSystem(
            initial_states={'x': [0., 1.]},
            deriv_fun=lambda x, u: {'x': A.dot(x) + B.dot(u)},
            name="linear_sys"
        )

        # control gain
        Q = np.identity(2)
        R = np.identity(1)
        P = lin.solve_continuous_are(A, B, Q, R)
        self.K = np.linalg.inv(R).dot(B.transpose().dot(P))

        self._add_sim_objs([self.linear_sys])

    # implement
    def _forward(self):
        x = self.linear_sys.state('x')
        u_lqr = -self.K.dot(x)
        self.linear_sys.forward(u=u_lqr)


def main():
    model = ClosedLoopSys()
    simulator = Simulator(model)
    simulator.propagate(dt=0.01, time=10., save_history=True)
    model.linear_sys.default_plot(show=True)


if __name__ == "__main__":
    main()

The following figures show the simulation result.

Example 3: LQR(Linear Quadratic Regulator) control of a linear system-Part 2

In this example, we will learn the following things.

• Defining a controller using StaticObject.
• Modifying the sampling interval of the controller.

We begin with importing modules required for the simulation.

import numpy as np
import scipy.linalg as lin
from pysimenv.core.base import StaticObject, SimObject, DynSystem
from pysimenv.core.simulator import Simulator

We follow the same procedure as in Example 2. The difference is that now we use an object of StaticObject class instead of simply defining a control gain.

        # controller
        Q = np.identity(2)
        R = np.identity(1)
        P = lin.solve_continuous_are(A, B, Q, R)
        K = np.linalg.inv(R).dot(B.transpose().dot(P))

        self.lqr_control = StaticObject(interval=0.05, eval_fun=lambda x: -K.dot(x))

To illustrate the usefulness of StaticObject class, the sampling interval (interval property) is intentionally set as 0.2(seconds) corresponding to sampling frequency of 5(Hz).

DynSystem and StaticObject both inherit SimObject class. Therefore, add both the system object self.linear_sys and controller object self.lqr_control to the list of simulation objects.

        self._add_sim_objs([self.linear_sys, self.lqr_control])

Then, we define the system structure by implementing _forward method.

    # implement
    def _forward(self):
        x = self.linear_sys.state('x')
        u_lqr = self.lqr_control.forward(x=x)
        self.linear_sys.forward(u=u_lqr)

The rest of the code is similar to that of Example 2. The entire code is written as

import numpy as np
import scipy.linalg as lin
from pysimenv.core.base import StaticObject, SimObject, DynSystem
from pysimenv.core.simulator import Simulator


class ClosedLoopSys(SimObject):
    def __init__(self):
        super(ClosedLoopSys, self).__init__()

        # open-loop system
        zeta = 0.1
        omega = 1.
        A = np.array([[0., 1.], [-omega ** 2, -2 * zeta * omega]])
        B = np.array([[0.], [omega ** 2]])

        self.linear_sys = DynSystem(
            initial_states={'x': [0., 1.]},
            deriv_fun=lambda x, u: {'x': A.dot(x) + B.dot(u)},
            name="linear_sys"
        )

        # controller
        Q = np.identity(2)
        R = np.identity(1)
        P = lin.solve_continuous_are(A, B, Q, R)
        K = np.linalg.inv(R).dot(B.transpose().dot(P))

        self.lqr_control = StaticObject(interval=0.2, eval_fun=lambda x: -K.dot(x))

        self._add_sim_objs([self.linear_sys, self.lqr_control])

    # implement
    def _forward(self):
        x = self.linear_sys.state('x')
        u_lqr = self.lqr_control.forward(x=x)
        self.linear_sys.forward(u=u_lqr)


def main():
    model = ClosedLoopSys()
    simulator = Simulator(model)
    simulator.propagate(dt=0.01, time=10., save_history=True)
    model.linear_sys.default_plot(show=True)


if __name__ == "__main__":
    main()

The following figure shows the simulation result.

Example 4: Simulation of a projectile

In this example, we will learn the following things.

• Defining a dynamic system with multiple state variables.
• Defining a stop condition for simulation.

An object that is projected near the ground moves along a curved path under the action of gravity, which is called projectile motion. We examine the effect of air resistance to the projectile through numerical simulation.

The equations of motion can be written as $$ \begin{align*} \dot{p} &= v \ \dot{v} &= a_{\text{grav}} + a_{\text{air}} = \begin{bmatrix} 0 \ -g \end{bmatrix} - \mu \Vert v \Vert v \end{align*} $$ where $p=[p_x, p_{y}]^{T}$ is the position, $v=[v_{x}, v_{y}]^{T}$ is the velocity, $g=9.807,(\rm{m/s^2})$ is the gravitational acceleration, and $\mu$ is a constant related to air resistance. $\mu$ is determined by $$ \mu = \frac{c\rho A}{2m} $$ where $c$ is the drag coefficient, $\rho$ is the air density, $A$ is the cross sectional area of the projectile, and $m$ is the mass of the projectile. A parameter set of $c=0.6$, $\rho=1.204 ,(\rm{kg/m^3})$, $A=0.065 ,(\rm{m^2})$, and $m=0.056, (\rm{kg})$ is used in the simulation, which approximately models the aerodynamic effect on the tennis ball. The resulting value of $\mu$ is $\mu=0.0214, (\rm{m^{-1}})$.

Import DynObject class from pysimenv.core.base.

import numpy as np
import matplotlib.pyplot as plt
from pysimenv.core.base import DynSystem
from pysimenv.core.simulator import Simulator

Define a class called Projectile that inherits DynSystem.

class Projectile(DynSystem):
    def __init__(self, p_0, v_0, mu=0.):
        super(Projectile, self).__init__(initial_states={'p': p_0, 'v': v_0})
        self.mu = mu

Projectile class is defined to have two state variables with keys p and v, and no input variables. The equations of motion is translated into _deriv method as

    # implement
    def _deriv(self, p, v):
        p_dot = v.copy()
        v_dot = np.array([0., -9.807]) - self.mu*np.linalg.norm(v)*v
        return {'p': p_dot, 'v': v_dot}

We want to finish the simulation when an instance of Projectile reaches the ground ($p_y < 0$). Therefore, implement _stop_condition method as

    # implement
    def _check_stop_condition(self):
        p_y = self.state('p')[1]
        return p_y < 0.

Define a simulator and perform the numerical simulation in the main function. We perform the numerical simulation twice, one without considering any effect of the drag ($\mu=0$) and one with considering the effect of the drag ($\mu=0.0214$).

def main():
    mu_list = {'no_drag': 0., 'air': 0.0214}
    data_list = dict()

    theta_0 = np.deg2rad(30.)
    p_0 = np.zeros(2)
    v_0 = np.array([20.*np.cos(theta_0), 20.*np.sin(theta_0)])
    for key, mu in mu_list.items():
        projectile = Projectile(p_0=p_0, v_0=v_0, mu=mu)
        simulator = Simulator(projectile)
        simulator.propagate(dt=0.01, time=10., save_history=True)

        data = projectile.history('t', 'p', 'v')
        data_list[key] = data

After the simulation, the history of the time, position, and velocity is obtained by calling history method and passing the corresponding keys as arguments. Note that history method returns the data as a dictionary variable when more than one argument are passed.

The entire code is written as

import numpy as np
import matplotlib.pyplot as plt
from pysimenv.core.base import DynSystem
from pysimenv.core.simulator import Simulator


class Projectile(DynSystem):
    def __init__(self, p_0, v_0, mu=0.):
        super(Projectile, self).__init__(initial_states={'p': p_0, 'v': v_0})
        self.mu = mu

    # implement
    def _deriv(self, p, v):
        p_dot = v.copy()
        v_dot = np.array([0., -9.807]) - self.mu*np.linalg.norm(v)*v
        return {'p': p_dot, 'v': v_dot}

    # implement
    def _check_stop_condition(self):
        p_y = self.state('p')[1]
        return p_y < 0.


def main():
    mu_list = {'no_drag': 0., 'air': 0.0214}
    data_list = dict()

    theta_0 = np.deg2rad(30.)
    p_0 = np.zeros(2)
    v_0 = np.array([20.*np.cos(theta_0), 20.*np.sin(theta_0)])
    for key, mu in mu_list.items():
        projectile = Projectile(p_0=p_0, v_0=v_0, mu=mu)
        simulator = Simulator(projectile)
        simulator.propagate(dt=0.01, time=10., save_history=True)

        data = projectile.history('t', 'p', 'v')
        data_list[key] = data

    color = {'no_drag': 'k', 'air': 'b'}
    fig, ax = plt.subplots(3, 1)
    for key in data_list.keys():
        data = data_list[key]
        ax[0].plot(data['p'][:, 0], data['p'][:, 1], color=color[key], label=key)
        ax[1].plot(data['t'], data['v'][:, 0], color=color[key], label=key)
        ax[2].plot(data['t'], data['v'][:, 1], color=color[key], label=key)

    ax[0].set_xlabel("Distance (m)")
    ax[0].set_ylabel("Height (m)")
    ax[0].set_title("Trajectory")
    ax[0].grid()
    ax[0].legend()

    ax[1].set_xlabel("Time (s)")
    ax[1].set_ylabel("v_x (m/s)")
    ax[1].set_title("Horizontal velocity")
    ax[1].grid()
    ax[1].legend()

    ax[2].set_xlabel("Time (s)")
    ax[2].set_ylabel("v_y (m/s)")
    ax[2].set_title("Vertical velocity")
    ax[2].grid()
    ax[2].legend()
    fig.tight_layout()
    plt.show()


if __name__ == "__main__":
    main()

The following figures show the simulation result.

Example 5: Cruise control of a vehicle using PI control

In this example, we will learn the following things.

• Defining a controller with internal state.
• Logging history of a variable using a Logger object.

The contents of this example is based on K. J. Astrom, 2002, Control System Design Lecture Notes for ME 155A, lecture notes, Simple Controllers, University of California.

The purpose of the cruise control is to keep the speed of a car constant. We will define a model for simulating the cruise control of a car moving on the road. The dynamic model is modeled based on a momentum balance. Let $v$ denote the velocity (speed) of the car, $F$ the force generated by the engine, and $\theta$ the slope of the road. The momentum balance can be written as $$ m\frac{dv}{dt}+cv = F - mg\theta $$ where $m$ is the mass of the car, $c$ is the coefficient for air resistance, and $g$ is the gravitational acceleration. The equation can be rewritten as $$ \frac{dv}{dt} = -\frac{c}{m}v + \frac{F}{m} - g\theta \ $$ Normalize each parameter and variable using the mass as $$ \frac{dv}{dt} = -\hat{c}v + u - g\theta $$ Values of parameters are $\hat{c}=0.02$, $g=9.81$. The control signal $u$ is limited to be in the interval $0 \le u \le 1$ and the slope of the road is assumed to be less than 10%, which corresponds to $\theta=0.1$.

The following PI control law will be implemented $$ u = k_{p}e + k_{i} \int_{0}^{t}e(\tau) ,d\tau $$ where $e=v_{r} - v$ is the velocity error between the commanded velocity $v_{r}$ and actual velocity $v$.

Import SimObject class from pysimenv.core.base.

import numpy as np
import matplotlib.pyplot as plt
from pysimenv.core.base import SimObject, DynSystem
from pysimenv.core.simulator import Simulator

Define a class PIController that inherits from SimObject class. PIController class is defined to have a state named as e_i because we need to keep track the integral value of the error. The initial integral value is set as zero.

class PIController(SimObject):
    def __init__(self, k_p, k_i):
        super(PIController, self).__init__()
        self._add_state_vars(e_i=np.array([0.]))
        self.k_p = k_p
        self.k_i = k_i

The _forward method of the class is written as

    # implement
    def _forward(self, e):
        self.state_vars['e_i'].set_deriv(deriv=e)

        e_i = self.state('e_i')
        u_pi = self.k_p*e + self.k_i*e_i
        return u_pi

Note that set_deriv method of the state variable should be called at each time when _forward method is called as in the following to properly update the integral value.

        self.state_vars['e_i'].set_deriv(deriv=e)

Now define the closed-loop system. The closed-loop system includes the open-loop dynamic model of the car and the PI controller.

class CCCar(SimObject):
    """
    Cruise-controlled car
    """
    def __init__(self, k_p, k_i, v_0):
        super(CCCar, self).__init__()
        # mass-normalized parameters
        c = 0.02
        g = 9.81

        def deriv_fun(v, u, theta):
            v_dot = -c*v + u - g*theta
            return {'v': v_dot}

        self.vel_dyn = DynSystem(
            initial_states={'v': v_0},
            deriv_fun=deriv_fun
        )

        # PI Controller
        self.pi_control = PIController(k_p=k_p, k_i=k_i)

        self._add_sim_objs([self.vel_dyn, self.pi_control])

Implement _forward method of the closed-loop system.

    # implement
    def _forward(self, v_r, theta):
        # tracking error
        v = self.vel_dyn.state('v')
        e = v_r - v

        # PI control input
        u_pi = self.pi_control.forward(e=e)
        u = np.clip(u_pi, 0., 1.)

        # update dynamic
        self.vel_dyn.forward(u=u, theta=theta)

        # log velocity error and control signal
        self._logger.append(t=self.time, e=e, u=u)

The reference speed v_r and the slope of the road theta are passed from the outside of the system. The PI control input u_pi is clipped to have a value between 0 and 1. A property named as _logger is used for logging the history of the time, velocity error, and control signal as in the following.

        # log velocity error and control signal
        self._logger.append(t=self.time, e=e, u=u)

_logger property is a Logger object defined in every instance of SimObject class. The history of any variable can be logged just by calling append method and passing the corresponding key-value as an argument.

To determine the appropriate value for k_p and k_i, derive the equation for the error dynamics. Assuming that $v_{r}$ is constant and using the relation $e=v_{r}-v$, the error dynamics can be expressed as $$ \dot{e} + \left(\hat{c} + k_{p} \right)e + k_{i} \int_{0}^{t}e(\tau),d\tau=g\theta + \hat{c}v_{r} $$ The characteristic polynomial of the closed-loop system is $$ s^{2} + \left(\hat{c} + k_{p} \right)s + k_{i} = 0 $$ Specify the desired response of the closed-loop system by choosing the value of natural frequency $\omega_{0}$ and damping ratio $\zeta$. Then, choose values of $k_{p}$ and $k_{i}$ as $$ k_{p} = 2\zeta\omega_{0} - \hat{c},, k_{i} = \omega_{0}^{2} $$ We are ready to code the main function. The main function simulates the controller for various values of $\omega_{0}=0.05, 0.1, 0.2$ with $\zeta=1$ and compare the results. The initial velocity and reference velocity of the car is set as 5(m/s). The slope of the road is assumed to suddenly change by 4% from a flat surface. The following code is portion of the main function.

def main():
    zeta = 1.
    omega_0_list = [0.05, 0.1, 0.2]
    v_r = 5.
    theta = 0.04  # 4% slope

    data_list = []

    for omega_0 in omega_0_list:
        k_p = 2*zeta*omega_0 - 0.02
        k_i = omega_0**2

        car = CCCar(k_p=k_p, k_i=k_i, v_0=v_r)
        simulator = Simulator(car)
        simulator.propagate(dt=0.01, time=100, save_history=True, v_r=v_r, theta=theta)

        data = car.history('t', 'e', 'u')  # returns dictionary when the number of variable is greater than 1
        data_list.append(data)

After each simulation, the history of the time, velocity error, and control signal is obtained by calling history method and passing the corresponding keys as arguments. history method returns a dictionary variable when the number of arguments is greater than 1.

        data = car.history('t', 'e', 'u')  # returns dictionary when the number of variable is greater than 1

The entire code is written as

import numpy as np
import matplotlib.pyplot as plt
from pysimenv.core.base import SimObject, DynSystem
from pysimenv.core.simulator import Simulator


class PIController(SimObject):
    def __init__(self, k_p, k_i):
        super(PIController, self).__init__()
        self._add_state_vars(e_i=np.array([0.]))
        self.k_p = k_p
        self.k_i = k_i

    # implement
    def _forward(self, e):
        self.state_vars['e_i'].set_deriv(deriv=e)

        e_i = self.state('e_i')
        u_pi = self.k_p * e + self.k_i * e_i
        return u_pi


class CCCar(SimObject):
    """
    Cruise-controlled car
    """

    def __init__(self, k_p, k_i, v_0):
        super(CCCar, self).__init__()
        # mass-normalized parameters
        c = 0.02
        g = 9.81

        def deriv_fun(v, u, theta):
            v_dot = -c * v + u - g * theta
            return {'v': v_dot}

        self.vel_dyn = DynSystem(
            initial_states={'v': v_0},
            deriv_fun=deriv_fun
        )

        # PI Controller
        self.pi_control = PIController(k_p=k_p, k_i=k_i)

        self._add_sim_objs([self.vel_dyn, self.pi_control])

    # implement
    def _forward(self, v_r, theta):
        # tracking error
        v = self.vel_dyn.state('v')
        e = v_r - v

        # PI control input
        u_pi = self.pi_control.forward(e=e)
        u = np.clip(u_pi, 0., 1.)

        # update dynamic
        self.vel_dyn.forward(u=u, theta=theta)

        # log velocity error and control signal
        self._logger.append(t=self.time, e=e, u=u)


def main():
    zeta = 1.
    omega_0_list = [0.05, 0.1, 0.2]
    v_r = 5.
    theta = 0.04  # 4% slope

    data_list = []

    for omega_0 in omega_0_list:
        k_p = 2 * zeta * omega_0 - 0.02
        k_i = omega_0 ** 2

        car = CCCar(k_p=k_p, k_i=k_i, v_0=v_r)
        simulator = Simulator(car)
        simulator.propagate(dt=0.01, time=100, save_history=True, v_r=v_r, theta=theta)

        data = car.history('t', 'e', 'u')  # returns dictionary when the number of variable is greater than 1
        data_list.append(data)

    fig, ax = plt.subplots(2, 1)
    lines = [':', '-', '--']
    for i in range(3):
        data = data_list[i]
        ax[0].plot(data['t'], data['e'], color='b', linestyle=lines[i])
        ax[1].plot(data['t'], data['u'], color='b', linestyle=lines[i])

    ax[0].set_xlabel("Time")
    ax[0].set_ylabel("Velocity error")
    ax[1].set_xlabel("Time")
    ax[1].set_ylabel("Control signal")
    ax[0].set_xticks(np.linspace(0., 100., 11))
    ax[0].set_yticks(np.linspace(0., 4., 5))
    ax[1].set_xticks(np.linspace(0., 100., 11))
    ax[1].set_yticks(np.linspace(0., 0.6, 7))
    fig.tight_layout()
    plt.show()


if __name__ == "__main__":
    main()

The following figures show the simulation result.

The dotted line is for the case where $\omega_{0}=0.05$, the full line is for $\omega_{0}=0.1$, and the dashed line is for $\omega_{0}=0.2$.

Overview

Important classes and methods

Main components for modeling dynamic systems are summarized in the following diagram. Only the essential attributes and methods are listed in the figure. Mostly used attributes and methods when modeling dynamic systems are expressed in bold font.

System classes inheriting SimObject are StaticObject, DynSystem, TimeVaryingDynSystem.