Communication Satellite (ComSat) — Longitudinal Dynamics¶

A communications satellite operates in orbit to relay and process radio signals. This page mirrors the ELV layout: quick start, math model, derivative tables, and API.

Quick start

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Model API

Python class documentation for ComSat.

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Theory

State equations and numerical parameters.

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Control object structure¶

The model is defined in the state space:

\[\dot{x} = A x + B u, \quad y = C x + D u\]

where:

\[ x = \begin{bmatrix} x_1 \\ x_3 \\ x_4 \end{bmatrix} = \begin{bmatrix} \rho \\ \dot{\rho} \\ \dot{\theta} \end{bmatrix}, \quad u = u_2 \]

The linearized system:

\[ \begin{bmatrix} \dot{x}_1 \\ \dot{x}_3 \\ \dot{x}_4 \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 \\ 0.01036 & 0 & 0.7753 \\ 0 & -0.01775 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_3 \\ x_4 \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ 0.1513 \end{bmatrix} u_2 \]

State VariablesControl InputSystem Coefficients

x₁ = ρ: radial position - distance from Earth center, km
x₃ = ρ̇: radial velocity, m/s
x₄ = θ̇: angular velocity, rad/s

u₂: tangential thrust, N
- u₂ > 0 — thrust in direction of motion (acceleration)
- u₂ < 0 — thrust against direction of motion (deceleration)
- u₂ = 0 — no thrust

a₁₃ = 1.0 — radial position changes with radial velocity
a₃₁ = 0.01036 — radial acceleration component from position
a₃₄ = 0.7753 — radial acceleration component from angular velocity
a₄₃ = -0.01775 — angular acceleration component from radial velocity
b₄ = 0.1513 — tangential thrust influence on angular acceleration

Units

Angular rates are in radians. Position in km, velocity in m/s. API methods can convert units.

Mathematical model¶

\[ \dot{x} = A x + B u, \qquad y = C x + D u \]

Numerical matrices (linearized system):

\[ \begin{bmatrix} \dot{x}_1 \\ \dot{x}_3 \\ \dot{x}_4 \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 \\ 0.01036 & 0 & 0.7753 \\ 0 & -0.01775 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_3 \\ x_4 \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ 0.1513 \end{bmatrix} u_2 \]

Expanded form: [ \begin{aligned} \dot{x}_1 &= x_3 \ \dot{x}_3 &= 0.01036 \cdot x_1 + 0.7753 \cdot x_4 \ \dot{x}_4 &= -0.01775 \cdot x_3 + 0.1513 \cdot u_2 \end{aligned} ]

Derivatives (numerical values)¶

Matrix A (state derivatives):

Coefficient	Value	Physical Meaning
a₁₃ (∂ẋ₁/∂x₃)	1.0	Radial position rate = radial velocity
a₃₁ (∂ẋ₃/∂x₁)	0.01036	Position effect on radial acceleration
a₃₄ (∂ẋ₃/∂x₄)	0.7753	Angular velocity effect on radial acceleration
a₄₃ (∂ẋ₄/∂x₃)	-0.01775	Radial velocity effect on angular acceleration

Matrix B (control input):

Coefficient	Value	Physical Meaning
b₄ (∂ẋ₄/∂u₂)	0.1513	Tangential thrust effect on angular acceleration

Actuator limits

Default control limits inside the model (normalized):

Maximum magnitude: \(\pm 25^\circ\)
Maximum rate: \(60^\circ/\text{s\)

Internal computations use radians; limits are converted accordingly.

Sources¶

Santosh Kumar Choudhary (2015). Design and Analysis of an Optimal Orbit Control for a Communication Satellite. INTERNATIONAL JOURNAL OF COMMUNICATIONS. Volume 9, 2015

Reward¶

The default reward function returns the negative absolute tracking error for the radial velocity:

\[r_t = -|\dot{\rho}(t) - \dot{\rho}_{\text{ref}}(t)|\]

Higher reward (closer to 0) indicates better tracking performance. A custom reward function can be passed via the reward_func parameter.

Quick start¶

GymnasiumModel only

import gymnasium as gym
import numpy as np

from tensoraerospace.envs import ComSatEnv
from tensoraerospace.utils import generate_time_period
from tensoraerospace.signals.standard import unit_step

dt = 0.01
tp = generate_time_period(tn=20, dt=dt)
number_time_steps = len(tp)
# Reference signal for angular velocity control
reference_signals = unit_step(degree=0.1, tp=tp, time_step=10, output_rad=True).reshape(1, -1)

env = gym.make(
    'ComSatEnv-v0',
    number_time_steps=number_time_steps,
    initial_state=[[6371.0], [0.0], [0.001]],  # [rho (km), rho_dot (m/s), theta_dot (rad/s)]
    reference_signal=reference_signals,
)
state, info = env.reset()
for _ in range(200):
    action = np.array([[0.1]])  # Tangential thrust u2
    state, reward, terminated, truncated, info = env.step(action)
    if terminated or truncated:
        break

import numpy as np
from tensoraerospace.aerospacemodel import ComSat

dt = 0.01
number_time_steps = 200

# Initial state: [rho (km), rho_dot (m/s), theta_dot (rad/s)]
x0 = np.array([6371.0, 0.0, 0.001])

model = ComSat(
    x0=x0,
    number_time_steps=number_time_steps,
    selected_state_output=["rho", "rho_dot", "theta_dot"],
    dt=dt,
)

for t in range(number_time_steps - 1):
    u = np.array([[0.05]])  # Tangential thrust u2
    x_next = model.run_step(u)

# Get state history
rho_history = model.get_state('rho')
rho_dot_history = model.get_state('rho_dot')
theta_dot_history = model.get_state('theta_dot')

Python API¶

ModelGymnasium environment

`ComSat(x0, number_time_steps, selected_state_output=None, t0=0, dt=0.01)` ¶

Bases: ModelBase

Communication satellite in longitudinal control channel.

Parameters:

Name	Type	Description	Default
`x0`	`ndarray \| list[float]`	Initial state of the control object.	required
`number_time_steps`	`int`	Number of time steps.	required
`selected_state_output`	`optional`	Selected states of the control object. Defaults to None.	`None`
`t0`	`int`	Initial time. Defaults to 0.	`0`
`dt`	`float`	Discretization frequency. Defaults to 0.01.	`0.01`

Action space

u2: tangential thrust (N) - positive accelerates satellite, negative decelerates

State space

rho: radial position - distance from Earth center [km] rho_dot: radial velocity [m/s] theta_dot: angular velocity [rad/s]

Output space

rho: radial position - distance from Earth center [km] rho_dot: radial velocity [m/s] theta_dot: angular velocity [rad/s]

Initialize ComSat instance.

Parameters:

Name	Type	Description	Default
`x0`	`ndarray \| list[float]`	Initial state of the control object.	required
`number_time_steps`	`int`	Number of time steps.	required
`selected_state_output`	`list[str] \| None`	Selected states of the control object. Defaults to None.	`None`
`t0`	`float`	Initial time. Defaults to 0.	`0`
`dt`	`float`	Discretization frequency. Defaults to 0.01.	`0.01`

`import_linear_system()` ¶

Load (set) stored linearized system matrices.

State vector: x = [x₁, x₃, x₄]ᵀ = [rho, rho_dot, theta_dot]ᵀ Control: u = u₂ (tangential thrust)

Equations: ẋ₁ = x₃ (radial position rate = radial velocity) ẋ₃ = 0.01036·x₁ + 0.7753·x₄ (radial acceleration) ẋ₄ = -0.01775·x₃ + 0.1513·u₂ (angular acceleration)

`initialise_system(x0, number_time_steps)` ¶

Initialize the system and allocate history buffers.

Parameters:

Name	Type	Description	Default
`x0`	`ndarray \| list[float]`	Initial state.	required
`number_time_steps`	`int`	Number of simulation steps.	required

`run_step(ut_0)` ¶

Run one discrete-time simulation step.

Parameters:

Name	Type	Description	Default
`ut_0`	`ndarray`	Control vector.	required

Returns:

Type	Description
`ndarray`	np.ndarray: Next state at time t+1.

`update_system_attributes()` ¶

Update time-dependent attributes after each simulation step.

`get_state(state_name, to_deg=False, to_rad=False)` ¶

Return the time history of a state.

Parameters:

Name	Type	Description	Default
`state_name`	`str`	State name (e.g., `rho`, `rho_dot`, `theta_dot`).	required
`to_deg`	`bool`	Convert radians to degrees.	`False`
`to_rad`	`bool`	Convert degrees to radians.	`False`

Returns:

Type	Description
`ndarray`	np.ndarray: State history array.

`get_control(control_name, to_deg=False, to_rad=False)` ¶

Return the time history of a control input.

Parameters:

Name	Type	Description	Default
`control_name`	`str`	Control name (e.g., `u2`).	required
`to_deg`	`bool`	Convert radians to degrees.	`False`
`to_rad`	`bool`	Convert degrees to radians.	`False`

Returns:

Type	Description
`ndarray`	np.ndarray: Control history array.

`get_output(state_name, to_deg=False, to_rad=False)` ¶

Return the time history of an output signal.

Parameters:

Name	Type	Description	Default
`state_name`	`str`	Output name.	required
`to_deg`	`bool`	Convert radians to degrees.	`False`
`to_rad`	`bool`	Convert degrees to radians.	`False`

Returns:

Type	Description
`ndarray`	np.ndarray: Output history array.

`plot_output(output_name, time, lang='rus', to_deg=False, to_rad=False, figsize=(10, 10))` ¶

Plot an output signal over time.

Parameters:

Name	Type	Description	Default
`output_name`	`str`	Output name.	required
`time`	`ndarray`	Time vector.	required
`lang`	`str`	Axis label language ('rus' or 'eng'). Defaults to 'rus'.	`'rus'`
`to_deg`	`bool`	Convert radians to degrees.	`False`
`to_rad`	`bool`	Convert degrees to radians.	`False`
`figsize`	`tuple`	Figure size.	`(10, 10)`

Returns:

Type	Description
`Figure`	matplotlib.figure.Figure: Figure object.

`ComSatEnv(initial_state, reference_signal, number_time_steps, tracking_states=None, state_space=None, control_space=None, output_space=None, reward_func=None, render_mode=None)` ¶

Bases: Env

Gymnasium environment for a communication satellite longitudinal model.

Parameters:

Name	Type	Description	Default
`initial_state`	`ndarray \| list[float]`	Initial state.	required
`reference_signal`	`ndarray \| Callable`	Reference signal.	required
`number_time_steps`	`int`	Number of simulation steps.	required
`tracking_states`	`list[str] \| None`	Tracked states.	`None`
`state_space`	`list[str] \| None`	State space.	`None`
`control_space`	`list[str] \| None`	Control space.	`None`
`output_space`	`list[str] \| None`	Full output space (including noise).	`None`
`reward_func`	`Callable \| None`	Reward function (WIP status).	`None`

Initialize communication satellite environment.

`reward(state, ref_signal, ts)` `staticmethod` ¶

Compute tracking reward (negative absolute error).

`step(action)` ¶

Run one environment step (Gymnasium API).

`reset(seed=None, options=None)` ¶

Reset environment to the initial state (Gymnasium API).

`render(mode=None)` ¶

Render a lightweight telemetry snapshot.

The legacy ComSat environment does not ship a graphical viewer. Human mode prints one concise state line; ansi returns it as a string for tests and logging.

Communication Satellite (ComSat) — Longitudinal Dynamics¶

Control object structure¶

Mathematical model¶

Derivatives (numerical values)¶

Sources¶

Reward¶

Quick start¶

Python API¶

ComSat(x0, number_time_steps, selected_state_output=None, t0=0, dt=0.01) ¶

import_linear_system() ¶

initialise_system(x0, number_time_steps) ¶

run_step(ut_0) ¶

update_system_attributes() ¶

get_state(state_name, to_deg=False, to_rad=False) ¶

get_control(control_name, to_deg=False, to_rad=False) ¶

get_output(state_name, to_deg=False, to_rad=False) ¶

plot_output(output_name, time, lang='rus', to_deg=False, to_rad=False, figsize=(10, 10)) ¶

ComSatEnv(initial_state, reference_signal, number_time_steps, tracking_states=None, state_space=None, control_space=None, output_space=None, reward_func=None, render_mode=None) ¶

reward(state, ref_signal, ts) staticmethod ¶

step(action) ¶

reset(seed=None, options=None) ¶

render(mode=None) ¶

`ComSat(x0, number_time_steps, selected_state_output=None, t0=0, dt=0.01)` ¶

`import_linear_system()` ¶

`initialise_system(x0, number_time_steps)` ¶

`run_step(ut_0)` ¶

`update_system_attributes()` ¶

`get_state(state_name, to_deg=False, to_rad=False)` ¶

`get_control(control_name, to_deg=False, to_rad=False)` ¶

`get_output(state_name, to_deg=False, to_rad=False)` ¶

`plot_output(output_name, time, lang='rus', to_deg=False, to_rad=False, figsize=(10, 10))` ¶

`ComSatEnv(initial_state, reference_signal, number_time_steps, tracking_states=None, state_space=None, control_space=None, output_space=None, reward_func=None, render_mode=None)` ¶

`reward(state, ref_signal, ts)` `staticmethod` ¶

`step(action)` ¶

`reset(seed=None, options=None)` ¶

`render(mode=None)` ¶