Question

This is a continuation of Subsection 2.4.2. Consider a satellite of unit mass in earth orbit specified by its position and velocity in polar coordinates \(r, \dot{r}, \theta, \dot{\theta} .\) The input functions are a radial thrust \(u_{1}(t)\) and a tangential thrust of \(u_{2}(t) .\) Newton's laws yield $$ \vec{r}=r \dot{\theta}^{2}-\frac{g}{r^{2}}+u_{1} ; \quad \ddot{\theta}=-\frac{2 \dot{\theta} \dot{r}}{r}+\frac{1}{r} u_{2} . $$ (Compare (2.6) and take \(m_{\mathrm{s}}=1\) and rewrite \(G m_{\mathrm{e}}\) as \(g .\) ) Show that, if \(u_{1}(t)=\) \(u_{2}(t)=0, r(t)=\sigma\) (constant), \(\theta(t)=\omega t\) ( \(\omega\) is constant) with \(\sigma^{3} \omega^{2}=g\) is a solution and that linearization around this solution leads to (with \(x_{1}=r(t)-\) \(\left.\sigma ; x_{2}=\dot{r} ; x_{3}=\sigma(\theta-\omega t) ; x_{4}=\sigma(\dot{\theta}-\omega)\right)\) $$ \frac{d x}{d t}=\left(\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 3 \omega^{2} & 0 & 0 & 2 \omega \\ 0 & 0 & 0 & 1 \\ 0 & -2 \omega & 0 & 0 \end{array}\right) x+\left(\begin{array}{ll} 0 & 0 \\ 1 & 0 \\ 0 & 0 \\ 0 & 1 \end{array}\right) u $$

Short answer

The linearized system is a 4-dimensional state system with two input functions.

Step-by-step solution

Understanding the Initial Condition

The problem provides that when the thrusts are zero, the position and velocity function solutions are constant: \( r(t) = \sigma \) (a constant radius) and \( \theta(t) = \omega t \) (a constant angular velocity). These indicate a stable circular orbit, fulfilling the condition \( \sigma^3 \omega^2 = g \). This balances the centripetal force and gravitational pull.

Verifying the Solution for Zero Thrust

Substitute \( r(t) = \sigma \) and \( \theta(t) = \omega t \) into the given equations:1. \( r\ddot{\theta} = \sigma \omega^2 \) and \( r\dot{\theta}^2 = \sigma \omega^2 \), which satisfy the conditions when thrusts are zero.2. From \( \sigma^3 \omega^2 = g \), this balance satisfies \( r \, \dot{\theta}^2 = \frac{g}{r^2} \) and \( \ddot{\theta} = -\frac{2 \dot{r} \dot{\theta}}{r} \) inherently, due to definitions of \( \dot{r} = 0 \).

Setting Up the Linearization

Given small perturbations around these constant values, define the deviations:- \( x_1 = r(t) - \sigma \)- \( x_2 = \dot{r} \)- \( x_3 = \sigma (\theta - \omega t) \)- \( x_4 = \sigma (\dot{\theta} - \omega) \)

Computing the Linearized Equations

Derive \( \frac{dx}{dt} \) in terms of these deviations' definitions:- Differentiate \( x_1, x_2, x_3, x_4 \) with respect to time using chain rules and substitute into the equations for \( \vec{r} \) and \( \ddot{\theta} \), set \( u_1 = 0 \) and \( u_2 = 0 \).- Find the Jacobian around the steady state of the 2nd-order non-linear system.

Expressing the Linearized System

The resulting system in state space form becomes:\[\frac{d}{dt}\begin{pmatrix} x_1 \ x_2 \ x_3 \ x_4 \end{pmatrix} = \begin{pmatrix} 0 & 1 & 0 & 0 \ 3 \omega^2 & 0 & 0 & 2 \omega \ 0 & 0 & 0 & 1 \ 0 & -2 \omega & 0 & 0 \end{pmatrix} \begin{pmatrix} x_1 \ x_2 \ x_3 \ x_4 \end{pmatrix} + \begin{pmatrix} 0 & 0 \ 1 & 0 \ 0 & 0 \ 0 & 1 \end{pmatrix} \begin{pmatrix} u_1 \ u_2 \end{pmatrix}\]

Key concepts

Orbital Mechanics

Orbital mechanics refers to the study of the motion of objects, like a satellite, as it travels around another body, like the Earth. In this exercise, we are considering the simple case of a satellite in a stable circular orbit. This means the satellite's velocity and position remain constant over time, creating a perfect balance of forces.- **Centripetal Force and Gravity**: The centripetal force required to keep the satellite in its path matches the gravitational pull exerted by the Earth. This balance can be shown as \[ \sigma^3 \omega^2 = g \] where \(\sigma\) is the orbit's radius, \(\omega\) is the angular velocity, and \(g\) represents Earth's gravitational pull.Understanding this balance helps in developing control strategies where minimal thrust only adjusts the orbit slightly, instead of significant maneuvers.

Polar Coordinates

Polar coordinates are a way to represent a point in the plane using a distance and an angle. Unlike Cartesian coordinates that use x and y values, polar coordinates use r (distance from the origin) and \(\theta\) (angle from a reference direction). This system is particularly useful in problems involving circular or orbital motion, such as our satellite orbiting the Earth.In this exercise, the satellite's position is described through:

• \(r(t) = \sigma\): the radius (distance from the center of the earth)
• \(\theta(t) = \omega t\): the angle increasing over time, depicting rotational motion

By using polar coordinates, it becomes straightforward to manipulate equations related to rotational dynamics, making calculations simpler when dealing with orbits and rotation.

State Space Representation

State space representation is a mathematical modeling method to represent all the possible states of a system. It uses state variables to describe the system's behavior in terms of a set of possibly differential equations, especially useful in control systems.In our exercise, we linearize the satellite's movement around a balanced point using state space representation, which simplifies the analysis of complex or nonlinear systems:

• Define states: \(x_1, x_2, x_3, x_4\) representing small deviations in position and velocity.
• State equations: Represent these deviations in terms of matrix equations, helping predict future states with or without inputs.
• Control inputs: Allow for designing strategies to change the system's state, influencing the satellite's orbit with minimal thrusts.

This approach breaks down the orbital problem into manageable parts that can be adjusted independently creating efficient control over the satellite's orbit.

Newton's Laws of Motion

Newton's Laws of Motion provide the foundation for understanding dynamics in control systems, especially in orbital mechanics.
- **First Law** states that an object will continue in its state of motion unless acted upon by an external force. For a stable orbit, no extra force is needed beyond gravity.- **Second Law**, expressed as \( F = ma \), relates to our exercise as it dictates the change in motion of the satellite due to the forces involved.- **Third Law** reminds us that for every action, there is an equal and opposite reaction, influencing how our thrusts affect the satellite's path.By applying these laws, we can derive how the satellite should behave when different forces act upon it, and determine the necessary thrust directions and magnitudes to maintain or alter its orbit effectively.