Rotational control of a satellite

1. Erik_at_DTU

28
Differential equations!

:rofl: Hey,

We're working on a mathematic project, which is about rotational control of a satellite. It's in some ways connected to physics,because we're supposed to analyse the dynamics of a satellite. First a one-axis, then a three axis model.

ONE-AXIS MODEL - problem formulation
A simplified model is given by:
$$I \dot{ \omega } (t) = N_{ctrl} + N_{ext} \qquad (1)$$

where $$\omega(t)$$ is the angular speed as a function of time, $$\dot{\omega}(t)$$ is the derived function. $$N_{ctrl}$$ is the control torque generated by a momentum wheel (reaction wheel), and $$N_{ext}$$ is the disturbance. $$I$$ is the inertiemomentum for the satellite. Finally we set the angular position to be $$g(t)$$, so we have:
$$\dot{g}(t)=\omega(t) \qquad (2)$$

The disturbance $$N_{ext}$$ on the satellite is a torque caused by air resistance. It is modelled by the following expression:
$$N_{ext}=N_0(1+\sin^2(2 \pi \frac {t}{T_0})) \qquad (3)$$

Where $$T_0$$ is the periode and $$N_0$$ is a constant. This can be rewritten by knowing:
$$\sin^2(2 \pi \frac{t}{T_0})=\frac{1}{2} (1-\cos(4 \pi \frac{t}{T_0})) \qquad (4)$$

So $$N_{ext}$$ is a constant influence superimposed with a periodical influence with the periode $$2 T_0$$

The reaction wheels influence on the system is described by a constant $$N_{nom}$$ multiplied with the control signal u :
$$N_{ctrl}=N_{nom}u \qquad (5)$$

We have the following data:

$$I=14.33$$ $$kgm^2$$
$$T_0=90$$ $$min = 5400$$ s
$$N_0=2 \cdot 10^{-4}$$ $$Nm$$
$$N_{nom}=0.010$$ $$Nm$$

Now we are ready to start on the tasks. There will be around 10 tasks, but we won't ask any questions before running into problems, so some of the tasks (like the first one) will just be uploaded with a solution. So if we make any mistakes, please make a point of it :)

Find the complete solution to the homogene differential equation $$I \cdot \ddot{g}(t)=N_{ctrl}+N_{ext}$$, when $$N_{ctrl}=0$$ and $$N_{ext}=0$$. Then find solutions to the following initial conditions:

$$1: \qquad \omega(0)=0, \qquad g(0)= \frac{1}{2}$$
$$2: \qquad \omega(0)=0.01, \qquad g(0)= \frac{1}{2}$$

SOLUTION
Complete solution: $$g(t)=C_1 \cdot t+C_2$$, where $$C_1$$, $$C2$$ are constants.

$$1: \qquad \frac{1}{2}$$
$$2:\qquad \frac{1}{100} \cdot t+\frac{1}{2}$$

The satellite can be regulated with an regulator, given by:
$$u=k_1 \cdot \omega(t)+k_2 \cdot g(t) \qquad (6)$$

Where $$k_1$$ and $$k_2$$ are regulator parameters. Set up a condition (state) model (a linear differential equation of second order) for the closed loop system given by equations (1), (2) and (6).

Show that it's possible to write the model as:
$$I \cdot \ddot{g}(t)-N_{nom} \cdot k_1 \cdot \dot{g}(t)-N_{nom} \cdot k_2 \cdot g(t)=\frac{3}{2} \cdot N_0-\frac{1}{2} \cdot N_0 \cdot \cos(4\pi \frac{t}{T_0}) \qquad (7)$$.

SOLUTION
Can somebody help us with figuring out what a condition model is?

$$I \cdot \ddot{ g } (t) = N_{ctrl} + N_{ext}$$
Which if we move $$N_{ctrl}$$ over to the left side, and use equations (3)-(5), would give us equation (7).