(a) In the system of Problem \(1.4-1, J=0.6\) and \(K=12.4\), in appropriate units. The attitude of the satellite is initially at \(0^{\circ}\). At \(t=0\), the attitude is commanded to \(40^{\circ}\); that is, a \(40^{\circ}\) step is applied at \(t=0\). Find the response \(\theta(t)\).

(b) Repeat part (a), with the initial conditions \(\theta(0)=10^{\circ}\) and \(\dot{\theta}(0)=30^{\circ} / \mathrm{s}\). Note that we have assumed that the units of time for the system is seconds.

(c) Verify the solution in part (b) by first checking the initial conditions and then substituting the solution into the system differential equation.

Problem 1.4-1:

The satellite of Section 1.4 is connected in the closed-loop control system shown in Fig. P1.4-1. The torque is directly proportional to the error signal.

• Derive the transfer function \(\Theta(s) / \Theta_{c}(s)\), where \(\theta(t)=\mathscr{L}^{-1}[\Theta(s)]\) is the commanded attitude angle.

• The state equations for the satellite are derived in Section 1.4. Modify these equations to model the closed-loop system of Fig. P1.4-1.

Fig. P1.4-1

Transcribed Image Text:

Thrusters (a) 0(1) Thrusters