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Problem: Analyze the behavior of a simplified satellite attitude control system

Figure 1: Communication Satellite

Figure 2: Attitude Control Block Diagram

Known: D(s) is the compensator, G(s) is the plant, and (s) is the attitude angle from the z-axis.

Given: D(s) = K, D(s) = K(1+s+1/4s), D(s) = K(s+0.1)/(s+0.7), G(s) = 1/Js², J=10 kg m²

Objective: Using MATLAB, analyze the behavior of a simplified satellite attitude control system (single axis control) as shown in Figure 2.

1. Determine its closed-loop transfer function with various controllers
   - D(s) is a proportional gain
   - D(s) is a PID controller
   - D(s) is a Lead-Lag Compensator
2. Find the poles and zeros, plot them, and determine whether the system is stable
   - When D(s) is a PID Controller
3. Plot its system step response and find: Rise Time (t_r), Settling Time (t_s), Overshoot (M_p)
   - When D(s) is a lead-lag Compensator
4. Plot the root locus and determine the gain value for pole values at: -1.4±0.4i
   - When D(s) is a PID Controller
5. Plot the Bode and Nyquist as another means of determining system stability
   - When D(s) is a lead-lag Compensator

Note: The problem is simplified by considering only one axis of motion of the satellite, which is around the y-axis as shown in Figure 1. The reason for analyzing only one axis of the satellite and assuming similar behavioral characteristics from the other two axes (x and z), is because small attitude (or pointing angle) changes are considered. With small attitude changes, the satellite motions are considered uncoupled and therefore independent.

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