The dynamics of a controlled submarine are significantly different from those of an aircraft, missile, or surface ship. This difference results primarily from the moment in the vertical plane due to the buoyancy effect. Therefore, it is interesting to consider the control of the depth of a submarine. The equations describing the dynamics of a submarine can be obtained by using Newton’s laws and the angles defined in Figure 1. To simplify the equations, we will assume that  is a small angle and the velocity  is constant and equal to 25 ft/s. The state variables of the submarine, considering only vertical control, are  and  , where   is the angle of attack.

Fig.1: Submarine depth control

Thus the state vector differential equation for this system, when the submarine has an Albacore type hull, is

where  , the deflection of the stern plane.

1. Applying the state-space theories to determine the transition matrix of this system.
2. Find the transfer matrix of the system.
3. Test the system controllability and observability of the system.
4. Design state-feedback gains to locate the desired closed loop poles at .
5. Find the response of the system due to a unit step input.
6. Draw the block diagram for the designed system.
7. It should be submitted in a form of a report.

• Single student or group of students can perform the case study.
• The student should post his report through Moodle.
• The project report must contain the followings:
• Title of the Project
• Objective of the project Summary Abstracts
• Introduction and Literature Review
• Design and analysis
• Results and discussion
• Conclusions
• References