This is a very interesting problem that I've encounter and I would like some guidance that would help me to solve the problem.  The problem pertains to the analysis and control of the Manned Maneuvering Unit System (MMUS). This system utilizes small thrusters, which discharge gas—typically nitrogen—to generate thrust, facilitating astronaut mobility in zero- gravity environments. The scenario considers a combined mass of 100 kilograms, encompassing the astronaut, the MMUS, and any associated equipment. It is assumed that the thrusters function ideally, delivering the specified reactive force instantaneously and with precision. The primary focus is on controlling the astronaut's vertical position to align with the entrance of a space station. This involves an autopilot control strategy that incorporates a velocity control loop within a position control loop. The design of these loops employs proportional gain controllers, denoted as Kv for velocity and Kp for position, respectively. The tasks are as follows: 1. Establish the transfer function Gv(s) that describes the relationship between the thrust force F(s)and the vertical velocity Vz(s).  2. Draw the control diagramfor vertical positioning, including both the velocity and position control loops. 3. Formulate the closed-loop transfer function T(s) that connects the position command Zr(s) to the astronaut's vertical position Zc(s).  4. Given a settling time (Ts)requirement of 1 second and an overshoot (OS) limit of 2%, calculate the necessary damping ratio and natural frequency.  5. Determine the values of Kv and Kp based on the derived damping ratio and natural frequency

This is a very interesting problem that I've encounter and I would like some guidance that would help me to solve the problem. 

The problem pertains to the analysis and control of the Manned Maneuvering Unit System (MMUS). This system utilizes small thrusters, which discharge gas—typically nitrogen—to generate thrust, facilitating astronaut mobility in zero- gravity environments.

The scenario considers a combined mass of 100 kilograms, encompassing the astronaut, the MMUS, and any associated equipment. It is assumed that the thrusters function ideally, delivering the specified reactive force instantaneously and with precision.

The primary focus is on controlling the astronaut's vertical position to align with the entrance of a space station. This involves an autopilot control strategy that incorporates a velocity control loop within a position control loop. The design of these loops employs proportional gain controllers, denoted as Kv for velocity and Kp for position, respectively.

The tasks are as follows:

1. Establish the transfer function Gv(s) that describes the relationship between the thrust force F(s)and the vertical velocity Vz(s). 

2. Draw the control diagramfor vertical positioning, including both the velocity and position control loops.

3. Formulate the closed-loop transfer function T(s) that connects the position command Zr(s) to the astronaut's vertical position Zc(s). 

4. Given a settling time (Ts)requirement of 1 second and an overshoot (OS) limit of 2%, calculate the necessary damping ratio and natural frequency. 

5. Determine the values of Kv and Kp based on the derived damping ratio and natural frequency