1. Develop a transfer function for the circuit below.

2. Given the transfer functions and input functions, F, use a numerical method to calculate the output of the system as a function of time for 0 to 0.5 seconds in 0.05 second intervals. Record the values in a table.

3. Simplify the following block diagram.

4. Simplify the following block diagram.

5. Simplify the following block diagram.

6. Simplify the following block diagram.

7. Simplify the following block diagram.

8. Simplify the following block diagram for a crane gantry position control system. The system also includes a negative feedback loop to control the sway of an attached load

9. Write a C subroutine to implement the control system that is shown inside the dashed line. The subroutine arguments are the setpoint, C, and the feedback value, F. the subroutine returns the new controller output value, U.

10. Write C subroutines to implement the control system that is shown inside the dashed line. The subroutine arguments are the setpoint, C, and the feedback value, F. the subroutine returns the new controller output value, U.

11. Given the following negative feedback control system, calculate a new controller transfer function (Gc) to get the desired response for the system.

12. Write a C subroutine that implements the control function below. (Hint: Convert it to state equations first.)

13. Develop a transfer function for the following system. The input is Vd and the output is the motor shaft speed. Assume all components are ideal. The motors are identical with a resistance of 10 ohms. With an input voltage of 4V the motors spin at 4000RPM (steady state), and have a time constant of 0.1s.

14. Draw the block diagram equivalent of the C subroutine.

15. Select a controller transfer function, Gc, that will reduce the system to a first order system with a time constant of 0.5s, as shown below.

16. A feedback control system is shown below. The system incorporates a PID controller. The closed loop transfer function is given.

a) Verify the closed loop controller function given.

b) For the given transfer function select controller values that will result in a natural frequency of 2 rad/sec and damping factor of 0.5. Verify that the controller will be stable. (Hint: assume Kd=0.)

c) For the given transfer function, if the values of Kp=1 and Ki=Kd=0, find the response equation to a unit ramp input (i.e., X=t) as a function of time by solving the differential equation.

d) For the given transfer function, if the values of Kp=1 and Ki=Kd=0, find the response to a unit ramp input (i.e., X=t) as a function of time using state equations and a numerical method. Give the values in a table from 0 to 1.0s in 0.1s intervals.

17. Draw FBDs for a system described with the equations below.