3. Given the transfer function below, and the input ‘x(s)’, find the output ‘y(t)’ as a function of time.

8. Draw a detailed root locus diagram for the transfer function below. Be careful to specify angles of departure, ranges for breakout/breakin points, and gains and frequency at stability limits.

10. Draw the root locus diagram for the transfer function below,

11. Draw the root locus diagram for the transfer function below,

12. The block diagram below is for a motor position control system. The system has a proportional controller with a variable gain K.

a) Simplify the block diagram to a single transfer function.

b) Draw the Root-Locus diagram for the system (as K varies). Use either the approximate or exact techniques.

c) Select a K value that will result in an overall damping factor of 1. State if the Root-Locus diagram shows that the system is stable for the chosen K.

13. Draw a Bode Plot for either one of the two transfer functions below.

15. Given the system transfer function below.

a) Draw the root locus diagram and state what values of K are acceptable.

b) Select a gain value for K that has either a damping factor of 0.707 or a natural frequency of 3 rad/sec.

c) Given a gain of K=10 find the steady state response to an input step of 1 rad.

d) Given a gain of K=10 find the response of the system as

17. The equation below describes a dynamic system. The input is ‘F’ and the output is ‘V’. It has the initial values specified. The following questions ask you to find the system response to a unit step input using various techniques.

a) Find the response using Laplace transforms.

b) Find the response using the homogenous and particular solutions.

c) Put the equation is state variable form, and solve it using your calculator. Sketch the result accurately below.

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

a) Verify the close loop controller function given.

b) Draw a root locus plot for the controller if Kp=1 and Ki=1. Identify any values of Kd that would leave the system unstable.

c) Draw a Bode plot for the feedback system if Kd=Kp=Ki=1.

d) 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.

e) For the parameters found in the last step find the initial and final values.

f) If the values of Kd=1 and Ki=Kd=0, find the response to a ramp input as a function of time.

19. The following system is a feedback controller for an elevator. It uses a desired heigh ‘d’ provided by a user, and the actual height of the elevator ‘h’. The difference between these two is called the error ‘e’. The PID controller will examine the value ‘e’ and then control the speed of the lift motor with a control voltage ‘c’. The elevator and controller are described with transfer functions, as shown below. all of these equations can be combined into a single system transfer equation as shown.

a) Find the response of the final equation to a step input. The system starts at rest on the ground floor, and the input (desired height) changes to 20 as a step input.

b) Write find the damping factor and natural frequency of the results in part a).

c) verify the solution using the initial and final value theorems.