1. Draw the root locus diagram for the system below. specify all points and values.

2. 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.

3. 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=0.01 find the response of the system to an input step of 0.1rad.

4. 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 can the initial values be found?

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

5. Draw a root locus plot for the control system below and determine acceptable values of K, including critical points.

6. The feedback loop below is for controlling a DC motor with a PID controller.

a) Find the transfer function for the system.

b) Draw a root locus diagram for the variable parameter ‘P’.

c) Find the response of the system in to a unit step input using explicit integration.