1. Obtain the system models of both the cart and the pendulum in transfer functions G1 (s)  X(s)

1. Obtain the system models of both the cart and the pendulum in transfer functions  G1 (s)  X(s) 
  (s) F(s) 
and  G2 (s)  
F (s) , at equilibrium position of  around zero (0).   2. Use root-locus to design a PID controller such that the pendulum displacement  can be controlled to respond to a step input with settling time ts = 1.25 seconds (2% criterion) and a maximum percentage overshoot of Mp = 1.52%. 
3. Show step and impulse response of your designed system using ‘Step’ and ‘impulse’ command in MATLAB and comment on the performance.  
4. Using Simulink to simulate the behavior of the system for the following situations – plot the desired and actual response of the pendulum angle  :  Show that your controller can control the pendulum to follow a step signal  d =0.1 (rad).  When the desired pendulum position is a sine wave  d = 0.1sin(t) (rad), what is the response of your control system.   
5. What is the response of the pendulum if  d  

realistic outcome in real life situation? 
(rad)? Why do you have this outcome? Is this a   
6. What is the position of the cart when you are controlling the pendulum? Explain why such an output is obtained? Can you control both the position of the cart and the pendulum position using your current controller? **The control results should be plotted using MATLAB plots that shows the desired position and actual position of the pendulum with All MATLAB codes. 

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