An electric rail car is powered through its roof using a contact assembly to high power voltage lines. The assembly dynamics are shown below, where the input to the system is the Force Up,fup(t), and the output,y, is the distance between the Head Mass and the Contact Wire,y(t) =yh(t) ycat(t), yielding a transfer functionGp(t) =Y(s). The objective is to maintain a constant distance from the Fup(s) wire assembly. The system parameters that are known are summarized in the table below (Kaveis the average of the tension wire spring constants). (a) (10 points) Model the system in Matlab. You were able to characterize the frequency response of the system using a vector signal analyzer as shown in the bode plot below, is your model accurate? (Explain with plots and analysis. The values for the plot are available on the Course Website) (b) (15 points) After characterizing the model, use root locus techniques to design a PID controller that yields: •0.5 second settling time•No more than 50% overshoot •Zero Steady State Error In order to remove high frequency oscillations use a notch filter with the following transfer function: GN(s) =(s+ 8 + 95.5824j)(s+ 8 95.5824j) (s+ 60)2 For the following figure, the values are:
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