**High-Speed Rail Pantograph Modeling and Analysis **

Some high-speed rail systems are powered by electricity supplied to a pantograph on the train’s roof from a catenary overhead, show in the figure below.

The force applied by the pantograph to the catenary is regulated to avoid loss of contact due to excessive transition motion. The diagram for the pantograph and catenary coupling is shown in the figure below.

(a) Using the simplified model on the right to find the differential equations for the rail pantograph, where *y*cat(*t*) is the catenary displacement, *y*h(*t*) is the pantograph head displacement, and *f*up(*t*) is the upward force applied to the pantograph under active control.

(b) Use the transfer function to estimate maximum overshoot, damping ratio, undamped natural frequency, settling time, peak time, and rise time. Obtain the step response of *G*(*s*) and compare with the hand calculations.

(c) The feedback block diagram of the rail pantograph system is shown below. Obtain the closed-loop transfer function. If K=1, draw the closed-loop step response, ramp response, and acceleration response in MATLAB.

(d) Find the value of the proportional controller K to ensure the stability. Sketch different the step responses in MATLAB to verify your design. Evaluate the steady-state error for these responses and draw your conclusion on how the steady-state errors are changing.

(e) Sketch the root locus by hand. Plot the root locus in MATLAB as well to verify your plot.

(f) Find the constant gain K to yield the closed-loop step response of 38% overshoot. Using the second order approximation to estimate the settling time and peak time. Plot the step response in MATLAB to see whether your estimations are valid, and explains your observations.

(g) In order to eliminate the high frequency oscillation, we can cascade a notch filterwith the plant, where the notch filter is given as. Using root locus method to design a PD controller to yield a settling time of 0.3 seconds with no more than 60% overshoot for step inputs. Plot the step response in MATLAB. After your design, furthermore, add a PI controller to yield zero steady-state error for step inputs. Plot the step response for the PID/notch- compensated closed-loop system in MATLAB.

(h) Continue (g) to realize your PID controller using an op-amp circuit with appropriate values of resistances, inductances, and capacitances.

(i) Repeat part (g) using a lag-lead compensator to yield ess=0.01 of the steady state error for step inputs. Plot the step response for the PID/notch-compensated closed-loop system in MATLAB.

(j) Continue (i) to realize the lag-lead compensator using an op-amp circuit appropriate values of resistances, inductances, and capacitances.

Write a clear project report in a Word file, including detailed calculations, your observations and conclusions, MATLAB codes, Simulink diagrams, and figures.

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