System and control

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HW #7, ECE 356 – Systems and Control, spring 2017, Due: April 17, 2017 Show all the details of your work. 1.) (a) Sketch the positive root locus of the system shown below using the (2, 2) Pade approximation for the delay. State the asymptote angles and their centroid, the arrival and departure angles at any complex pole or zero, the frequencies of any imaginary axis crossings, and the locations of any break-in or break-away points. (b) Use Matlab to plot the positive root locus of the system shown below using the (2, 2) Pade approximation for the delay. Your sketch and the Matlab results should be displayed at similar scales. (c) Use Matlab to plot the step response of the system shown below using the (2, 2) Pade approximation for the delay and K = 3. You should see the 0.5 second delay in the step response. + Σ R e s / 2 s 2  4s K − 2. (a) Calculate the magnitude and phase of G ( s )  Y 6401 for ω = 10, 20, 50, 100, 200, 500, s  2s  6401 2 and 1000 rad/sec. (b) Use Matlab (bode) to plot the Bode plot for G(s). This can also be done using the LTIviewer. (c) Calculate the exact frequency at which |G(jω)| reaches its maximum value. Also, what is the phase at this frequency? (d) Use Matlab to plot the step response. Indicate the rise-time, maximum overshoot, and settlingtime on your plot. Question 3 – 5: Determine the following for the three open-loop transfer functions given below: (a) Draw the asymptotic Bode plots on logarithmic paper with a ruler (no free-hand sketches). (b) Then use Matlab to obtain the exact Bode plots. Your sketches and the Matlab results should be plotted on similar scales. Also, indicate the gain and phase margins as well as the crossover frequencies. (c) Use these values to estimate the rise-time, overshoot, and the settling-time of the closed-loop step responses. (d) Use Matlab to obtain the step responses (with K = 1) of the closed-loop systems indicating the rise-time, overshoot, and the settling-time. Compare your estimates with the exact values obtained with Matlab. (e) Sketch the Nyquist plots and indicate the range of K for stability. 10( s  200) 3.) G ( s)  s( s  4)( s 2  2s  401) + 10000( s  10)( s  80) Σ K G (s) 4.) G ( s)  2 s ( s  3)( s  200) − 2(100  s) 5.) G ( s)  ( s  0.1)( s  10)
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