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Questiоn 1 (18 Pоints) Shоw your work. A correct аnswer without supporting work mаy receive little or no credit. Consider the feedbаck control system shown below. Without any block diagram reduction, provide concise answer to the following questions: A) [2 points] The feedforward transfer function is (expressed in terms of (s)) _____ B) [2 points] The open-loop transfer function is (expressed in terms of (s)) _____ C) [2 points] The closed-loop transfer function is(expressed in terms of (s)) _____ D) [2 points] The characteristic equation of the system is _____ E) [2 points] The differential equation that describes the system is _____ F) [2 points] The time constant of the system is ____ G) [2 points] The undamped natural frequency of the system is ____ H) [2 points] The damped natural frequency of the system is ____ I) [2 points] The type number of the system is ____ Question 2 (8 Points) (I) Consider the unity-feedback control system shown below with a unit-step input (r(t)). Sketch each stated system's closed-loop pole locations on the (s)-plane and its corresponding unit-step response. Question 3 (30 Points) In each case sketch the Nyquist plot and determine the number of unstable closed loop characteristic roots. (a) [10 Points] [G(s)=frac{s-1}{s(s+1)}.] (b) [10 Points] [G(s)=frac{s+5}{s^{2}(s+2)}.] (c) [10 Points] [G(s)=frac{s-3}{(s+1)(s+2)}.] Question 4 (12 Points) Determine whether each of the following statements is True or False (T/F). No justification is necessary. (a) (4 Points) The Nyquist plot of an open-loop stable system can encircle (-1+j0). (i) Clockwise. T/F (ii) Counterclockwise. T/F (b) (4 Points) The Nyquist plot corresponding to a stable closed-loop system can encircle (-1+j0). (i) Clockwise. T/F (ii) Counterclockwise. T/F (c) (4 Points) The Nyquist plot of a system encircles (-1+j0) clockwise. (i) Does this imply the open-loop system is unstable? T/F (ii) Does this imply the closed-loop system is unstable? T/F Question 5 (32 Points) Consider the compensator-design problem for a feedback control system shown below. It is working in a noise-free environment. The output of the Bode plot of (G(s)) when (K=1) is put into this table: Problem5_Tables.pdf DO NOT interpolate between the points in the table.Just pick the nearest point that satisfies your design criterion from the table.Perform your calculations up to 2--3 significant decimal places. The desired design specifications of the compensated system are: - Desired phase margin [ phi_{PM}^{G_d}geq 45^circ. ] - In order to satisfy the desired (K_v), the gain of (G(s)) must be increased by 3 times. (A) The phase margin, the gain crossover frequency, the gain margin, and the phase crossover frequency of the uncompensated system are: [10 points] (omega_c) (in rad/sec) (=) _____ Phase margin (in degrees) (=) ______ (omega_pi) (in rad/sec) (=) _____ Gain margin (in dB) (=) _____ Is the uncompensated system stable (yes/no and why)? _____ (B) Design a lead compensator for this system with (varepsilon=8^circ) added. [6 points total] (a) What is the phase lead angle from the lead compensator? What is the (alpha) value of the compensator to satisfy the design specification? [2 points] (b) What is the new gain crossover frequency (omega_c^{*})? [2 points] (c) What are the locations of the pole and zero of the lead compensator? What is the resultant transfer function of the compensator (G_c(s)) from this design? [2 points] (C) With your designed lead compensator (G_c(s)), please explain whether the compensated system (G_d(s)) will or will not satisfy the desired design specification. Clearly justify your answer with your phase margin of the compensated system! [5 points] (D) Next, design a lag compensator for this system with (varepsilon=8^circ) added. [6 points total] (a) What is the new gain crossover frequency (omega_c^{*})? [2 points] (b) What is the (alpha) value of the compensator to satisfy the design specification? [2 points] (c) What are the locations of the pole and zero of the lag compensator? What is the resultant transfer function of compensator (G_c(s)) from this design? [2 points] (E) With your designed lag compensator (G_c(s)), please explain whether the compensated system (G_d(s)) will or will not satisfy the desired design specification. Clearly justify your answer with your phase margin of the compensated system! [5 points] Congratulations, you are almost done with this exam. DO NOT end the Honorlock session until you have submitted your work to Gradescope. When you have answered all questions: Use your smartphone to scan your answer sheet and save the scan as a PDF. Make sure your scan is clear and legible. Submit your PDF to Gradescope as follows: Email your PDF to yourself or save it to the cloud (Google Drive, etc.). Click this link to go to Gradescope to submit your work: Final Exam Return to this window and click the button below to agree to the honor statement. Click Submit Quiz to end the exam. End the Honorlock session.
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