Blank ExamSolutionsEE 350 EXAM IV 14 December 2011Last Name (Print):First Name (Print):ID number (Last 4 digits):Section:DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO D O SOProblem Weight Score1 252 253 254 25Total 100Te st Form AINSTRUCTIONS1. You have one hour and fifty minutes to complete this exam.2. This is a closed book exam. You may use one 8.5” × 11” note sheet.3. Calculators are not allowed.4. Solve each part of the problem in the space following the question. If you need more space, continue your solutionon the reverse side labeling the page with the question number; for example, Problem 1.2 Continued. NOcredit will be given to solutions that do not meet this requirement.5. DO NOT REMOVE ANY PAGES FROM THIS EXAM. Loose papers will not be accepted and agrade of ZERO will be assigned.6. The quality of your analysis and eva luation is as important as your answers. Your reasoning must be preciseand clear; your complete English sentences should convey what you are doing. To receive credit, you mustshow your work.1Problem 1: (25 Points)1. (9 points) Using the method of Laplace transforms, determine the transfer function H(s) of the passive filterin Figure 1, where f(t) is the input and y(t) is the output. Express your answer in the standard formH(s) =bmsm+ · ·· + b1s + bosn+ an−1sn−1+ ·· · + a1s + a0.Figure 1: Passive ci rcuit with input voltage f(t) and output voltage y(t).232. (9 points) The circuit in Figure 2 has input f(t) and output y(t). The comp onent values are R =52Ω, L =12H,and C =15F. Given thatf(t) = cos√10 tu(t),determine the zero-state response of the circuit using the method of Laplace transforms.Figure 2: Passive RLC circuit w ith output voltage y(t).453. (7 points) The MATLAB command res idue converts between the partia l fraction expansion and polynomialcoefficients. For example, in problem set 1 0 you determined the partial fraction expansion using the MATLABcommand>> [r, p, k] = resi due(P, Q)One can a lso use the MATLAB command residue to determine the polynomial coefficients from the partialfraction expansion, in specific>> [P,Q] = residue(r,p,k)Consider a BIBO system whose transfer function H(s) has the partial transfer expansionr = [8,-3];p = [-4,-3];k = 1;(a) (3 points) What is the DC gain of the system?(b) (4 points) Determine the ODE representation of the system and express your result in the standard formdnydtn+ an−1dn−1ydtn−1+ · ·· + aoy = bmdmfdtm+ bm−1dm−1fdtm−1+ · ·· + bof,6Problem 2: (25 points)1. (10 points) Fig ure 3 shows the block diagram representation of a closed-loop system where R(s) is the referenceinput, D(s) is a disturbance input, and Y (s) is the system output. In order to determine the effect of thedisturbance input on the system output, a control engineer asks you to set R(s) to zero, and determine thetransfer function from the disturbance input to the system output. Specify your result using the standard formY (s)D(s)=bmsm+ ·· · + b1s + bosn+ an−1sn−1+ ·· ·+ a1s + a0.Figure 3: Feedback control system with reference input r(t), disturbance input d(t), and controlled output y(t).782. (5 points) Another feedback control system, different from the one considered in part 1, has the closed-looptransfer function representationY (s)R(s)=s + 5s(s + 2),where R(s) is the command input and Y (s) is the controlled output. Determine the steady-state value of theclosed-loop output,yss= limt→∞y(t),for the command inputr(t) = t e−5tu(t).93. (10 p oints) A third feedback control system, different from the ones considered in parts 1 and 2, has theclosed-loop transfer function representationY (s)R(s)=s + 6s2+ α s + β.Choose the values of the controller gains α and β so that the zero-state uni t- step response of the closed-loopsystem has the formy(t) =A + Be−2t+ Ce−3t u(t),and determine the numeric value of the parameters A, B, and C.10Problem 3: (25 points)1. (15 points) A system has the transfer function representationH(s) =s2+ 100ss2+ 20s + 100.Construct the Bode magnitude and phase plots using the semilog graphs provided in Figure 4 (a duplicate copyappears in Figure 5).In order to receive credit:• In both your magnitude and phase plots, indicate each term separately using dashed lines.• Indicate the slope of each straight-line segment and the corner frequencies of the final magnitude andphase plots.• Do not show the 3 dB corrections in the magnitude plot.11Figure 4: Semi log paper for Bode magnitude and phase plots.12Figure 5: Semi log paper for Bode magnitude and phase plots.132. (10 points) Figure 6 shows the straight-line magnitude and phase plo ts of a transfer function G(s). Determinethe transfer function G(s) and place your answer i n the standard fo rmG(s) =bmsm+ · ·· + b1s + bosn+ an−1sn−1+ · ·· + a1s + a0.10−1100101102103104105020406080Magnitude [dB]10−1100101102103104105−4504590135Phase [Deg]frequency [rad/sec]Figure 6: Straight-line approxima tion of the magnitude and phase plot of H(s).1415Problem 4: (25 points)1. (7 points) The impulse response of a certain linear time-invaria nt system ish(t) = e−5t[3 cos(12 t) − sin(12 t)] u(t).Determine the ODE representation of the system in place your answer in the standard formdnydtn+ an−1dn−1ydtn−1+ ·· ·+ aoy = bmdmfdtm+ bm−1dm−1fdtm−1+ ·· ·+ bof,where f(t) and y(t) represent the system input and output respectively.162. (6 poi nts) A linear time-invariant system with input f(t) a nd output y(t) is represented by the ordinarydifferential equation¨y − ˙y − 6y =˙f − 3f.(a) (3 points) Determine whether or not the system is asymptotically stable. Justify your work by showingappropriate calculations.(b) (3 points) Determine whether or not the system is bounded-input bounded-output stable. Justify yourwork by showing appropriate calculations.173. (12 points) Another linear time-invariant system, different from the one in part 1, has the pole-zero map shownin Figure 7, and the straight-line magnitude and phase plots shown in Figure 8.Figure 7: Pole-zero map of the linear time-invariant system.−40−20020Magnitude [dB]4590135180225270Phase [Deg]frequency [rad/sec]Figure 8: Straight-line approxima tion of the magnitude and phase plot of the system frequency response function.18(a) (2 points) Specif y the D C gain of the system.(b) (4 points) Determine the corner frequencies of the straight-line magnitude plot, and label them in themagnitude plot shown in Figure 8.(c) (6 poi
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