EE 350 Problem Set 10 Cover Sheet Fall 2014Last Name (Print):First Name (Print):ID number (Last 4 digits):Section:Submission deadl ines:• Turn in the written solutions by 4:00 pm on Tuesday D ecember 9 in the homework slot outside 1 21 EE East.Problem Weight Score47 2048 2049 2050 2051 20Total 100The solution submitted for grading represents my own analysis of the problem, and not that of another student.Signature:Neatly print the name(s) of the students you coll aborated with on this assignment.Reading assignment:• Lathi Chapter 7: Sections 7.1 and 7.2• Priemer Chapter 12: Section 12.3.7 pages 53 6 through 54 0Problem 47: (20 points)The MATLAB command resid ue generates a partial fraction expansion of the ratio of two polynom ials N(s)/D(s).First, learn to use this command by typing help residue at the MATLAB command prompt. Second, consider alinear time-invariant system whose transfer function representation isH(s) =3s2+ 27s + 60s3+ 6s2+ 11s + 61. (7 points) Use the command residue to find the impulse response function h(t) of the system.2. (5 points) Represent the system as an ordinary differential equation, a nd express your answer in the standardformdnydtn+ an−1dn−1ydtn−1+ · · · + aoy = bmdmfdtm+ bm−1dm−1fdtm−1+ · · · + bof.3. (8 points) Use the command MATLAB command re sidue to determine the zero-state response of the systemto the inputf(t) =h1 + e−3tiu(t).In order to determine the numerator and denominator polynomials of Y (s) = F (s)H(s), recall from problemset 2, problem 1 0, that you can multipl y polynomials using the command conv.To receive credit for parts 1 and 3, include an m-file with your name a nd section that shows the MATLABcommands used to generate your solution.Problem 48: (20 points)Figure 1 shows the passive RL low-pass filter from problem set 3, problem 11, part 1.Figure 1: Passive RL low-pass filter.1. (2 points) In order to cal culate the transfer function of the circuit, assume that all ini tial conditions are zeroand sketch the circuit in the frequency domain.2. (5 points) Write the node equations (in the frequency domain) associated with the node voltages X(s) andY (s).3. (5 points) Using your results in part 2, find the transfer f unction H(s) relati ng the output y(t) to the inputf(t).4. (2 points) From the transfer function, write the differential equation relating y(t) to f(t), and verify that youranswer agrees with that in problem 12, part 1 .5. (6 points) Suppose that R1= R2= 1 Ω, L1= 1/3 H, and L2= 1/2 H. Find the zero-state response y(t) if theinput voltag e f(t) = 10 u(t). In order to receive credit, show a ll steps and do no t solve this problem using theMAT L AB command re sidue, as you must be able to sol ve this type of probl em by hand.Problem 49: (20 points)Figure 2 shows the switched RLC circuit that you investigated in problem set 1, problem 3. You may assume thatthe switch has been closed long enough for the voltages and currents to reach steady-state values. At time t = 0 theswitch is opened.Figure 2: The switch in the RLC circuit is opened at ti me t = 0.1. (5 p oints) Assuming that the switch is opened, sketch the circuit i n the s-domain and express the initialcondition generators in terms of v(0−) and i(0−).2. (6 points) Determine an expression fo r I(s) in terms of the initial conditions v(0−) and i(0−).3. (9 points) Suppose that Vs= 8 V, R = Rs= 2 Ω, L = 0.5 H, and C = 2 µF . Determine the response i(t)for t ≥ 0. It i s important to note, that unlike the cla ssical solution method considered earlier in the semester,it is not necessary to first determine the values of i(t) and di/dt at t = 0+befo re solving for i(t). In order toreceive credit, show all steps and do not solve thi s problem using the MATLAB command residue, as youmust be a ble to solve this type of problem by hand.Problem 50: (20 points)1. (8 points) It is common to draw block diagrams representing the flow of signals in systems, for example, considerthe block diagram representation in Figure 3 of a system with input f(t) and output y(t). Represent the systemby a single transfer function and express your answer in the standard formY (s)F (s)=bmsm+ · · · + b1s + bosn+ an−1sn−1+ · · · + a1s + a0.Figure 3: Block diagram representation of a system.2. (12 points) The feedback control system in Figure 4 has a single control gai n K that can be chosen by thecontrol engineer.Figure 4: Block diagram representation of a feedback control system.(a) (2 points) Determine the close-loop transfer function and place your answer in the standard formY (s)R(s)=bmsm+ · · · + b1s + bosn+ an−1sn−1+ · · · + a1s + a0.(b) (2 points) Determine the range of values of K for which the closed-loop systems is BIBO stable.(c) (2 points) Given that the closed-loop system is BIBO stable, determine the DC gain of the closed-loopsystem.(d) (2 points) For what value(s) of the control gain K, if any, is the uni t- step response of the closed-loopsystem overdamped?(e) (2 points) For what value(s) of the control ga in K, if any, is the unit-step response of the closed-loopsystem critically da mped?(f) (2 points) For what value(s) of the control gain K, if any, is the unit-step response of the clo sed-loo psystem underdamped?Problem 51: (20 points)Figure 5 is an example of a feedback control system that i s designed to regulate the angular position θ(t) of a motorshaft to a desired value θr(t). The signal e(t) represents the error between the measured shaft angle θ(t) and thedesired shaft angle θr(t). The Laplace transforms of θr(t), θ(t), and e(t) are denoted as ΘR(s), Θ(s), and E(s),respectively. The control gains K1and K2are chosen by the control engineer to achieve a desired transient responseand steady-state accuracy.Figure 5: Block diagram of a feedback control system.1. (4 points) Cal culate the closed-loop transfer function Θ(s)/ΘR(s).2. (4 points) Cal culate the closed-loop transfer function E(s)/ΘR(s).3. (4 points) Assume that the controller gains K1and K2are chosen so that the closed-loop system is BIBO stable.Use the final val ue theorem to determine the steady-state value of the error e(t) for the following commandinputs.(a) The unit-step input θr(t) = u(t).(b) T he uni t-ramp input θr(t) = t u(t).4. (8 points) Assume that the controller gains K1and K2are chosen so that the closed-loop system is BIBOstable. In response to the input θr(t) = πu(t) and some initi al conditions θ(0) and˙θ(0), it is observed that
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