6.01, Spring Semester, 2008—Final exam announcement and practice final, Revised May 12 1MASSACHVSETTS INSTITVTE OF TECHNOLOGYDepartment of Electrical Engineering and Computer Science6.01—Introduction to EECS ISpring Semester, 2008Final exam announcement and practice final, Revised May 12This is a sample final to help you study and to give you a sense of what kinds of problems to expecton the final. The final will be given the afternoon of Wednesday, May 21. You can pick it up in34-501 (the 6.01 lab) at 1:30PM, and you must turn it in by 7:30PM. You may turn in your solutionearly. You may not turn it in late.Ground rules for taking the finalThese instructions apply to taking the real fine—not this practice exam:• Read through the entire final and ask any questions you have before you leave. If you find thatyou have a question after that, try to pick your own answer, and write down what assumptionyou’re making.• Your paper must have all the sheets stapled together, and must have your name on each sheet.In the midterm, some papers came in as separated sheets and without names. We took thetrouble then to identify the anonymous papers and reassemble the separated pieces. We willnot do that in grading the final: we’ll simply not grade any unidentified work.• For the problems that ask you to write Python code, you should just think it through andwrite down the code. Don’t try to actually debug it (it will take too long). Don’t spend timetrying to be sure the syntax is correct. It’s more important to communicate your ability tothink algorithmically than to worry about the fine-grained details of syntax. Be sure to addcomments that explain what you’re trying to do.• Your answers may be handwritten or typed; they should be legible in either case. We reservethe right to refuse to grade illegible papers.• You answers must include coherent explanations. Simply writing down equations with noexplanations, or attaching pages of uncommented program output, will get no credit. Ingrading, we will make no attempt to guess what you are thinking by trying to piece togetherfragments of phrases and equations and printout.• You may read anything you like (labs, books, the Web) during the exam; you may notcommunicate with anyone else.• Take above prohibition on collaboration seriously. On the midterm, we found somepapers (a very small number) where people had collaborated on problems. We handled this bygiving the students involved grades of zero for the midterm. The consequences of collaboratingon the final would be at least as severe and, at a minimum, result in failing the course. Avoidtemptation: find some place to work on the final without other people in around.• You can use any of the software we’ve provided or you’ve written for lab. But it’s yourresponsibility to make sure that your computer runs the code. Tales of woe of the form“I couldn’t get the code to run,” may receive sympathetic sighs, but that won’t make anydifference in grading.6.01, Spring Semester, 2008—Final exam announcement and practice final, Revised May 12 21. Robot in a corridorFor this problem, please consider a long corridor of 100 alternating black and white squares, labeled0 through 99. Your robot is initially known to be in square 0, which is black and at the far left.The robot can move only right. When you ask the robot to move right one square, it will actuallymove right one square only 50% of the time; 40% of the time it will not move at all; and 10% ofthe time it will move right two squares.Suppose first that the robot can perfectly reliably observe whether it is on a black square or a whitesquare.(a) Suppose you ask the robot to move one square to the right twice, but do not make any obser-vations until after the second move is completed, at which point the robot observes that it ison a black square. What is the probability that the robot is on square 0? What is the entirebelief state distribution for the robot’s position?(b) Suppose you want the robot to end up on square 10. A simple strategy would be to commandthe robot to move right ten times and hope for the best. Devise a more reliable strategy thathas the robot perform an observation after every move. Describe your strategy in English.(c) Write a Python program that implements your strategy from part (b). Assume you haveprocedures moveRight(), which commands the robot to move right, and look(), which returnseither “black” or “white”. You do not need to debug your code.(d) Suppose now that the robot’s vision is faulty. The observation model is this: If the robot isactually on a black square, it will see black with 80% probability and white with 20% probability.If the robot is actually on a white square, it will see black with 35% probability and white with65% probability.Suppose as in part (a) above, the robot starts on square 0 and you command it to move onesquare to the right twice, not making any observations until after the second move is completed.If the robot now sees that it is on a black square, what is the the probability that the robot ison square 0? What is the entire belief state distribution for the robot’s position?6.01, Spring Semester, 2008—Final exam announcement and practice final, Revised May 12 32. Motor controllerThis problem is about using a computer to precisely control the speed of a spinning motor. Themotor has a tachometer that provides very precise speed measurements, and you can control thevoltage across the motor. The motor speed S is given byS = JVwhere J is a constant associated with the motor and V is the voltage across the motor. Unfortu-nately, not all all motors have the same J : the nominal value of J is 10, but J can actually be aslow as 9 and as high as 11.(a) Your computer samples the motor velocity 10 times a second and can change the motor voltagebased on readings from previous samples. One suggested control strategy is to determine themotor voltage using the nominal value of J (J = 10):V [n + 1] =110Sdesiredwhere Sdesiredis a given desired motor speed. In the worst case, considering the variation in J ,how far off will the motor speed be from the desired speed?(b) Suppose we use a feedback control system to reduce the difference between the actual motorspeed and the desired speed as follows:V [n] =110Sdesired+ K (Sdesired− S[n − 1])where S is the actual motor speed as measured by the tachometer and K is the feedback gain.Draw a delay-adder-gain block
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