6 01 Spring Semester 2008 Assignment 5 Issued Tuesday Mar 4 1 MASSACHVSETTS INSTITVTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6 01 Introduction to EECS I Spring Semester 2008 Assignment 5 Issued Tuesday Mar 4 Overview of this week s work In software lab Work through the software lab Paste your solution to Question 7 into the tutor Before the start of your design lab on Mar 4 or 5 Read the class notes and review the lecture handout Do the on line tutor problems in section PS 6 2 Read the entire description of the design lab so that you will be ready to work on it when you get to lab In design lab Take the nanoquiz in the first 15 minutes don t be late Work through the design lab with a partner and take good notes on the results of your work At the beginning of your next software lab on Mar 11 or 12 Submit online tutor problems in section PS 6 3 Submit written solutions to questions 7 9 11 14 19 and 20 All written work must conform to the homework guidelines on the web page Get the lab5 files via athrun 6 01 update or from the home page 6 01 Spring Semester 2008 Assignment 5 Issued Tuesday Mar 4 2 Software Lab System Functions for Robot Controller We saw in lecture this week that the behavior of a linear time independent system can be completely characterized using a ratio of two polynomials in R the delay operator We call this characterization the system function Further we saw that systems composed of multiple LTI components can be described using system functions that are relatively simple compositions of the system functions of the components It is possible to solve such problems by hand with a paper and pencil but why do that when we can get a computer to do the work for us In this software lab and the next we will build and use a set of tools for analyzing simulating and combining system functions The file SystemFunctionSkeleton py contains the definition of the class SystemFunction with some methods filled in and others just documented but with pass instead of the method body We ll fill in its methods this week and next We ll also give you SystemFunction pyc which is a complete compiled version of that module Making System Functions The module SystemFunction contains a function that constructs a new instance of the SystemFunction class given a characterization of that system function as a difference equation Assume that the difference equation has the form a0 y n a1 y n 1 ak y n k b0 x n b1 x n 1 bj x n j where X is the input signal and Y is the output signal It is crucial that the first coefficient of both polynomials be from the same time step but they can go back different depths in history To make a new system function say corresponding to the difference equation 9y n 4y n 1 5y n 3 2x n 1 we d make the call sf s y s t e m F u n c t i o n F r o m D i f f e r e n c e E q u a t i o n 9 4 5 0 2 You can see from printing out the system function that we store it as a ratio of polynomials in R sf SF 2 000 R 5 000 R 2 4 000 R 9 000 Question 1 Describe in English what this system does s y s t e m F u n c t i o n F r o m D i f f e r e n c e E q u a t i o n 1 0 0 1 Question 2 What call would you make to generate the system function for a system with this difference equation y n 2y n 2 3x n 3 Understanding a system A crucial thing to understand about a system is whether even in the absence of input its output will go to zero or whether it will grow unboundedly We can find this out by determining the poles of the system So for example if you ask for the poles of a system function you ll get a report on the poles of the system and their implications for the stability of the system 6 01 Spring Semester 2008 Assignment 5 Issued Tuesday Mar 4 3 sf s y s t e m F u n c t i o n F r o m D i f f e r e n c e E q u a t i o n 9 4 5 0 2 sf poles Magnitude of poles 0 7453559924999299 0 7453559924999299 0 22222222222222227 0 71145824860364981 j 0 2 2 2 2 2 222222222218 0 71145824860364981 j sf2 s y s t e m F u n c t i o n F r o m D i f f e r e n c e E q u a t i o n 3 4 5 0 2 sf poles Magnitude of poles 1 2909944487358056 1 2909944487358056 Danger Warning Unstable system 0 66666666666666663 1 1055415967851332 j 0 6 66 6 6 666666666663 1 1055415967851332 j Another way to get an understanding of how a system is working is to start it at some initial conditions and simulate it by running the associated difference equation forward We have built facilities for doing this into the SystemFunction class Let k be the order of the polynomial on the Y terms also called the order of the system and j be the order of the polynomial on the X terms We need j initial input values and k initial output values in order to be able to specify all future values of the system In addition we need to provide an input sequence for generality instead of providing a list we use a function from n to x n So given initial values the lists x 0 x j 1 and y 0 y k 1 and an input source the SystemFunction plotSequence method plots the sequence of Y values So for example you can do sf2 s y s t e m F u n c t i o n F r o m D i f f e r e n c e E q u a t i o n 3 4 5 0 2 sf2 plotSequence 1 0 1 lambda x 1 n 30 idle True and get the plot shown in figure 1 The last two arguments are optional n controls the number of points that are plotted idle controls whether the plotting will work nicely with Idle If you call this with idle set to True then you will have to close the plotting window before your program that called this method or the Python shell can continue to run Question 3 Experiment with plots of sf2 above with different initial conditions and inputs Robot meets wall Now let s return to …
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