CALIFORNIA INSTITUTE OF TECHNOLOGYControl and Dynamical SystemsCDS 101/110R. M. MurrayFall 2004Homework Set #1 Issued: 27 Sep 04Due: 4 Oct 04Note: In the upper left hand corner of the first page of your homework set, pleaseput the class you are taking (CDS 101, CDS 110) and the number of hours that youspent on this homework set (including reading).All students should complete the following problems:1. Identify 5 feedback systems that you encounter in your everyday environment. For eachsystem, identify the sensing mechanism, actuation mechanism, and control law. Describe theuncertainty that the feedback system provides robustness with respect to and/or the dynamicsthat are changed through the use of feedback. At least one example should correspond to asystem that comes from your own discipline or research activities.(Hint: If you can’t think of any examples, look in the index of Control in an InformationRich World, available on the course homepage.)2. (MATLAB/SIMULINK) Download the file “hw1cruise.mdl” from the course web page, whichcontains a SIMULINK model of a simple cruise controller, similar to the one described in class(see documentation from web page for more details). Figure out how to run the example andplot the vehicles speed as a function of time.(a) Leaving the control gains at their default values, plot the response of the system to astep input and measure the time it takes for the system error to settle to within 95% ofcommanded change in speed (i.e., ±0.5 miles/hour).(b) By manually tuning the control gains, design a controller that settles 50% faster thanthe default controller. Include the gains you used, a plot of the closed loop response,and describe any undesireable features in the solution you obtain.All plots should included a title, labeled axes (with units), and reasonable axis limits.Note: The purpose of this problem is to give you some familiarity with MATLAB andSIMULINK. Don’t worry if you don’t yet know how the control law works or why it doeswhat it does. We’ll go over this in class later in the term.3. Complete the course survey attached to this homework set. Please turn this in at the sametime as your homework, but do not attach to your homework set.Only CDS 110a students need to complete the following additional problems:4. On the course homepage under “Reading” you will find a collection of articles from the popularpress about feedback and control systems. Choose any one of the articles and describe thefeedback system using the terminology given in class and the course text. In particular,identify the control system in the article and describe (a) the underlying process or systembeing controlled, along with the (b) sensor, (c) actuator and (d) computational element. Ifthe some of the information is not available in the article, indicate this and take a guess atwhat might have been used.5. (MATLAB/SIMULINK) Download the file “hw1ballbeam.mdl” from the course web page,which contains a SIMULINK model of a “ball and beam” experiment, in which you apply atorque to a beam and try to balance a ball that rolls along the beam (see course web pagefor more documentation).(a) Run the simulation with default parameters and create a plot of the ball position versustime. Note that the desired action of the system is to move the ball from its initialposition at the center of the beam to a new resting point at r = 0.25 m.(b) While keeping the gain on ˙α fixed at its default value, vary the gain on α from 75% tothree times the default value. Plot the “overshoot” (the maximum amount by which theball goes past the desired resting point, expressed as a percentage of the commandedposition) as a function of this gain for stable cases.(c) While keeping gain on α fixed at its default value, vary the gain on ˙α from zero to twiceits default value. Give the numerical range of this gain for which the system is stable.Plot the “settling time” (amount of time required for the system to get within 5% of thedesired resting point) as a function of this gain for the stable cases.All plots should included a title, labeled axes (with units), and reasonable axis limits.Note: It is not necessary for you to understand the dynamics of the system and control lawat this point. This model makes use of some advanced features in MATLAB and SIMULINK,which you may need to spend some time
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