Bfh Ti Control F Ws 2008/2009 Lab Matlab-1

Transcription

1 Bfh Ti Control F Ws 2008/2009 Lab Matlab-1 Theme: The very first steps with Matlab. Goals: After this laboratory you should be able to solve simple numerical engineering problems with Matlab. Furthermore, you will have a basis to successfully apply Matlab to problems in control, signal-processing, and many other engineering branches. Documentation. We propose that you work with the on-line documentation coming with the Matlab system. You may obtain that documentation as follows: 1. In a shelltool (a Terminal window) start Matlab by the command matlab; you are prompted with a window named MATLAB. Depending on the window system that you use, you may alternatively start Matlab from a button. 2. In the window named MATLAB use from the Help button the menu MATLAB Help. There are two types of documentation sets; first, there is a html documentation; second, all paper-type manuals are likewise available in pdf format (see Printable Documentation (in PDF)). 3. From the html documentation set we propose to begin with Getting Started. From the pdf manuals we propose to start with Getting Started with MATLAB. Hints: During your work with Matlab you will find the help utilities convenient (at the Matlab prompt >> type help help and help lookfor). Nevertheless, you may want to read more in the various manuals accompanying the Matlab system. Some of the problems are easier to solve if you follow the explanations given during the lectures. This advise does not mean, however, that it is impossible to achieve the goals in a self-study manner. Assignments: You should solve all of the problems below except Problem 5. If you wish to also solve Problem 5, you should contact me (gtj); I will give you some additional explanations. Problem 1 Just for fun: Enjoy some demos; if you read the accompanying texts, you will surely better understand the aim of particular demos. Hint: Type demo into the Matlab command window to obtain the Matlab Demo Window. 1

2 Problem 2 Some simple commands: Assign values to some variables such as a =1.0 b =2.0 c =3.0 then save these variables and exit Matlab. Restart Matlab and load the variables again. Use the command who to see your variables; use the command whos to get more information about them. Type the names of the variables to get their values. clc Use the following commands in sequence: computer eps (this gives you the machine epsilon) who clear who Type the command help and subsequently try some helps such as help general help who help load help save help demo Hint: You may want to type the command more on to obtain page formatting. As in the Unix tc shell (tcsh), you then advance for one single line by typing the <Return> key, and you advance a whole page by typing the <SpaceBar> key. The command more off inhibits page formatting. Problem 3 Plotting: The numbers below may be data collected from a wind tunnel test; each line contains a flight path angle in degrees and a corresponding coefficient of lift: 2

3 Flight Path Angle [degrees] Coefficient of Lift Enter the matrix wind as wind = [ then save these data with ]; save wind to create the file wind.mat. With the data from wind.mat create an xy-plot with the Coefficient of Lift as a function of the Flight Path Angle. Give a title to the plot, label the axis, and set the grid; also use the command axis. Finally, write an m-file which automatically does the job of reading-in the data from wind.mat, creating the plot, setting the grid and the axis, and labeling it. Problem 4 Printing, the colon operator: In the following problems, use the colon operator to generate the vectors specified, then print the resulting tables. Include a table heading and a column heading. 1. Generate a table of conversions from degrees to radians in 10 steps. (The first line should contain 0,10,...,360, the second line the corresponding numbers in radians. Recall that π radians = 180.) 3

4 2. Generate a table of conversions from radians to degrees with radian increments of π/ Generate a table of conversions from inches to centimeters from 0.0 inch to 20.0 inch in steps of 0.5inch. 4. Generate a table of conversions from centimeters to inches from 0.0 cm to 50.0 cm in 2.0 cm steps. 5. Assume the following currency conversions: Generate various tables of conversion. $1 = 1.3sFr, 1yen = $0.0079, 1.5DM = $1. Problem 5 Plotting, computing difference equations: We consider a voltage divider circuit see explanations during the lecture. 1. For the linear voltage divider, that is, the voltage divider where both resistors R 1 and R 2 are linear resistors, plot the lines f l (v 2 ) and f r (v 2 ) for numerical values of R 1 and R 2 such that a ˆ= R 2 /R 1 < 1 and a > 1. Indicate in these plots the iterations according to v 2 [k + 1] = αv 2 [k] + βv 1 where α is either a or 1/a. Assume that v 1 = For the linear voltage divider, program the difference equations x[k + 1] = ax[k] + b (v 1 = 1), z[k + 1] = 1 a z[k] + b a (v 1 = 1), where v 2 [k] is either x[k] or z[k] (normal time and inversed time, respectively). Experiment with these equations, that is, for values a > 1 and a < 1 compute the sequences x[ ] and z[ ] starting from various initial conditions x[0] and z[0]. 3. Repeat items 1) and 2) for a nonlinear voltage divider circuit resulting if we replace the resistor R 2 by a diode, whose characteristic equation is i Diode = I s (exp { vdiode V T } ) 1. For simplicity assume the numerical values R 1 = 1, I s = 1, V T = 1, and v 1 = 1. 4

5 Problem 6 Linear systems in Matlab simple commands useful in control problems: We consider the simple resistor-inductor-capacitor (Rlc) circuit of Figure 1. Thereby, we assume that all three elements, the resistor, the inductor and the capacitor, are linear elements. i S(t) G C L i G i C i L Figure 1: A simple Rlc-circuit. Obviously, the input to the circuit is the current i S (t) impressed by the independent current source; as output we consider the inductance current i L (t). Because the circuit is linear, its behavior may be described in the Laplace domain by way of its transfer function: I L (s) I S (s) = n 0 d 2 s 2 + d 1 s + d 0 ˆ= H(s), where I L (s) is the Laplace-domain representation of the output time-domain current signal i L (t), I S (s) is correspondingly the Laplace-domain representation of the input timedomain current signal i S (t), and where the coefficients n 0, d 0, d 1, and d 2 are given as functions of the circuit elements G, C, and L. 1. Obtain the transfer function H(s) symbolically, that is, expressed as a function in the symbols G, C, and L. 2. Select numerical values for the circuit elements. (To start with, you may use R = 1/G = 1Ω, C = 1F, and L = 1H.) Obtain the transfer function H(s) numerically with your values selected. 3. One way to specify a linear system to Matlab is to do it by way of its transfer function. Assuming that the transfer function H(s) is in numerical form, an example being H(s) = 2 3s 2 + 5s + 7, then we could define two vectors numerator and denominator as and specify the system to Matlab by 1 numerator = [2] ; denominator = [3 5 7] ; mysystem = tf(numerator,denominator) ; Obtain the Bode diagrams describing your circuit. (Hint: At the Matalb prompt >>, use help bode). 1 Note that the names you give to the vectors is not important; you could likewise use, for example, gugus and gaga. It is the order in which you specify the vectors to the function tf() that counts: tf() interprets the first given vector as the vector of numerator-polynomial coefficients, and the second given vector as the vector of denominator-polynomial coefficients. 5

6 4. Obtain the step response describing your circuit. (Hint: At the Matalb prompt >>, use help step). 5. Obtain the impulse response describing your circuit. (Hint: At the Matlab prompt >>, use help impulse). 6. Now use more realistic numerical values for your circuit elements; repeat the above items 2 to 5 for the new values. (Hint: We have selected two sets of values, the first set being R = 1/G = 4Ω, C = 1nF, and L = 7mH; the second set being R = 1/G = 1kΩ, C = 100nF, and L = 7mH.) 7. When using help on a Matlab command, you see towards the end of the given description a section called see also. Study some of the mentioned commands found in the see also sections of help bode and help step and experiment with these commands. (Hint: You may want to test commands such as ltiview, freqresp, initial, and lsim.) Problem 7 Linear continuous-time filters in Matlab simple commands useful in filter-design problems: We start with considering the transfer function of a first-order system such as an Rc circuit: 1 H 1 (s) = 1 + st, (1) where T is the time constant and 1/T gives the (3dB) cut-off frequency in rad/sec. 1. Select the time constant T such the the (3dB) cut-ff frequency is 10Hz. Obtain the Bode diagrams of your filter. (Hint: Pay attention to the difference between natural frequencies coming in Hz and radian frequencies coming in rad/sec.) 2. Generate a signal x(t) as the superposition of a 1Hz and a 100Hz sinusoid: x(t) = sin(2π 1 t) + sin(2π 100 t). (2) Plot that signal x(t). (Hint: You have to generate a vector of time points and a second vector of signal values at these time points. To obtain a smooth plot, you need about 20 samples per period; this rule of thumb together with the larger of the two frequencies indicates you how to select the interval t between two samples.) 3. Filter the signal x(t) from item 2 above with the filter from item 1. (Hint: Use help on lsim.) Display input- and output-signals in a plot containing two subplots. (Hint: Use help on subplot.) 4. The transfer function (1) is the lowest-order transfer function in the Butterworth family of filter functions. For designing Butterworth filters, Matlab knows the command butter. Verify that you indeed re-obtain the filter from item 1 using the command butter. (Hint: Use help on butter. Pay attention to using the command butter with the option s the usually used Laplace variable indicating that you are designing a continuous-time filter.) 5. Design a second-order Butterworth filter realizing the same cut-off frequency as the first-order Butterworth filter from item 4. Obtain the Bode diagrams. Filter the signal x(t) in (2) with this second-order Butterworth filter. Display input- and output-signals in a plot containing two subplots. 6

7 6. Obtain the step-responses of your two Butterworth filters. Plot them in a single plot. 7. Experiment with higher-order Butterworth filters. Repeat items 5 and 6 for Butterworth filters of orders three, four, and five. 8. When you read the answer to >> help butter, you find in the see also section other filter types that can be designed. Design Bessel-, Chebyshev-, and elliptic filters and experiment with them. What about designing other than low-pass filters? Design high-pass, band-pass, and band-stop filters. 7