Version 1.1 1 of BEFORE YOU BEGIN PREREQUISITE LABS ECE 01 and 0 Labs EXPECTED KNOWLEDGE ECE 03 LAB 1 MATLAB CONTROLS AND SIMULINK Linear systems Transfer functions Step and impulse responses (at the level covered in ECE ) EQUIPMENT Intel PC MATERIALS Formatted 1.44 3¼ floppy diskette (optional) OBJECTIVES After completing this lab you should be more familiar with MATLAB. Specifically, you should be able to use MATLAB to analyze transfer functions and know how to use some of the functions provided in the Control Toolbox and Simulink. INTRODUCTION MATLAB has many tools to help you analyze and design control systems. These tools allow you to determine the response of systems, measure the performance of various controls, and visualize the system dynamics. PRELAB Answer Questions 1 3. MATLAB DEMOS MATLAB comes with many demos that can help you get a grasp of how MATLAB can assist in the design and analysis of control systems. Three of these demos are the Response Demo (respdemo), the RLC Demo (rlcdemo), and the Gain and Phase Margins Demo (margindemo). Typing respdemo, rlcdemo and margindemo at the MATLAB command prompt can access these demos.
Version 1.1 of Take some time to explore these three demos to get and idea of MATLAB s capabilities. In this lab, you will be doing simulations similar to the ones found in these demos. Answer Questions 4 6. MATLAB STRUCTURES Structures allow the grouping of dissimilar data types into a single variable. This is similar to creating a new data type. For example, we can define a variable called curve that can store information about a curve to be plotted in MATLAB. For example, a curve can be represented as the values for the abscissa, ordinate, and line color. For this data, we would want to use vectors to store the information for the abscissa and the ordinate, and a string to hold the information about the line color. We will use the curve defined by y = x + x 1. 1 + We start by declaring the abscissa with the following command: >> x = -:0.001:; >> curve.abscissa = x; This gives us an abscissa in the range of x with a step x = 0. 001. Next, we will define the ordinate as follows: >> curve.ordinate = x.^ +.*x + 1; and the line color as: >> curve.line = 'r'; Now if we type in the name of the structure, in this case curve, MATLAB displays the name of the structure, followed by the structure s elements, which are called fields. >> curve curve = abscissa: [1x4001 double] ordinate: [1x4001 double] line: 'r' The fields for our data structure, curve, are abscissa, ordinate, and line. We can tell from the information above that abscissa and ordinate are vectors (with 4001 elements) and that line is a string. We can access the individual fields of the structure by using the syntax struct.field. Notice that this is the same syntax we used to enter the information when we created the structure. For example, to plot the curve, we would type in the following command: >> plot(curve.abscissa, curve.ordinate, curve.line); The resulting curve is shown in Figure 1. We can store more than one element in our structure curve. We can add a second curve defined by y = by typing the following commands: 3 x
Version 1.1 3 of >> curve().abscissa = x; >> curve().ordinate = x.^3; >> curve().line = 'b'; Figure 1. Plot of Curve x = x = 1. Notice that we had to use an index ( in this case) to add another element to the structures. We have now created a cell array containing the structure curve that has two elements in it: curve(1) and curve(). To access any element in the array, the index must be used. For example, to access the second curve in the array we would have to type curve(). Furthermore, we can no longer access the first element by simply typing curve; we must now type curve(1). Answer Questions 7 8. CELL ARRAYS Cell arrays are arrays whose elements are cells. Each cell in an array can hold any MATLAB data type. Figure shows an example of a x cell array matrix. Each cell in the array matrix contains a different data type. The cell array represented in Figure can be created with the following commands: >> C(1,1) = {magic(3)};
Version 1.1 4 of >> C(1,) = {['Evans' 'Turner' 'McNames']}; >> C(,1) = {7 + 13i}; >> C(,) = {-:}; Cell 1,1 Cell 1, 8 3 4 1 5 9 6 7 'Evans' 'Turner' 'McNames' Cell,1 Cell, 7 + 13i -, -1, 0, 1, Figure. Example of a X Cell Array. The same cell array could also have been entered with the following commands: >> C{1,1} = magic(3); >> C{1,} = ['Evans' 'Turner' 'McNames']; >> C{,1} = 7 + 13i; >> C{,} = -:; To display the contents of an individual cell, we would use the syntax cellarray{i, j} i is the row number j is the column number For example, to display the complex number in row, column 1, we would type >>C{,1} Answer Question 9.
Version 1.1 5 of THE STRUCT COMMAND There may be times when we want to create a structure with data already contained in a cell array. This can be accomplished with the STRUCT command. The syntax for the STRUCT command is s = struct( field1,values1, field,values, ) s = struct( field1,{},field,{}, ) s is the name of the structure to create field1, field, are the names of the structure fields values1, values, are the cell arrays containing the values of the different structure elements The second syntax example creates an empty structure. Answer Questions 10 1. WORKING WITH POLYNOMIALS The POLY Command The POLY command will return the polynomial coefficients of a polynomial equation given a vector that contains the roots of the polynomial. The syntax for the POLY command is p = poly(r) p is a vector containing the polynomial coefficients r is a vector containing the roots of the polynomial For example, suppose we want to determine the polynomial with roots at and 1. Using the POLY command we would type >> p = poly([- -1]) p = 1 3 Therefore, the polynomial with roots at and 1 is x + x + 3. Answer Questions 13 14.
Version 1.1 6 of The ROOTS Command The ROOTS command is the opposite of the POLY command. The ROOTS command returns the roots of a polynomial given a vector that contains the polynomial coefficients. The syntax for ROOTS is r = roots(p) r is a vector containing the roots of the polynomial p is a vector containing the polynomial coefficients If we use the polynomial x + x + 3 from the POLY example, we can determine its roots by typing >> r = roots([1 3 ]) r = - -1 which match the values of the roots we used with the POLY command above. Answer Question 15. The POLYVAL Command The POLYVAL command evaluates a polynomial at a given value or set of values. The syntax for the POLYVAL command is y = polyval(p,x) y is the value of the polynomial at x p is a vector containing the polynomial coefficients x is a vector of values at which to evaluate the polynomial For example, if we wanted to find the value of the polynomial x + 3x + when x = 3, we would type the commands >> p = [1 3 ]; >> y = polyval(p, 3) y = 0 The results tells us that when x = 3, the polynomial equates to 0.
Version 1.1 7 of If we wanted to find the values of the same polynomial for the values x = {1,3,5,7 }, we would type the commands >> y = polyval(p, [1 3 5 7]) y = 6 0 4 7 Answer Question 16. The RESIDUE Command The RESIDUE command converts a quotient of polynomials to pole-residue representation. In other words, it does partial fraction expansion given the ratio of two polynomials. The syntax for the RESIDUE command is [r p k] = residue(num, den) num is a vector containing the polynomial coefficients of the numerator den is a vector containing the polynomial coefficients of the denominator r is a vector containing the residue values p is a vector containing the poles k is a vector of direct terms The best way to describe the RESIDUE command is with an example. Suppose we want to x + 1 perform a partial fraction expansion of. At the command prompt we would type x + 3x + 1 >> num = [1 0 1]; >> den = [ 3 1]; >> [r p k] = residue(num, den) r = -.0000e+000 1.500e+000 p = -1.0000e+000-5.0000e-001 k = 5.0000e-001 From the values of r, p and k we would write the partial fraction expansion as 1.5 0.5 +. x + 1 x + 0.5 The RESIDUE command can also be used to convert the expansion back to a ratio of two polynomials. The syntax for this is [num den] = residue(r, p, k)
Version 1.1 8 of Answer Question 17. TRANSFER FUNCTION MODELS The TF and the TFDATA Commands The TF command is used to create MATLAB models of transfer functions. The most commonly used syntax for TF is sys = tf(num, den) sys is the transfer function num is a vector containing the polynomial coefficients of the numerator den is a vector containing the polynomial coefficients of the denominator s 1 For example, we can create the transfer function H () s = with the following command s + 3s + >> sys = tf([1-1],[1 3 ]) and we get the following results: Transfer function: s - 1 ------------- s^ + 3 s + The TFDATA command undoes what the TF command does. Given a transfer function model, TFDATA returns the coefficient polynomials for the numerator and the denominator. The syntax for the TFDATA command is [num, den] = tfdata(sys) [num, den] = tfdata(sys, v ) num is a cell array containing the polynomial coefficients of the numerator den is a cell array containing the polynomial coefficients of the denominator sys is the transfer function model v returns the polynomial coefficients of the numerator and the denominator as vectors If TFDATA is used without the v argument, then num and den will be cell arrays. If you want num and den to be vectors, the v argument must be used. Answer Questions 18-19
Version 1.1 9 of The ZPK and the ZPKDATA Commands The ZPK is used to convert a transfer function to the zero-pole-gain form. The most commonly used syntax for the ZPK command is sys = zpk(z,p,k) sys = zpk(s) sys is the transfer function z is a vector containing the real or complex zeros p is a vector containing the real or complex poles k is the real valued scalar gain s is a transfer function model (such as one created by the TF command) ( )( ) 3 s 1 s + We can create the pole-zero-gain model of the transfer function H () s = with the ( s + 1) ( s 5) following command: >> sys = zpk([1 -],[-1-1 5],3) Zero/pole/gain: 3 (s-1) (s+) ------------- (s+1)^ (s-5) s 1 We could also create a zero-pole-gain model of the transfer function H () s = with the s + 3s + following commands (and results): >> s = tf([1-1],[1 3 ]); >> sys = zpk(s) Zero/pole/gain: (s-1) ----------- (s+) (s+1) The ZPKDATA returns information about the zero-pole-gain model created by the ZPK command (similar to the TFDATA command). The syntax for ZPKDATA is [z, p, k] = zpkdata(sys) [z, p, k] = zpkdata(sys, v ) z is a cell array containing the zeros
Version 1.1 10 of p is a cell array containing the poles k is the real valued scalar gain sys is the transfer function model v returns the zeros and poles as vectors Just like TFDATA, if ZPKDATA is used without the v argument, then z and p will be cell arrays. If you want z and p to be vectors, the v argument must be used. Answer Questions 0 1. The POLE and ZERO Commands The POLE and ZERO commands are used to extract the poles and zeros of transfer function models. The syntax for the two commands are: p = pole(sys) z = zero(sys) [z, k] = zero(sys) p is a vector containing the real or complex poles z is a vector containing the real or complex zeros k is the gain sys is the transfer function model Answer Question. THE PZMAP COMMAND PZMAP is a command used to crate a pole-zero plot of a linear time invariant system. The syntax for PZMAP is pzmap(sys) pzmap(sys1, sys, ) sys, sys1 and sys are linear time invariant transfer function models Figure 3 shows a pole-zero plot created using PZMAP of the transfer function s 1 H () s =. The Xs represent the poles and the Os represent the zeros. To add plot lines s + 3s + of constant damping ratio and natural frequency, the command SGRID (for s-domain) is used in conjunction with PZMAP.
Version 1.1 11 of PZMAP can also be used to extract the poles and zeros of a linear time invariant system. The syntax for this is [p, z] = pzmap(sys) p is a vector containing poles z is a vector containing the zeros When used to extract the poles and zeros the pole-zero map is not created. Answer Question 3. s 1 Figure 3. Pole-Zero Plot of H () s =. s + 3s +
Version 1.1 1 of LTIVIEW LTIVIEW is used to create various plots to a linear time invariant (LTI) system. The syntax for LTIVIEW is ltiview(sys) sys is a transfer function model When executed, LTIVIEW opens up a viewing window, as shown in Figure 4. Various plots can be selected for viewing by right clicking on the current plot and then selecting Plot Type and then clicking on the desired plot. Experiment around in LTIVIEW by looking at all the available plots it has to offer. Answer Questions 4 5. Figure 4. LTI Viewer.
Version 1.1 13 of LINEAR TIME INVARIANT SYSTEMS THE LSIM COMMAND The LSIM command is used to simulate and plot the response of linear time invariant (LTI) systems to arbitrary inputs. The syntax for LSIM is lsim(sys,u,t) sys is a transfer function model u is a vector of (signal) inputs and must be the same length as t t is a vector of time samples Figure 5 shows an example of a plot created with LSIM. The top plot is a square wave used as the input to the LTI model. The bottom plot is the response to the square wave of the LTI model. s + 5 The transfer function H () s = was used as the LTI system. s + s + 5
Version 1.1 14 of Figure 5. Responce of LTI System to Square Wave. Figure 5 was created with the following commands: >> sys=tf([1 5],[1 5]); >> [u,t]=gensig('square',1,10); >> subplot(,1,1) >> plot(t,u) >> set(gca,'ylim',[-0.1 1.1]) >> subplot(,1,) >> lsim(sys,u,t) Answer Question 6. THE STEP AND IMPULSE COMMANDS STEP and IMPULSE commands are used to plot the response of a system to a step and inpulse signal respectively. The syntax for these two commands are step(sys) step(sys,t) impulse(sys) impulse(sys,t) sys is a transfer function model t is a time vector If the variable t is omitted, the range will be chosen automatically. Figure 6 shows the step and impulse response to the system H () s = s created with the following commands: >> subplot(,1,1) >> step(sys,10) >> subplot(,1,) >> impulse(sys,10) Answer Question 7. s + 5. The figure was + s + 5
Version 1.1 15 of Figure 6. The Response of an LTI System to a Step and Impulse Signal. CONTROL DESIGN THE BODE COMMAND You should already be familiar with MATLAB s BODE command (ECE 0, Lab Experiment 4). The BODE command can also be used with transfer function models to create magnitude and phase Bode plots. The syntax is quite simple: bode(sys) bode(sys1,sys, ) sys, sys1, sys are transfer function models Figure 7 is an example of Bode plots created with the BODE command of the transfer function s + 5 H () s =. s + s + 5
Version 1.1 16 of The BODE command can also be used to return the magnitude and the phase of the transfer function model. [mag, phase, w] = bode(sys) Recall that mag is the absolute magnitude of the transfer function; it is not in units of decibels. Answer Question 8. THE RLOCUS COMMAND MATLAB s RLOCUS command is used to create the root locus of a transfer function model. The syntax for the rlocus command is rlocus(sys) rlocus(sys1, sys, ) sys, sys1 and sys are transfer function models Figure 7. Bode Plots of an LTI System.
Version 1.1 17 of s + 5 Figure 8 shows the root locus plot for the transfer function H () s =. s + s + 5 Answer Question 9. THE RLOCFIND COMMAND The RLOCFIND command can be used to find the gain k at a selected pole position from a plot created using the RLOCUS command. The syntax for the RLOCFIND command is rlocfind(sys) [k, poles] = rlocfind(sys) k is the gain of the system poles is a list of all the system poles for the gain k sys is a transfer function model
Version 1.1 18 of Figure 8. Root Locus Plot of LTI System. After the command has been entered, the figure window will open with a set of crosshairs (Figure 9). Move the cross hairs to the position on the root locus plot you wish to find the gain and poles for and click the left mouse button. RLOCFIND will display the results at the command prompt. For example, the position of the crosshairs in Figure 9 resulted in the values selected_point = -5.9747e+000 +4.3511e+000i ans = 9.9553e+000 Answer Questions 30 31. Figure 9. RLOCFIND with Cross Hairs.
Version 1.1 19 of SIMULINK Simulink is a MATLAB modeling tool which can be used to model control systems. We will be s s + 6 using Simulink to model the transfer function H () s = and determine the step 3 s + s + 5s response. To enter Simulink, type simulink at the MATLAB command prompt. This will open up the Simulink Library Browser (Figure 10). The lower right pane shows the libraries available to Simulink. Take a few minutes to explore the different links to get familiar with what s available to Simulink. Create a new model by selecting File New Model. This will open the New Model window. Now we will begin to build our model. To add a part to the model, all we have to do is drag and drop the part from the library to the model window. We will start by adding the step function. The step function can be found in the Source Library. Open the Source Library by double clicking on its icon. Scroll down until you see the Step icon. Drag and drop the Step icon into the model window. Next, add the Sum and the Slider Gain controls to the model. Both of these items can be found in the Math Library. Now add the Transfer Function control from the Continuous Library and the Scope control from the Sinks Library. Your model should start to look like the one in Figure 11.
Version 1.1 0 of Figure 10. Simulink Library Browser. In order to get the proper results, we have to change the values of the symbols in the model. Start by double clicking on the Sum icon. This will open up the Block Parameters dialog box. In the List of signs: text box you will see two plus signs (++). Change the contents of this box to a plus sign and a minus sign (+-) and then click OK. See Figure 1. Double click the Slider Gain icon and change the value in the middle text box from 1 to 0.38 and click OK. Notice that this is the value, or close to the value, you found using the RLOCFIND command. Double-click on the Transfer Function icon and type in the polynomial coefficients for the s s + 6 numerator and the denominator for the transfer function H () s = (see Figure 13) 3 s + s + 5s and click on OK. Now that all the blocks are in place and all the values are set, we need to connect the blocks.
Version 1.1 1 of Figure 11. Simulink Model of Control System. Figure 1. Block Parameters: Sum Dialog Box. If we look at the Step icon, we will notice a > symbol pointing out of the block; this is an output port. Similarly, looking at the Scope icon we will see a > pointing into the block; this is an input port. If we look at all the other blocks, we notice that they have various input and output ports as well. The input and output ports are we will be connecting the blocks together. To connect the Step icon to the Sum icon, place the cursor over the output port of the Step icon. Notice how the cursor becomes a crosshair. Hold down the left mouse button and move the crosshairs over to the left input port of the Sum icon. When you reach the Sum icon s input port, the crosshairs change to a double-lined crosshairs. Release the mouse button to complete the
Version 1.1 of connection. Repeat this procedure to connect the rest of the model. Your model should now look like the one in Figure 11. Now we are ready to run the simulation. Figure 13. Block Parameters: Transfer Function Dialog Box. From the menu toolbar, select Simulation Start. After the simulation is complete, double-click on the Scope icon to see the response of the system. Your plot should look like that of Figure 14. Answer Questions 3 38. Figure 14. Output of Scope From Simulink Model. Before exiting Simulink, you may want to save your model (it may be useful for future labs).