ECE 351 – Linear Systems II MATLAB Tutorial #5 Modeling Discrete Time Systems in Simulink This tutorial describes the use of Simulink, a graphical user interface within MATLAB. Using Simulink, the block diagrams of discrete time systems may be entered graphically into MATLAB and analyzed. Getting Started with Simulink In this section we will illustrate a very simple use of Simulink to display the sum of two sequences. 1. Open MATLAB and in the command window, type: simulink at the prompt. Alternatively, there is a Simulink icon in the menu bar. 2. After a few seconds Simulink will open and the Simulink Library Browser will open as shown in figure 1. It is important to note that the list of libraries may be different on your computer. The libraries are a function of the toolboxes that you have installed. At a minimum you should have the Simulink and Simulink Extras libraries on your list. Figure 1. Simulink Library Browser3. Click on the Create a New Model icon in the Library Browser window. An additional window will open. This is where you will build your Simulink models. 4. Click on the “+” sign next to “Simulink” in the Library Browser. A list of sub-libraries will appear including Continuous, Discrete, etc. These sub-libraries contain the most common Simulink blocks. 5. Click once on the “Sources” sub-library. You should see a listing of blocks as shown in the right column of figure 2. Figure 2. Source Blocks in the Simulink Library 6. Drag the simin block from the Simulink: Sources library into the model window. This source block is used to bring data from the MATLAB workspace into the Simulink environment. 7. Return to the MATLAB command window. Type the following commands to generate two discrete time sequences, x1 and x2.>> clear >> k=0:10; >> x1=(1/2).^k; >> x2=(-1/2).^k; 8. In order to transfer the sequence x1 into the Simulink model, double click on the simin block and enter the values as shown in figure 3. The data field input is a matrix with the time or sequence index (column 1) and values (column 2) vectors that were set up in the workspace. Additionally, the time sampling rate that was used to generate these values is entered. Note that the k and x1 vectors are converted to column vectors in the parameters window. Figure 3. Parameters for simin Block 9. Add a second simin block to the workspace and apply the sequence x2 to this input. 10. Open the Math Operations Library and drag the sum block into the model window. The model window should now appear similar to the one shown in Figure 4.Figure 4. Simulink Model Window 11. Next we want to connect the inputs to the summer. Move the mouse over the output terminal of the top simin block until it becomes a crosshair. Click and drag the wire to one of the summer input terminals. The cursor will become a double cursor when it is in the correct position. Release the mouse button and the wire will snap into place. Repeat to connect the second simin block to the second summer input. Your model should now appear as shown in figure 5. Figure 5. Simulink Model Window 12. In order to display the sum of the two sequences we want to export the output of this model back to the MATLAB workspace. Open the Sinks Library and drag the simout block to model window. Connect to the output of the summer as shown in figure 6.Figure 6. Simulink Model Window 13. Double click on the simout sink and change the save format selection to: Array. 14. Select Simulation à Configuration Parameters from the model window. Note that the default Stop Time is 10, which is fine for our example. Under “Solver Options” set the Type to Fixed-Step and the Solver to Discrete. Set the Fixed Step Size to 1 to match the increment between input values. Click OK and run the simulation by selecting Simulation à Start. 15. Return to the MATLAB command window. Note that a sequence simout is now present in the workspace. Plot the two inputs and the output with the following MATLAB commands: >> subplot(3,1,1) >> stem(k,x1) >> subplot(3,1,2) >> stem(k,x2) >> subplot(3,1,3) >> stem(k,simout) Modeling a Discrete Time System in Simulink 1. Open a new Simulink model window. Input the diagram shown in figure 7. Note the following: a. The unit delay is found in the Discrete Library. b. After dragging a summer to the model window, double click on its icon. Modify the List of Signs in the window that opens to be: +- instead of the default ++. c. The Gain block is found in the Math Operations Library. After dragging the Gain block to the model window, select Format à Flip block to change its orientation as shown. d. Double click on the Gain block to change its gain to ½.Figure 7. Discrete Time System 2. Return to the MATLAB command window and type the following commands to set the input sequence to be a unit step sequence from 0 ≤ k ≤ 10. >> clear >> k=0:10; >> x=ones(1,11); 3. Apply this input to the simin block as described in the previous section. 4. Set the simulation parameters as described in the previous section and run the simulation. Again, the stop time should be 10. Return to the MATLAB command window and plot the output sequence. Confirm that your output is as shown in figure 8 below. Figure 8. Output of Discrete Time System State Variable Representation of Discrete Time Systems The discrete time system in figure 7 can be represented using the following state variable equations:qk +1= Aqk+ Bxkyk= Cqk+ Dxk where A=[-1/2] B = [1] C=[-1/2] D = [1] Note that in this simple case A, B, C, and D are simple 1x1 elements, but in general they will be matrices. Before performing the steps below, DO NOT clear the MATLAB workspace as you will continue to need the inputs and output found in the previous section. 1. From the MATLAB command window, enter the following commands to input these four matrices: >> A=[-1/2]; >> B=[1]; >> C=[-1/2]; >> D=[1]; 2. Open a new Simulink window and create the model shown below in figure 9. Note that the Discrete State-Space block is found in the Discrete Library. Figure 9. State Variable Simulink Model 3. Double click on the simin block and apply the same step sequence input as used in the previous example. 4. Double click on the Discrete State Space block and set the parameters as shown in figure 10 below. Note that rather than inputting the A, B, C, D matrices directly we are