automatic control system lab
Laboratory Exercise 3A
Simulink Modeling – Train System
Introduction: Simulink Modeling
In Simulink, it is very straightforward to represent and then simulate a mathematical model representing a physical system. Models are represented graphically in Simulink as block diagrams. A wide array of blocks is available to the user in provided libraries for representing various phenomena and models in a range of formats. One of the primary advantages of employing Simulink (and simulation in general) for the analysis of dynamic systems is that it allows us to quickly analyze the response of complicated systems that may be prohibitively difficult to analyze analytically. Simulink is able to numerically approximate the solutions to mathematical models that we are unable to, or don't wish to, solve "by hand."
In general, the mathematical equations representing a given system that serves as the basis for a Simulink model can be derived from physical laws. In this page we will demonstrate how to derive a mathematical model and then implement that model in Simulink.
Contents
· Train system
· Free-body diagram and Newton's second law
· Constructing the Simulink model
· Running the model
In this example, we will consider a toy train consisting of an engine and a car. Assuming that the train only travels in one dimension (along the track), we want to apply control to the train so that it starts and comes to rest smoothly, and so that it can track a constant speed command with minimal error in steady state.
The mass of the engine and the car will be represented by M1 and M2, respectively. Furthermore, the engine and car are connected via a coupling with stiffness k. In other words, the coupling is modeled as a spring with a spring constant k. The force F represents the force generated between the wheels of the engine and the track, while μ represents the coefficient of rolling friction.
Free-body diagram and Newton's second law
The first step in deriving the mathematical equations that govern a physical system is to draw the free-body diagram(s) representing the system. This is done below for our train system.
From Newton's second law, we know that the sum of the forces acting on a body is equal to the product of the mass of the body and its acceleration. In this case, the forces acting on the engine (M1) in the horizontal direction are the spring force, the rolling resistance, and the force generated at the wheel/track interface. The forces acting on the train car (M2) in the horizontal direction are the spring force and the rolling resistance. In the vertical direction, the weight forces are balanced by the normal forces applied by the ground (N = mg). Therefore, there will be no acceleration in the vertical direction.
We will model the spring as generating a force that is linearly proportional to the deformation of the spring, k (x1 – x2), where x1 and x2 are the displacements of the engine and car, respectively. Here it is assumed that the spring is un-deformed when x1 and x2 equal zero. The rolling resistance forces are modeled as being linearly proportional to the product of the corresponding velocities and normal forces (which are equal to the weight forces).
Applying Newton's second law in the horizontal direction based on the above free-body diagrams leads to the following governing equations for the train system.
Week One: Constructing the Simulink model
This set of system equations can now be represented graphically without further manipulation. Specifically, we will construct two copies (one for each mass) of the general expression or .
1. Open the Simulink library, and open a new model window by clicking the” New Model” icon.
2. Drag two Sum blocks (from the Simulink Library Browser) into your model window and place them approximately as shown in the figure below.
The outputs of each of these Sum blocks represent the sum of the forces acting on each mass.
Tip: You can obtain any blocks needed for this lab by typing its name into the Search box in the Library Browser window, then drag and drop the block into the model window.
3. Multiplying each output signal by 1/M will give us the corresponding acceleration of each mass. To do this, drag two Gain blocks into your model and attach each one with a line from the output of one of the Sum blocks.
4. Label these two signals as "Sum_F1" and "Sum_F2" in order to make your model clearer. This is accomplished by double-clicking in the space above each of the two signal lines and entering the desired label.
5. These Gain blocks should contain 1/M for each of the masses. Double-click on the upper Gain block and enter "1/M1" into the Gain field. Similarly, enter "1/M2" in the Gain field of the second Gain block.
6. You will notice that the gains did not appear in the image of the Gain blocks, rather the blocks display a value of -K-. This is because the blocks are too small on the screen to show the full variable name inside the triangle. The blocks can be resized so that the actual gain value can be seen. To resize a block, select it by clicking on it once. Small squares will appear at the corners. Drag one of these squares to stretch the block. Your model should appear as below.
7. The outputs of these gain blocks are the accelerations of each of the masses (the train engine and car). The governing equations we derived above depend on the velocities and displacements of the masses. Since velocity can be determined by integrating acceleration, and position can be determined by integrating velocity, we can generate these signals employing integrator blocks. Drag a total of four Integrator blocks from the Continuous library into your model, two for each of our two accelerations.
8. Connect these blocks and label the signals as shown below. Specifically, the first integrator takes the acceleration of mass 1 ("x1_ddot") as an input and generates the velocity of mass 1 ("x1_dot"). The second integrator then takes this velocity and outputs the displacement of the first mass ("x1"). The same pattern holds for the integrators for the second mass.
9. Now, drag two Scopes from the Sinks library into your model and connect them to the outputs of these integrators. Label them "x1" and "x2".
10. Now we are ready to add the forces acting on each mass. First, we need to adjust the inputs on each Sum block to represent the proper number of forces (we will worry about the signs later). Since there is a total of three forces acting on mass 1, double-click on the corresponding Sum block and change the List of signs field to "|+++". The symbol "|" serves as a spacer. There are only 2 forces acting on mass 2, therefore, we can leave that Sum block alone for now.
11. The first force acting on mass 1 is just the input force, F. Drag a Signal Generator block from the Sources library and connect it to the uppermost input of the corresponding Sum block. Label this signal as "F".
12. The next force acting on mass 1 is the rolling resistance force. Recall that this force is modeled as follows:
To generate this force, we can tap off the velocity signal and multiply by an appropriate gain. Drag a Gain block into your model window.
13. Tap off the "x1_dot" signal and connect it to the input of this new Gain block (draw this line in several steps if necessary).
14. Connect the output of the Gain block to the second input of the Sum block.
15. Double-click the Gain block and enter "mu*g*M1" into the Gain field.
16. The rolling resistance force, however, acts in the negative direction. Therefore, change the list of signs of the Sum block to "|+--". The screen shot below has “+-+”, which is wrong. Next, resize the Gain block to display the full gain and label the output of the Gain block "Frr1". Your model should now appear as follows.
17. The last force acting on mass 1 is the spring force. Recall that this force is equal to the following.
Therefore, we need to generate a signal which we can then be multiplied by a gain k to create the force. Drag a Subtraction block (or a Sum block or an Addition block) below the rest of your model. In order to change the direction of this block, right-click on the block and choose Format > Flip block from the resulting menu. Alternatively, you can select the block then hit Ctrl-I.
18. Now, tap off the "x2" signal and connect it to the negative input of the Subtract block. Also, tap off the "x1" signal and connect it to the positive input. This will cause signal lines to cross. Lines may cross, but they are only actually connected where a small block appears (such as at a tap point).
19. Now, we can multiply this difference by the spring constant to generate the spring force. Drag a Gain block into your model to the left of the Subtraction block.
20. Change the value of the Gain block to "k" and connect the output of the Subtract block to its input. Then connect the output of the Gain block to the third input of the Sum block for mass 1 and label the signal "Fs". Your model should appear as follows:
21. We can now apply forces to mass 2. For the first force, we will use the same spring force we just generated, except that it is applied to mass 2 in the positive direction. Simply tap off the spring force signal "Fs" and connect it to the first input of the Sum block for mass 2.
22. The last force applied to mass 2 is its rolling resistance force. This force is generated in an analogous manner to the rolling resistance force applied to mass 1. Tap off the signal "x2_dot" and multiply it by a Gain block with value "mu*g*M2". Then connect the output of the Gain block to the second input of the corresponding Sum block and label the signal "Frr2". Next, change the second input of the F2 Sum block to be negative, which will lead to the following model:
23. Now the model is complete. We simply need to supply the proper input and define the output of interest. The input to the system is the force
24. Tap a line from the "x1_dot" signal and connect it to the Scope block. Label this scope as "x1_dot" and your model should appear as in the following:
Now, the model is complete and should be saved . Provide a screen capture with your name embedded in the diagram for the lab report.
Before running the model, we need to assign numerical values to each of the variables used in the model. For the train system, we will employ the following values.
· M1 = 2 kg
· M2 = 1 kg
· k = 1 N/sec
· F = 1 N
· μ = 0.02 sec/m
· g = 9.8 m/s^2
25. Create a new m-file (by going to the MATLAB Command window and clicking ‘New Script” on the top lift of the window) and enter the following commands.
M1 = 2;
M2 = 1;
k = 1;
F = 1;
mu = 0.02;
g = 9.8;
Save and run your m-file to define these values, making sure that the m-file is in the same directory as the model. Simulink will recognize these MATLAB variables for use in the model.
26. Now, we need to give an appropriate input to the engine. Double-click on the Signal Generator block (outputs "F").
27. Select square from the Wave form drop-down menu and set the Frequency field to equal "0.001". You may leave the Units as the default Hertz. Also enter "-1" into the Amplitude field (positive amplitude steps negative before stepping positive).
28. The last step before running the simulation is to select an appropriate simulation time. To view one cycle of the 0.001 Hz square wave, we should simulate the model for 1000 seconds. Select Parameters from the Simulation menu at the top of the model window and change the Stop Time field to "1000". Close the dialog box.
29. Now, run the simulation and open the "x1_dot" scope to examine the velocity output (hit autoscale). The input was a square wave with two steps, one positive and one negative. Physically, this means that the engine first went forward, then backward. The velocity output reflects this.
Provide a screen capture with your name embedded in the diagram for the lab report.
Save your results.
Questions :
1. Simulate and describe the impact on your model when mass of the engine only is increased by factor of 5?
2. Simulate and describe the effect if the same system is to be simulated in a different environment where gravitational constant is three times of what is on earth?
3. For your lab report, section data and results section, include the values of variables used to generate plot shown in Step 25.
End Week One
Here is Part A screenshots pictures of the work done in matlab
Part B
Laboratory Exercise 3B
Simulink Modeling – Train System
Simulink Control
Week of 2 of this lab, we will look at the following:
· The open-loop plant model
· Implementing a PID controller in Simulink
· Running the closed-loop model
· Extracting a model into MATLAB
The open-loop plant model
In week 1 of this lab, we demonstrated how Simulink can be employed to simulate a physical system. More generally, Simulink can also simulate the complete control system, including the control algorithm in addition to the physical plant. As mentioned previously, Simulink is especially useful for generating the approximate solutions of mathematical models that may be prohibitively difficult to solve "by hand." For example, consider that you have a nonlinear plant. A common approach is to generate a linear approximation of the plant and then use the linearized model to design a controller using analytical techniques. Simulink can then be employed to simulate the performance of your controller when applied to the full nonlinear model. Simulink can be employed for generating the linearized model and MATLAB can be employed for designing the controller as described in the other Introduction pages. Various control design facilities of MATLAB can also be accessed directly from within Simulink. We will demonstrate both approaches in this page.
Recall the Simulink model of the toy train system that was built in part 1 and pictured below.
Assuming that the train only travels in one dimension (along the track), we want to apply control to the train engine so that it starts and comes to rest smoothly, and so that it can track a constant speed command with minimal error in steady state.
Week Two: Implementing a PID controller in Simulink
1. Open the model created last week.
2. Let us first create the structure for simulating the train system in unity feedback with a PID controller. In order to make our Simulink model more understandable, we will first save the train model into its own subsystem block. To accomplish this, delete the three scope blocks and replace each one by an Out1 block from the Sinks library.
3. Label each Out1 block with the corresponding variable name, "x1_dot", "x1", and "x2".
4. Delete the Signal Generator block and replace it with an In1 block from the Sources library.
5. Label this input "F" for the force generated between the train engine and the railroad track. Your model should now appear as follows:
6. Next select all of the blocks in your model (Ctrl A) and right click from the model to choose Create Subsystem (Ctrl G). With a little rearranging and relabeling, your model will appear as shown below.
7. Now we can add a controller to our system. We will employ a PID controller which can be implemented using a PID Controller block from the Continuous library. Placing this block in series with the train subsystem, your model will appear as follows.
8. In the following, we model the controller as generating the force "F" directly. This neglects the dynamics with which the train engine generates the torque applied to the wheels, and subsequently neglects the dynamics of how the force is generated at the wheel/track interface. This simplified approach is taken at this point since we only wish to introduce the basic functionality of Simulink for controller design and analysis.
Double-clicking on the PID Controller block, we will initially set the Integral (I) gain field equal to 0 and will leave the Proportional (P) and Derivative (D) gains as their defaults of 1 and 0, respectively.
9. Add a Sum block from the Math Operations library.
10. Double-click on this block and modify the List of signs field to "|+-".
11. Since we wish to control the velocity of the toy train engine, we will feedback the engine's velocity. This is accomplished by 4ring a line off of the "x1_dot" signal and connecting it to the negative sign of the Sum block. The output of the Sum block will be the velocity error for the train engine and should be connected to the input of the PID Controller block. Connecting the blocks as described and adding labels, your model should appear as follows.
12. Add a Signal Builder block from the Sources library to represent the velocity commanded to the train. Since we wish to design a controller to bring the train smoothly up to speed and smoothly to rest, we will test the system with a velocity command that steps up to 1 m/s followed by a step back down to 0 m/s (recall that our system is a toy train). To generate this type of command signal, double-click on the Signal Builder block. Then choose Change time range from the Axes menu at the top of the block's dialog window.
13. Set the Max time field to "300" seconds.
14. Set the step up to occur at 10 seconds and the step down to occur at 150 seconds. This is accomplished by clicking on the corresponding portions of the signal graph (left and right vertical lines) and either dragging the line to the desired position, or entering the desired time in the T field at the bottom of the window. When done, your signal should appear as follows.
15. Also add a Scope block from the Sinks library and use it to replace the Out1 block for the train's velocity. Relabeling the blocks, your model will appear as follows.
We are now ready to run the closed-loop simulation.
Running the closed-loop model
Before running the model, we need to assign numerical values to each of the variables used in the model. For the train system, we will employ the following values.
· M1 = 2 kg
· M2 = 1 kg
· k = 1 N/sec
· F = 1 N
· μ = 0.02 sec/m
· g = 9.8 m/s^2
16. Create a new m-file and enter the following commands.
M1 = 2;
M2 = 1;
k = 1;
F = 1;
mu = 0.02;
g = 9.8;
17. Execute your m-file in the MATLAB command window to define these values. Simulink will recognize these MATLAB variables for use in the model.
18. We need to set the time for which our simulation will run to match the time range of the command from the Signal Builder block. This is accomplished by selecting Model Configuration Parameters from the Simulation menu at the top of the model window and changing the Stop Time field to "300".
19. Run the simulation and open the "x1_dot" scope to examine the velocity output (hit autoscale). The result as shown below demonstrates that the closed-loop system is unstable for this controller.
(Capture this for Lab Report)
20. Since the performance achieved above is unsatisfactory, we need to redesign our controller. We will first demonstrate how to extract a model from Simulink into MATLAB for analysis and design. Then we will demonstrate how to design the control from directly within Simulink.
Extracting a model into MATLAB
The Simulink Control Design toolbox offers the functionality to extract a model from Simulink into the MATLAB workspace. This is especially useful for complicated or nonlinear simulation models. This is also useful for generating discrete-time (sampled) models. For this example, let us extract a continuous-time model of our train subsystem.
21. First we need to identify the inputs and outputs of the model we wish to extract. The input to the train system is the force F. We can designate this fact by right-clicking on the signal representing "F" (output of the PID block) and choosing Linearization Points > Input Point from the resulting menu.
22. Likewise, we can designate the output of the train system by right-clicking on the "x1_dot" signal and choosing Linearization Points > Output Point from the resulting menu. These inputs and outputs will now be indicated by small arrow symbols as shown in the following figure.
23.
Note: the figure above was likely generated in an earlier version of MATLAB. Your input and output symbols may look different.
24. We can now extract the model by opening the Linear Analysis Tool. This is accomplished by selecting Control Design > Linear Analysis from under the Analysis menu at the top of the model window. Following these steps will open the window shown below.
25. This tool generates an LTI object from a (possibly nonlinear) Simulink model and allows you to specify the point about which the linearization is performed. Since our Simulink model is already linear, our choice of operating point will have no effect and we can leave it as the default Model Initial Condition. In order to generate the linearized model, select the Linearize button in the above figure, which is indicated by the green triangle. The Linear Analysis Tool window should now appear as shown below.
Capture the Step Plot below for report:
26. Inspecting the above, the step response of the linearized model was automatically generated. Comparing this step response to the one generated by the simulation of the open-loop train system in the Introduction: Simulink Modeling page, you can see that the responses are identical. This makes sense since the simulation model was already linear. Additionally, the linearization process generated the object linsys1 shown in the Linear Analysis Workspace above. This LTI object can be exported for use within MATLAB by simply dragging the object into the MATLAB Workspace window immediately above the Linear Analysis Workspace window.
27. Having extracted this model, we can now employ all of the facilities that MATLAB offers for controller design. For example, let us employ the following commands to generate and analyze the closed-loop system reflecting the Simulink model created above. Report your results below:
sys_cl = feedback(linsys1,1);
pole(sys_cl)
ans =
-0.9747 + 0.0000i
0.0000 + 0.0000i
-0.2087 + 1.1332i
-0.2087 - 1.1332i
Examination of the above demonstrates that the closed-loop system in its current state has poles with positive real part and, therefore, is unstable. This agrees with the result of our closed-loop simulation from above. We can then employ MATLAB or Simulink to design a new controller.
Questions :
1. In step 19, how can you tell that system is unstable from the plot?
2. Use Simulink scope block and plot acceleration, velocity and displacement of the two individual masses in the train model. There should only be two plots.
3. Write out the mathematical relation between acceleration, velocity and displacement?
4. In the train model, we are taking into account force along the horizontal axis but why not along the vertical axis?
Here is screenshots for the work done in part B by matlab
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