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Rose-Hulman ECE 320 - Quadratic Optimal Control

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Lab 4: ITAE, Deadbeat, and Quadratic Optimal Control Overview In this lab you will be controlling the systems you previously modeled using an ITAE, a deadbeat, and quadratic optimal controller (one of each type of controller.). For each system you modeled, you need to plot and examine the difference between how the model predicts the response and how the system actually responds. You will need to set up a folder for Lab 4 and copy all files from the folder basic_files into this folder. You will need your Simulink model and the closedloop_driver.m file from your homework for this lab. Special note: When comparing the predicted response and the actual response, I do not want to see a graph that shows very little. For example, if both the predicted response and the real response have settled within 0.5 seconds, I do not want to see a graph that goes out to 4 seconds. Design Specifications: For each of your systems, you should try and adjust your parameters until you have achieved the following: Torsional Systems (Model 205) • Settling time less than 0.5 seconds. • Absolute value of the steady state error less than 2 degrees for a 15 degree step, and less than 1 degree for a 10 degree step (the input to the Model 205 must be in radians!) • Percent Overshoot less than 10% Rectilinear Systems (Model 210) • Settling time less than 0.5 seconds. • Absolute value of the steady state error less than 0.1 cm for a 1 cm step, and less than 0.05 cm for a 0.5 cm step • Percent Overshoot less than 10% You should start with the lower step amplitudes, and work your way up to the higher step amplitudes. Not all systems or controllers will work for a 1 cm step or a 15 degree step. Your real systems may oscillate a bit. If this happens try to reduce the input level, since this limits the allowed control effort. It may not be possible to eliminate all of the oscillations, since these types of controllers depend on canceling the plant dynamics, and if your model is not accurate enough the controller will not cancel the real plant well enough. Do the best you can, as though your grade depended on it.PART A: Changes to closedloop_driver.m Note: You may not need to make all of the changes below. It depends on how much you may have modified this file. It will speed up lab time if these changes are made before lab. You should end up with a section of code which includes all the different types of controllers. You will comment out the types you are not using, and will be adding to these in the next few labs. Step 1: Modify closedloop_driver.m to read in your state model file. For example, if the model file was named state_model_1dof.mat you need the following near the beginning of your Simulink file load state_model_1dof C = [1 0]; [num_Gp,den_Gp] = ss2tf(A,B,C,D); Step 2: Be sure closedloop_driver.m contains the following lines. num_Gp = (abs(num_Gp) > tol*ones(1,length(num_Gp))).*num_Gp; den_Gp = (abs(den_Gp) > tol*ones(1,length(den_Gp))).*den_Gp; Step 3: Modify the saturation_level variable in closedloop_driver.m for the appropriate system. saturation_level = 1000/2196; % (rectilinear system, Model 210) saturation_level = 1000/2546; % (torsional system, Model 205) Comment out these lines until you need them. To comment out use a ‘%’ Step 4: Modify closedloop_driver.m for utilizing ITAE model matching control. Here, the desired closed loop transfer function is: 222()1.4ooooGsssωωω=++ You should enter this into closedloop_driver.m, but leave the frequency oωa variable. Step 5: Modify closedloop_driver.m for utilizing deadbeat model matching control. Here the desired closed loop transfer function is: 222()1.82ooooGsssωωω=++ You should enter these into closedloop_driver.m, but leave the frequency oωa variable.Step 6: Modify closedloop_driver.m to utilize quadratic optimal control, similar to your homework. You should add the lines q = 1; % or whatever you want Go = solve_quadratic(Gp,q); Step 7: Modify closedloop_driver.m to determine the controller. For model matching control, this will be Gc = minreal(Go/(Gp*(1-Go)) Be sure to use the minreal command. Step 8: Be sure closedloop_driver.m computes the prefilter gain, Gpf. You did this in Lab 1. PART B: Creating Closed Loop Simulink Systems for the ECEp Systems Torsional Systems: Open Model205_Openloop.mdl and save it as Model205_Closedloop.mdl. Then modify Model205_Closedloop.mdl so it looks like the following This is a cross between closedloop.mdl and Model205_Openloop.mdl., so you can pretty much just cut and paste. Note that if you want to scale to degrees, you must do this after the feedback branch. In the step function, be sure the Final value is set to Amp, the Step time is set to zero. Also set the Stop time (Select Simulation then Simulation Parameters) to Tf.Rectilinear Systems: Open Model210_Openloop.mdl and save it as Model210_Closedloop.mdl. Then modify Model210_Closedloop.mdl so it looks like the following This is a cross between closedloop.mdl and Model210_Openloop.mdl., so you can pretty much just cut and paste. In the step function, be sure the Final value is set to Amp, the Step time is set to zero. Also set the Stop time (Select Simulation then Simulation Parameters) to Tf. PART C: Control of One Degree of Freedom Systems For each of your two 1 dof systems, you will need to go through the following steps: Step 1: Set up the 1 dof system exactly the way it was when you determined its model parameters. Step 2: Modify closedloop_driver.m to read in the correct model file. You may have to copy this model file to the current folder. Step 3: Modify closedloop_driver.m to use the correct saturation_level for the system you are using. Step 4: ITAE Control. (Be sure to record the value of oωyou use.) • Vary oωuntil you meet the design specs with the simulation (closedloop_driver.m). The larger the value ofoω, the faster your system will respond and the closer your model will predict the response of the real system. Be sure you do not reach the limiter on the control effort, unless you really like restarting your computer. At this point all of the variables you should need are in your current Matlab workspace. • Compile the correct closed loop ECP Simulink driver, connect to the system, and run the simulation.• Use the compare1.m file (or a modification of it) to plot the results of both the simulation and the real system on one nice, neatly


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