ECE-320: Linear Control Systems Homework 9 Due: Friday May 14 at the (very) beginning of lab In this homework we will simulate and study integral control, full order observers coupled with state variable feedback, and then we will combine them. Having an integrator in our system is generally preferable to using a prefilter for achieving a zero steady state error for a step input, since it will help overcome errors in the plant model. Some of the things we will do in this homework are a bit weird, but they are being done for the sake of demonstrating that your programs are working. For all of these problems, use the state models available on the class website. Both systems are torsional systems. Be sure you are using a variable step solver, and the maximum time step is something like 1e-5 (look under Simulation-> Configuration Parameters). Also, set the initial conditions on the integrators to just 0 (a scalar). Some useful Matlab commands are [n,m] = size(A); zeros(n,m); eye(n) You need to save plots of every case you are told to run. Save them in a word document with appropriate figure captions and turn that in (or e-mail it). This entire homework is a pre-lab. 1) In this problem we will incorporate integral control for a one degree of freedom system and run some test cases to verify your code is working. a) Copy your program Basic_1dof_State_Variable_Model_driver.m to a new file sv1_integral_driver.m, then copy the Simulink file Basic_1dof_State_Variable_Model.mdl to sv1_integral.mdl. Modify these new files to implement state variable feedback using integral control. The basic integral control scheme is shown below in Figure 1. Note that the prefilter is equal to 1. Figure 1. Integral control structure. 1b) Modify sv1_integral_driver.m to plot the two states and the control effort (the control ) Set the step amplitude to 10 degrees (but convert to radians), the final time to 1.5 seconds, Figure 2. Initial results for the 1 dof system with integral control (problem 1c). ) At this point there is no real reason for us to use integral control, since we know everything tion, erything ) Systems with these integrators have an interesting characteristic. Change the poles to -5, -remove the line A = A*0.25, or you will have trouble in lab! effort does not get scaled to degrees). cplace the poles at -5, -10, -15, and run the simulation. You should get results like those shownin Figure 2. 0 0.5 1 1.50510Disk Position (deg)0 0.5 1 1.50102030Disk Velocity (deg/sec)0 0.5 1 1.500.020.040.060.08Control Effort dexactly. Let’s assume your lab partner is way off on the system model. In your code, just before your run the simulation (just before the sim command), type A = A*0.25, which basically says you’re a matrix is way off. If you make this change just before the simulayour gains will be chosen assuming your model is correct, and we will be running the simulation using a different system. Change the final time to 2.5 seconds, and leave evelse the same as before. You should get results like those shown in Figure 3, which look pretty hopeless, but we can improve them. e10, and -1500 and rerun the system. You should get the results shown in Figure 4, which are much better. Note that the control effort did not change much. Before you go on, be sure to 2Figure 3. Integral control of a 1 dof system with bad data (problem 1d) 0 0.5 1 1.5 2 2.505101520Disk Position (deg)200Disk Velocity (deg/sec)1000-1000 0.5 1 1.5 2 2.50.030 0.5 1 1.5 2 2.500.010.02Control Effort0 0.5 1 1.5 2 2.5051015Disk Position (deg)0 0.5 1 1.5 2 2.5-50050100Disk Velocity (deg/sec)0 0.5 1 1.5 2 2.500.0050.010.0150.02Control Effort3Figure 4. Integral control of a 1 dof system with bad data, but a better result (problem 1e). 2) n some test cases to verify your code is working. _Model_driver.m to a new file v2_integral_driver.m, then copy the Simulink file Basic_2dof_State_Variable_Model.mdl In this problem we will incorporate integral control for a two degree of freedom system andru a) Copy your program Basic_2dof_State_Variablesto sv2_integral.mdl. Modify these new files to implement state variable feedback usintegral control. b) Modify sv2_ining tegral_driver.m to plot the four states and the control effort (the control ffort does not get scaled to degrees). final time to 2 seconds, and place the poles at -5, -10, -5, -20, -25. We are trying to control the position of the second disk. Run the simulation and Figure 5. Integral control of a two degree of freedom system. We care controlling the position of the second disk (part 2c). e c) Set the amplitude to 10 degrees, the1you should get results like those shown in Figure 5. 0 0.5 1 1.5 20102030 10Disk 1 Position (deg)0 0.5 1 1.5 20204060Disk 1 Velocity (deg/sec)0 0.5 1 1.5 205Disk 2 Position (deg)30Disk 2 Velocity (deg/sec)20100-100 0.5 1 1.5 20.060 0.5 1 1.5 200.020.04Control Effort4d) Assume your lab partner screwed up gain, so you model is not accurate. To see what appens when the model does not match the real system, just before the simulation type A = , you igure 6. Integral control of a 2 dof system with bad data. We are controlling the position of e second disk (problem 2d). e same as in part d, but now we want to control the position of e first disk. You should get results like those in Figure 7. If your results agree, remove the hA*0.25, place the poles at -5, -10, -15, -20, -2500, and run the simulation for 10 secondsshould get a plot like that in Figure 6. 0 2 4 6 8 10010203040 20Fth e) Now assume everything is ththline A = A*0.25 from your code. Disk 1 Position (deg)0 2 4 6 8 10-1000100200Disk 1 Velocity (deg/sec)0 2 4 6 8 10051015Disk 2 Position (deg)Disk 2 Velocity (deg/sec)100500-500 2 4 6 8 100.0150 2 4 6 8 1000.0050.01Control Effort5Figure 7. Integral control of a 2 dof system with bad data. We are controlling the position of the first disk (problem 2e). 0 2 4 6 8 1005101520Disk 1 Position (deg)0 2 4 6 8 10-50050100Disk 1 Velocity (deg/sec)0 2 4 6 8 1002468Disk 2 Position (deg)0 2 4 6 8 10-40-2002040Disk 2 Velocity (deg/sec)0 2 4 6 8 1002468x 10-3Control Effort 63) In this problem we will incorporate a full order observer for a one degree of freedom system and run some test cases to verify your code is working. a) Copy your program Basic_1dof_State_Variable_Model_driver.m to a new file sv1_observer_driver.m, then copy the Simulink file Basic_1dof_State_Variable_Model.mdl to sv1_observer.mdl. Modify