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ME 2045 –Linear Control Systems

Final Project

Due 8:00 pm Wednesday, December 14, 2022

(submit by the CanvasAssignment link)

The objective of this project is to design control systems to move the quadcopter of the midterm project to specified positions and orientations. Using the full model of the system, you are to:

1. For the full system (12 states that were used in the midterm project: 3 orientation angles,

3 positions of the center of mass, and their derivatives – –use the same order for the state vector as in the midterm: [xb, yb, zb, qr, qp, qy, derivatives]T) and assuming full state feedback, determine controllability and observability. Design a full-state feedback controller for the system. Clearly state the objective(s) of the controller in terms of desired transient response Show the gain set and provide details for how you derived it. Show the desired closed-loop eigenvalues and show the resulting theoretical closed-loop eigenvalues for the controlled system. If the objectives cannot be achieved, explain why not.

2. Now consider the system in which the positions are constrained to be fixed and only the angles are allowed to change. Show the model (A, B, C, and D matrices) for the constrained system (only the orientation angles and their derivatives remain as states). Keep the same order for the state vector as in part 1: [qr, qp, qy, derivatives]T.

3. Assuming full state feedback for the constrained system of part 2, design a full-state feedback controller for the system. Clearly state the objective(s) of the controller in terms of desired transient response (you are encouraged to use the same objectives for all controllers in this project, to the extent that it is feasible, understanding that in some cases the objectives may not be achievable). Show the gain set and provide details for how you derived it. Show the desired closed-loop eigenvalues and show the resulting theoretical closed-loop eigenvalues for the controlled system. If the objectives cannot be achieved, explain why not.

4. For the constrained system of part 2 and assuming partial-state feedback where only the three angles and are measured, determine controllability and observability. Design a partial-state feedback controller (output feedback) for the system. Clearly state the objective(s) of the controller in terms of desired transient response. Show the gain set and provide details for how you derived it. Show the desired closed-loop eigenvalues and show the resulting theoretical closed-loop eigenvalues for the controlled system. If the objectives cannot be achieved, explain why not. Note that it might not be possible to design a stable system in this case.

5. For the constrained system of part 2 and assuming only the three angles are measured (same as in part 4), design an observer to recreate the missing state(s) in the controller so that the full-state feedback controller of part 2 may be used. Show the observer gain set and provide details for how you derived it. Show the desired observer eigenvalues and show the resulting observer eigenvalues for the controlled system. If the objectives cannot be achieved, explain why not.

6. Choose a drone maneuver that you will simulate in the next part of the project. The maneuver must involve at least two angles. Clearly state what that maneuver is. For example, you may plan to cause the drone to undergo a combined step change in roll and pitch angles, or to experience a sequence of yaw and then roll changes, or some other maneuver.

7. Numerically simulate the response of the constrained system of part 2 with each of your controllers (from parts 3, 4, and 5) being implemented. Enforce the motor voltage limits (24V) in your simulation –that is, do not let the motor voltage exceed 24V (this is called saturation). You may wish to adjust your controller objectives or your maneuver if the motor voltage is too often saturated. Plot the system responses (angles only) as well as plots of motor voltages (you can decide how the is best done to assist with the following discussion–that is, multiple cases on one plot, all on separate plots, combine states with motor voltage on the plots, etc.). Provide a short discussion comparing each controlled case to the others and explain how each controlled case fares in terms of your goals. Point out any notable differences in behavior.

8. Optionalpending availability of the test rig: Implement your partial-state controller on the quadcopter rig in the SB27 lab.

To be submitted:

Submit a report (.pdf) for this project that contains a step-by-step discussion of your work on parts 1-8, in the order stated above. You may wish to use a Matlab Live Script

(https://www.mathworks.com/help/matlab/matlab_prog/what-is-a-live-script-or-function.html), which allows all parts to be presented in order, with code and results. For each part, state the problem statement and then provide your answer, including derivations, schematics (hand-written derivations and schematics are fine), code, results, response plots, etc. Do not use an appendix –instead provide all relevant material in each section. If you do not use a Matlab Live Script, in a separate file provide your Matlab code (.m script) or other code, e.g. Python.