EML 4314C - Fall 2023 - Lab 4 Assignment - V2 (1)

*Lab #4 reports will be accepted, without late penalty, any time before 11 pm on Dec 10, 2023. NO reports will be accepted after that time. 1 EML 4314C Fall 2023 Lab #4 Assignment – Version 1 Due: Dec 6, 2023* GOALS: The goals for this lab are for you to design a full-state feedback controller for the rotational pendulum system that you characterized in Lab #3 and to implement & test controllers on the system for a range of conditions. GROUPS: Groups of 1 or 2 students. ALL students need to register for a FP group in order to upload a Lab #4 report. All exercises should be completed on a single hardware setup. TIMEFRAME: You will have ~4 weeks to complete this assignment. BACKGROUND: The hardware from Lab #3 will be used for this exercise. Theory and examples: See class notes for state-space review and pole-placement/Ackerman, the inverted pendulum model, pole-placement and LQR design examples, etc. Assumptions: We are implementing a full-state feedback system with four states. The description below assumes your states are ordered the same as the Cazzolato paper: X = [ θ 1 , θ 2 , θ 1_dot , θ 2_dot ] T , and that the calculated gains K = [K 1 , K 2 , K 3 , K 4 ] T are ordered the same way, i.e. K 1 corresponds to the first state. Our design ASSUMES that all states are measured, which is not true: We will use numerical derivatives of the angles to get both rotational velocities. TASKS 1. Immobilize the pendulum arm in the downward position with adhesive tape so that the platen can rotate freely but the pendulum does not rotate. Use a manual tuning approach to get the base link to track a square wave ±60° at 0.25Hz as accurately as possible (tune gains K 1 and K 3 , use K 2 = K 4 = 0). Record these gains and the system performance (desired θ 1 , θ 2 , and actual θ 1 , θ 2 , motor command signal before dithering). Remove the tape so that the pendulum now swings freely and, with exactly the same gains, again record the system performance. 2. With the pendulum in the downward position, manually tune the gains K 1 through K 4 to minimize the pendulum motion while the base link follows the ±60° at 0.25Hz square wave as accurately as possible. Note: You may wish to make K 1 =K 3 =0 while you tune K 2 /K 4 to damp out pendulum excursions, noting that K 2 /K 4 may be negative! Record these gains and the system performance. 3. With your system model from Lab 3 (in LabVIEW or Matlab), design a full-state feedback controller for the single input (one motor control), multiple output (motor rotation and pendulum rotation) system. Use Ackerman’s, pole -placement, or LQR methods to design your controller. Be sure to document and justify your choice of closed-loop pole locations, or Q and R matrix values, that are used in your model-based designs. Implement these gains on your pendulum-down system and assess their performance for stabilizing the pendulum during the ±60° at 0.25Hz square wave base-link tracking task. Record these gains and the system performance.

*Lab #4 reports will be accepted, without late penalty, any time before 11 pm on Dec 10, 2023. NO reports will be accepted after that time. 2 4. Choose either two (2) of the following tasks a-d or JUST e to complete the lab exercise (note to choose as a single option, you must successfully balance the inverted pendulum): a. Using the same process as Task 3 above, change the desired poles or Q values to achieve smaller pendulum angle deviations during the ±60° at 0.25Hz square wave base-link tracking task. Record these gains and the system performance. b. If you used LQR in Task 3, change the R value for computing your gains by a factor of 0.1x and a factor of 10x and recompute new controller gains. Test the performance of the system for the two additional sets of gains for the ±60° at 0.25Hz square wave base- link tracking task. Record these gains and the system performance. c. Significantly modify the mass and inertial properties of your pendulum (e.g. change m 2 , l 2 and J 2hat by at least a factor of 2), update your system model, and then calculate gains using the same desired poles or Q/R values used in Task 3. Test the performance of the system for the ±60° at 0.25Hz square wave base-link tracking task. Record these gains and the system performance. d. Implement a full-state observer for your system. Using the controller gains from Task 3, record system performance during the ±60° at 0.25Hz square wave base-link tracking task. Record the controller and observer gains, system performance AND be sure to record the measured rotation angles, numerical angle derivatives and the estimated states produced by the observer. e. Using pole-placement or LQR, compute gains to control the inverted pendulum system (Pendulum UP). Test these gains for the condition where you try to maintain the base link angle at 0deg, and the pendulum balanced upright. If you are able to stabilize the pendulum upright, then assess the performance of the system as you try to have the platen angle track a small (5-30 degree) square wave at 0.1Hz while the systems keeps the pendulum inverted. To receive credit for this task, the pendulum must be balanced for 10 seconds, platen position uncontrolled. No guarantees are made that this is achievable. Proof is to be live in lab. On completion, significant extra credit may be applied to the Final Project grade. UPDATE: 10 points extra credit will be awarded to the lab 4 report score if the student’s controller can balance the pendulum in the inverted position (+- 15 degrees) for 10 seconds unaided. REPORT: Make a report (4 pages maximum) following the standard IEEE report format, and be sure to include: 1. Abstract - (as usual) 2. Introduction – Introduce the Furuta pendulum & dynamics (cite some of the literature and types of activities it has been used for). Introduce pole-placement and LQR methods if you have used them in your assessments. Describe the goals for this lab (manual tuning, pole- placement/LQR, plus whatever optional choices you decided to include).