2024A-MEAM513-HW06

pdf

School

University of Pennsylvania *

*We aren’t endorsed by this school

Course

513

Subject

Mechanical Engineering

Date

Apr 3, 2024

Type

pdf

Pages

Uploaded by SuperHumanWaspMaster925

Black Box Lab, State-Space Representation & PD Design 2024A MEAM 513/ESE 505 – Homework #6 – Upload by Friday 2024-03-22 @ 11:59PM 1 1. Black Box Lab Analysis Make a pretty bode plot that shows your data from the lab as markers (no line). If you missed lab or you did the analog feedback project instead of measuring the frequency response, just make a note of that on your graph and use the data provided here. Now develop a dynamic model for the system that is inside the black box. You should be able to make a reasonable guess about the order of the system based on clues like the slope of the gain curve and the phase at high frequency. Do there appear to be any lightly damped modes in the system? Write the model in both state-space form and as a transfer function. Then add the bode plot that represents the model to the graph with your data. Be sure to use lines with no markers for the model. You should adjust the model until you get a decent fit. As a bonus for anyone who understands op-amps and RC circuits, here is a simple circuit diagram of the contents of the black box. Even a rudimentary understanding of the circuits might allow you to quickly create a reasonable state-space model of the system. Again, state models are often a very natural and intuitive way to write differential equations to describe system dynamics. If we design a proportional feedback controller, what gain would result in a phase margin of 60 degrees? Use the state-space representation of the system to determine the position of the closed-loop poles corresponding to this gain. (Hint: You can do this with a single call to the Matlab eig command, once you have defined the A, B, and C matrices.) SUBMIT : Your state matrices, your pretty graph, the value of the proportional gain you chose, and the locations of the closed-loop poles with that gain. Freq [Hz] Ranalog [db] PhaAnalog [deg] 0.1 6.1 -6 0.5 5.7 -25 1 5 -47 2 2.3 -88 3 -1.1 -120 4 -4.7 -142 6 -11.5 -178 8 -17.1 -190 10 -20.6 -221

Black Box Lab, State-Space Representation & PD Design 2024A MEAM 513/ESE 505 – Homework #6 – Upload by Friday 2024-03-22 @ 11:59PM 2 2. Robotic Car with Trailer In class, we derived a linear state-space model for a car-like robot with two trailers. The state matrices can be written as follows: 𝐴 = ൦ 0 0 0 0 𝑎 −𝑎 0 0 0 𝑎 −𝑎 0 𝑎 0 0 0 ൪ 𝐵 = ቎ 𝑎 0 0 0 ቏ 𝐶 = [ 0 −1 −1 1 ] where 𝑎 = ௏ ௅ . The (4,1) element of A and the (1,4) element of C are different from what we wrote in class because distances have been normalized by the length of the car, L . Show that this model corresponds to the following scalar transfer function: 𝐺 ௉ (𝑠) = 𝑎 ସ 𝑠 ଶ (𝑠 + 𝑎) ଶ where the output is the normalized distance of the second trailer from the wall and the input is the steering angle on the car. Both system representations have been implemented in the Simulink model provided on Canvas, using (a = 1.0/sec), which corresponds to the robot driving at a speed that takes one second to travel the length of the body 1 . Why didn’t we just use a regular unit step as the input to the open-loop system? Oops! There is at least one mistake in the state-space implementation that makes the two outputs differ. Find and fix it and submit the resulting step-response graph. Also be sure to check out the Scope that shows the state response so you understand what is going on. 3. Classical Control Design Let’s design a feedback controller for the plant using the “classical control” tools we learned in the first half of the class. Suppose we have a measurement of the distance of the second trailer from the wall, y , and we want it to track a desired distance, y d . a. Let’s see if a PD controller might work. In other words, let’s try 𝑢 = 𝐾 ௉ (𝑦 ௗ − 𝑦) + 𝐾 ஽ (𝑦 ௗ ̇ − 𝑦̇) Let’s write the feedback in the following form : 𝑢 = 𝐾𝐺 ௖ (𝑠)(𝑦 ௗ − 𝑦) = 𝐾(𝑇 ஼ 𝑠 + 1)(𝑦 ௗ − 𝑦) where 𝐾 = 𝐾 ௉ and 𝑇 ஼ = ௄ ವ ௄ ು . For part (a), the only thing you have to submit is a short explanation of why it is reasonable not to use integral feedback for this problem. 1 You could also think of this as normalizing by the time it takes the robot to travel the body length, so instead of using “seconds” we are using “body_length_times”.

Black Box Lab, State-Space Representation & PD Design 2024A MEAM 513/ESE 505 – Homework #6 – Upload by Friday 2024-03-22 @ 11:59PM 3 b. Okay, we have a two-parameter design problem. Solve it. Choose values of K and 𝑇 ஼ in the controller to get the system to behave in a way that you think is reasonable. You may use whatever tools you want to do the design, including simulation, root locus and bode. But you don’t have to give details of the process. When you are done, implement the design in Simulink and show a closed-loop step response. c. Once you have the final values of K and 𝑇 ஼ , make a nice root locus (varying K with 𝑇 ஼ fixed at the value of you chose), and use it to explain how your value of K makes sense. d. Finally, make a loop bode plot of 𝐾𝐺 ௖ (𝑠)𝐺 ௉ (𝑠) and use it to explain why your design makes sense (hint: margins). SUBMIT : An explanation of why we didn’t use integral feedback; a closed-loop step response; root locus with explanation; and a bode plot with explanation.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version