AE470F23Homework10

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University of Michigan *

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Course

470

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Mechanical Engineering

Date

Dec 6, 2023

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pdf

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Uploaded by MasterIron5790

AE470 Fall 2023 Homework #10 Be sure to check for the latest version of the course notes in case I post an update. Due Wednesday, December 6, by 11:00 PM uploaded to Canvas as a single PDF file . NOTE: There is a 45-minute grace period for late uploads. If you try to upload your homework past 11:45 PM, you will not be able to submit it. However, I will drop the lowest HW score, so if you miss just one homework, it will not affect your grade. Instructions: i ) Be sure to follow the honor code guidelines in the course information posted on the Canvas site. In particular, conceptual discussion is allowed, but all detailed work must be your own. ii ) Reminder: No use of solutions from prior offerings of this course is allowed. iii ) Symbolic computing is not allowed. iv ) Homework must be neat and professional in appearance. You may type it in Word or Latex if you wish (Professor Bernstein uses Latex.) Use a ruler to draw all lines and diagrams. Messy homework will not be graded. No crossouts of any kind may appear anywhere. v ) Put a box around your final answer to help the grader. vi ) Label your file as: LastNameAE470F23HW10.pdf vii ) Please upload a single pdf file that contains the following: (a) HW solutions (b) Figures with axes labels, captions and legends (in the case of multiple curves) (c) PDF published from Matlab script (d) Simulink report (must contain ONLY the chapters named root system, subsystems, and system design variables) 1

Problem 1. [15 points] Consider the system given by the transfer function G ( s ) = s + 2 ( s + 0 . 5)( s + 3) with input u ( t ) = sin ωt . The harmonic steady-state output of the system is y ( t ) = α ( ω ) sin( ωt + ϕ ( ω )). (a) Derive expressions for α ( ω ) and ϕ ( ω ). Use them to compute α (1) and ϕ (1). (b) For ω = 1 rad/sec, simulate the system in Simulink and plot the input and output signals on the same figure. Verify that the amplitude of the harmonic steady-state output is α (1). (c) Use data from your plot to compute the phase shift between the input and the harmonic steady-state output signals and verify that the phase shift is ϕ (1). Problem 2. [10 points] Use the Matlab command nyquist to plot the Nyquist plots for the following loop transfer functions and determine the stability of the closed-loop system using the Nyquist stability criterion. Verify your answers by computing the poles of the closed-loop system. (a) L ( s ) = s − 2 s 2 + 4 s + 1 (b) L ( s ) = s 2 + 2 s + 1 s 3 + 0 . 2 s 2 + s + 1 Problem 3. [15 points] Consider the loop transfer function L ( s ) = 60 ( s − 1)( s + 5)( s + 10) . (a) Draw Bode and Nyquist plots of L using the bode and nyquist commands in Matlab. (b) Using the Nyquist stability criterion, explain why the closed-loop system is asymptotically stable. (c) Consider kL , where k > 0, and determine the range of values of k for which the closed-loop system is asymptotically stable by using the following methods: 1) Routh test. 2) Root locus plot. (Include the root locus plot in your homework solutions.) 3) Nyquist plot drawn in part (a). Hint: To understand how to determine the range of values of k from the Nyquist plot for k = 1, draw Nyquist plots for various values of k (please don’t include these plots in your homework solutions) and notice how the Nyquist plot of kL changes as you vary k. Double hint: Professor Bernstein likes to think of the effect of k on the Nyquist plot as blowing air into or letting air out of a balloon. Problem 4. [15 points] Consider the loop transfer function L ( s ) = 3 ( s + 1) 3 . (a) Draw Bode and Nyquist plots of L using the bode and nyquist commands in Matlab. (b) Using the Nyquist stability criterion, explain why the closed-loop system is asymptotically stable. (c) Superimpose the unit circle on the Nyquist plot and mark the points where the Nyquist plot cuts the unit circle and the negative real axis. Finally, determine numerical values for the upward and downward gain margins (in dB) and the phase margin from the Nyquist plot. 2

(d) Use the Routh test to verify the gain margins obtained in part (c). Problem 5. [20 points] Suppose you are given the system ˙ x = 1 2 3 4 x + 0 1 u, y = 1 0 x (a) Assuming all states are measured, design and implement an LQRI (LQR + integrator) controller u = K x x + K I z , where K △ = [ K x K I ], to track the constant reference command r = − 2 ∗ 1 ( t ) . Let R 2 = 0 . 1 and R 1 = R x 0 2 × 1 0 1 × 2 R I , where R x = 2 I 2 and R I = 3. Report your gain matrix K and closed-loop eigenvalues of your augmented system. Simulate your closed-loop response for t ∈ [0 , 10] seconds by initializing your system at x (0) = 1 2 T and your integrator at z (0) = 0. Plot your output and your error in the same figure. (b) Given the observer dynamics ˙ ˆ x = A ˆ x + Bu + F ( y − ˆ y ) , ˆ y = C ˆ x. Design and implement an LQE observer letting V 1 = 0 . 01 I 2 and V 2 = 10. Use the same initial conditions of your system as in part (a), initialize your observer at ˆ x (0) = 0 0 T , and assume u = 0. Simulate your observer for t ∈ [0 , 0 . 5] seconds. Plot the states and their respective estimates versus time. There must be two figures with two curves on each. (c) Combine the LQRI from part (b) and the LQE from part (c) to implement an LQG controller with an integrator (LQGI), where u = K x ˆ x + K I z . Simulate the closed-loop response for t ∈ [0 , 15] seconds where the initial conditions x (0), z (0), ˆ x (0), and reference command r remain the same as in parts (a) and (b). Plot the output, reference, and estimated output in a single figure. In a separate figure, plot the error and control input u . 3

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