METR4201: Introduction to Control Systems
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METR4201: Introduction to Control Systems
Assignment Aligned to Practical 2
Learning objectives
• LO1.3 Represent and simulate systems models by block diagrams and transfer functions in MATLAB and Simulink.
• LO1.5 Find the Laplace Transform of common mathematical functions and linear ordinary differential equations using both first principles and mathematical tables.
• LO1.6 Construct transfer functions for linear dynamic systems from (i) differential equations and (ii) reduction of block diagrams.
• LO1.7 Determine the response of a system to an arbitrary input and having arbitrary initial conditions using the Laplace Transform.
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LO 1.3 |
LO 1.5 |
LO 1.6 |
LO 1.7 |
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Q1: Compute Laplace transform |
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Q2: Construct transfer function |
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Q3: Obtain response in Simulink |
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Q4: Obtain response via inverse Laplace transform |
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Introduction
This assignment is aligned with the work you will complete in Practical 2. It involves
• Forming a transfer function for a mechanical system from provided equations of motion
• Developing a Simulink model of that system.
• Using Simulink to predict the time response of the plant to impulse, step, and sinusoidal inputs.
• Using analytical methods to validate and interpret the time response.
You should submit your solution by Turn-it-in on Blackboard. Your report should be kept brief but readable. It will be marked against the provided rubric. Your report should be computer readable but you do not need to overproduce it. Marks are not given for executive summaries, bibliographies, or tables of contents.
Question 1 (4 marks)
The following figure shows a mass-spring-damper system that will be used in
Practical 2.
x! (t)
x2 (t)
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f(t) |
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c2 |
Two masses, m! and m2, are connected with a spring, k. A force, f(t), is applied on the first mass. Both masses experience viscous damping, C! and C2, through the surface that they sit on.
The equations of motion that describe the system dynamics are:
m!x…! (t) = f(t) 一 C!x.! (t) 一 k(x! (t) 一 x2 (t))
m2x…2 (t) = 一C2x.2 (t) 一 k(x2 (t) 一 x! (t))
x.! (0) = a
x! (0) = b
x.2 (0) = C
x2 (0) = d
Represent these ODEs with initial conditions in the Laplace Domain.
Question 2 (4 marks)
Assuming zero initial conditions, rearrange the two equations of motion to find the response for X! (S) and X2 (S) due to the input F (S).
Question 3 (4 marks)
Use the transfer function block in Simulink to simulate the response of )! (%) and )2to an impulse of 20N acting for 25 ms. (The description document for Practical 1 gives instruction on how to implement an impulse in Simulink.).
Parameter values for the system as configured for this practical are given as follows.
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Parameter |
Value and units |
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!! |
2.77 kg |
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!2 |
2.59 kg |
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(! |
17 N/(m/s) |
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(2 |
1.2 N/(m/s) |
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" |
390 N/m |
A plot of this response should be provided in your report. Be sure to include units, axis labels, appropriate time range, i.e. not a printscreen of a scope output.
Question 4 (12 marks)
Determine the forced response of mass 1 position in the time domain, )! (%) for an applied impulse force input, 6 (5) = 1. Interpret the results using the principle of superposition to guide your interpretation.
HINT: The coverup rule with non-linear factors.
Question 4 asks you to find the impulse response. This is too complex to do algebraically so it is recommended that you first substitute in parameter values. The response factors as
4! (5) = 
+ 
+ 
Note that this includes a second order denominator term in addition to the two linear factors. The cover-up rule can be used to find coefficients by two strategies.
1. Factor the second-order denominator into linear factors, using complex coefficients, and then use the cover-up method, but with complex numbers. At the end, conjugate complex terms have to be combined in pairs to produce real summands. The calculations are sometimes longer, and require skill with complex numbers.
2. Find coefficients A and B using the cover-up rule in the usual way and then find coefficients C and D using the method of undetermined coefficients. This is less work that using the method of undetermined coefficients to find all four coefficients.
If you pursue the second strategy, having found the coefficients C and D, the easiest way to proceed is to rearrange the second order term so that the following Laplace transform pair can be used.
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Function |
Time domain |
Laplace s-domain |
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Exponentially decaying cosine wave |
exp(−G%) cos(K%) ⋅ M(%) |
5 + G
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Exponentially decaying sine wave |
exp(−G%) sin(K%) ⋅ M(%) |
K
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2023-03-13