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. LO 1.3 LO 1.5 LO 1.6 LO 1.7
Q1: Compute Laplace transform •
Q2: Construct transfer function •
Q3: Obtain response in Simulink •
Q4: Obtain response via inverse Laplace transform •
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.
Two masses, ! and ", are connected with a spring, . A force, (), is applied
on the first mass. Both masses experience viscous damping, ! and ", through the
surface that they sit on.
The equations of motion that describe the system dynamics are: !!̈() = () − !!̇() − .!() − "()/ ""̈() = −""̇() − ."() − !()/
The initial conditions are: !̇(0) = !(0) = "̇(0) = "(0) =
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 !() and "() due to the input ().
() !
!() "()
" ! "
Question 3 (4 marks)
Use the transfer function block in Simulink to simulate the response of !() and "to 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.
Parameter Value and units ! 2.77 kg " 2.59 kg ! 17 N/(m/s) " 1.2 N/(m/s) 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, () = 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
!() = + + 3.49 + + " + 3.11 + 283.36
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.
Function Time domain Laplace s-domain
Exponentially decaying
cosine wave
exp(−) cos() ⋅ () + ( + )" + "
Exponentially decaying
sine wave
exp(−) sin() ⋅ () ( + )" + "