University of Warwick
School of Engineering
ES3C8 Systems Modelling and Control
Assignment:
Modelling and Control of Electro-Mechanical System
The purpose of this assignment is to use Matlab/Simulink to analyse and simulate a
mathematical model of an electromechanical system. This system comprises two component
subsystems consisting of the electrical circuit and translational mechnical system shown in
Figure 1.
(a) Electrical subsystem (b) Mechnical subsystem
Figure 1: The two component subsystems of an electromechanical system.
The coupling is through the voltage source em(t) (and current i) and the applied force fE(t)
(and the velocity v1) and will be considered later in the assignment. We will first analyse the
two subsystems separately.
For the entire assignment let α denote the last digit of your ID number (the large seven digit
number across the front of your university card). This parameter will be used to personalise
some of the constants used in the assignment. Ensure that you use the last digit of your
own ID number!
1. (25% credit) Considering the electrical system in Figure 1(a) in isolation first: There
are two inputs to the system consisting of the voltage sources ei(t) and em(t) and the
output is the current i(t).
(a) Obtain a system model of this electrical system as a single differential equation (in i,
em and e2) coupled with a single alegbraic equation in i, ei and e2, where e2 denotes
the voltage across the resistor R2. Construct a continuous-time Simulink model
for the electrical system when the two inputs are step functions of the form:
ei(t) =
{
0 t < τ0
e0 t ≥ τ0
and em(t) =
{
0 t < τ1
e1 t ≥ τ1
.
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(b) Simulate your model using a unit step input at time t = 1 s for ei(t) only (e0 = 1,
e1 = 0, τ0 = 1), investigating the transient and steady state parts of the response
(for current, i), using the appropriate parameter values from Table 1 (use the
values in the column corresponding to your value for α). Then explore the effects
Table 1: Parameter values for Part 1(b–e).
α 0 1 2 3 4 5 6 7 8 9
R1 (Ω) 1 2 3 4 5 6 7 8 9 10
R2 (Ω) 3 6 9 12 15 18 21 24 27 30
L (H) 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
of varying R1 and L independently. Explain your results.
(c) Simulate your model using a unit step input at time t = 1 s for em(t) only (e1 = 1,
e0 = 0, τ1 = 1), investigating the transient and steady state parts of the response
(for current, i), using the the appropriate parameter values from Table 1. Then
explore the effects of varying R1 and L independently. Explain your results and
compare them with those obtained in Part 1(b).
(d) Simulate your model using a unit step input at time t = 1 s for e0(t) and a step to
0.75 V at time t = 3 s for em(t) (e0 = 1, e1 = 0.75, τ0 = 1, τ1 = 3), investigating
the transient and steady state parts of the response (for current, i), using the
appropriate parameter values from Table 1. Explain your results. Then explore
the effects of varying τ1 — what general principle (for linear systems) do your
results illustrate?
(e) In the case where the input voltage em(t) = 0 derive the transfer function G1(s)
relating the input voltage Ei(s) to the output current I(s) (i.e., I(s) = G1(s)Ei(s)).
Then plot the unit step response using theMatlab command step and verify your
result for Part 1(b).
2. (16% credit) Now considering the mechanical system in Figure 1(b) in isolation: The
input to the system is the applied force fE(t) and the output is the velocity v2(t) (though
we will also need the velocity v1(t) when we couple the two subsystems).
(a) Obtain a system model of this mechanical system as a coupled pair of differential
equations in v1(t) and v2(t) (with input fE(t)). Construct a continuous-time
Simulink model for the mechanical system when the input is a step function of
the form:
fE(t) =
{
0 t < τ
f0 t ≥ τ
(b) Simulate your model using a unit step input at time t = 1 s (f0 = 1, τ = 1),
investigating the transient and steady state parts of the responses for the velocities
v1 and v2 using the following parameter values:
M1 = M2 = 2 kg, b1 = b2 = 5 N s/m and b12 = (1 + α) N s/m.
Explore the effects of varying b12 and explain your results.
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(c) Derive the transfer function Gm(s) relating the input force FE(s) to the output
velocity V2(s) (i.e., V2(s) = Gm(s)FE(s)). Then plot the unit step response using
the Matlab command step and verify your result for Part 2(b).
(d) Determine the damping ratio and undamped natural frequency using Gm(s) and
hence characterise the original response in Part 2(b).
3. (10% credit) The coupling between the electrical and mechanical subsystems in Figure 1
is via the magnetic field generated as current flows through the inductor. As current
flows through the inductor a force, fE, is generated that is directly proportional to the
current. As the coupled mass moves a voltage is generated in the electrical circuit, em,
that is directly proportional to the velocity. With respect to the subsystems in Figure 1
we have the following additional equations:
fE(t) = ki(t) and em(t) = kv1(t).
(a) Construct a continuous-time Simulink model for the full electro-mechanical
system; you are strongly advised to reuse your previous models in subsystems
blocks. This system has the supplied voltage ei(t) as input and the velocity v2(t)
as output. Implement in your model a step input of the form:
ei(t) =
{
0 t < τ
e0 t ≥ τ
(b) Simulate your model using a unit step input at time t = 1 s (e0 = 1, τ = 1),
investigating the transient and steady state parts of the response for the velocity
v2 using the following parameter values:
R1 = 5 Ω, R2 = 15 Ω, L = 0.05 H, M1 = M2 = 2 kg,
b1 = b2 = 5 N s/m, b12 = 1 N s/m
exploring different (positive) values for k. Comment on your results.
(c) Show that the transfer function Gp(s) relating the input voltage ei(t) to the output
velocity v2(t) (i.e., V2(s) = Gp(s)Ei(s)) is given by:
Gp(s) =
15k
4s3 + 324s2 + (1835 + 40k2)s+ (2625 + 120k2)
Now fix the following parameters within the electro-mechanical system (these val-
ues are to be used throughout the remainder of the assignment):
R1 = 5 Ω, R2 = 15 Ω, L = 0.05 H, M1 = M2 = 2 kg,
b1 = b2 = 5 N s/m, b12 = 1 N s/m and k = 1 +
α
10
Wb/m.
Determine, to three decimal places, the three real negative poles of the transfer
function Gp(s). (Hint: What does the Matlab command roots do?)
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For the final part of the assignment we will investigate a feedback system in which the above
(full) system is the plant. Therefore the plant transfer function is Gp(s) determined above
(with your value for k). We will assume that v2 is fed back unaltered/processed so that the
sensor transfer function is H(s) = 1:
4. (25% credit) Start by assuming that D(s) = 0 and Gc(s) = Kc (a proportional con-
troller).
(a) What are the forward transfer function, feedback transfer function, open-loop
transfer function and closed-loop transfer function?
(b) Plot the closed-loop root-locus as Kc varies and comment on your result.
(c) Determine from your plot the critical value of Kc for stability of the closed-loop
system and verify one stable and one unstable case using the Nyquist Stability
Criterion.
(d) Generate Bode plots for the open-loop system using a stable value for Kc and
determine the corresponding phase and gain margins. Determine the multiplicative
gain at the crossover frequency and hence provide another estimate for the critical
value for Kc.
(e) What is the response to a unit step reference signal (with zero disturbance)? What
is the response to a unit step disturbance signal (with zero reference)? Explore
different values for Kc, both above and below the critical value.
5. (24% credit) Suppose that the controller is PD: Gc(s) = Kc(KDs+ 1).
(a) Investigate the positioning of the open-loop zero at s = −1/KD, with respect to
the three open-loop poles p1 < p2 < p3 < 0, on the closed-loop root-locus. (Hint:
Consider the four cases −1/KD < p1, p1 < −1/KD < p2, p2 < −1/KD < p3 and
p3 < −1/KD < 0.)
(b) Determine a value for Kc such that the steady state error for a reference step input
(no disturbance) is just less than 2% of the input.
(c) Determine a value for KD such that the closed-loop transfer function TR(s) has
poles with time constants less than 0.1 s for the exponential factors in the tran-
sient response, and have the least amount of oscillatory behaviour. (Hint: Use
Matlab’s commands rlocus and/or rlocfind for different values of KD, based
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on your findings from Part 5(a), to find the closed-loop poles for the value of Kc
determined above in Part 5(b).)