ENGR 110 Engineering Modelling and Design Control Systems Modelling II .
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Transcript of ENGR 110 Engineering Modelling and Design Control Systems Modelling II .
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ENGR 110 Engineering Modelling and Design
Control Systems Modelling II
https://www.youtube.com/watch?v=u_0yR3kCR2s
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Lecture Plan
1. Braitenberg VehiclesOpen and Closed Loop SystemsFeedbackWhy use control?
2. Transfer functions Transfer functions to Time response Methods to integrate
3. ControlPID control
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System Modelling
System to Model
Simplify
PlantInput OutputStart with a
single input - single output model
O(t)= I(t).G(t)
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Modelling
In order to model a system:1. We identify input signals [variables]
2. Identify components [things that manipulate variables]– Add/subtract them– Multiply/divide – Integrate/differentiate– Duplicate/merge – …
3. We combine internal signals [modified variables]
4. Produce the output signal [another variable].
The Input-Output relationship may then be determined
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Components of a model:
Combine into single system linking input to output:
=?
tkxtf s
dt
tdxctfd
2
2
dt
txdmtfm
tf
tx
tf
tx
tf
tx tftftftf mdkt
2
2 )(
dt
txdm
dt
tdxctkxtft
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Convenience of ‘s’
To make life easier, we replace the differential term by ’s’
We’ll give the maths next year
This must have a variable (signal) to make sense:xNote: if s is the differential, then must be the integrator!
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Simplify
Combine into single system linking input force to output distance:
Or
2
2 )(
dt
txdm
dt
tdxctkxtft
)()( 2 sxmsscsxskxsft
xmscsxkxft2
)( 2mscskxft
Time domain
s domain
Can leave (s) off as implied when we see an ‘s’ term
𝑂𝑢𝑡𝑝𝑢𝑡𝐼𝑛𝑝𝑢𝑡
=𝑥𝑓 𝑡
=1
𝑚𝑠2+𝑐𝑠+𝑘x
mscsk
ft )( 2
rearrange
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Transfer Function
The concept of a transfer function is vital for control systems modelling!
Must have both LHS and RHS!
Variables on LHS - output and input
Constants and ‘s’ on RHS
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Transfer Function
note single input, single output
Linear Time Invariant Systems
PlantInput Output
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Transfer Function
Describe how the system is changing in an instant.
f(t)
t
y ∝ x
y d∝ x/dt
y d∝ 2x/dt2
y ∝….
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Transfer Function
Can be spatial:
Or temporal:
[ most systems we model are temporal- both input and output variables vary with time]
f(x)
xf(t)
t
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Transfer Function to Time Response
Have how a system changes in an instant
Want how the system changes over time:
Must sum up each of these instantaneous changesIntegrate!
f(t)
t
f(t)
t
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Input Function Sketch s-domain
Ramp tu(t)
Sinusoid sin t
Input Function Sketch s-domain
Impulse (t)
Step u(t)
Types of Input
f(t)
t
f(t)
t
1
1/s
22s
f(t)
t
f(t)
t
1/s2
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Can be a unit step inpute.g. 1V
Can be multiple-unit step inpute.g. 2.5V 2.5
[What would you used to model an input from an Arduino port?]
Input Function Sketch s-domain
Step u(t)
Very common input to systems:• switch being closed (on)• new value being set• DC signal• ...
Step Input
f(t)
time t
1/s
f(t)
1
time t
1/s
f(t)
1
time t
2.5/s
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We know that V=IRwhere R is a constant value
Let us set R to 400 ohmsthen connect the 5 V signal from the Arduino
What happens?
Input voltage from the Arduino
Step Input – Example 1
f(t)
5
time t
5/s
f(t)
5
time t
Input
Output???
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Integration!
I=V/R
Step Input - Examplef(t)
5
t = 1
f(t)
5
time t
Input
Output???
I=V/R
f(t)
5
t = 2
I=V/R
f(t)
5
t = 3
I=V/R
f(t)
5
t = 4
I=V/R
f(t)
5
t = 5
I=V/R
f(t)
5
t = 6
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Input force on the mass
Step Input – Example 2
f(t)
5
time t
5/s
f(t)
5
time t
Input
Output???
f(t)
5
t = 1
)( 2mscsk
fx t
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How to integrate?
NumericallyGraphically
Mathematically Look up table
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clf; %clear all graphs K = 10 %Spring constantC = 3 %Damping constantm = 1 %mass (constant)
t = [0: 0.01: 20];%set up the time incrementsstept = 1 + 0*t; %graph to show step responseplot(t,stept,'m');xlabel('Time t (s)')ylabel('Distance x (m)')
hold on % put each graph on top of each other
for C = 1.0: 1: 10.0d = tf(9,[m C K]) [y,t]=step(d,T);%step response over one secondplot(t,y,'k');pause(2)
end
0 2 4 6 8 10 12 14 16 18 200
1
2
3
4
5
6
7
8
9
Time t (s)
Dis
tanc
e x
(m)
0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5
Time t (s)
Dis
tanc
e x
(m)
0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5
Time t (s)
Dis
tanc
e x
(m)
Numerical in Matlab