Post on 14-Dec-2015
transcript
ENGR 110 Engineering Modelling and Design
Control Systems Modelling II
https://www.youtube.com/watch?v=u_0yR3kCR2s
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
System Modelling
System to Model
Simplify
PlantInput OutputStart with a
single input - single output model
O(t)= I(t).G(t)
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
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
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!
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
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
Transfer Function
note single input, single output
Linear Time Invariant Systems
PlantInput Output
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 ∝….
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
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
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
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
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???
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
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
How to integrate?
NumericallyGraphically
Mathematically Look up table
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