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