An Introduction to Cybernetics

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Nov 30, 2013 (3 years and 11 months ago)

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1

An Introduction to Cybernetics

Robert Oates

Room B50

rxo@cs.nott.ac.uk

2

Overview


What is “Cybernetics”?


Control Theory and Cybernetics


Ordinary Differential Equations
(ODEs) for Simulation


ODEs & Isoclines


ODEs vs Agent Based Simulation

3

Before we start… calculus!


Integration


Calculates the area under a curve


Just adds up at each ‘sample’



Differentiation


Calculates the gradient of a curve


The difference between each ‘sample’



Differentiation is to integration what
division is to multiplication

4

Calculus

t

x

T

5

The Original Interdisciplinary Research
Topic!


Product of The Macy Conferences
(1946


1953)


Contributors include


Norbert Weiner


John Von Neumann


Claude Shannon


Warren McCulloch


Walter Pitts

6

What is Cybernetics?

a)
The study of systems where the
input affects the output


b)
The study of control and
communication in man and
machine


c)
The study of sailors

7

The Steersman (
Κυβερνήτης)

8

Block Diagram Representation of a
Control System

System

-

Control

System

Input

Output

+

9

Block Diagram Representation of a
Control System

Boat

-

Steersman

Input / Desire

Output

+

Transducer (Eyes)


Error

10

Block Diagram Representation of a
Control System

Motor

-

Proportional

Integral

Differential

Controller

Input (
θ
B
)

Output (
θ
A
)

+


Encoder

PID

Controller

11

PID Controller

Error

K
P

K
I

K
D

∫e.dt

de/dt

+

+

+

Input

To

System

12

Cybernetics vs Control Theory


Control Theory


Control
!


Manipulate inputs


Negative feedback is good


Positive feedback is bad



Cybernetics


Understand, characterise and unite


Feedback is feedback!


13

Is Positive Feedback Really That Bad?

Negative

Feedback

Positive

Feedback

Combining

Feedback

14

Ordinary Differential Equations (ODEs)
for System Representation

y
x
dt
dx
21
2
13
2




y

x

2
3
8
13
y
x
dt
dy




-

-

+

+

15

Numerical Simulation Based on
Differential Equations


Euler’s Method



n
n
t
n
n
y
x
x
x
21
2
13
2
1







y
x
dt
dx
21
2
13
2




)
(
x
f
dt
dx



)
(
1
n
t
n
n
x
f
x
x




16

Euler’s Method

t

x

)
(
x
f
dt
dx

x
0

x
1

x
2

x
3

x
4

17

A Quick Aside


Better numerical integration
techniques exist



The best one in general is Fourth
-
Order Runge
-
Kutta. The wikipedia
page is actually very good!

18

Differential Equations for System
Representation

y
x
dt
dx
21
2
13
2




y

x

2
3
8
13
y
x
dt
dy




-

-

+

+

}

But where do we start?



This technique can only

comment on systems

once we know the initial

conditions

19

Isoclines


There are techniques that allow us
to examine a system without
knowing the initial conditions



Examine the isoclines!

20

Isoclines

x

y


Assessing stability and “flow”

dx/dt = 0

dy/dt = 0

(5,3)

(2,1)

+
-

++

-
+

--

--

21

Sea Angels (Cliones)

http://www.youtube.com/watch?v=vB5recdpPaI

Dorsal

Ventral

External Stimulus

+

-

-

Muscle output

+

+

22

Clione Neuron Interaction

Taken from Hugh R Wilson’s “Spikes, Decisions and Actions”,

Oxford University Press, 1999

23

Isoclines in the Clione Nervous System

Time (ms)

0

5

10

V(mv)

-
100

-
50

0

50

dv/dt=0

dR/dt=0

X

Y

-
1

-
0.5

0

0.5

V(V)

R

0

1

0.5

dR/dt and dV/dt models taken from Nagumo et al (1962)

24

Simulation


ODEs are not the only way to
perform simulation



Many other techniques exist



It would be interesting to compare
ODEs to agent
-
based simulation



25

Daisyworld


An Investigation into
ODE’s vs Agent
-
Based Simulations


The Parable of Daisyworld


James Lovelock and Andrew Watson


Designed to illustrate “Gaia Theory”



Grey planet


Two species of daisy


black and
white


A sun getting hotter

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Daisy Fitness

β
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
Temperature
Fitness
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Population Dynamics

)
(




x
a
dt
da
P
P
Fitness

Death

Rate

28

Agent
-
Based System

29

Rules

Occupied?

P(Death) =
γ

P(Growth of daisy type p)

=

a
p
β

yes

no

30

References


Watson, A. J. and J. E. Lovelock (1983).
Biological homeostasis of the global
environment: the parable of Daisyworld.
Tellus 35B, 284
-
289.



Isoclines example taken from Dr Richard
Mitchell’s lecture notes (1999)