An Introduction to Cybernetics

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30 Νοε 2013 (πριν από 3 χρόνια και 11 μήνες)

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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
yx
dt
dx
21213
2
+−−=
y
x
2
3813 yx
dt
dy
−+−=
?
?
+
+
15
Numerical Simulation Based on
Differential Equations
￿
Euler’s Method
(
)
nntnn
yxxx 21213
2
1
+−−+=
+
yx
dt
dx
21213
2
+−−=
)(xf
dt
dx
=
(
)
)(
1 ntnn
xfxx

+
=
+
16
Euler’s Method
t
x
)(xf
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
yx
dt
dx
21213
2
+−−=
y
x
2
3813 yx
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)
I100
I50
0
50
dv/dt=0
dR/dt=0
X
Y
I1 I0.5 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 AgentIBased 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
26
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
27
Population Dynamics
)(
γβ
−= xa
dt
da
P
P
Fitness
Death
Rate
28
AgentIBased 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)