University of Zagreb, Faculty of Electrical Engineering & Computing
Department of Control and Computer Engineering
SWAN06
ARRI, December 8, 2006
Fuzzy Controller Design Based on Fuzzy
Lyapunov Stability
Stjepan Bogdan
University of Zagreb
•
•
•
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Fuzzy Lyapunov stability
Fuzzy numbers and fuzzy arithmetic
Cascade fuzzy controller design
Experimental results
– ball and beam
– 2DOF airplane
• Fuzzy Lyapunov stability and occupancy grid –
implementation to formation control
Laboratory for Robotics and Intelligent Control Systems
1
University of Zagreb, Faculty of Electrical Engineering & Computing
Department of Control and Computer Engineering
SWAN06
ARRI, December 8, 2006
Fuzzy Lyapunov stability
operator can define stabilizing (allowed) and destabilizing (forbidden)
actions in linguistic form
QUESTION : if we replace a crisp mathematical definition of Lyapunov
stability conditions with linguistic terms, can we still treat these
conditions as a valid test for stability?
Answer to this question was proposed by M. Margaliot and G.Langholz
in “Fuzzy Lyapunov based approach to the design of fuzzy controllers”
and L.A. Zadeh in “From computing with numbers to computing with
words”.
Laboratory for Robotics and Intelligent Control Systems
2
University of Zagreb, Faculty of Electrical Engineering & Computing
Department of Control and Computer Engineering
SWAN06
ARRI, December 8, 2006
Fuzzy Lyapunov stability
2nd order system Lyapunov function sample:
dx1/dt=x2 and dx2/dt~u
pos*pos + pos*u = neg => u = ?
X1
X2
U
V’
Positive
Positive
Negative big
Negative
Positive
Negative
Zero
Negative
Negative
Positive
Zero
Negative
Negative
Negative
Positive big
Negative
Laboratory for Robotics and Intelligent Control Systems
3
University of Zagreb, Faculty of Electrical Engineering & Computing
Department of Control and Computer Engineering
SWAN06
ARRI, December 8, 2006
Fuzzy numbers and fuzzy arithmetic
• linguistic terms in a form of fuzzy numbers
• fuzzy number - fuzzy set with a bounded support + convex and normal
membership function μς(x):
• triangular fuzzy number (L-R fuzzy number):
Laboratory for Robotics and Intelligent Control Systems
4
University of Zagreb, Faculty of Electrical Engineering & Computing
Department of Control and Computer Engineering
SWAN06
ARRI, December 8, 2006
Fuzzy numbers and fuzzy arithmetic
• fuzzy arithmetic
Facts against
intuition in fuzzy
arithmetic:
0
Fuzzy zero ?
Laboratory for Robotics and Intelligent Control Systems
5
University of Zagreb, Faculty of Electrical Engineering & Computing
Department of Control and Computer Engineering
SWAN06
ARRI, December 8, 2006
Fuzzy numbers and fuzzy arithmetic
Definition: greater then or equal to
b
a
?
a >=< b
b
bα
aα
bα
a
aα
aα
bα
Laboratory for Robotics and Intelligent Control Systems
6
University of Zagreb, Faculty of Electrical Engineering & Computing
Department of Control and Computer Engineering
SWAN06
ARRI, December 8, 2006
Cascade fuzzy controller design
Known facts about the system:
- the range of the beam angle θ is ±π/4,
- the range of the ball displacement from
center of the beam is ± 0.3 [m]
- the ball position and the beam angle
are measured.
r


Even though we assume that an exact physical law of motion is unknown,
from the common experience we distinguish that the ball acceleration
increases as the beam angle increases, and that angular acceleration of
the beam is somehow proportional to the applied torque.
Laboratory for Robotics and Intelligent Control Systems
7
University of Zagreb, Faculty of Electrical Engineering & Computing
Department of Control and Computer Engineering
SWAN06
ARRI, December 8, 2006
Cascade fuzzy controller design
Task: determine fuzzy controller that stabilizes the system
- consider the Lyapunov function of the following form:
4 state variables, 3
linguistic values each 
81 rules
Observe each of two terms separately
and
Laboratory for Robotics and Intelligent Control Systems
8
University of Zagreb, Faculty of Electrical Engineering & Computing
Department of Control and Computer Engineering
SWAN06
ARRI, December 8, 2006
Cascade fuzzy controller design
Observe each of two terms separately
and
er
edr
N
Z
P
e
ed
N
Z
P
N
LP
P
Z
N
LN
N
Z
Z
P
Z
N
Z
N
Z
P
P
Z
N
LN
P
Z
P
LP
Laboratory for Robotics and Intelligent Control Systems
only 9+9=18 rules
9
University of Zagreb, Faculty of Electrical Engineering & Computing
Department of Control and Computer Engineering
SWAN06
ARRI, December 8, 2006
Experimental results – ball and beam
Experimental results – ball and beam
Laboratory for Robotics and Intelligent Control Systems
10
University of Zagreb, Faculty of Electrical Engineering & Computing
Department of Control and Computer Engineering
SWAN06
ARRI, December 8, 2006
Experimental results – ball and beam
V
0
Laboratory for Robotics and Intelligent Control Systems
11
University of Zagreb, Faculty of Electrical Engineering & Computing
Department of Control and Computer Engineering
SWAN06
ARRI, December 8, 2006
Experimental results – ball and beam
V
0
Laboratory for Robotics and Intelligent Control Systems
12
University of Zagreb, Faculty of Electrical Engineering & Computing
Department of Control and Computer Engineering
SWAN06
ARRI, December 8, 2006
Experimental results – 2 DOF airplane
Laboratory for Robotics and Intelligent Control Systems
13
University of Zagreb, Faculty of Electrical Engineering & Computing
Department of Control and Computer Engineering
SWAN06
ARRI, December 8, 2006
Fuzzy Lyapunov stability and occupancy grid –
implementation to formation control
Wifibot – Robosoft, France
I2C bus
Ethernet
SC12 (BECK)
IR sensors
encoders
Web cam DCS-900
Laboratory for Robotics and Intelligent Control Systems
14
University of Zagreb, Faculty of Electrical Engineering & Computing
Department of Control and Computer Engineering
SWAN06
ARRI, December 8, 2006
Fuzzy Lyapunov stability and occupancy grid –
implementation to formation control
0
Visual feedback – web cam DCS-900


320:240 or 640:480
Rj
46o
75o
Wide angle lens (Sony 0.6x)
Laboratory for Robotics and Intelligent Control Systems
15
University of Zagreb, Faculty of Electrical Engineering & Computing
Department of Control and Computer Engineering
markers
SWAN06
ARRI, December 8, 2006
Fuzzy Lyapunov stability and occupancy grid –
implementation to formation control
formation definition - graph (Desai et al.)
v j   v lij , lij , ij , ij 
 j   lij , lij , ij , ij 
fuzzy controllers
Formation requires
increasing order of IDs!
set of predefined rules for formation change
possible collisions during formation change
Laboratory for Robotics and Intelligent Control Systems
16
University of Zagreb, Faculty of Electrical Engineering & Computing
Department of Control and Computer Engineering
SWAN06
ARRI, December 8, 2006
Fuzzy Lyapunov stability and occupancy grid –
implementation to formation control
Occupancy grid with time windows:
• each cell represents resource used by mobile agents,
• formation change => path planning and execution for each mobile
agent => missions (with priorities?),
• one mobile agent per resource is allowed => dynamic scheduling =>
time windows.
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Wedge formation
to T formation
b – 32 => 55 (43,54)
c – 34 => 51 (33,42)
d – 51 => 33 (52,43)
e – 55 => 53 (54)
Laboratory for Robotics and Intelligent Control Systems
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University of Zagreb, Faculty of Electrical Engineering & Computing
Department of Control and Computer Engineering
SWAN06
ARRI, December 8, 2006
Fuzzy Lyapunov stability and occupancy grid –
implementation to formation control
b – 32 => 55 (32,43,54,55)
c – 34 => 51 (34,33,42,51)
d – 51 => 33 (51,52,43,33)
e – 55 => 53 (55,54,53)
43, 54 - shared
resources
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Laboratory for Robotics and Intelligent Control Systems
18
University of Zagreb, Faculty of Electrical Engineering & Computing
Department of Control and Computer Engineering
SWAN06
ARRI, December 8, 2006
Fuzzy Lyapunov stability and occupancy grid –
implementation to formation control
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t0
Laboratory for Robotics and Intelligent Control Systems
tf
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