Fuzzy Controller Design Based on Fuzzy Lyapunov Stability

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Fuzzy Controller Design Based on Fuzzy Lyapunov
Stability
Stjepan Bogdan University of Zagreb

• 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

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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.
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Fuzzy Lyapunov stability
2nd order system Lyapunov function sample
dx1/dtx2 and dx2/dtu
pospos posu neg gt u ?
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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 µ?(x)

• triangular fuzzy number (L-R fuzzy number)

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Fuzzy numbers and fuzzy arithmetic

• fuzzy arithmetic

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Fuzzy numbers and fuzzy arithmetic
Definition greater then or equal to
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Cascade fuzzy controller design
Known facts about the system - the range of the
beam angle ? is p/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.
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.
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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
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Cascade fuzzy controller design
Observe each of two terms separately
and
only 9918 rules
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Experimental results ball and beam
Experimental results ball and beam
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Experimental results ball and beam
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Experimental results ball and beam
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Experimental results 2 DOF airplane
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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
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Fuzzy Lyapunov stability and occupancy grid
implementation to formation control
Visual feedback web cam DCS-900
320240 or 640480
46o
75o
Wide angle lens (Sony 0.6x)
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Fuzzy Lyapunov stability and occupancy grid
implementation to formation control
markers
fuzzy controllers
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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 gt path planning and execution
  for each mobile agent gt missions (with
  priorities?),
• one mobile agent per resource is allowed gt
  dynamic scheduling gt time windows.

Wedge formation to T formation
b 32 gt 55 (43,54) c 34 gt 51 (33,42) d 51
gt 33 (52,43) e 55 gt 53 (54)
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Fuzzy Lyapunov stability and occupancy grid
implementation to formation control
b 32 gt 55 (32,43,54,55) c 34 gt 51
(34,33,42,51) d 51 gt 33 (51,52,43,33) e 55
gt 53 (55,54,53)
43, 54 - shared resources
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Fuzzy Lyapunov stability and occupancy grid
implementation to formation control