Fuzzy Controller Design Based on Fuzzy Lyapunov Stability. Stjepan Bogdan University of Zagreb. F uzzy Lyapunov stability F uzzy numbers and fuzzy arit h metic C ascade fuzzy controller design E xperimental results ball and beam 2DOF airplane - PowerPoint PPT Presentation
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University of Zagreb, Faculty of Electrical Engineering & Computing SWAN06 Department of Control and Computer Engineering ARRI, December 8, 2006 Laboratory for Robotics and Intelligent Control Systems 1 Fuzzy Controller Design Based on Fuzzy Lyapunov Stability • 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 Stjepan Bogdan University of Zagreb
Transcript
University of Zagreb, Faculty of Electrical Engineering & Computing SWAN06 Department of Control and Computer Engineering ARRI, December 8, 2006
Laboratory for Robotics and Intelligent Control Systems 1
Fuzzy Controller Design Based on Fuzzy Lyapunov Stability
• Fuzzy Lyapunov stability and occupancy grid – implementation to formation control
Stjepan BogdanUniversity of Zagreb
University of Zagreb, Faculty of Electrical Engineering & Computing SWAN06 Department of Control and Computer Engineering ARRI, December 8, 2006
Laboratory for Robotics and Intelligent Control Systems 2
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”.
Fuzzy Lyapunov stability
University of Zagreb, Faculty of Electrical Engineering & Computing SWAN06 Department of Control and Computer Engineering ARRI, December 8, 2006
Laboratory for Robotics and Intelligent Control Systems 3
X1 X2 U V’
Positive Positive Negative big Negative
Positive Negative Zero Negative
Negative Positive Zero Negative
Negative Negative Positive big Negative
2nd order system Lyapunov function sample:
dx1/dt=x2 and dx2/dt~u
Fuzzy Lyapunov stability
pos*pos + pos*u = neg => u = ?
University of Zagreb, Faculty of Electrical Engineering & Computing SWAN06 Department of Control and Computer Engineering ARRI, December 8, 2006
Laboratory for Robotics and Intelligent Control Systems 4
• fuzzy number - fuzzy set with a bounded support + convex and normal membership function μς(x):
• triangular fuzzy number (L-R fuzzy number):
• linguistic terms in a form of fuzzy numbers
Fuzzy numbers and fuzzy arithmetic
University of Zagreb, Faculty of Electrical Engineering & Computing SWAN06 Department of Control and Computer Engineering ARRI, December 8, 2006
Laboratory for Robotics and Intelligent Control Systems 5
Facts against intuition in fuzzy arithmetic:
• fuzzy arithmetic
Fuzzy numbers and fuzzy arithmetic
Fuzzy zero ?
0
University of Zagreb, Faculty of Electrical Engineering & Computing SWAN06 Department of Control and Computer Engineering ARRI, December 8, 2006
Laboratory for Robotics and Intelligent Control Systems 6
Definition: greater then or equal to
ab
aα
bα
aα aαbαbα
ab
a b>=<?
Fuzzy numbers and fuzzy arithmetic
University of Zagreb, Faculty of Electrical Engineering & Computing SWAN06 Department of Control and Computer Engineering ARRI, December 8, 2006
Laboratory for Robotics and Intelligent Control Systems 7
r
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.
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.
Cascade fuzzy controller design
University of Zagreb, Faculty of Electrical Engineering & Computing SWAN06 Department of Control and Computer Engineering ARRI, December 8, 2006
Laboratory for Robotics and Intelligent Control Systems 8
Task: determine fuzzy controller that stabilizes the system
4 state variables, 3 linguistic values each
81 rules
Observe each of two terms separately
and
Cascade fuzzy controller design
- consider the Lyapunov function of the following form:
University of Zagreb, Faculty of Electrical Engineering & Computing SWAN06 Department of Control and Computer Engineering ARRI, December 8, 2006
Laboratory for Robotics and Intelligent Control Systems 9
Observe each of two terms separately
and
N Z P
N
Z
P
eredr
LP P Z
P Z N
Z N LN
N Z P
N
Z
P
eed
LN N Z
N Z P
Z P LP
only 9+9=18 rules
Cascade fuzzy controller design
University of Zagreb, Faculty of Electrical Engineering & Computing SWAN06 Department of Control and Computer Engineering ARRI, December 8, 2006
Laboratory for Robotics and Intelligent Control Systems 10
Experimental results – ball and beam
Experimental results – ball and beam
University of Zagreb, Faculty of Electrical Engineering & Computing SWAN06 Department of Control and Computer Engineering ARRI, December 8, 2006
Laboratory for Robotics and Intelligent Control Systems 11
Experimental results – ball and beam
V 0
University of Zagreb, Faculty of Electrical Engineering & Computing SWAN06 Department of Control and Computer Engineering ARRI, December 8, 2006
Laboratory for Robotics and Intelligent Control Systems 12
Experimental results – ball and beam
V 0
University of Zagreb, Faculty of Electrical Engineering & Computing SWAN06 Department of Control and Computer Engineering ARRI, December 8, 2006
Laboratory for Robotics and Intelligent Control Systems 13
Experimental results – 2 DOF airplane
University of Zagreb, Faculty of Electrical Engineering & Computing SWAN06 Department of Control and Computer Engineering ARRI, December 8, 2006
Laboratory for Robotics and Intelligent Control Systems 14
Fuzzy Lyapunov stability and occupancy grid – implementation to formation control
I2C bus
Ethernet
SC12 (BECK)
IR sensors
encoders
Web cam DCS-900
Wifibot – Robosoft, France
University of Zagreb, Faculty of Electrical Engineering & Computing SWAN06 Department of Control and Computer Engineering ARRI, December 8, 2006
Laboratory for Robotics and Intelligent Control Systems 15
Visual feedback – web cam DCS-900
320:240 or 640:480
Wide angle lens (Sony 0.6x)
46o 75o
Rj
0
Fuzzy Lyapunov stability and occupancy grid – implementation to formation control
University of Zagreb, Faculty of Electrical Engineering & Computing SWAN06 Department of Control and Computer Engineering ARRI, December 8, 2006
Laboratory for Robotics and Intelligent Control Systems 16
markers
formation definition - graph (Desai et al.)
, , ,
, , ,
j v ij ij ij ij
j ij ij ij ij
v l l
l l
fuzzy controllers Formation requiresincreasing order of IDs!
set of predefined rules for formation changepossible collisions during formation change
Fuzzy Lyapunov stability and occupancy grid – implementation to formation control
University of Zagreb, Faculty of Electrical Engineering & Computing SWAN06 Department of Control and Computer Engineering ARRI, December 8, 2006
Laboratory for Robotics and Intelligent Control Systems 17
61 6562 63 64
51
d55
e52 53 54
41 4542 43 44
31 3532
b 3334
c
21 2522 23 24
11 151213
a 14
61 6562 63 64
51d
55e
52 53 54
41 4542 43 44
31 3532b 33 34
c
21 2522 23 24
11 1512 13a 14
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.
