Mathematics of human brain & human language
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THE MATHEMATICS OF HUMAN BRAIN &
HUMAN LANGUAGEWith Applications
Madan M. Gupta Intelligent Systems Research Laboratory
College of Engineering, University of SaskatchewanSaskatoon, SK., Canada, S7N 5A9
1-(306) [email protected]
http//:www.usask.ca/Madan.Gupta
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MATHEMATISC OF HUMAN BRAIN :
THE NEURAL NETWORKS
Computer
Brain:The carbon based
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BIOLOGICAL AND ARTIFICIAL NEURONS
W
b
W
b
Input
Hidden layersOutput layer
Output
W
bNeu
ral in
pu
t
Neu
ral
ou
tpu
t
W
b
W
b
Input
Hidden layersOutput layer
Output
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MATHEMATISC OF HUMAN LANGUAGE
THE FUZZY LOGIC
- Today the weather is very good. - This tea is very tasty. - This fellow is very rich.
- If I have some money
and the weather is good
then I will go for shopping.
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A BIOLOGICAL NEURONS & ITS MODEL
W
bNeu
ral in
pu
t
Neu
ral
ou
tpu
t
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OUTLINE
Introduction Motivations Important remarks Examples Conclusions
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SOME KEY WORDS:
-- PERCEPTION,-- Cognition-- Neural network-- Uncertainty-- Randomness-- Fuzzy-- Quantitative-- Qualitative-- Subjective-- Reasoning-- ------- - --- ----- etc.
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Brain:The carbon based computer
Vision (perception)
Hand (actuator)
Brain (computer)
FeedbackCog
nitio
n
(inte
llige
nce)
ISRL
INTE
LLIGENT SYSTEM
S
RESEARCH LABORATO
RY
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A BIOLOGICAL MOTIVATION: THE HUMAN CONTROLLER: A ROBUST NEURO- CONTROLLER
by googling
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APPLICATIONS OF NN & FL IN AGRICULLTURE:
- Control of farm machines: speed and spray control in a tractor
- Drying of grains, fruits and vegetables
- Irrigation - etc.etc.
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EXAMPLES OF OPTIMAL DESIGN OF
MACHINE CONTOLLERS
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ON THE DESIGN OF ROBUST ADAPTIVE CONTROLLER: A NOVEL PERSPECTIVE
Dynamic pole-motion based controller : A robust control design approach
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AN EXAMPLE:
A typical second-order system with position (x1) and velocity (x2) feedback controller with parameters K1 and K2
Controller
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DEFINITION OF THE VARIOUS PARAMETERS IN THE COMPLEX S-PLANE
js
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SOME IMPORTANT PARAMETERS IN A STEP RESPONSE OF A SECOND-ORDER SYSTEM
15
Settling time:
Peak time: Peak overshoot:
Rise time:
,
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A TYPICAL SYSTEM RESPONSE TO A UNIT-STEP INPUT
xx
Underdamped system
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SYSTEM RESPONSE TO A UNIT STEP INPUT WITH DIFFERENT POLE LOCATIONS
xx
A:Underdampted system ( )
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SYSTEM RESPONSE TO A UNIT STEP INPUT WITH DIFFERENT POLE LOCATIONS
x xB: Overdampted system ( )
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SYSTEM RESPONSE TO A UNIT STEP INPUT WITH DIFFERENT POLE LOCATIONS
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SYSTEM RESPONSE TO A UNIT STEP INPUT WITH DIFFERENT POLE LOCATIONS
xx
A:Underdampted system ( )
x xB: Overdampted
system ( )
A desired system response
(a marriage between A & B)
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DEVELOPMENT OF AN ERROR-BASED ADAPTIVE CONTROLLER DESIGN APPROACH
xx
Underdampted
x x
Overdampted
A desired error response
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For initial large errors: the system follows the underdamped response curve.
And for small errors:the system follows the overdamped response curveand then settles down to a steady-state value.
y(t)
t
e(t)
SYSTEM RESPONSE TO A UNIT STEP INPUT WITH DYNAMIC POLE MOTIONS
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Remark 1: A design criterion for the adaptive
controller:
(i) If the system error is large, then make the damping ratio, ζ, very small and natural frequency, ωn, very large.
(ii) If the system error is small, then make damping ratio, ζ, large and natural frequency, ωn, small.
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Remark 2: Design of parameters for the
adaptive controller:(i) Position feedback Kp controls the
natural frequency of the system ωn.
i.e. , the bandwidth of the system is determined by the system natural frequency ωn;
(ii) Velocity feedback Kv controls the damping ratio ζ;
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Thus, we can design the adaptive controller parameters for position feedback Kp(e,t) and velocity feedback Kv(e,t) as a function of the error e(t):
“As error changes from a large value to a small value, Kp(e,t) is varied from a very large value to a small value, and simultaneously, Kv(e,t) is varied from a very small value to a large value”.
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System error:
System output:
Controller parameters
Position feedback:
Velocity feedback:
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STRUCTURE OF THE PROPOSED ADAPTIVE CONTROLLER
Neuro-ControllerError-Based Adaptive Controller
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SOME SUGGESTED FUNCTIONS FOR THE POSITION, KP(E,T), AND VELOCITY, KV(E,T), FEEDBACK GAINS
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SOME EXAMPLES FOR THE DESIGN OF A ROBUST NEURO-CONTROLLER
Example1: Satellite positioning control system
Example2: An undrerdamped second-order system
Example3: A third-order system
Example4: A nonlinear system with hysteresis
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EXAMPLE1: SATELLITE POSITIONING CONTROL SYSTEM
Satellite positioning systemBlock diagram of the satellite positioning system
J
R2s
1
s
1F 1x2x
12
for ,)(
)(2)(
2
2
J
R
s
sFJs
sRFs
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Example1: Satellite Positioning Control System (cont.)(An overdamped system)
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)](),()(),([)()( 21 txteKtxteKtftu vp
32
)]( 1[),( 2 teKteK pfp
)]( exp[),( 2 teKteK vfv
)()()( 1 txtfte
Neuro-controller
Example1: Satellite Positioning Control System (cont.)
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Neuro-controller
)](),()(),([)()( 21 txteKtxteKtftu vp
)]( 1[),( 2 teKteK pfp
)]( exp[),( 2 teKteK vfv
)()()( 1 txtfte
])}()({1[)( 21 txtfKt pfn
])}()({1[2
])}()({exp[)(
21
21
txtfK
txtfKt
pf
vf
How to
choose Kpf & Kvf ,andα & β
Example1: Satellite Positioning Control System (cont.)
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How to choose Kpf & Kvf ,and α & βIn the design of controller, the parameters are
chosen using the following two criteria:
1. α & β : initial position of the poles should have very small damping (ζ) and large bandwidth (ωn).
2. Kpf & Kvf : final position of the poles should have large damping (ζ) and
small bandwidth (ωn).
Example1: Satellite Positioning Control System (cont.)
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(final poles are at -1 and -3)
(initial poles are at -0.1±j2)
Tr1 Tr2
For neuro-control system : Tr1 = 1.26 (sec)For overdamped system: Tr2 = 2.67 (sec)
Example1: Satellite Positioning Control System (cont.)
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t
y(t)
O
1
Example1: Satellite Positioning Control System (cont.)(dynamic pole motion and output)
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t
e(t)
O
1
Example1: Satellite Positioning Control System (cont.)(dynamic pole motion and error)
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Example1: Satellite Positioning Control System (cont.)
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initial values
final values
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versus
Example1: Satellite Positioning Control System (cont.)
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As the error
decreases, the poles
move from the initial
underdamped
positions (-0.1 ± j2)
to the final
overdamped
positions (-1 and -3).
Dynamic pole movement as a function of error
Example1: Satellite Positioning Control System (cont.)
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EXAMPLE2: AN UNDERDAMPED SYSTEM
with open-loop poles at -0.1±j2
42.0
4)(
2
sssGp
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y(t) (Neuro-Control)
y(t) (Overdamped Control)
r(t) (reference input)
Example2: An Underdamping System (cont.)
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Example2: An Underdamping System (cont.)
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Example2: An Underdamping System (cont.)
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Example2: An Underdamping System (cont.)
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EXAMPLE 3: THIRD-ORDER SYSTEM
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Example3: Third-Order System (cont.)
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Example3: Third-Order System (cont.)
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Dynamic pole zero movement (DPZM) as a function of error
Example3: Third-Order System (cont.)
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Dynamic pole zero movement (DPZM) as a function of error
Example3: Third-Order System (cont.)
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EXAMPLE4: NONLINEAR SYSTEM WITH HYSTERESIS
mass with hysteretic spring
robust adaptive controller -+r
y
eu
Er[v]: stop operatorp(z): density function
Y. Peng et. al. (2008)
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Example4: Non-linear System with Hysteresis (cont.)
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Example4: Non-linear System with Hysteresis (cont.)
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Example4: Non-linear System with Hysteresis (cont.)
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Example4: Non-linear System with Hysteresis (cont.)
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CONCLUSUONS
In this work we have presented a novel approach for the design of a robust neuro-controller for complex dynamic systems.
Neuro == learning & adaptation,
The controller adapts the parameter as a function of the error yielding the system response very fast with no overshoot.
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FURTHER WORK
We are in the process of designing the neuro-controller for non-linear and only partially known systems with disturbances.
This new approach of dynamic motion of poles leads us to investigate the stability of nonlinear and timevarying systems in much easier way.
Same approach will be extended for discrete systems as well.
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!!! THANK YOU!!!
Any Questions or
comments???