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) 966-5451Madan.Gupta@usask.ca

http//:www.usask.ca/Madan.Gupta

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MATHEMATISC OF HUMAN BRAIN :

THE NEURAL NETWORKS

Computer

Brain:The carbon based

BIOLOGICAL AND ARTIFICIAL NEURONS

W

b

W

b

Input

Hidden layersOutput layer

Output

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bNeu

ral in

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Neu

ral

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b

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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

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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

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SOME IMPORTANT PARAMETERS IN A STEP RESPONSE OF A SECOND-ORDER SYSTEM

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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

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R2s

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s

1F 1x2x

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for ,)(

)(2)(

2

2

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R

s

sFJs

sRFs

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Example1: Satellite Positioning Control System (cont.)(An overdamped system)

)](),()(),([)()( 21 txteKtxteKtftu vp

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)]( 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.)

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

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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

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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???