THE MATHEMATICS OF HUMAN BRAIN              &       HUMAN LANGUAGE            With Applications              Madan M. Gupt...
MATHEMATISC OF HUMAN BRAIN :   THE NEURAL NETWORKS         Brain:         The carbon based          Computer              ...
BIOLOGICAL AND ARTIFICIAL NEURONS                    Neural                                Neural                         ...
MATHEMATISC OF HUMAN LANGUAGETHE FUZZY LOGIC- Today the weather is very   good.- This tea is very tasty.- This fellow is v...
A BIOLOGICAL NEURONS & ITS MODEL        Neural                                   Neural                                   ...
OUTLINE Introduction Motivations Important remarks Examples Conclusions                      6
SOME KEY WORDS:-- PERCEPTION,-- Cognition-- Neural network-- Uncertainty-- Randomness-- Fuzzy-- Quantitative-- Qualitative...
Brain:   The carbon based computerBrain(computer)    Vision    (perception) Feedback                                ISRL  ...
A BIOLOGICAL MOTIVATION:          THE HUMAN CONTROLLER:        A ROBUST NEURO- CONTROLLER                                 ...
APPLICATIONS OF NN & FL IN AGRICULLTURE:   - Control of farm machines:        speed and spray control in a tractor   - D...
EXAMPLES OFOPTIMAL DESIGN OF    MACHINE   CONTOLLERS                    11
ON THE DESIGN OF ROBUST ADAPTIVECONTROLLER: A NOVEL PERSPECTIVE   Dynamic pole-motion based controller :     A robust cont...
AN EXAMPLE:                                      ControllerA typical second-order system with position (x1) and velocity (...
DEFINITION OF THE VARIOUS PARAMETERSIN THE COMPLEX S-PLANE                      s    j                                ...
SOME IMPORTANT PARAMETERS INA STEP RESPONSE OF A SECOND-ORDER SYSTEM     ,                                           15
A TYPICAL SYSTEM RESPONSE TO A UNIT-STEPINPUT                          x                          x                       ...
SYSTEM RESPONSE TO A UNIT STEP INPUTWITH DIFFERENT POLE LOCATIONS                           x                           x ...
SYSTEM RESPONSE TO A UNIT STEP INPUTWITH DIFFERENT POLE LOCATIONS                          x x                        B: O...
SYSTEM RESPONSE TO A UNIT STEP INPUTWITH DIFFERENT POLE LOCATIONS                                       19
SYSTEM RESPONSE TO A UNIT STEP INPUTWITH DIFFERENT POLE LOCATIONS                       x                       x         ...
DEVELOPMENT OF AN ERROR-BASEDADAPTIVE CONTROLLER DESIGN APPROACH             x x            Overdampted                   ...
SYSTEM RESPONSE TO A UNIT STEP INPUTWITH DYNAMIC POLE MOTIONS                   For initial large errors:                 ...
Remark 1:A design criterion for the adaptive   controller:  (i) If the system error is large, then make the      damping r...
Remark 2:Design of parameters for the adaptive    controller:  (i) Position feedback Kp controls the natural     frequency...
Thus, we can design the adaptivecontroller parameters for positionfeedback Kp(e,t) and velocity feedbackKv(e,t) as a funct...
System error:    System output:Controller parametersPosition feedback:Velocity feedback:                        26 26
STRUCTURE OF THE PROPOSED ADAPTIVECONTROLLER                                     27
SOME SUGGESTED FUNCTIONS FORTHE POSITION, KP(E,T), AND VELOCITY, KV(E,T),FEEDBACK GAINS                                   ...
SOME EXAMPLES FOR THE DESIGN OF A    ROBUST NEURO-CONTROLLER   Example1: Satellite positioning control system   Example2...
EXAMPLE1: SATELLITE POSITIONING CONTROLSYSTEM                                           2R           1     x2     1     x1...
Example1: Satellite Positioning Control System (cont.)             (An overdamped system)                                 ...
Example1: Satellite Positioning Control System (cont.)u (t )  f (t )  [ K p (e, t ) x1 (t )  K v (e, t ) x 2 (t )]     ...
Example1: Satellite Positioning Control System (cont.)                  u (t )  f (t )  [ K p (e, t ) x1 (t )  K v (e, ...
Example1: Satellite Positioning Control System (cont.)In the design of controller, the parameters arechosen using the foll...
Example1: Satellite Positioning Control System (cont.)                                         (final poles are at        ...
Example1: Satellite Positioning Control System (cont.)         (dynamic pole motion and output)   1y(t)   O               ...
Example1: Satellite Positioning Control System (cont.)           (dynamic pole motion and error)       1e(t)   O          ...
Example1: Satellite Positioning Control System (cont.)                                                         38
Example1: Satellite Positioning Control System (cont.)                   initial values                              versu...
Example1: Satellite Positioning Control System (cont.)       Dynamic pole movement as a function of error                 ...
EXAMPLE2: AN UNDERDAMPED SYSTEMwith open-loop poles at -0.1±j2                      4         Gp( s)  2                 s...
Example2: An Underdamping System (cont.)                        r(t) (reference input)                                    ...
Example2: An Underdamping System (cont.)                                           43
Example2: An Underdamping System (cont.)                                           44
Example2: An Underdamping System (cont.)                                           45
EXAMPLE 3: THIRD-ORDER SYSTEM                                46
Example3: Third-Order System (cont.)                                       47
Example3: Third-Order System (cont.)                                       48
Example3: Third-Order System (cont.)Dynamic pole zero movement (DPZM) as a function of error                              ...
Example3: Third-Order System (cont.)Dynamic pole zero movement (DPZM) as a function of error                              ...
EXAMPLE4: NONLINEAR SYSTEM WITHHYSTERESIS                                          mass with hysteretic spring            ...
Example4: Non-linear System with Hysteresis (cont.)                                                      52
Example4: Non-linear System with Hysteresis (cont.)                                                      53
Example4: Non-linear System with Hysteresis (cont.)                                                      54
Example4: Non-linear System with Hysteresis (cont.)                                                      55
CONCLUSUONSIn this work we have presented a novel approach for the  design of a robust neuro-controller for complex dynami...
FURTHER WORK   We are in the process of designing the    neuro-controller for non-linear and only    partially known syst...
!!! THANK YOU!!! Any Questions      or comments???                   58
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Mathematics of human brain & human language

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Lect. by Prof. Madan M. Gupta,
Intelligent Systems Research Laboratory College of Engineering, University of Saskatchewan Saskatoon, SK., Canada, S7N 5A9 1-(306) 966-5451

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Mathematics of human brain & human language

  1. 1. THE MATHEMATICS OF HUMAN BRAIN & HUMAN LANGUAGE With Applications Madan M. Gupta Intelligent Systems Research LaboratoryCollege of Engineering, University of Saskatchewan Saskatoon, SK., Canada, S7N 5A9 1-(306) 966-5451 Madan.Gupta@usask.ca http//:www.usask.ca/Madan.Gupta 1
  2. 2. MATHEMATISC OF HUMAN BRAIN : THE NEURAL NETWORKS Brain: The carbon based Computer 2
  3. 3. BIOLOGICAL AND ARTIFICIAL NEURONS Neural Neural output input
  4. 4. MATHEMATISC OF HUMAN LANGUAGETHE 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. 4
  5. 5. A BIOLOGICAL NEURONS & ITS MODEL Neural Neural output input 5
  6. 6. OUTLINE Introduction Motivations Important remarks Examples Conclusions 6
  7. 7. SOME KEY WORDS:-- PERCEPTION,-- Cognition-- Neural network-- Uncertainty-- Randomness-- Fuzzy-- Quantitative-- Qualitative-- Subjective-- Reasoning-- ------- - --- ----- etc. 7
  8. 8. Brain: The carbon based computerBrain(computer) Vision (perception) Feedback ISRL Hand (actuator) 8 8
  9. 9. A BIOLOGICAL MOTIVATION: THE HUMAN CONTROLLER: A ROBUST NEURO- CONTROLLER 9 by googling
  10. 10. 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. 10
  11. 11. EXAMPLES OFOPTIMAL DESIGN OF MACHINE CONTOLLERS 11
  12. 12. ON THE DESIGN OF ROBUST ADAPTIVECONTROLLER: A NOVEL PERSPECTIVE Dynamic pole-motion based controller : A robust control design approach 12
  13. 13. AN EXAMPLE: ControllerA typical second-order system with position (x1) and velocity (x2)feedback controller with parameters K1 and K2 13
  14. 14. DEFINITION OF THE VARIOUS PARAMETERSIN THE COMPLEX S-PLANE s    j 14
  15. 15. SOME IMPORTANT PARAMETERS INA STEP RESPONSE OF A SECOND-ORDER SYSTEM , 15
  16. 16. A TYPICAL SYSTEM RESPONSE TO A UNIT-STEPINPUT x x Underdamped system 16
  17. 17. SYSTEM RESPONSE TO A UNIT STEP INPUTWITH DIFFERENT POLE LOCATIONS x x A:Underdampted system ( ) 17
  18. 18. SYSTEM RESPONSE TO A UNIT STEP INPUTWITH DIFFERENT POLE LOCATIONS x x B: Overdampted system ( ) 18
  19. 19. SYSTEM RESPONSE TO A UNIT STEP INPUTWITH DIFFERENT POLE LOCATIONS 19
  20. 20. SYSTEM RESPONSE TO A UNIT STEP INPUTWITH DIFFERENT POLE LOCATIONS x x A:Underdampted system ( ) x x B: Overdampted system ( ) 20
  21. 21. DEVELOPMENT OF AN ERROR-BASEDADAPTIVE CONTROLLER DESIGN APPROACH x x Overdampted x x Underdampted 21
  22. 22. SYSTEM RESPONSE TO A UNIT STEP INPUTWITH DYNAMIC POLE MOTIONS For initial large errors: the system follows they(t) underdamped response curve. t And for small errors: the system follows the overdamped response curvee(t) and then settles down to a steady-state value. 22
  23. 23. 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. 23
  24. 24. 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 ω n; (ii) Velocity feedback Kv controls the damping ratio ζ; 24
  25. 25. Thus, we can design the adaptivecontroller parameters for positionfeedback Kp(e,t) and velocity feedbackKv(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”. 25
  26. 26. System error: System output:Controller parametersPosition feedback:Velocity feedback: 26 26
  27. 27. STRUCTURE OF THE PROPOSED ADAPTIVECONTROLLER 27
  28. 28. SOME SUGGESTED FUNCTIONS FORTHE POSITION, KP(E,T), AND VELOCITY, KV(E,T),FEEDBACK GAINS 28
  29. 29. 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 29
  30. 30. EXAMPLE1: SATELLITE POSITIONING CONTROLSYSTEM 2R 1 x2 1 x1 F  J s s Block diagram of the satellite positioning systemSatellite positioning system 2 RF ( s )  (s)  Js 2 F (s)  2R   2 ,  for  1 s  J  30
  31. 31. Example1: Satellite Positioning Control System (cont.) (An overdamped system) 31
  32. 32. Example1: Satellite Positioning Control System (cont.)u (t )  f (t )  [ K p (e, t ) x1 (t )  K v (e, t ) x 2 (t )] K p (e, t )  K pf [1   e (t )] 2 K v (e, t )  K vf exp[ e (t )] 2 e(t )  f (t )  x1 (t ) 32
  33. 33. Example1: Satellite Positioning Control System (cont.) u (t )  f (t )  [ K p (e, t ) x1 (t )  K v (e, t ) x 2 (t )] K p (e, t )  K pf [1   e 2 (t )] n (t )  K pf  [1  { f (t )  x1 (t )}2 ]Kv (e, t )  Kvf exp[   e2 (t )] K vf exp[ { f (t )  x1 (t )}2 ] How to choose  (t )  Kpf & Kvf , e(t )and (t )  x1 (t ) f 2 K pf  [1  { f (t )  x1 (t )}2 ] α&β 33
  34. 34. Example1: Satellite Positioning Control System (cont.)In the design of controller, the parameters arechosen 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). 34
  35. 35. Example1: Satellite Positioning Control System (cont.) (final poles are at -1 and -3) (initial poles are at -0.1 j2) For neuro-control system : Tr1 = 1.26 (sec) For overdamped system: Tr2 = 2.67 (sec) Tr1 Tr2 35
  36. 36. Example1: Satellite Positioning Control System (cont.) (dynamic pole motion and output) 1y(t) O t 36
  37. 37. Example1: Satellite Positioning Control System (cont.) (dynamic pole motion and error) 1e(t) O t 37
  38. 38. Example1: Satellite Positioning Control System (cont.) 38
  39. 39. Example1: Satellite Positioning Control System (cont.) initial values versus final values 39
  40. 40. Example1: Satellite Positioning Control System (cont.) Dynamic pole movement as a function of error 40
  41. 41. EXAMPLE2: AN UNDERDAMPED SYSTEMwith open-loop poles at -0.1±j2 4 Gp( s)  2 s  0.2s  4 41
  42. 42. Example2: An Underdamping System (cont.) r(t) (reference input) y(t) (Overdamped Control) y(t) (Neuro-Control) 42
  43. 43. Example2: An Underdamping System (cont.) 43
  44. 44. Example2: An Underdamping System (cont.) 44
  45. 45. Example2: An Underdamping System (cont.) 45
  46. 46. EXAMPLE 3: THIRD-ORDER SYSTEM 46
  47. 47. Example3: Third-Order System (cont.) 47
  48. 48. Example3: Third-Order System (cont.) 48
  49. 49. Example3: Third-Order System (cont.)Dynamic pole zero movement (DPZM) as a function of error 49
  50. 50. Example3: Third-Order System (cont.)Dynamic pole zero movement (DPZM) as a function of error 50
  51. 51. EXAMPLE4: NONLINEAR SYSTEM WITHHYSTERESIS mass with hysteretic spring y u e - r robust adaptive controller + Er[v]: stop operator p(z): density function Y. Peng et. al. (2008) 51
  52. 52. Example4: Non-linear System with Hysteresis (cont.) 52
  53. 53. Example4: Non-linear System with Hysteresis (cont.) 53
  54. 54. Example4: Non-linear System with Hysteresis (cont.) 54
  55. 55. Example4: Non-linear System with Hysteresis (cont.) 55
  56. 56. CONCLUSUONSIn 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. 56
  57. 57. 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. 57
  58. 58. !!! THANK YOU!!! Any Questions or comments??? 58

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