Lect. by Prof. Madan M. Gupta, Intelligent Systems Research Laboratory College of Engineering, University of Saskatchewan Saskatoon, SK., Canada, S7N 5A91-(306) 966-5451

1. THE MATHEMATICS OF HUMAN BRAIN
&
HUMAN LANGUAGE
With Applications
Madan M. Gupta
Intelligent Systems Research Laboratory
College 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. MATHEMATISC OF HUMAN BRAIN :
THE NEURAL NETWORKS
Brain:
The carbon based
Computer
2

3. BIOLOGICAL AND ARTIFICIAL NEURONS
Neural
Neural
output
input

4. 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.
4

5. A BIOLOGICAL NEURONS & ITS MODEL
Neural
Neural
output
input
5

6. OUTLINE
 Introduction
 Motivations
 Important remarks
 Examples
 Conclusions
6

7. SOME KEY WORDS:
-- PERCEPTION,
-- Cognition
-- Neural network
-- Uncertainty
-- Randomness
-- Fuzzy
-- Quantitative
-- Qualitative
-- Subjective
-- Reasoning
-- ------- - --- ----- etc.
7

8. Brain:
The carbon based computer
Brain
(computer)
Vision
(perception)
Feedback
ISRL
Hand
(actuator)
8 8

9. A BIOLOGICAL MOTIVATION:
THE HUMAN CONTROLLER:
A ROBUST NEURO- CONTROLLER
9
by googling

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. EXAMPLES OF
OPTIMAL DESIGN OF
MACHINE
CONTOLLERS
11

12. ON THE DESIGN OF ROBUST ADAPTIVE
CONTROLLER: A NOVEL PERSPECTIVE
Dynamic pole-motion based controller :
A robust control design approach
12

13. AN EXAMPLE:
Controller
A typical second-order system with position (x1) and velocity (x2)
feedback controller with parameters K1 and K2
13

14. DEFINITION OF THE VARIOUS PARAMETERS
IN THE COMPLEX S-PLANE
s    j
14

15. SOME IMPORTANT PARAMETERS IN
A STEP RESPONSE OF A SECOND-ORDER SYSTEM
,
15

16. A TYPICAL SYSTEM RESPONSE TO A UNIT-STEP
INPUT
x
x
Underdamped
system
16

17. SYSTEM RESPONSE TO A UNIT STEP INPUT
WITH DIFFERENT POLE LOCATIONS
x
x
A:Underdampted system ( )
17

18. SYSTEM RESPONSE TO A UNIT STEP INPUT
WITH DIFFERENT POLE LOCATIONS
x x
B: Overdampted system ( )
18

19. SYSTEM RESPONSE TO A UNIT STEP INPUT
WITH DIFFERENT POLE LOCATIONS
19

20. SYSTEM RESPONSE TO A UNIT STEP INPUT
WITH DIFFERENT POLE LOCATIONS
x
x
A:Underdampted
system ( )
x x
B: Overdampted
system ( )
20

21. DEVELOPMENT OF AN ERROR-BASED
ADAPTIVE CONTROLLER DESIGN APPROACH
x x
Overdampted
x
x
Underdampted
21

22. SYSTEM RESPONSE TO A UNIT STEP INPUT
WITH DYNAMIC POLE MOTIONS
For initial large errors:
the system follows the
y(t)
underdamped response
curve.
t And for small errors:
the system follows the
overdamped response curve
e(t)
and then settles down to a
steady-state value. 22

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. 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. 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”.
25

26. System error:
System output:
Controller parameters
Position feedback:
Velocity feedback:
26 26

27. STRUCTURE OF THE PROPOSED ADAPTIVE
CONTROLLER
27

28. SOME SUGGESTED FUNCTIONS FOR
THE POSITION, KP(E,T), AND VELOCITY, KV(E,T),
FEEDBACK GAINS
28

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. EXAMPLE1: SATELLITE POSITIONING CONTROL
SYSTEM
2R 1 x2 1 x1
F 
J s s
Block diagram of the satellite positioning system
Satellite positioning system
2 RF ( s )
 (s) 
Js 2
F (s)  2R 
 2 ,  for  1
s  J 
30

31. Example1: Satellite Positioning Control System (cont.)
(An overdamped system)
31

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. 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. Example1: Satellite Positioning Control System (cont.)
In 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).
34

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. Example1: Satellite Positioning Control System (cont.)
(dynamic pole motion and output)
1
y(t)
O
t
36

37. Example1: Satellite Positioning Control System (cont.)
(dynamic pole motion and error)
1
e(t)
O
t
37

38. Example1: Satellite Positioning Control System (cont.)
38

39. Example1: Satellite Positioning Control System (cont.)
initial values
versus
final values
39

40. Example1: Satellite Positioning Control System (cont.)
Dynamic pole movement as a function of error
40

41. EXAMPLE2: AN UNDERDAMPED SYSTEM
with open-loop poles at -0.1±j2
4
Gp( s)  2
s  0.2s  4
41

42. Example2: An Underdamping System (cont.)
r(t) (reference input)
y(t)
(Overdamped Control)
y(t)
(Neuro-Control)
42

43. Example2: An Underdamping System (cont.)
43

44. Example2: An Underdamping System (cont.)
44

45. Example2: An Underdamping System (cont.)
45

46. EXAMPLE 3: THIRD-ORDER SYSTEM
46

47. Example3: Third-Order System (cont.)
47

48. Example3: Third-Order System (cont.)
48

49. Example3: Third-Order System (cont.)
Dynamic pole zero movement (DPZM) as a function of error
49

50. Example3: Third-Order System (cont.)
Dynamic pole zero movement (DPZM) as a function of error
50

51. EXAMPLE4: NONLINEAR SYSTEM WITH
HYSTERESIS
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. Example4: Non-linear System with Hysteresis (cont.)
52

53. Example4: Non-linear System with Hysteresis (cont.)
53

54. Example4: Non-linear System with Hysteresis (cont.)
54

55. Example4: Non-linear System with Hysteresis (cont.)
55

56. 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.
56

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. !!! THANK YOU!!!
Any Questions
or
comments???
58