1. Delayed feedback of sampled higher derivatives Tamas Insperger€, Gabor Stepan€, Janos Turi$ €Department of Applied Mechanics Budapest University of Technology and Economics $Programs in Mathematical Sciences University of Texas at Dallas

2. Contents Stability gained with time-periodic parameters
Human balancing (delay and threshold)
The labyrinth and the eye – a mechanical view
Robotic balancing (sampling and round-off)
Micro-chaos (stable & unstable)
Segway – without gyros
Retarded, neutral and advanced FDEs (linear)
Stability achieved with sampled higher derivatives
Conclusions

3. The delayed Mathieu equation
Analytically constructed stability chart for testing numerical methods and algorithms
Time delay and time periodicity are equal:
Mathieu equation (1868)
Delayed oscillator (1941)

4. Stability chart – Mathieu equation
Floquet (1883)
Hill (1886)
Rayleigh(1887)
van der Pol &
Strutt (1928)
Strutt – Ince diagram (1956)
Swing (2000BC)
Stephenson (1908), Swinney (2004), Zelei (2005)

5. Stability chart – delayed oscillator
Vyshnegradskii…
Pontryagin (1942)
Nyquist (1949)
Bellman &
Cooke (1963)
Hsu & Bhatt (1966) Olgac (2000)

6. The delayed Mathieu – stability charts
ε= ε=0

7. Stability chart of delayed Mathieu
Insperger, Stepan Proc Roy Soc A (2002)

8. Chaos is amusing
Unpredictable games – strong nonlinearities: throw dice, play cards/chess, computer games ball games (football, soccer, basketball… impact) plus nonlinear rules (tennis 6/4,0/6,6/4, snooker) balancing (skiing, skating, kayak, surfing,…)
Ice-hockey (one of the most unpredictable games) - impacts between club/puck/wall - impacts between players/wall - self-balancing of players on ice (non-holonomic) - continuous and fast exchanging of players

9. Stabilization (balancing)
Control force: Q = – Px – Dx
Large delays can destroy this simple strategy, but time-periodic parameters can help…

10. Balancing inverted pendulum
Higdon, Cannon (1962) …10-20 papers / year
n = 2 DoF  , x ; x – cyclic coordinate
linearization at  = 0

11. Human balancing
Analogous or digital? Winking, eye-motion – ‘self-sampling’ plus neurons firing… still, not ‘digital’
1) Q(t) = P(t) + D(t) (PD control)
 ≡ 0 is exponentially stable  D > 0, P > mg
2) Q(t) = P(t – ) + D(t – ) (with ‘reflex’ delay  )

13. Schurer Math Nachr 1948 … Stepan Ret Dyn Syst 1989… Sieber Krauskopf Phys D 2004

14. Stability chart & critical delay
 instability

15. Stability chart & critical reflex delay
 instability

16. Experimental observations
Kawazoe (1992) untrained manual control
(Dagger, sweep, pub)
Self-balancing: Betzke (1994) target shooting 0.3 – 0.7 [Hz]
(Daffertshofer 2009)

17. Stability is the art of keeping the balance

18. Labyrinth – human balancing organ
Dynamic receptor
Static receptor
Both angle and angular velocity signals are needed!

19. Vision and balancing
Vision can help balancing even when labyrinth does not function properly (e.g., ‘dry ear’ effect)
The visual system also provides the necessary angle and angular velocity signals!
But: the vertical direction is needed (buildings, trees), otherwise it fails…
Delay in vision and ‘thinking’

20. Tactile / auditory / visual ~ sensors / cortex
organ
effect
overall performance
cortex
brain
small
large
skin
pressure
object
fast
medium
ear
sound
eye
light
slow
delay
distance
Lynx ~ Italian (National) Academy

21. Medial Temporal Loop τ ~ 0.1s
brain
Colliculus superior
τ > 0.6 s
Medial Temporal Loop
MTL
τ ~ 0.1s
arm
eyes

22. Human balancing – some conclusions
We could reduce the delay below critical value through the MTL (Medial Temporal Loop)
But we cannot reduce much the thresholds of our sensory system (glasses...)
Both delay and threshold increase with age – see increasing number of fall-overs in elderly homes
Reduce gains, add stochastic perturbation to signal to decrease threshold at a 3rd sensory system – our feet (Moss, Milton, Nature, 2003)
Delay & threshold lead to chaos… (stochastic nature)

23. Digital balancing 1) Q = 0 – no control   = 0 is unstable
2) Q(t) = P(t) + D(t) (PD control)
 = 0 is exponentially stable  D > 0, P > mg
3) Q(t) ≡ P (tj – ) + D (tj – ) (with sampling  )

24. Alice’s Adventures in Wonderland
Lewis Carroll (1899)
Lewis Carroll, 1899

25. Sampling delay of digital control
ZOH

26. Digitally controlled pendulum,
(Claussen)

27. Stability of digital control – sampling
Hopf
pitchfork

30. ABB
Sampling frequency of industrial robots ~ 30 Hz for the years 1990 – above 100 Hz recently
Force control (EU 6FP RehaRob project), and balancing (stabilization-)tasks
RehaRob
Balancing

31. Random oscillations of robotic balancing
sampling time and
quantization (round-off)

32. Stability of digital control – round-off
h – one digit converted to control force
det(I – B) = 0 
1 = e >1, 2 = e–, 3 = 0

33. 1D cartoon – the micro-chaos map
Drop 2 dimensions, rescale x with h  a  e, b  P
A pure math approach ( p > 0 , p < q )
solution with xj = y(j) leads to -chaos map,
a = ep, b = q(ep – 1)/p  a > 1, (0 <) a – b < 1
small scale: xj+1= a xj , large scale: xj+1= (a – b) xj

34. Micro-chaos map large scale small scale
Typical in digitally controlled machines

35. Csernak,Stepan (Int J Bif Chaos ’09)
2D micro-chaos map
ZOH + delay, and round-off for 1st order process:
(p > 0, p < q)
Solution and Poincare lead to
(a >1, a – b < 1)
Linearization at fixed points leads to eigenvalues
So in 1 step the solution settles at an attractor that has a graph similar to the 1D micro-chaos map
Csernak,Stepan (Int J Bif Chaos ’09)

36. Enikov,Stepan (J Vib Cont, 98)
3D micro-chaos
Enikov,Stepan (J Vib Cont, 98)

38. Vertical direction?

43. Segway – mechanical model
accelerometer

44. Segway control with delay
Analog case
Advance DDE …unstable for any “time delay”.
Digital case

45. Retarded DDE
Analog (Hayes, 1951) Digital

46. Neutral DDE
Analog (Kolmanovski, Nosov 1986) Digital

47. Advanced DDE
Analog (El’sgolt’c 1964) Digital

52. Balancing the self-balanced
Warning: only fathers have the right to do this…
Thank you for your attention!
Delay effects in brain dynamics Phil. Trans. R. Soc. A 367 (2009) doi: /rsta
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