Delayed feedback of sampled higher derivatives Tamas Insperger

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Delayed feedback of sampled higher
derivativesTamas Insperger, Gabor Stepan,
Janos TuriDepartment of Applied
MechanicsBudapest University of Technology and
Economics Programs in Mathematical
SciencesUniversity of Texas at Dallas
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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

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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)

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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)
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Stability chart delayed oscillator

• 
  Vyshnegradskii
• 
  Pontryagin (1942)
• 
  Nyquist (1949)
• 
  Bellman
• 
  Cooke (1963)
• Hsu Bhatt (1966)
  Olgac (2000)

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The delayed Mathieu stability charts

• 
  b0
• 
  e1
  e0

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Stability chart of delayed Mathieu

• 
  
• 
  Insperger, Stepan
  
  Proc Roy Soc A
  (2002)

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Chaos is amusing

• Unpredictable games strong nonlinearitiesthrow
  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

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Stabilization (balancing)

• Control forceQ Px Dx
• Large delays can destroy this simple strategy,
  buttime-periodic parameters can help

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Balancing inverted pendulum

• Higdon, Cannon (1962) 10-20 papers / year
• n 2 DoF ? ?, x x cyclic coordinate
• linearization at ?
  0

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Human balancing

• Analogous or digital?Winking, eye-motion
  self-samplingplus neurons firing still, not
  digital
• 1) Q(t) P?(t) D?(t) (PD control)
• ? 0 is exponentially stable ? D gt 0, P gt
  mg
• 2) Q(t) P?(t ?) D?(t ?) (with reflex
  delay ? )

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Schurer Math Nachr 1948 Stepan Ret Dyn Syst
1989 Sieber Krauskopf Phys D 2004
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Stability chart critical delay

• 
  ? instability

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Stability chart critical reflex delay

• 
  ? instability

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Experimental observations

• Kawazoe (1992)untrained manual control
• (Dagger, sweep, pub)
• Self-balancingBetzke (1994)target shooting0.3
  0.7 Hz
• (Daffertshofer 2009)

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Stability is the art of keeping the balance
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Labyrinth human balancing organ
Dynamic receptor
Static receptor
Both angle and angular velocity signals are
needed!
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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

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

• Colliculus superior

t gt 0.6 s
Medial Temporal Loop
MTL
t 0.1s
arm
eyes
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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)

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Digital balancing

• 1) Q 0 no control
• ? ? 0 is unstable
• 2) Q(t) P?(t) D?(t) (PD control)
• ? 0 is exponentially stable ? D gt 0, P gt
  mg
• 3) Q(t) P? (tj ?) D? (tj ?) (with
  sampling ? )

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Alices Adventures in Wonderland

• Lewis Carroll (1899)

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Sampling delay of digital control
delay
ZOH
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Digitally controlled pendulum

• ,
• 
  

(Claussen)
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Stability of digital control sampling

• 
  Hopf
• 
  pitchfork

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ABB

• Sampling frequency of industrial robots 30 Hz
  for the years 1990 2005 above 100 Hz recently
• Force control (EU 6FP RehaRob project),and
  balancing (stabilization-)tasks

Balancing
RehaRob
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Random oscillations of robotic balancing

• 
  sampling time
  and
• quantization (round-off)

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Stability of digital control round-off

• h one digit converted to control force
• det(?I
  B) 0 ?
• ?1 e?
  gt1, ?2 e?, ?3 0

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1D cartoon the micro-chaos map

• Drop 2 dimensions, rescale x with h ? a ? e?,
  
  b ? P
• A pure math approach ( p gt 0 , p lt q )
• solution with xj y(j) leads to ?-chaos map,
• a ep, b q(ep 1)/p ? a gt 1, (0 lt) a b
  lt 1
• small scale xj1 a xj , large scale xj1 (a
  b) xj

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Micro-chaos map

• large scale
• small scale
• Typical in digitallycontrolled machines

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2D micro-chaos map

• ZOH delay, and round-off for 1st order process
• 
  (p gt 0, p lt q)
• Solution and Poincare lead to
• 
  (a gt1, a b lt 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)
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3D micro-chaos

Enikov,Stepan (J Vib Cont, 98)
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Vertical direction?
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Segway mechanical model
accelerometer

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Segway control with delay

• Analog case
• Advance DDE unstable for any time delay.
• Digital case

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Retarded DDE

• Analog (Hayes, 1951)
  Digital

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Neutral DDE

• Analog (Kolmanovski, Nosov 1986)
  Digital

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Advanced DDE

• Analog (Elsgoltc 1964) Digital

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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 10.1098/rsta.2008.0279

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