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11.12.2009 - BIRS: Noise, Time Delay and Balance Control

Tamas Insperger
  • System under consideration
    • \dot{x} = Ax + Bu
    • y = Cx
      • y is only known by y(t-\tau)
    • u = f(r-y)
      • u = Dy(t-\tau)
    • \ddot{x}(t) +a_1\dot{x}(t)+a_0x(t)=u(t)
      • linear approximation u(t)=px(t-\tau)+p\tau\dot{x}(t-\tau)
      • secant approximation u(t)=2px(t-\tau)+p\tau x(t-2\tau)
      • integral approximation u(t)=\int_0^{t-\tau} w_0 x(s) + w_1\dot{x}(s) ds
  • Smith predictor
    • x = \frac{1}{s-a}u
    • s-a+b e^{\tau s} or \dot{x}(t) + a x(t) - b x(t-\tau)
    • becomes \ddot{x}(t) +(b-2a)\dot{x}(t) + a(a-b)x(t) if estimated delay and state are same as the real delay and state
    • Smith predictor is sensitive to parameters and not necessarily good to stabilize unstable systems
  • Predictive control
    • Use the available delayed state to predict the current state
  • Delayed feedback
    • Use the delayed output directly
  • Classification of linear systems
    • Time invariant ODE
      • n dimensions = n eigen-values
    • Time invariant DDE
      • n dimensions = infinite number of eigen-values
    • Time periodic ODE
      • On one period (need to know the map) then the there are n eigen-values for n-dimensions of the period
      • \dot{x}(t)=A(t)x(t)
      • A(t+T) = A(t)
      • x(T) = \Phi x(0)
    • Time periodic DDE
      • Infinite number of eigen-values
  • Brockett problem
    • \dot{x}(t) = A x(t) + BG(t)Cx(t-\tau)
  • Act and wait
    • G(t) = 0 during waiting period, which is longer than delay
    • G(t) = gg(t) during acting period, which is shorter than delay
    • Step-by-step solution
      • \dot{x}(t)=Ax(t)
      • x(t)=\Phi^1(t)x(0)
      • \Phi^1(t)=e^{At}
      • \Phi^2(t)=e^{At}+\int_{tw}^t e^{A(t-s)} B \Gamma(s) C e^{A(s-/tau)} ds
      • \Phi^3(t)=\Phi^2(t)+stuff
      • x(T)=\Phi^3(T)x(0)
        • This ends up with in n-poles.
Francisco Valero-Cuevas
  • Internal models
  • Predictive strategies
  • State estimators
  • Development in childhood
  • Biological computation
James Finley and Eric Perreault
  • Feedforward vs feedback control during balance
  • Heightened co-contraction (Hogan 1984; Milner 2002)
  • Larger gain on stretch reflexes during a "compliant" environment
  • Co-contraction strategy
    • see co-contraction increase in "unstable" condition
    • co-contraction appears to inhibit stretch reflex
    • Ankle stiffness increases as stability of joint decreases
    • What about signs on net stiffness? -Kank-Kenv+mgl
  • There appears to be a bias towards feedforward over feedback control
Tim Kiemel and John Jeka
  • Linearized "unstable" pendulum model with delayed PD control
    • Plant is the mapping from EMG to body segment angles
    • Feedback is the mapping between changes in body angles into EMG
    • Intrinsic stiffness vs ankle stiffness
      • stability achieved with a combination of hip and ankle stiffness
    • Feedback in the nervous system is probably not PD, most likely something better
Meeting thoughts
  • Intermittent vs Continuous and Linear vs Non-linear
    • How do these fundamental questions help/hurt modeling of posture?
  • Kleinman DL, "Optimal control with time delay and observation noise." IEEE Trans Automatic Control (15)524-527, 1969
  • Palmor ZJ (1996) Time delay compensation smith predictor and its modifications. Levine "The control handbook", CRC Press
Gabor Stepan
  • Chaos is amusing. This means that there needs to be large non-linearities
  • Digital control systems introduce "spatial" and "temporal" delays.
  • There is a point where there is "micro-chaos" due digitization where the system is caught in an oscillation before it gets to the regulated point.
  • Are the delays in the human system a continuous delay or a discrete delay?
  • Delayed feedback on jerk!
Questions for future math problems
  • Instead of linearizing equations that have noise in them transfer the system into probability space where the equations are linear in probability
  • Try and determine if you can tell if a system is linear or non-linear from time-series data
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