Tamas Insperger  System under consideration
 y = Cx
 y is only known by y(t\tau)
 u = f(ry)
 \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(t2\tau)
 integral approximation u(t)=\int_0^{t\tau} w_0 x(s) + w_1\dot{x}(s) ds
 Smith predictor
 x = \frac{1}{sa}u
 sa+b e^{\tau s} or \dot{x}(t) + a x(t)  b x(t\tau)
 becomes \ddot{x}(t) +(b2a)\dot{x}(t) + a(ab)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 eigenvalues
 Time invariant DDE
 n dimensions = infinite number of eigenvalues
 Time periodic ODE
 On one period (need to know the map) then the there are n eigenvalues for ndimensions 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 eigenvalues
 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
 Stepbystep 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(ts)} 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 npoles.
Francisco ValeroCuevas  Internal models
 Predictive strategies
 State estimators
 Development in childhood
 Biological computation
James Finley and Eric Perreault  Feedforward vs feedback control during balance
 Heightened cocontraction (Hogan 1984; Milner 2002)
 Larger gain on stretch reflexes during a "compliant" environment
 Cocontraction strategy
 see cocontraction increase in "unstable" condition
 cocontraction appears to inhibit stretch reflex
 Ankle stiffness increases as stability of joint decreases
 What about signs on net stiffness? KankKenv+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 Nonlinear
 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)524527, 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 nonlinearities
 Digital control systems introduce "spatial" and "temporal" delays.
 There is a point where there is "microchaos" 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 nonlinear from timeseries data
