Research‎ > ‎Seminars‎ > ‎Seminars 2009‎ > ‎

### 11.12.2009 - BIRS: Noise, Time Delay and Balance Control

 Tamas InspergerSystem under consideration\dot{x} = Ax + Buy = Cxy 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) dsSmith predictorx = \frac{1}{s-a}us-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 stateSmith predictor is sensitive to parameters and not necessarily good to stabilize unstable systemsPredictive controlUse the available delayed state to predict the current stateDelayed feedbackUse the delayed output directlyClassification of linear systemsTime invariant ODEn dimensions = n eigen-valuesTime invariant DDEn dimensions = infinite number of eigen-valuesTime periodic ODEOn 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 DDEInfinite number of eigen-valuesBrockett problem\dot{x}(t) = A x(t) + BG(t)Cx(t-\tau)Act and waitG(t) = 0 during waiting period, which is longer than delayG(t) = gg(t) during acting period, which is shorter than delayStep-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)+stuffx(T)=\Phi^3(T)x(0)This ends up with in n-poles.Francisco Valero-CuevasInternal modelsPredictive strategiesState estimatorsDevelopment in childhoodBiological computationJames Finley and Eric PerreaultFeedforward vs feedback control during balanceHeightened co-contraction (Hogan 1984; Milner 2002)Larger gain on stretch reflexes during a "compliant" environmentCo-contraction strategysee co-contraction increase in "unstable" conditionco-contraction appears to inhibit stretch reflexAnkle stiffness increases as stability of joint decreasesWhat about signs on net stiffness? -Kank-Kenv+mglThere appears to be a bias towards feedforward over feedback controlTim Kiemel and John Jeka Linearized "unstable" pendulum model with delayed PD controlPlant is the mapping from EMG to body segment anglesFeedback is the mapping between changes in body angles into EMGIntrinsic stiffness vs ankle stiffnessstability achieved with a combination of hip and ankle stiffnessFeedback in the nervous system is probably not PD, most likely something betterMeeting thoughtsIntermittent vs Continuous and Linear vs Non-linearHow 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, 1969Palmor ZJ (1996) Time delay compensation smith predictor and its modifications. Levine "The control handbook", CRC PressGabor StepanChaos is amusing. This means that there needs to be large non-linearitiesDigital 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 problemsInstead of linearizing equations that have noise in them transfer the system into probability space where the equations are linear in probabilityDelayed Langevin equationTry and determine if you can tell if a system is linear or non-linear from time-series data