Identification of multiple-input, single-output, discrete transfer function models. Application to ankle stiffness.

Dynamic ankle joint stiffness defines the relationship between the position of the ankle and the torque acting about it and can be separated into intrinsic and reflex components.Under stationary conditions, intrinsic stiffness can be described by a linear second order system while reflex stiffness is described by a Hammerstein system whose input is delayed velocity.Given that reflex and intrinsic torque cannot be measured separately, there has been much interest in the development of system identification techniques to separate them analytically.To date, most methods have been nonparametric and as a result there is no direct link between the estimated parameters and those of the stiffness model.This thesis demonstrates that the ankle stiffness model can be approximated by a discrete-time, multiple input, single output linear transfer function model and introduces a novel algorithm for the identification of this class of models.As the algorithm is novel, proofs of the convergence and the existence of the solution are provided.Through simulations we show that the algorithm gives unbiased results even in the presence of large non-white noise.Application of the method to experimental data demonstrates that it produces results consistent with previous findings.3-5 A) Input position.B) Output torque, continuous-time simulation (blue line) and discrete-time simulation (red line).Zoom-in in the selected are of the C) Input position and, D) Output torque. . . .40 3-6 A) Input position.B) Difference between the output of the continuous and discrete time models. . . . . . . . . . . . . . . . . . . . . . . .40 4-1 Block diagram of a Multiple-Input Single Output transfer function model with non-white disturbances . . . . . . . . . . . . . . . . . .43 6-1 Different components of the noise signal.a) GWN signal, b) 1 Hz low-pass filtered GWN signal, c) 60Hz sinusoidal and d) total noise added to the torque signal. . . . . . . . . . . . . . . . . . . . . . . 100 6-2 a) 10 s segment of the position input signal used in simulations, b) Amplitude distribution of the input signal. . . . . . . . . . . . . .101 x 6-3 a) Intrinsic Torque and b) Reflex torque. . . . . . . . . . . . . . . .102 6-4 a) Position input signal, b) total torque elicited by the position input signal displayed in panel a. . . . . . . . . . . . . . . . . . . . . . .103 6-5 Total Torque (red line) and Observer Torque (blue line). . . . . . . .104 6-6 Identified intrinsic parameters K, B and I with MISO SRIV algorithm (panels a, b and c), Naive IV algorithm (panels d, e and f) and Matlab's PEM algorithm (panels g, h and i).Red line in each panel is the true value, gray dots are the results of the 100 simulated experiments, blue line is mean, the red shading represents one standard deviation and the blue shading is the 90% range. . . . . .105 6-7 Identified intrinsic parameters g, ω and ζ with MISO SRIV algorithm (panels a, b and c), Naive IV algorithm (panels d, e and f) and Matlab's PEM algorithm (panels g, h and i).Red line in each panel is the true value, gray dots are the results of the 100 simulated experiments, blue line is mean, the red shading represents one standard deviation and the blue shading is the 90% range.Note that in panels h) and i) the 95th percentile is out of the figure limits.1066-8 Identified shape of the nonlinearity in the 100 simulated experiments (blue line) and true nonlinearity used in simulation (red line).Algorithms : a) MISO SRIV, b) Naive IV, c) Matlabs's PEM and d) Parallel-Cascade. . . . . . . . . . . . . . . . . . . . . . . . . . .108 6-9 Identified intrinsic compliance with a) MISO SRIV, b) Naive IV, c) Matlab's PEM and d) Parallel-Cascade algorithms.Red lines represent the true values, blue lines the mean of the 100 simulated experiments and the blue shading are the 90% range (5th percentile to 95th percentile). . . . . . . . . . . . . . . . . . . . . . . . . . . .109 6-10 Identified linear element of the reflex stiffness with a) MISO SRIV, b) Naive IV, c) Matlab's PEM and d) Parallel-Cascade algorithms.Red lines represent the true values, blue lines the mean of the 100 simulated experiments and the state blue shadows are the 90% range (5th percentile to 95th percentile). . . . . . . . . . . . . . . .110 xi 6-11 %VAF between the reflex torque obtained with the true model and with the identified model.Algorithms : a) MISO SRIV, b) Naive IV, c) Matlabs's PEM and d) Parallel-Cascade. . . . . . . . . . . .111 6-12 %VAF between the intrinsic torque obtained with the true model and with the identified model.Algorithms : a) MISO SRIV, b) Naive IV, c) Matlabs's PEM and d) Parallel-Cascade. . . . . . . . . . . .112 6-13 %VAF between the total torque obtained with the true model and with the identified model.Algorithms : a) MISO SRIV, b) Naive IV, c) Matlabs's PEM and d) Parallel-Cascade. . . . . . . . . . . .113 6-14 %VAF between the total torque obtained with the true model and with the identified model using Kukreja's parametric algorithm . .114 6-15 Experimental apparatus.Subjects lay supine while their left foot is perturbed by an electrohydraulic actuator.Ankle position and torque are acquired and used to estimate reflex and intrinsic stiffness in real-time.Feedback information is displayed on LCD monitor hung over the subjects head. . . . . . . . . . . . . . . . . . . . . . .

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