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I'm trying to reproduce and extend figure 9 results in

Nonlinear dynamical system identification from uncertain and indirect measurements"HU Voss, J Timmer, J Kurths - International Journal of Bifurcation and Chaos, Vol. 14, No. 6 (2004) 1905-1933.

In this work, Voss et.al explains how to use unscented Kalaman filter for parameter estimation and parameter tracking. They have considered FitzHugh - Nagumo system for this calculation. FitzHugh - Nagumo system is given by following set of differential equations; \begin{aligned} \dot{x}_{1} &=c\left(x_{2}+x_{1}-\frac{x_{1}^{3}}{3}+z(t)\right) \\ \dot{x}_{2} &=-\frac{x_{1}-a+b x_{2}}{c} \end{aligned}

The parameters are $a = 0:7, b = 0:8$, and $c = 3$.The variable $x_1$ is a membrane voltage which can be measured, and $x_2$ is a usually unobserved lumped variable that describes the combined effect of different kinds of ionic currents. The external voltage $z(t)$ influences the dynamics. Here we assume that $z(t)$ is varying slowly in comparison with the other two variables; it is treated here as the parameter to be tracked.

The measurements are constructed by corrupting the component $x_1$; $t$ with Gaussian noise $\eta = N(0; R)$ at 800 equidistant $(t = 0:2)$ sample times. The standard deviation of $\eta$, $\sqrt{R}$, is 20% of the standard deviation of the $x_1$-component of the system. The external variable $z(t)$ is constructed as a cosine-halfwave, up to an additive constant. The initial guesses of $z$ and $x_2$ are set to zero, the initial guess of $x_1$ is set to the measured value. For parameter tracking, the augmented state vector approach is used with a constant covariance matrix $Q = 0:015$.

I need to track the parameter $a$ together with $z(t)$, but I don't understand how to change the algorithm and unscented Kalaman filter equations according to this. Can someone please explain this? Thank you very much!

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  • $\begingroup$ "The initial guesses of $z$ and $x_2$ are set to zero..." But you say that $z(t)$ is a driven variable -- is this a typo? $\endgroup$ – TimWescott Nov 26 '20 at 21:43
  • $\begingroup$ @TimWescott Initial guesses are for the numerical integration using Runge-Kutta Method. $\endgroup$ – ccc Nov 26 '20 at 21:50
  • $\begingroup$ But $z$ is known -- maybe I need to brush up on Runge-Kutta, but you shouldn't have to guess. $\endgroup$ – TimWescott Nov 26 '20 at 21:52
  • $\begingroup$ @TimWescott Thanks for that. Do you have any idea about this question if we set tha $z(t)$ is unknown? If so can you provide it? $\endgroup$ – ccc Nov 26 '20 at 21:55

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