Parameter tracking using Augmented state vector approach and unscented Kalman filter

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!

• "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? Nov 26, 2020 at 21:43
• @TimWescott Initial guesses are for the numerical integration using Runge-Kutta Method.
– ccc
Nov 26, 2020 at 21:50
• But $z$ is known -- maybe I need to brush up on Runge-Kutta, but you shouldn't have to guess. Nov 26, 2020 at 21:52
• @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?
– ccc
Nov 26, 2020 at 21:55