Monte Carlo method for adaptively estimating the unknown parameters and the dynamic state of chaotic systems

We propose a Monte Carlo methodology for the joint estimation of unobserved dynamic variables and unknown static parameters in chaotic systems. The technique is sequential, i.e., it updates the variable and parameter estimates recursively as new observations become available, and, hence, suitable for online implementation. We demonstrate the validity of the method by way of two examples. In the first one, we tackle the estimation of all the dynamic variables and one unknown parameter of a five-dimensional nonlinear model using a time series of scalar observations experimentally collected from a chaotic ${\text{CO}}_2$ laser. In the second example, we address the estimation of the two dynamic variables and the phase parameter of a numerical model commonly employed to represent the dynamics of optoelectronic feedback loops designed for chaotic communications over fiber-optic links.

Figure 1

(Color online) Experimental setup for a ${\text{CO}}_2$ laser with modulated losses. EOM: intracavity electro-optic modulator, R: reflecting grating, M2: partial reflecting mirror, D: fast infrared detector, FG: function generator, and DO: digital oscilloscope.

Figure 2

(a) Experimentally observed laser intensity and the estimate of $x_{1,n}$. (b) Convergence of the estimated parameter ${\widehat{\gamma }}_1$ toward the correct value ${\gamma }_1=10.043$. (c) Phase-space diagram obtained by integrating the model equations (in the absence of noise). (d) The same phase-space diagram obtained from the estimates of $x_{2,n}$ versus the estimates of $x_{1,n}$. The number of particles in the applied SMCO algorithm was $M=2000$.

Figure 3

(Color online) (a) The upper axes depict the evolution of variable $x_1$ and the associate observation $y=x_1+m$, where $m$ is a Gaussian zero-mean random process, for $1800\le t\le 2100\text{ns}$. The variance of the Gaussian perturbations in $y$ is chosen to ensure a signal-to-noise ratio of 20 dB and the discrete-time data rate is $1/N=0.1$ (one observation per ns). The estimate of $x_1$ computed recursively by the SMCO algorithm is shown in the lower axes. (b) The upper axes depict the evolution of the unobserved dynamic variable $x_2$ for $22.5\le t\le 2977.5\text{ns}$, while the lower plot shows its estimate.

Figure 4

Square error in the estimation of the phase parameter ${\psi }_0$ in the same simulation of Fig. 3. The feedback strength is $\beta =3$ and the observation rate is $1/N=0.1$.

Figure 5

(a) MLE of the delay-feedback system versus the feedback strength $\beta$. (b) Normalized MSE attained by the SMCO algorithm with observation rate $1/N=0.1$.

Figure 6

(Color online) (a) The upper axes depict the evolution of variable $x_1$ and the associate observation $y=x_1+m$, where $m$ is a Gaussian zero-mean random process, for $1800\le t\le 2100\text{ns}$. The variance of the Gaussian perturbations in $y$ is chosen to ensure a signal-to-noise ratio of 20 dB and the discrete-time data rate is $1/N=0.5$ (five observations per ns). The estimate of $x_1$ computed recursively by the SMCO algorithm is shown in the lower axes. (b) The upper axes depict the evolution of the unobserved dynamic variable $x_2$ for $22.5\le t\le 2977.5\text{ns}$, while the lower plot shows its estimate.