Parameter and state estimation of experimental chaotic systems using synchronization

We examine the use of synchronization as a mechanism for extracting parameter and state information from experimental systems. We focus on important aspects of this problem that have received little attention previously and we explore them using experiments and simulations with the chaotic Colpitts oscillator as an example system. We explore the impact of model imperfection on the ability to extract valid information from an experimental system. We compare two optimization methods: an initial value method and a constrained method. Each of these involves coupling the model equations to the experimental data in order to regularize the chaotic motions on the synchronization manifold. We explore both time-dependent and time-independent coupling and discuss the use of periodic impulse coupling. We also examine both optimized and fixed (or manually adjusted) coupling. For the case of an optimized time-dependent coupling function $u\left(t\right)$ we find a robust structure which includes sharp peaks and intervals where it is zero. This structure shows a strong correlation with the location in phase space and appears to depend on noise, imperfections of the model, and the Lyapunov direction vectors. For time-independent coupling we find the counterintuitive result that often the optimal rms error in fitting the model to the data initially increases with coupling strength. Comparison of this result with that obtained using simulated data may provide one measure of model imperfection. The constrained method with time-dependent coupling appears to have benefits in synchronizing long data sets with minimal impact, while the initial value method with time-independent coupling tends to be substantially faster, more flexible, and easier to use. We also describe a method of coupling which is useful for sparse experimental data sets. Our use of the Colpitts oscillator allows us to explore in detail the case of a system with one positive Lyapunov exponent. The methods we explored are easily extended to driven systems such as neurons with time-dependent injected current. They are expected to be of value in nonchaotic systems as well. Software is available on request.

Figure 1

Colpitts circuit.

Figure 2

Bifurcation diagram calculated by numerically integrating the model equations over a range of values for $R$. The value of $V_{CE}\left(t_n\right)$ is plotted for all $t_n$ where $V_E\left(t_n\right)=-0.6\text{V}$ with $V_E$ increasing. There is a period doubling bifurcation at $R\approx 59\Omega$ and again at $R\approx 48\Omega$.

Figure 3

Two-dimensional view of the attractor: (top panel) $V_E\left(t\right)$ versus $I_L\left(t\right)$ calculated by integrating the model equations; (bottom panel) $V_E\left(t\right)$ versus $I_L\left(t\right)$ as observed in our experiments.

Figure 4

(Color online) Cost [Eq. (11)] as a function of $R^{\prime }$ for four different coupling strengths (decreasing from bottom to top). The calculation was done over 10 ms, with the two systems starting at the same point in phase space and all parameters identical except $R\ne R^{\prime }$. Increasing $u$ smoothes out the surface, but it also makes the minimum less well defined as $u$ becomes large.

Figure 5

(Color online) Largest conditional Lyapunov exponent (CLE) as a function of coupling strength. With coupling to $V_E$ or $V_{CE}$ the largest CLE is reduced from positive to negative, but for coupling to $I_L$ the largest CLE is actually increased.

Figure 6

Optimized rms error as a function of fixed time-independent coupling strength using 10 ms of real data and analyzed with the standard model. The result is anomalous—the initial increase in error with increasing coupling was unexpected. Note that, for these results, the coupling strength is the same for all time steps and is set to the values on the horizontal axis. Note also that the model parameters are reoptimized for each point on the curves to minimize the rms error, so these calculated parameters are all functions of the coupling strength. In most cases we will use the parameter values obtained for zero coupling (when available) as our best estimate of the experimental parameters.

Figure 7

Optimized rms error as a function of fixed time-independent coupling strength using 60 ms of real data and analyzed with the standard model. Due to the length of the time series it is not possible to reduce the coupling strength to zero without a loss of synchronization. The points were found by moving downward slowly in $u$ and reoptimizing after each step. It is not possible to continue the curve below the marked end point—instead the optimization is forced to jump to a very unfavorable state with at least ten times the rms error.

Figure 8

(Color online) Time-dependent coupling $u\left(t\right)$ for $\eta =0.10\left[\text{V}\text{ms}\right]$ (middle) and $\eta =0.14\left[\text{V}\text{ms}\right]$ (bottom). The structure is very similar in both cases, but the magnitude scales approximately as $u\left(t\right)\propto {\eta }^{-2}$. When $u\left(t\right)$ is allowed to be positive or negative, additional structure becomes visible (top).

Figure 9

(Color online) Localization of peaks of $u\left(t\right)$ in phase space. Phase space was divided into $40\times 40\times 1$ cells (40 in the $I_L$ and $V_E$ directions, 1 in the $V_{CE}$ direction), and the average value of $u\left(t\right)$ in each cell is indicated by the shading. The calculated phase space orbit is overlaid for reference. The coupling was either to $V_E$ (top panel) or to $V_{CE}$ (bottom panel). In both cases $\eta =0.01\left[\text{V}\text{ms}\right]$ and 1800 data points were used.

Figure 10

(Color online) Using 1000 measurements of $V_E\left(t\right)$ over 10 ms (dots in top panel), we estimate the other two state variables, $V_{CE}\left(t\right)$ and $I_L\left(t\right)$ (solid lines). Also shown for comparison is the measurement of $V_{CE}\left(t\right)$ and $I_L\left(t\right)$ (dots in middle and bottom panels).

Figure 11

(Color online) Using the first 10 ms of data for $V_E\left(t\right)$, sampled every $10\mu \text{s}$, we estimated the parameters and the other state variables $V_{CE}\left(t\right)$ and $I_L\left(t\right)$ over the interval $0\le t\le 10\text{ms}$. Then using the equations of motion with estimated parameters and the values of $V_E\left(t=10\text{ms}\right)$ (as measured) and $V_{CE}\left(t=10\text{ms}\right)$ and $I_L\left(t=10\text{ms}\right)$ (as estimated), we predicted the three state variables for 10 ms $<t<25\text{ms}$. The predictions (solid lines) are compared to the measured data (dots). Initially the prediction matches the measurement, but the two diverge after about 8 ms because the dynamics is chaotic.

Figure 12

(Color online) The consistency check $R^2\left(t\right)$ defined in Eq. (14). Since $R^2\left(t\right)\approx 1$ the model is consistent with the data. In this calculation $\eta =0.1\left[\text{V}\text{ms}\right]$.

Figure 13

Optimized rms error as a function of fixed time-independent coupling strength. (a) Real data analyzed with standard model. (b) Real data analyzed with improved model. (c) Simulated data generated from standard model plus noise and analyzed with standard model. (d) Simulated data generated from improved model plus noise and analyzed with improved model. (e) Simulated data generated from improved model and analyzed with standard model.

Figure 14

(Color online) Optimized coupling strength $u\left(t\right)$ as a function of time with $\eta =1.0$. (a) Jagged curve from experimental data analyzed without emitter resistance. (b) Smooth curve for simulated data with parameters matching the experimental data that was generated using an emitter resistance but analyzed without (imperfect model).