Nonlinear stability of traffic models and the use of Lyapunov vectors for estimating the traffic state

Valuable information for estimating the traffic flow is obtained with current GPS technology by monitoring position and velocity of vehicles. In this paper, we present a proof of concept study that shows how the traffic state can be estimated using only partial and noisy data by assimilating them in a dynamical model. Our approach is based on a data assimilation algorithm, developed by the authors for chaotic geophysical models, designed to be equivalent but computationally much less demanding than the traditional extended Kalman filter. Here we show that the algorithm is even more efficient if the system is not chaotic and demonstrate by numerical experiments that an accurate reconstruction of the complete traffic state can be obtained at a very low computational cost by monitoring only a small percentage of vehicles.

Figure 1

Calculation of the Lyapunov exponents according to the algorithm [44] as a function of time over which the exponents are averaged. $N=150$, $\beta =0.4$, $L=200$. The three short time intervals that will be used in the assimilation experiments are shown in the figure and are labeled (a), (b), and (c). The top thick line labeled as $1/t$ is plotted for comparison with the asymptotic behavior of the exponents.

Figure 2

Calculation of the Lyapunov exponents according to the algorithm [44] as a function of time over which the exponents are averaged. $N=150$, $\beta =0.4$, $L=200$. Different from Fig. 1, here the averaging procedure starts at $t=5\times {10}^6$.

Figure 3

${\mathbf{P}}^a$ eigenvalues as a function of time in the assimilation experiment performed in the time interval (a) using the EKF-AUS algorithm run with $m=13$, ${\sigma }_0={10}^{-2}$, $\tau =180$, $p=150$ (all cars monitored).

Figure 4

${\mathbf{P}}^a$ eigenvalues as a function of time in the assimilation experiment performed in the time interval (b) using the EKF-AUS algorithm run with $m=8$, ${\sigma }_0={10}^{-2}$, $\tau =180$, $p=150$ (all cars monitored).

Figure 5

${\mathbf{P}}^a$ eigenvalues as a function of time in the assimilation experiment performed in the time interval (c) using the EKF-AUS algorithm run with $m=5$, ${\sigma }_0={10}^{-2}$, $\tau =180$, $p=150$ (all cars monitored).

Figure 6

Time evolution of the error (the rms difference between the truth and the analysis states normalized by the number of degrees of freedom, $2N$) during the assimilation performed in the time interval (b) for $m=8,7,6,5,4$. Error values relative to $m=6,7,8$ are so similar that they cannot be distinguished in the figure.

Figure 7

Time evolution of the error (the rms difference between the truth and the analysis states normalized by the number of degrees of freedom, $2N$) during the assimilation performed in the time interval (b) for $m=6$ and for different values of the number of monitored cars $p=30,15,10,5,3$.