Error bounds for Kalman filters on traffic networks

Figure 1.  The triangular fundamental diagram in (10)

Figure 2.  A diverge and a merge junction connected by three cells indexed by $i$, $j$, and $l$

Figure 3.  Three scenarios in the junction solver [24] for the diverge junction shown in Figure 2a, where cell $l$ diverges to cell $i$ and cell $j$. The blue vertical (resp. horizontal) solid line denotes the receiving capacity of cell $i$ (resp. cell $j$). The intercepts of the blue dashed line denote the sending capacity of cell $l$. The shaded area denotes the feasible values of the flux from cell $l$ to $i$ and the flux from cell $l$ to $j$. The slope of the black dotted line is the prescribed distribution ratio $\alpha_{\text{d}}$. The fluxes computed by the junction solver is marked by the red dot, whose horizontal axis and vertical axis values are the obtained ${\rm{f}}(\rho^{l}_{k}, \rho^{i}_{k})$ and ${\rm{f}}(\rho^{l}_{k}, \rho^{j}_{k})$, respectively. Note that in diverge case Ⅱ and diverge case Ⅲ, the receiving capacities of cell $i$ and cell $j$ are not necessarily smaller than the sending capacity of cell $l$, and the graphical illustration of the flux solutions is also applicable for ${\rm{r}}(\rho_k^i)\ge {\rm{s}}(\rho_k^l)$ and/or ${\rm{r}}(\rho_k^j)\ge {\rm{s}}(\rho_k^l)$.

Figure 4.  A local section with $n$ cells, three links and a junction

Figure 5.  The evolutions of the estimation errors (A) and the trace of the error covariance (B) when using the KF to track the unobservable system (39)-(40)