# American Institute of Mathematical Sciences

June  2018, 13(2): 261-295. doi: 10.3934/nhm.2018012

## Error bounds for Kalman filters on traffic networks

 1 University of Illinois at Urbana Champaign, 205 N. Mathews Ave., Urbana, IL 61801, USA 2 Vanderbilt University, 1025 16th Ave. S., Suite 102, Nashville, TN 37212, USA

Received  May 2017 Revised  February 2018 Published  May 2018

Fund Project: This material is based upon work supported by the National Science Foundation under Grant No. CMMI-1351717

This work analyzes the estimation performance of the Kalman filter (KF) on transportation networks with junctions. To facilitate the analysis, a hybrid linear model describing traffic dynamics on a network is derived. The model, referred to as the switching mode model for junctions, combines the discretized Lighthill-Whitham-Richards partial differential equation with a junction model. The system is shown to be unobservable under nearly all of the regimes of the model, motivating attention to the estimation error bounds in these modes. The evolution of the estimation error is investigated via exploring the interactions between the update scheme of the KF and the intrinsic physical properties embedded in the traffic model (e.g., conservation of vehicles and the flow-density relationship). It is shown that the state estimates of all the cells in the traffic network are ultimately bounded inside a physically meaningful interval, which cannot be achieved by an open-loop observer.

Citation: Ye Sun, Daniel B. Work. Error bounds for Kalman filters on traffic networks. Networks & Heterogeneous Media, 2018, 13 (2) : 261-295. doi: 10.3934/nhm.2018012
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##### References:
The triangular fundamental diagram in (10)
A diverge and a merge junction connected by three cells indexed by $i$, $j$, and $l$
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 ${\rm{r}}(\rho_k^j)\ge {\rm{s}}(\rho_k^l)$.
A local section with $n$ cells, three links and a junction
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)
Mode definition and observability of the SMM-J
 Mode F/C$^1$ status of cell(s) Transition$^{3}$ on link Diverge case Obser-vability$^4$ $1$ $n_1+n_2$ $n$ near junction$^{2}$ 1 2 3 1 F F F F none none none Ⅱ O 2 F F F C Sh. Ep. Ep. Ⅰ U 3 F F F C Sh. Ep. Ep. Ⅱ U 4 F F F C Sh. Ep. Ep. Ⅲ U 5 C C C C none none none Ⅰ U 6 C C C C none none none Ⅱ U 7 C C C C none none none Ⅲ U 8 C C C F Ep. Sh. Sh. Ⅱ U 9 F C C C Sh. none none Ⅰ U 10 F C C C Sh. none none Ⅱ U 11 F C C C Sh. none none Ⅲ U 12 F C C F none Sh. Sh. Ⅱ U 13 C C F C none none Ep. Ⅰ U 14 C C F C none none Ep. Ⅱ U 15 C C F C none none Ep. Ⅲ U 16 C C F F Ep. Sh. none Ⅱ U 17 C F C C none Ep. none Ⅰ U 18 C F C C none Ep. none Ⅱ U 19 C F C C none Ep. none Ⅲ U 20 C F C F Ep. none Sh. Ⅱ U 21 C F F F Ep. none none Ⅱ O 22 C F F C none Ep. Ep. Ⅰ U 23 C F F C none Ep. Ep. Ⅱ U 24 C F F C none Ep. Ep. Ⅲ U 25 F C F F none Sh. none Ⅱ U 26 F C F C Sh. none Ep. Ⅰ U 27 F C F C Sh. none Ep. Ⅱ U 28 F C F C Sh. none Ep. Ⅲ U 29 F F C F none none Sh. Ⅱ U 30 F F C C Sh. Ep. none Ⅰ U 31 F F C C Sh. Ep. none Ⅱ U 32 F F C C Sh. Ep. none Ⅲ U 1 "F" and "C" stand for freeflow and congestion, respectively.2 Cells indexed by $n_1$, $n_1+1$ and $n_1+n_2+1$.3 "Sh." and "Ep." stand for shock (i.e., transition from freeflow to congestion) and expansion fan (i.e., transition from congestion to freeflow), respectively.4 "O" stands for uniformly completely observable and "U" stands for unobservable. Note that the observability results are derived under sensor locations shown in Figure 4.
 Mode F/C$^1$ status of cell(s) Transition$^{3}$ on link Diverge case Obser-vability$^4$ $1$ $n_1+n_2$ $n$ near junction$^{2}$ 1 2 3 1 F F F F none none none Ⅱ O 2 F F F C Sh. Ep. Ep. Ⅰ U 3 F F F C Sh. Ep. Ep. Ⅱ U 4 F F F C Sh. Ep. Ep. Ⅲ U 5 C C C C none none none Ⅰ U 6 C C C C none none none Ⅱ U 7 C C C C none none none Ⅲ U 8 C C C F Ep. Sh. Sh. Ⅱ U 9 F C C C Sh. none none Ⅰ U 10 F C C C Sh. none none Ⅱ U 11 F C C C Sh. none none Ⅲ U 12 F C C F none Sh. Sh. Ⅱ U 13 C C F C none none Ep. Ⅰ U 14 C C F C none none Ep. Ⅱ U 15 C C F C none none Ep. Ⅲ U 16 C C F F Ep. Sh. none Ⅱ U 17 C F C C none Ep. none Ⅰ U 18 C F C C none Ep. none Ⅱ U 19 C F C C none Ep. none Ⅲ U 20 C F C F Ep. none Sh. Ⅱ U 21 C F F F Ep. none none Ⅱ O 22 C F F C none Ep. Ep. Ⅰ U 23 C F F C none Ep. Ep. Ⅱ U 24 C F F C none Ep. Ep. Ⅲ U 25 F C F F none Sh. none Ⅱ U 26 F C F C Sh. none Ep. Ⅰ U 27 F C F C Sh. none Ep. Ⅱ U 28 F C F C Sh. none Ep. Ⅲ U 29 F F C F none none Sh. Ⅱ U 30 F F C C Sh. Ep. none Ⅰ U 31 F F C C Sh. Ep. none Ⅱ U 32 F F C C Sh. Ep. none Ⅲ U 1 "F" and "C" stand for freeflow and congestion, respectively.2 Cells indexed by $n_1$, $n_1+1$ and $n_1+n_2+1$.3 "Sh." and "Ep." stand for shock (i.e., transition from freeflow to congestion) and expansion fan (i.e., transition from congestion to freeflow), respectively.4 "O" stands for uniformly completely observable and "U" stands for unobservable. Note that the observability results are derived under sensor locations shown in Figure 4.
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