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On the Redundant Distributed Observability of Mixed Traffic Transportation Systems

M. Doostmohammadian, U. A. Khan, N. Meskin

TL;DR

The paper addresses robust, scalable state estimation for mixed-traffic ITS by deriving a distributed observable state-space model that couples HDV dynamics with a network of CAVs. It introduces a Kronecker-product formulation and proves that strong connectivity of the CAV communication graph $\mathcal{G}_W$ together with outputs from every parent SCC of the HDV dynamics graph $\mathcal{G}_A$ ensures observability of the pair $ (W \otimes A, D_C) $. To enhance resilience, the authors define $q$-node/$q$-link connectivity and show how redundancy preserves observability under faults, leveraging Menger’s theorem. A one-step-consensus distributed observer with block-diagonal gain matrices $K_i$ is proposed, designed via an LMI to ensure Schur stability of the error dynamics, and validated through simulations demonstrating bounded estimation error and fault-tolerant performance. The work advances scalable, fault-tolerant distributed sensing for mixed-traffic ITS, enabling CAVs to collectively estimate HDV states with localized communication and minimal coordination overhead.

Abstract

In this paper, the problem of distributed state estimation of human-driven vehicles (HDVs) by connected autonomous vehicles (CAVs) is investigated in mixed traffic transportation systems. Toward this, a distributed observable state-space model is derived, which paves the way for estimation and observability analysis of HDVs in mixed traffic scenarios. In this direction, first, we obtain the condition on the network topology to satisfy the distributed observability, i.e., the condition such that each HDV state is observable to every CAV via information-exchange over the network. It is shown that strong connectivity of the network, along with the proper design of the observer gain, is sufficient for this. A distributed observer is then designed by locally sharing estimates/observations of each CAV with its neighborhood. Second, in case there exist faulty sensors or unreliable observation data, we derive the condition for redundant distributed observability as a $q$-node/link-connected network design. This redundancy is achieved by extra information-sharing over the network and implies that a certain number of faulty sensors and unreliable links can be isolated/removed without losing the observability. Simulation results are provided to illustrate the effectiveness of the proposed approach.

On the Redundant Distributed Observability of Mixed Traffic Transportation Systems

TL;DR

The paper addresses robust, scalable state estimation for mixed-traffic ITS by deriving a distributed observable state-space model that couples HDV dynamics with a network of CAVs. It introduces a Kronecker-product formulation and proves that strong connectivity of the CAV communication graph together with outputs from every parent SCC of the HDV dynamics graph ensures observability of the pair . To enhance resilience, the authors define -node/-link connectivity and show how redundancy preserves observability under faults, leveraging Menger’s theorem. A one-step-consensus distributed observer with block-diagonal gain matrices is proposed, designed via an LMI to ensure Schur stability of the error dynamics, and validated through simulations demonstrating bounded estimation error and fault-tolerant performance. The work advances scalable, fault-tolerant distributed sensing for mixed-traffic ITS, enabling CAVs to collectively estimate HDV states with localized communication and minimal coordination overhead.

Abstract

In this paper, the problem of distributed state estimation of human-driven vehicles (HDVs) by connected autonomous vehicles (CAVs) is investigated in mixed traffic transportation systems. Toward this, a distributed observable state-space model is derived, which paves the way for estimation and observability analysis of HDVs in mixed traffic scenarios. In this direction, first, we obtain the condition on the network topology to satisfy the distributed observability, i.e., the condition such that each HDV state is observable to every CAV via information-exchange over the network. It is shown that strong connectivity of the network, along with the proper design of the observer gain, is sufficient for this. A distributed observer is then designed by locally sharing estimates/observations of each CAV with its neighborhood. Second, in case there exist faulty sensors or unreliable observation data, we derive the condition for redundant distributed observability as a -node/link-connected network design. This redundancy is achieved by extra information-sharing over the network and implies that a certain number of faulty sensors and unreliable links can be isolated/removed without losing the observability. Simulation results are provided to illustrate the effectiveness of the proposed approach.

Paper Structure

This paper contains 8 sections, 4 theorems, 27 equations, 17 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Main Results
  4. Network Design for Distributed Observability
  5. Redundant Survivable Network Design
  6. Redundant Distributed Observer Design
  7. Illustrative Example and Simulations
  8. Conclusion

Key Result

Theorem 1

A cyclic system graph $\mathcal{G}_A$ is observable (in the structural sense) if and only if it contains a directed path (or sequence of connected nodes) from every node $i$ to an output measurement.

Figures (17)

  • Figure 1: This figure illustrates the distributed observer design formulation in this paper. The system state to be estimated includes the state $\mathbf{x}_k$ of the HDVs. Each CAV aims to observe the entire state of the HDVs. The global state estimate dynamics can be represented as the Kronecker product of the vehicle's system matrix $A$ and the adjacency matrix $W$ of the communication network $\mathcal{G}_W$. Some nearby CAVs take measurements of the state of the HDV. Then, the observability of the global HDVs' dynamics depends on both the local system matrix $A$ and the graph topology $\mathcal{G}_W$ along with the local measurement matrix $C$ (or $D_C$).
  • Figure 2: This figure shows the system graph $\mathcal{G}_A$ associated with the NCA system matrix in Eq. \ref{['eq_nca']}. The partial order of SCCs (here as self-cycles) is given at the right of the figure. The position states $\mathbf{x}_5$ and $\mathbf{x}_6$ represent the parent SCCs (or parent nodes). Outputs $\mathbf{y}_1$ and $\mathbf{y}_2$ ensure structural observability of the system graph according to Theorem \ref{['thm_scc']}.
  • Figure 3: This figure presents the structure of $W \otimes A$ matrix. Assume that every reducible block of system matrix $A$ contains a smaller irreducible block $A_{ii}$ (representing a child SCC in $\mathcal{G}_A$) which forms a path to another irreducible block $A_{jj}$ (representing a parent SCC in $\mathcal{G}_A$). From the definition of $W \otimes A$, the blocks of $A$ matrices follow the irreducible structure of the adjacency matrix $W$ which represents the SC communication network $\mathcal{G}_W$. Following the dashed blue arrows, one can find a giant irreducible block formed by $A_{jj}$s, the irreducible block obtained after a suitable permutation that collects all blocks connected by parent-child SCC paths. This giant block forms a giant parent SCC in the network $\mathcal{G}_W \otimes \mathcal{G}_A$.
  • Figure 4: An example vehicle network of $2$-nearest neighour ring.
  • Figure 5: The position of the $i$-th HDV, $p_{x,i}$, $i=1,\dots,4$ and the estimated one, $\hat{p}^j_{x,i}$ by the $j$-th CAV $j=1,\dots,5$ using the distributed observer in Algorithm \ref{['alg_ac']}.
  • ...and 12 more figures

Theorems & Definitions (16)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Example 1
  • Theorem 3
  • proof
  • Remark 1
  • Definition 2
  • ...and 6 more