[Paper Review] Stability Analysis and Control Synthesis for Dynamical Transportation Networks

Enrico Lovisari, Giacomo Como+2 more|arXiv (Cornell University)|Oct 22, 2014

Traffic control and management27 references18 citations

TL;DR

This paper proposes a stability analysis and convex optimal control framework for dynamical transportation networks using a state-dependent dual graph and monotone ODE models based on the Cell Transmission Model. It establishes that connectivity of the dual graph ensures local stability of equilibria and periodic solutions, and shows that optimal control via speed limits, supply thresholding, and turning preferences can be cast as a convex optimization problem, guaranteeing free-flow equilibria.

ABSTRACT

We study dynamical transportation networks in a framework that includes extensions of the classical Cell Transmission Model to arbitrary network topologies. The dynamics are modeled as systems of ordinary differential equations describing the traffic flow among a finite number of cells interpreted as links of a directed network. Flows between contiguous cells, in particular at junctions, are determined by merging and splitting rules within constraints imposed by the cells' demand and supply functions as well as by the drivers' turning preferences, while inflows at on-ramps are modeled as exogenous and possibly time-varying. First, we analyze stability properties of dynamical transportation networks. We associate to the dynamics a state-dependent dual graph whose connectivity depends on the signs of the derivatives of the inter-cell flows with respect to the densities. Sufficient conditions for the stability of equilibria and periodic solutions are then provided in terms of the connectivity of such dual graph. Then, we consider synthesis of control policies that use a combination of turning preferences, speed limits, and ramp metering, in order to optimize convex objectives. We first show that, in the general case, the optimal control synthesis problem can be cast as a convex optimization problem, and that the equilibrium of the controlled network is in free-flow. If the control policies are restricted to speed limits and ramp metering, then the resulting synthesis problem is still convex for networks where every node is either a merge or a diverge junction, and where the dynamics is monotone. These results apply both to the optimal selection of equilibria and periodic solutions, as well as to finite-horizon network trajectory optimization. Finally, we illustrate our findings through simulations on a road network inspired by the freeway system in southern Los Angeles.

Motivation & Objective

• To analyze the stability of equilibria and periodic solutions in dynamical transportation networks with arbitrary topologies.
• To develop a convex optimal control synthesis framework that leverages speed limits, supply thresholding, and turning preferences to optimize network performance.
• To establish sufficient conditions for stability based on the connectivity of a state-dependent dual graph derived from inter-cell flow derivatives.
• To show that the optimal control problem remains convex under specific network structures (merge/diverge-only) and monotonicity constraints.
• To demonstrate the applicability of the framework through simulations on a Los Angeles freeway network.

Proposed method

• Modeling traffic dynamics as a system of ODEs based on mass conservation on a directed graph of cells, with demand and supply functions per cell.
• Defining inter-cell flows via merging/splitting rules that incorporate drivers' turning preferences and constraints from demand and supply functions.
• Introducing a state-dependent dual graph whose connectivity depends on the signs of the derivatives of inter-cell flows with respect to cell densities.
• Applying an ℓ₁ contraction principle for monotone dynamical systems to derive stability conditions based on dual graph connectivity.
• Formulating the optimal control synthesis problem as a convex optimization problem using control inputs such as speed limits, supply thresholds, and turning preferences.
• Proving that under monotonicity and merge/diverge-only network structure, the control synthesis problem remains convex and yields free-flow equilibria.

Experimental results

Research questions

• RQ1 Under what conditions is the equilibrium of a dynamical transportation network locally stable?
• RQ2 How does the connectivity of a state-dependent dual graph relate to the stability of equilibria and periodic solutions?
• RQ3 Can optimal control policies that include speed limits, supply thresholding, and turning preferences be formulated as a convex optimization problem?
• RQ4 What structural properties of the network (e.g., merge/diverge junctions) preserve convexity in the control synthesis problem?
• RQ5 Does the optimal control policy under convex objectives always result in a free-flow equilibrium?

Key findings

• Local stability of equilibria is guaranteed if the state-dependent dual graph is connected, under monotonicity of the dynamics.
• For periodic inflows, sufficient conditions for stability of periodic solutions are derived based on the connectivity of the dual graph.
• The optimal control synthesis problem is convex when using a combination of turning preferences, speed limits, and supply thresholding, ensuring global convergence to the optimal solution.
• When restricted to speed limits and supply thresholding, the synthesis problem remains convex for networks composed only of merge and diverge junctions with monotone dynamics.
• The equilibrium of the controlled network is always in free-flow, regardless of the control inputs, under the proposed convex framework.
• Simulations on a southern Los Angeles freeway network validate the theoretical findings, demonstrating effective stability and control performance.