Combined Route Choice and Adaptive Traffic Control in a Day-to-day Dynamical System

Abstract

We formulate a joint dynamical traffic system that encompasses both route choice and traffic control in which the signal controller interacts with and adapts to the route choices of travelers, and vice versa. Travelers’ route choice dynamics are captured by a recurrence function, which governs the system evolution from day to day. Traffic control also adjusts its plan from day to day in response to the route choices of travelers. We show that this joint dynamical system has a set of fixed points in which both the user equilibrium is achieved and the traffic signal converges to a fixed-time plan. We address equilibrium stability and convergence of this joint dynamical system through analyzing the Jacobian matrix associated with each fixed point. We demonstrate the attraction domains of this joint problem and illustrate their importance and relevance for designing adaptive traffic control through some numerical examples.

Appendixes

1.1 Appendix 1 The Recurrence Functions for the Joint Dynamical Traffic System

The dynamical traffic system is expressed as x ⁿ = y(x ⁿ⁻¹), or equivalently:

$$ \begin{array}{l}\begin{array}{l}{f}_1^n={y}_1\left({\mathbf{x}}^{n-1}\right)= D/\left(1+{e}^{C_1-{C}_2}+{e}^{C_1-{C}_3}\right)\hfill \\ {}{f}_2^n={y}_2\left({\mathbf{x}}^{n-1}\right)= D/\left(1+{e}^{C_2-{C}_1}+{e}^{C_2-{C}_3}\right)\hfill \\ {}{f}_3^n={y}_3\left({\mathbf{x}}^{n-1}\right)= D/\left(1+{e}^{C_3-{C}_1}+{e}^{C_3-{C}_2}\right)\hfill \\ {}{g}_1^n={y}_4\left({\mathbf{x}}^{n-1}\right)=\left( S- L\right){f}_1{\mu}_1^{-1}/\left({f}_1{\mu}_1^{-1}+{f}_2{\mu}_2^{-1}+{f}_3{\mu}_3^{-1}\right)\hfill \\ {}{g}_2^n={y}_5\left({\mathbf{x}}^{n-1}\right)=\left( S- L\right){f}_2{\mu}_2^{-1}/\left({f}_1{\mu}_1^{-1}+{f}_2{\mu}_2^{-1}+{f}_3{\mu}_3^{-1}\right)\hfill \\ {}{g}_3^n={y}_6\left({\mathbf{x}}^{n-1}\right)=\left( S- L\right){f}_3{\mu}_3^{-1}/\left({f}_1{\mu}_1^{-1}+{f}_2{\mu}_2^{-1}+{f}_3{\mu}_3^{-1}\right)\hfill \\ {}{C}_1^n={y}_7\left({\mathbf{x}}^{n-1}\right)=\left({f}_1+3{f}_2+1\right)+ k{\left( S-{g}_1\right)}^2{\mu}_1/\left( S\left({\mu}_1-{f}_1\right)\right)\hfill \\ {}{C}_2^n={y}_8\left({\mathbf{x}}^{n-1}\right)=\left(2{f}_1+{f}_2+2\right)+ k{\left( S-{g}_2\right)}^2{\mu}_2/\left( S\left({\mu}_2-{f}_2\right)\right)\hfill \\ {}{C}_3^n={y}_9\left({\mathbf{x}}^{n-1}\right)=\left({f}_3+1\right)+ k{\left( S-{g}_3\right)}^2{\mu}_3/\left( S\left({\mu}_3-{f}_3\right)\right)\hfill \end{array}\\ {}\end{array} $$

where we simplify x ⁿ⁻¹ as [f ₁, f ₂, f ₃, g ₁, g ₂, g ₃, C ₁, C ₂, C ₃] and set constant k to be 1/250.

1.2 Appendix 2 The Jacobian Matrix of the Joint Dynamical Traffic System

The Jacobian matrix of the dynamical system in this example is:

	$\raisebox{1ex}{$\partial $}\!\left/ \!\raisebox{-1ex}{$\partial {C}_1$}\right.$	$\raisebox{1ex}{$\partial $}\!\left/ \!\raisebox{-1ex}{$\partial {C}_2$}\right.$	$\raisebox{1ex}{$\partial $}\!\left/ \!\raisebox{-1ex}{$\partial {C}_3$}\right.$
y 1(x n−1)	$-{e}^{-{C}_1}\left({e}^{-{C}_2}+{e}^{-{C}_3}\right){\varDelta}_1$	${e}^{-{C}_1}{e}^{-{C}_2}{\varDelta}_1$	${e}^{-{C}_1}{e}^{-{C}_3}{\varDelta}_1$
y 2(x n−1)	${e}^{-{C}_1}{e}^{-{C}_2}{\varDelta}_1$	$-{e}^{-{C}_2}\left({e}^{-{C}_1}+{e}^{-{C}_3}\right){\varDelta}_1$	${e}^{-{C}_2}{e}^{-{C}_3}{\varDelta}_1$
y 3(x n−1)	${e}^{-{C}_1}{e}^{-{C}_3}{\varDelta}_1$	${e}^{-{C}_2}{e}^{-{C}_3}{\varDelta}_1$	$-{e}^{-{C}_3}\left({e}^{-{C}_1}+{e}^{-{C}_2}\right){\varDelta}_1$
	$\raisebox{1ex}{$\partial $}\!\left/ \!\raisebox{-1ex}{$\partial {f}_1$}\right.$	$\raisebox{1ex}{$\partial $}\!\left/ \!\raisebox{-1ex}{$\partial {f}_2$}\right.$	$\raisebox{1ex}{$\partial $}\!\left/ \!\raisebox{-1ex}{$\partial {f}_3$}\right.$
y 4(x n−1)	$\frac{\varDelta_2}{\mu_1}\left({f}_2{\mu}_2^{-1}+{f}_3{\mu}_3^{-1}\right)$	− Δ 2(μ 1 μ 2)− 1 f 1	− Δ 2(μ 1 μ 3)− 1 f
y 5(x n−1)	− Δ 2(μ 1 μ 2)− 1 f 2	$\frac{\varDelta_2}{\mu_2}\left({f}_1{\mu}_1^{-1}+{f}_3{\mu}_3^{-1}\right)$	− Δ 2(μ 2 μ 3)− 1 f 2
y 6(x n−1)	− Δ 2(μ 1 μ 3)− 1 f 3	− Δ 2(μ 2 μ 3)− 1 f 3	$\frac{\varDelta_2}{\mu_3}\left({f}_1{\mu}_1^{-1}+{f}_2{\mu}_2^{-1}\right)$
	$\raisebox{1ex}{$\partial $}\!\left/ \!\raisebox{-1ex}{$\partial {f}_1$}\right.$	$\raisebox{1ex}{$\partial $}\!\left/ \!\raisebox{-1ex}{$\partial {f}_2$}\right.$	$\raisebox{1ex}{$\partial $}\!\left/ \!\raisebox{-1ex}{$\partial {f}_3$}\right.$
y 7(xn−1)	$1+ k\frac{{\left( S-{g}_1\right)}^2{\mu}_1}{S{\left({\mu}_1-{f}_1\right)}^2}$	3	0
y 8(xn−1)	2	$1+ k\frac{{\left( S-{g}_2\right)}^2{\mu}_2}{S{\left({\mu}_2-{f}_2\right)}^2}$	0
y 9(xn−1)	0	0	$1+ k\frac{{\left( S-{g}_3\right)}^2{\mu}_3}{S{\left({\mu}_3-{f}_3\right)}^2}$
	$\raisebox{1ex}{$\partial $}\!\left/ \!\raisebox{-1ex}{$\partial {g}_1$}\right.$	$\raisebox{1ex}{$\partial $}\!\left/ \!\raisebox{-1ex}{$\partial {g}_2$}\right.$	$\raisebox{1ex}{$\partial $}\!\left/ \!\raisebox{-1ex}{$\partial {g}_3$}\right.$
y 7(xn−1)	$\frac{-2 k\left( S-{g}_1\right){\mu}_1}{S\left({\mu}_1-{f}_1\right)}$	0	0
y 8(xn−1)	0	$\frac{-2 k\left( S-{g}_2\right){\mu}_2}{S\left({\mu}_2-{f}_2\right)}$	0
y 9(xn−1)	0	0	$\frac{-2 k\left( S-{g}_3\right){\mu}_3}{S\left({\mu}_3-{f}_3\right)}$

where ${\varDelta}_1=\frac{D}{{\left({e}^{-{C}_1}+{e}^{-{C}_2}+{e}^{-{C}_3}\right)}^2}$ and ${\varDelta}_2=\frac{S- L}{{\left({f}_1{\mu}_1^{-1}+{f}_2{\mu}_2^{-1}+{f}_3{\mu}_3^{-1}\right)}^2}$. All other entries are 0.