Journal of Sensors

Volume 2017 (2017), Article ID 6290248, 11 pages

https://doi.org/10.1155/2017/6290248

## Continuous and Discrete-Time Optimal Controls for an Isolated Signalized Intersection

^{1}Beijing Key Lab of Urban Intelligent Traffic Control Technology, North China University of Technology, Beijing 100144, China^{2}Graduate School at Shenzhen, Tsinghua University, Shenzhen 518055, China^{3}The Institute for Transport Planning and Systems (IVT), ETH Zurich, Zurich, Switzerland^{4}School of Management Science and Engineering, Central University of Finance and Economics, Beijing 100081, China^{5}Beijing Key Laboratory for Cooperative Vehicle Infrastructure Systems and Safety Control, School of Transportation Science and Engineering, Beihang University, Beijing 100191, China

Correspondence should be addressed to Haiyang Yu

Received 13 March 2017; Revised 29 June 2017; Accepted 19 July 2017; Published 7 September 2017

Academic Editor: Jia Hu

Copyright © 2017 Jiyuan Tan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A classical control problem for an isolated oversaturated intersection is revisited with a focus on the optimal control policy to minimize total delay. The difference and connection between existing continuous-time planning models and recently proposed discrete-time planning models are studied. A gradient descent algorithm is proposed to convert the optimal control plan of the continuous-time model to the plan of the discrete-time model in many cases. Analytic proof and numerical tests for the algorithm are also presented. The findings shed light on the links between two kinds of models.

#### 1. Introduction

Along with the fast increase of auto population, urban streets are becoming more crowded nowadays. To relieve congestions and reduce accidents, various traffic control methods have been proposed since the late 1950s [1].

As a typical traffic scenario, oversaturated intersections attracted consistent interest during the last six decades [2, 3]. The term “oversaturated” means the following: the vehicles that remained since the last cycle plus the vehicles that newly arrived exceed the capacity of the intersection. This leads to the carryover of vehicle queues (at least in one leg of the intersection) to the next cycle.

Discussions on the road networks that consist of many oversaturated intersections can be found in researches done by Chang and Sun [4], Di Febbraro and Sacco [5], Dotoli and Fanti [6], Ma [7, 8], and Sun et al. [9, 10]. In the research done by Varaiya [11] and Le et al. [12], the study of pressure-based signal control developed stability properties of a decentralized signal timing policy for networks with stochastic arrivals. But for a real-time signal timing optimization problem, the data that could be used is the arriving information of vehicles in the recent several signal cycles (data could be gotten by connecting vehicle technology, etc.). The optimization objects and scenarios are different between the model in this paper and pressure-based policies. This paper will focus on the isolated oversaturated intersection.

Usually, researchers aim to find an optimal signal timing plan that minimizes the total delay of vehicles passing this intersection. The total delay is often defined as the time integral of the sum of all queue lengths for all legs of the intersection over a given time horizon. However, the total delay is a nonlinear and nonconvex function of control variables (e.g., green phases), which makes it difficult to optimize. One promising approach is to apply heuristic algorithms to solve the formulated optimization problem. For example, the genetic algorithm was applied by Park et al. [13] to optimize the total delay. An alternative approach is to first approximate the nonconvex total delay with some convex functions and then solve the newly formulated optimization problem. The rest of this paper will focus on the second approach within a typical traffic scenario: an isolated intersection with only two movements.

There are mainly two kinds of convexified models for this scenario. The first kind of models originated from Gazis [3] who used continuous-time differential equations to describe the traffic dynamics. The cycle length, departing flow rates, and arriving flow rates in Gazis [3] were all assumed to be constant. Michalopoulos and Stephanopoulos [14, 15] extended the continuous-time model by including the maximum queue lengths constraints and time-variant arrival flow rates. Such formulations led to a classical control problem that can be solved via the Pontryagin Maximum Principle (PMP) [16]. However, the obtained continuous-time signal timing plan should be discretized into the corresponding discrete-time signal timing plan that can be executed in practice.

The second kind of models uses discrete-time difference equations to describe the traffic dynamics [7, 8, 17–19]. The corresponding design problem can then be formulated as a linear programming (LP) problem [20–25]. One interesting question that naturally arises is how to depict the difference and connection between the discrete-time model and the continuous-time model.

Recently, Ioslovich et al. [26] studied the formulated LP problem by considering the corresponding continuous-time approximation model and gave an elegant approximate solution in continuous-time forms. However, it was not verified how this approximate solution differs from the accurate solution (the solution obtained by the continuous-time model). Whether a discretized version of this continuous-time approximate solution is still optimal to the LP problem also needs further discussions.

Zou et al. [27] have given a preliminary result on the relationship between the continuous-time model and the discrete-time model. Zou et al. applied a graphical method to adjust the continuous-time approximate solution to a discrete-time accurate solution. They assumed that both streams are dispatched simultaneously and the adjustment method does not influence the clearance cycle. It is shown that the approximate solution can be adjusted to the optimal solution by changing the green ratio in the switching cycle. However, the discrete-time LP problem in Ioslovich et al. [26] is more complex than the one considered in [27], since the two streams are allowed to be cleared at different times. Merely changing the green ratios at the switching cycle may not obtain the accurate solution.

In this paper, the LP problems proposed by Ioslovich et al. [26] are directly attacked using the strong duality theorem [28]. It is first shown that the approximate solution and the accurate solution do not coincide in many situations. Then, the relationship between the approximate solution and the accurate solution will be discussed. The errors introduced by discretization are carefully studied. It is shown that, in many cases, the discretized approximate solution can be converted to the accurate solution within a few gradient descent adjustments. Finally, an algorithm is proposed to implement this conversion. These findings shed light on the connection between the continuous-time and discrete-time signal timing models.

To give a detailed analysis, the rest of this paper is arranged as follows. Section 2 introduces the LP problem proposed by Ioslovich et al. [26] and lists the nomenclature used in this paper. Section 3 proposes two counterexamples to show that the optimality of the discretized approximate solution may not hold. Section 4 discusses the difference and connection between the discrete-time model and the continuous-time model. Finally, Section 5 concludes the paper.

#### 2. Problem Presentation

##### 2.1. Nomenclature and Assumptions

For presentation simplicity, the nomenclature in this paper follows Ioslovich et al. [26] as shown in Nomenclature List. Similar to Ioslovich et al. [26], consider an isolated intersection with two one-way streams and governed by signal lights, as shown in Figure 1. Note that the two-stream isolated intersection is chosen as an initial building block to understand the connection between the continuous-time model and the discrete-time model. This is a widely applied treatment even in recent works (e.g., [23]). Nevertheless, this assumption can be relaxed to consider general intersections. Both the continuous model and the discrete model are able to handle cases with more generalized intersections. Although it is hard to directly compare the two types of models in generalized intersections, we expect that the findings in this paper can shed light on more generalized intersections.