Optimal Control of Connected Automated Vehicles at Urban Traffic Intersections: A Feasibility Enforcement Analysis
Earlier work has established a decentralized optimal control framework for coordinating on line a continuous flow of Connected Automated Vehicles (CAVs) entering a “control zone” and crossing two adjacent intersections in an urban area. A solution, when it exists, allows the vehicles to minimize their fuel consumption while crossing the intersections without the use of traffic lights, without creating congestion, and under the hard safety constraint of collision avoidance. We establish the conditions under which such solutions exist and show that they can be enforced through an appropriately designed “feasibility enforcement zone” that precedes the control zone. The proposed solution and overall control architecture are illustrated through simulation.
Connected and automated vehicles (CAVs) provide significant new opportunities for improving transportation safety and efficiency using inter-vehicle as well as vehicle-to-infrastructure communication . To date, traffic lights are the prevailing method used to control the traffic flow through an intersection. More recently, however, data-driven approaches have been developed leading to online adaptive traffic light control as in . Aside from the obvious infrastructure cost and the need for dynamically controlling green/red cycles, traffic light systems also lead to problems such as significantly increasing the number of rear-end collisions at an intersection. These issues have provided the motivation for drastically new approaches capable of providing a smoother traffic flow and more fuel-efficient driving while also improving safety.
The advent of CAVs provides the opportunity for such new approaches. Dresner and Stone  proposed a scheme for automated vehicle intersection control based on the use of reservations whereby a centralized controller coordinates a crossing schedule based on requests and information received from the vehicles located inside some communication range. The main challenges in this case involve possible deadlocks and heavy communication requirements which can become critical. There have been numerous other efforts reported in the literature based on such a reservation scheme [4, 5, 6].
Increasing the throughput of an intersection is one desired goal which can be achieved through the travel time optimization of all vehicles located within a radius from the intersection. Several efforts have focused on minimizing vehicle travel time under collision-avoidance constraints [7, 8, 9, 10]. Lee and Park  proposed a different approach based on minimizing the overlap in the position of vehicles inside the intersection rather than their arrival times. Miculescu and Karaman  used queueing theory and modeled an intersection as a polling system where vehicles are coordinated to cross without collisions. There have been also several research efforts to address the problem of vehicle coordination at intersections within a decentralized control framework. A detailed discussion of the research in this area reported in the literature to date can be found in .
Our earlier work  has established a decentralized optimal control framework for coordinating online a continuous flow of CAVs crossing two adjacent intersections in an urban area. We refer to an approach as “centralized” if there is at least one task in the system that is globally decided for all vehicles by a single central controller. In contrast, in a “decentralized” approach, a coordinator may be used to handle or distribute information available in the system without, however, getting involved in any control task. The framework in  solves an optimal control problem for each CAV entering a specified Control Zone (CZ) which subsequently regulates the acceleration/deceleration of the CAV. The optimal control problem involves hard safety constraints, including rear-end collision avoidance. These constraints make it nontrivial to ensure the existence of a feasible solution to this problem. In fact, it is easy to check that the rear-end collision avoidance constraints cannot be guaranteed to hold throughout the CZ under an optimal solution unless the initial conditions (time and speed) of each CAV entering the CZ satisfy certain conditions. It is, therefore, of fundamental importance to determine these feasibility conditions and ensure that they can be satisfied.
The contributions of this paper are twofold. First, we study the feasibility conditions required to guarantee a solution of the optimal control problem for each CAV; these are expressed in terms of a feasible region defined in the space of the CAV’s speed and arrival time at the CZ. Second, we introduce a Feasibility Enforcement Zone (FEZ) which precedes the CZ and within which a CAV is controlled with the goal of attaining a point in the feasible region determined by the current state of the CZ. This subsequently guarantees that all required constraints are satisfied when the CAV enters the CZ under an associated optimal control. We emphasize again that the benefits of an optimal controller maximizing throughput and minimizing fuel consumption can only be realized subject to ensuring feasible initial conditions to the optimization problem under consideration.
The structure of the paper is as follows. In Section II, we review the model in  and its generalization in . In Section III, we present the CAV coordination framework and associated optimal control problems and solutions considering control/state constraints. In Section IV, we carry out the analysis necessary to identify a feasible region for the initial conditions of each CAV when entering the CZ. In Section V, we develop a design procedure for the FEZ and in Section VI, we include simulation results. We offer concluding remarks in Section VII.
Ii The Model
We briefly review the model introduced in  and  where there are two intersections, 1 and 2, located within a distance (Fig. 1). The region at the center of each intersection, called Merging Zone (MZ), is the area of potential lateral CAV collision. Although it is not restrictive, this is taken to be a square of side . Each intersection has a Control Zone (CZ) and a coordinator that can communicate with the CAVs traveling within it. The distance between the entry of the CZ and the entry of the MZ is , and it is assumed to be the same for all entry points to a given CZ.
Let be the cumulative number of CAVs which have entered the CZ and formed a first-in-first-out (FIFO) queue by time , . When a CAV reaches the CZ of intersection , the coordinator assigns it an integer value . If two or more CAVs enter a CZ at the same time, then the corresponding coordinator selects randomly the first one to be assigned the value . In the region between the exit point of a MZ and the entry point of the subsequent CZ, the CAVs cruise with the speed they had when they exited that MZ.
For simplicity, we assume that each CAV is governed by second order dynamics
where , , and denote the position, i.e., travel distance since the entry of the CZ, speed and acceleration/deceleration (control input) of each CAV . These dynamics are in force over an interval , where is the time that CAV enters the CZ and is the time that it exits the MZ of intersection .
To ensure that the control input and vehicle speed are within a given admissible range, the following constraints are imposed:
To ensure the absence of any rear-end collision throughout the CZ, we impose the rear-end safety constraint
where is the minimal safe distance allowable and is the CAV physically ahead of .
As part of safety considerations, we impose the following assumption (which may be relaxed if necessary):
The speed of the CAVs inside the MZ is constant, i.e., , , where is the time that CAV enters the MZ of the intersection. This implies that
The objective of each CAV is to derive an optimal acceleration/deceleration in terms of fuel consumption over the time interval while avoiding congestion between the two intersections. In addition, we impose hard constraints so as to avoid either rear-end collision, or lateral collision inside the MZ. In fact, it is shown in  that the centralized throughput maximization problem is equivalent to a set of decentralized problems whereby each CAV minimizes its fuel consumption as long as the safety constraints applying to it are satisfied. Thus, in what follows, we focus on these decentralized problems and their associated safety constraints.
Iii Vehicle Coordination and Control
Iii-a Decentralized Control Problem Formulation
Since the coordinator is not involved in any decision on the vehicle control, we can formulate and decentralized tractable problems for intersection 1 and 2 respectively that may be solved on line. When a CAV enters a CZ, , it is assigned a pair from the coordinator, where is a unique index and indicates the positional relationship between CAVs and . As formally defined in , with respect to CAV , CAV belongs to one and only one of the four following subsets: contains all CAVs traveling on the same road as and towards the same direction but on different lanes, contains all CAVs traveling on the same road and lane as CAV , contains all CAVs traveling on different roads from and having destinations that can cause lateral collision at the MZ, and contains all CAVs traveling on the same road as and opposite destinations that cannot, however, cause collision at the MZ. Note that the FIFO structure of this queue implies the following condition:
Under the assumption that each CAV has proximity sensors and can observe and/or estimate local information that can be shared with other CAVs, we define its information set , , as
where are the position and speed of CAV inside the CZ it belongs to, and is the subset assigned to CAV by the coordinator. The fourth element in is , the distance between CAV and CAV which is immediately ahead of in the same lane (the index is made available to by the coordinator). The last element above, , is the time targeted for CAV to enter the MZ, whose evaluation is discussed next. Note that once CAV enters the CZ, then all information in becomes available to .
The time that CAV is required to enter the MZ is based on maximizing the intersection throughput while satisfying (5) and the constraints for avoiding rear-end and lateral collision in the MZ. There are three cases to consider regarding , depending on the value of :
Case 1: in this case, none of the safety constraints can become active while and are in the CZ or MZ. This allows CAV to minimize its time in the CZ while preserving the FIFO queue through . Therefore, it is obvious that we should set
and since CAV speeds inside the MZ are constant (Assumption 1), both and will also be exiting the MZ at the same time by setting
where and . Note that, by Assumption 1, for all .
Case 2: in this case, only the rear-end collision constraint (3) can become active. In order to minimize the time CAV spends in the CZ by ensuring that (3) is satisfied over while is constant (Assumption 1), we set
and as in (8).
Case 3: in this case, only the lateral collision may occur. Hence, CAV is allowed to enter the MZ only when CAV exits from it. To minimize the time CAV spends in the CZ while ensuring that the lateral collision avoidance is satisfied over , we set
and as in (8).
Although (7), (9), and (10) provide a simple recursive structure for determining , the presence of the control and state constraints (2) may prevent these values from being admissible. This may happen by (2) becoming active at some internal point during an optimal trajectory (see  for details). In addition, however, there is a global lower bound to , hence also through (4), which depends on and on whether CAV can reach prior to or not: If CAV enters the CZ at , accelerates with until it reaches and then cruises at this speed until it leaves the MZ at time , it was shown in  that
If CAV accelerates with but reaches the MZ at with speed , it was shown in  that
where . Thus,
is a lower bound of regardless of the solution of the problem. Therefore, we can summarize the recursive construction of over as follows:
where can be evaluated from through (4), and thus, it is always feasible.
Note that at each time , each CAV communicates with the preceding CAV in the queue and accesses the values of , , , , from its information set in (6). This is necessary for to compute appropriately and satisfy (13) and (3). The following result is established in  to formally assert the iterative structure of the sequence of decentralized optimal control problems:
The decentralized communication structure aims for each CAV to solve an optimal control problem for the solution of which depends only on the solution of CAV -1.
The decentralized optimal control problem for each CAV approaching either intersection is formulated so as to minimize the -norm of its control input (acceleration/deceleration). It has been shown in  that there is a monotonic relationship between fuel consumption for each CAV , and its control input . Therefore, we formulate the following problem for each :
where is a factor to capture CAV diversity (for simplicity we set for the rest of this paper). Note that this formulation does not include the safety constraint (3).
Iii-B Analytical solution of the decentralized optimal control problem
An analytical solution of problem (14) may be obtained through a Hamiltonian analysis. The presence of constraints (2) and (13) complicates this analysis. Assuming that all constraints are satisfied upon entering the CZ and that they remain inactive throughout , a complete solution was derived in  and  for highway on-ramps, and in  for two adjacent intersections. This solution is summarized next (the complete solution including any constraint (2) becoming active is given in ). The optimal control input (acceleration/deceleration) over is given by
where and are constants of integration. The constants , , , can be computed by using the given initial and final conditions. The interdependence of the two intersections, i.e., the coordination of CAVs at the MZ of one intersection which affects the behavior of CAV coordination of the other MZ, is discussed in .
We note that the control of CAV actually remains unchanged until an “event” occurs that affects its behavior. Therefore, the time-driven controller above can be replaced by an event-driven one without affecting its optimality properties under conditions described in .
As already mentioned, the analytical solution (15) is only valid as long as all initial conditions satisfy (2) and (13) and these constraints continue to be satisfied throughout . Otherwise, the solution needs to be modified as described in .
Recall that the constraint (3) is not included in (14) and it is a much more challenging matter. To deal with this, we proceed as follows. First, we analyze under what initial conditions the constraint is violated upon CAV entering the CZ. This defines a feasibility region in the space which we denote by . Assuming the CAV has initial conditions which are feasible, we then derive a condition under which the CAV’s state maintains feasibility over . Finally, we explore how to enforce feasibility at the time of CZ entry, i.e., enforcing the condition . This is accomplished by introducing a Feasibility Enforcement Zone (FEZ) which precedes the CZ. If the FEZ is properly designed, we show that can be ensured.
Iv Feasibility enforcement analysis
We begin with a simple example of how the safety constraint (3) may be violated under the optimal control (15). This is illustrated in Fig. 2 with for two CAVs that follow each other into the same lane in the CZ. We can see that while (3) is eventually satisfied over the MZ, due to the constraints imposed on the solution of (14) through (13), the controller (15) is unable to maintain (3) throughout the CZ. What is noteworthy in Fig. 2 is that (3) is violated by CAV 3 at an interval which is interior to , i.e., the form of the optimal control solution (15) causes this violation even though the constraint is initially satisfied at in Fig. 2.
Recall that we use to denote the CAV physically preceding on the same lane in the CAV, and is the CAV preceding in the FIFO queue associated with the CAV, we have the following theorem.
There exists a nonempty feasible region of initial conditions for CAV such that, under the decentralized optimal control, for all given the initial and final conditions of CAV .
Proof: To prove the existence of the feasible region, there are two cases to consider, depending on whether any state or control constraint for either CAV or becomes active in the CZ.
where , , and are functions defined over . Recall that CAV is cruising in the MZ, so that (15) through (17) do not apply for over leading to different expressions for , , and . Therefore, we consider two further subcases, one for and the other for . For ease of notation, in the sequel we replace by .
Case 1.1: . In this case, is a cubic polynomial inheriting the cubic structure of (17). We can solve (16), (17) for the cofficients , , , , , , and using the initial and final conditions of CAVs and . Then, denoting , , and as , , and for , these are explicitly given by
Aside from , all remaining arguments are known to CAV and can be determined. Hence, varies only with and . First, observing that the first half of each of the coefficient expressions in (IV) (which is derived by solving (16) and (17) for CAV ) is a constant fully determined by information provided by CAV , we can rewrite these as , , , . Therefore, in (17) can be expressed as
Next, the second half of the coefficients can be expressed through polynomials in either or explicitly derived by solving (16) and (17) for CAV . We will use the notation , to represent polynomials of degree and . Similarly, we set . Thus, for the coefficients in Eq. (IV), we get
Note that in (19) involves only the terms, while the analogous cubic polynomial for involves only the and terms.
Our goal is to ensure that for all (recall that ). We can guarantee this by ensuring that . Thus, we shift our attention to the determination of . We can obtain expressions for the first and the second derivative of , and respectively, from (18), as follows:
Clearly, we can determine as the solution of with , unless occurs at the boundaries, i.e., or . Thus, there are three cases to consider:
Case 1.1.A: . In this case,
and we can satisfy for any as long as a feasible is determined. Since at , we have and using the definition of and (19),
Observe that if , then CAV enters the CZ at a safe distance from its preceding CAV and since , we have for all . Thus, it suffices to select
where is the smallest real root of .
Case 1.1.B: . In this case,
Thus, the feasibility region is defined by all such that in the space.
Case 1.1.C: . This case only arises if the determinant of (21) is positive, i.e.,
and we get
In addition, we must have
Therefore, the feasibility region is defined by all such that
Case 1.2: . Over this interval, by Assumption 1. Therefore, (15)-(17) no longer apply: (15) becomes , (16) becomes and (17) becomes . Evaluating in this case yields the following coefficients in (IV):
It follows that and in (20) should be modified accordingly, giving , and . Since we are assuming that no control or state constraints are active for CAV , the designated final time under optimal control satisfies (9), i.e., . Thus, we only need to consider the subcase where occurs in and we have , . Proceeding as in Case 1.1.C, the feasibility region is defined by all such that
Case 2: At least one of the state and control constraints is active over . The analysis for this case is similar and is omitted but it may be found in .
To complete the proof, we show that feasibility region is always nonempty. This is easily established by considering a point such that and : since and , it follows that . Obviously, any such is feasible.