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The online traffic signalization for intersection is addressed. A new model for intersection called positive switched system is presented. Then, based on the dissipativity analysis results for positive switched system, an online state-feedback control strategy for traffic signal in two-phase intersection section is proposed. A numerical example is provided to illustrate the effectiveness of our theoretical findings. Finally, in order to extend to more general cases, multiphase intersection is considered, and general dissipativity-based control is presented.

Traffic signal control is a long-lasting research problem in urban transportation network system [

Switched system can be efficiently used to model many practical systems which are inherently multimodel in the sense that several dynamical systems are required to describe their behavior. For more details of the recent results on the basic problems in stability and stabilization for switched systems, the reader is referred to survey papers [

The rest of this paper is organized as follows. In Section

The urban transportation system is composed of a network of intersections. Generally, an intersection is operated by a traffic signal that decides the movements of vehicles to pass the intersections or to stop to generate the queues. The movement may include vehicles going straight, turning left, turning right, or a combination of them.

In order to show our control idea clearly, we first consider the single intersection with four approaches and the traffic signal which has two phases, which are illustrated by Figure

Intersection with four approaches and two-phase signal.

It is noted that the movements 1 and 2 are supposed to have same characteristics, and same consideration holds for movements 3 and 4.

Considering the transportation status at the time of the traffic signal turning from one phase to another, and denoting the time instant as

Moreover, since the congestion situation is not considered in our model, the queue lengths are always beneath their capacities, which is denoted by

Based on the above definitions, we are now in the position to model the intersection system. Since there are two phases, it is natural that there are two dynamics for Phases

The queue length

And for movements 3 and 4, since the movement is stopped which implies

Following the same guideline in Phase

Among the two subsystems concerned with two phases, there is a switching signal

Augmenting the dynamics in Phases

The most reported model is considered to have a fixed cycle (one repetition of the basic series of signal phases combinations at a junction), which has to be prespecified appropriately, and an inappropriate choice of cycle could lead to a bad control performance. On the other hand, there is no cycle time constraint in our switched system model, or no cycle time has to be designed previously; only controllable green time needs to be considered.

The purpose of our control is to relieve the oversaturated queue by the traffic signal. With the controllable green time

At first, the lengths of queue

Then, to make the system nonaccumulative, a particular copositive storage function indicating the total lengths of the movement stored in the system is introduced as

In this section, we will investigate the dissipativity of switched positive system, which plays the key role in solving the control problem for intersection system.

Consider the following switched positive system composed by

System (

System (

The sufficient part is obvious. We consider the necessity. Denote

System (

In strictly dissipative case,

For switched positive system, the switched copositive function

Consider switched positive system (

Given a set of vectors

By the definition of dissipativity and supply rate

Theorem

Consider switched positive system (

Let

It is obvious to see that conditions in Theorem

Now, based on the formulated control problem and analysis results in previous section, considering the two-phase intersection system, the feedback controller (

Considering the closed loop system (

Then, the second point on dissipativity can be solved by Corollary

Moreover, to ensure dissipativity, the following condition has to be satisfied:

At last, the constraint on the feedback is considered. Note that queue length

Furthermore, the constraint set on the green time can be expressed by set

Summarizing above discussion, a solution for nonaccumulative feedback control is presented as follows.

Consider the two-phase intersection system (

In actual applications, a lower boundary for the effective green time has to be selected as

Proposition

Consider a two-phase intersection as

Evolution of queues within dissipative closed loop system.

From the simulation results in Figure

In this section, the results in previous section based on dissipativity and positivity of positive switched system will be generalized to multiphase intersection. At first, the system model for multiphase intersection will be presented as follows.

It is assumed that there exist

In accordance with the modeling procedure for two-phase intersection, in each phase

Similar to the derivation of two-phase intersection model, we have

The phases are working in turns in the intersection system model; the switching signal

With the aid of state feedback control as

Furthermore, following philosophy of positive and dissipative control for intersection in previous section, the

By

Summarizing the above steps, the nonaccumulative feedback control solution multiphase intersection is presented as follows.

Consider the

By modeling the intersection into positive switched system, a dissipativity-based control strategy is proposed for online traffic signalization in this paper. Through fulfilling the positivity, dissipativity, and control constraint, an LP problem based design method is presented. A numerical example is provided to illustrate our results, and, furthermore, the two-phase intersection results are extended to multiphase intersection. The positive switched system approach provides us with a new insight on modeling intersection; introducing other advanced control schemes from positive switched system to intersection system is our future work.

The authors declare that there is no conflict of interests regarding the publication of this paper.