# Signal Consensus in TSP of the Same Grid in Road Network
## 1 Introduction  
Transit signal priority (TSP) improves the operational efficiency of transportation and reduces overall delay at an intersection on a per-person basis. The priority of public transport vehicles is higher than that of ordinary social vehicles in TSP [1]. The strategies of TSP provide the signal priority for the priority vehicle by changing the current signal phase length, phase sequence or inserting dedicated phases without affecting the coordination relationship between traffic light signals [2]. The strategies of TSP include passive priority strategy, active priority strategy and real-time strategy [3; 4; 5; 6]. Passive priority [7; 8] is a strategy of signal timing at an intersection based on historical data without the vehicle detector. Active priority [9; 10; 11; 12] means that if a bus is detected close to the intersection, the method of extending, advancing, increasing or jumping phase is adopted to make the bus pass through the intersection smoothly. Through the GPS(Global Positioning System)and AVL(Automatic Vehicle Location) and other advanced information detection devices to estimate the status of the road network and vehicles, real-time strategy [13; 14] provides priority signal for buses by optimizing, adjusting the signal timing of the intersection or trunk according to the real-time traffic information. Table 1 shows the main strategies and methods of TSP [15].  
The TSP strategies have no significant difference between traffic signal control agents of intersections belonging to the same grid in the road network [16], but their differences will lead to poor driving experiences, and drivers tend to follow the same TSP strategy in the same grid, otherwise it will affect driving behavior and cause confusion. TSP strategies reaching consensus in the same grid play an important role in driving fluency and safety.  
The research of the consensus algorithm is widely used in control, traffic and distributed computation and many fields [17; 18; 19]. In 1995, a classical model to simulate the consensus phenomenon of particles was proposed by [PERSON] [20] et al., and some interesting results were obtained by simulation. After that, [PERSON] [21] et al. used the matrix method to analyze the model, and found that the system will eventually reach consensus as long as the network is connected. [PERSON] and [PERSON] [22] put forward the theoretical framework of the consensus problem based on the work of [PERSON] and [PERSON] [23; 24], designed the most general consensus algorithm, and found the algebraic connectivity of the network can characterize the convergence rate of the system, the condition that the algorithm achieves the average consistency is given and the result was extended to the symmetric consistency algorithm with time delay [25]. In the above study, the status updating of an agent is related to the difference between the neighbor agents, and it will change too fast if the difference is too large, the rapid change of TSP strategy leading to many problems in practice, such as traffic signal confusion and inaccurate timing. We focus on solving this problem in this paper, ensuring that the TSP strategy reaches consensus eventually with a smooth rate to avoid serious traffic problems.  
An outline of this paper is as follows: in Section 2, we introduce the basic knowledge of graph theory and matrix theory to better introduce the consensus algorithm, and the consensus algorithm without input constraints and its convergence conditions are demonstrated. In Section 3, we propose a consensus algorithm without input constraints; conditions and proof of its convergence are given, and stability of the algorithm is proved by the Lyapunov function. In Section 4, the application of the consensus algorithm in TSP and the reasons for adopting input constraint are described in detail. In Section 5, the simulation results are presented. Finally, in Section 6, we make our concluding remarks.