# Stochastic Model Predictive Control for Urban Traffic Networks
## 1 Introduction  
In many major metropolitan areas, inevitable delays and traffic congestion would be incurred by road users due to inappropriate design of traffic signal settings and uncertain traffic demand in road networks. Therefore, developing effective traffic signal control strategies for urban road networks has long been one of the imperative issues for traffic researchers and practitioners. In general, traffic signal control methodologies can be classified into three categories (Isolated Intersection Control, Fixed Time Coordinated Control and Coordinated Traffic-Responsive Control) by their characteristics [1]. As one of the earliest and simplest methods, isolated intersection control strategy is mainly used to access the optimal signal timing plan of an isolated intersection. Therefore, in many cases, it is a myopic control approach [2, 3]. Since their settings are based on history traffic flow information, fixed time coordinated control strategies can only provide good control performance under the condition of minor fluctuations in traffic demand [4]. As a result, coordinated traffic-responsive control strategies are proposed to overcome the drawbacks that existed in the previous two methods. Since the 1970s, lots of model-based traffic-responsive control strategies have been proposed. For example, SCOOT [5], SCATS [6], OPAC [7], PRODYN [8], and RHODES [9] are widely used in many cities. It is worth mentioning that the store-and-forward(SF) approach [10] is also one of the typical representatives in the family of coordinated traffic-responsive control strategies. Based on the SF approach, a number of traffic models and control strategies [11, 12, 13, 14] have been proposed. Without a doubt, a suitable traffic model can not only be convenient to assist traffic engineers to analyze the traffic dynamics, but also provides the basis for designing effective traffic signal control strategies. Since model predictive control (MPC) can deal with complex constraints and avoid myopic control schemes in large-scale control problems, MPC approaches are used to model and coordinate traffic signals in urban traffic networks [15; 16; 17; 18; 19; 20; 21; 22]. For instance, the authors in [15] proposed a linear model and a multi-agent control strategy for traffic signal coordination in urban traffic networks (UTN). In their model, by decomposing the centralized MPC problem into a number of coupled subproblems, the original control problem was solved by several distributed agents. Based on this work, a distributed interior-point algorithm was proposed to speed up the solving process in [16]. By using an improved nonlinear traffic flow model, [PERSON]. [17] proposed a centralized MPC model for network-wide traffic signal coordination. In addition, the optimization model reported in [17] was improved and replaced by a mixed-integer linear programming model in [18]. To enhance the computational efficiency and make the application of the MPC approach feasible for large scale urban traffic networks, hierarchical (or decentralized) and distributed MPC models can be found in later works [19; 20; 21; 22]. It is worth noting that the MPC methods aforementioned are all designed within a deterministic framework. However, uncertainties widely exist in reality and urban traffic networks are no exception. Hence, uncertainties (e.g., uncertain traffic demand, stochastic disturbances) have to be investigated in traffic model building. It is well known that using appropriate tools or methods to deal with uncertainties in traffic networks is very important for accessing the optimal signal timing plan of traffic networks. To our knowledge, so far, the studies that take these uncertainties into consideration on modeling of traffic signal control are rarely seen particularly under MPC framework. Aiming to minimize the average delays under changing traffic demand, Ref. [23] proposed three models to determine optimal signal timing plan. Ref.