Distributed-information neural control: the case of dynamic routing in traffic networks

Abstract

Large-scale traffic networks can be modeled as graphs in which a set of nodes are connected through a set of links that cannot be loaded above their traffic capacities. Traffic flows may vary over time. Then the nodes may be requested to modify the traffic flows to be sent to their neighboring nodes. In this case, a dynamic routing problem arises. The decision makers are realistically assumed 1) to generate their routing decisions on the basis of local information and possibly of some data received from other nodes, typically, the neighboring ones and 2) to cooperate on the accomplishment of a common goal, that is, the minimization of the total traffic cost. Therefore, they can be regarded as the cooperating members of informationally distributed organizations, which, in control engineering and economics, are called team organizations. Team optimal control problems cannot be solved analytically unless special assumptions on the team model are verified. In general, this is not the case with traffic networks. An approximate resolutive method is then proposed, in which each decision maker is assigned a fixed-structure routing function where some parameters have to be optimized. Among the various possible fixed-structure functions, feedforward neural networks have been chosen for their powerful approximation capabilities. The routing functions can also be computed (or adapted) locally at each node. Concerning traffic networks, we focus attention on store-and-forward packet switching networks, which exhibit the essential peculiarities and difficulties of other traffic networks. Simulations performed on complex communication networks point out the effectiveness of the proposed method.

Error Estimates for Approximate Optimization by the Extended Ritz Method

“…The extended Ritz method with such bases was successfully tested on a variety of problems with admissible solutions dependent on a large number of variables: stochastic optimal control [64,65,66,80] and optimal estimation of state variables [3] in nonlinear dynamic systems with a large number of state variables, team optimal control [8], optimal control of freeway traffic [81], routing in large-scale communication networks [9,10], optimal fault diagnosis [5], etc. In these applications, admissible sets of variable-basis functions were used, for which the degree n necessary to guarantee a fixed approximation accuracy grows only polynomially with the number of variables of admissible solutions.…”

“…Also, a new branch of nonlinear approximation theory investigating approximation capabilities of neural networks was developed [11,12,21,28,38,45,50,51,52,53,54,55,56,57,58]. In a series of papers [3,5,8,9,10,64,65,66,80,81], a new method of approximate optimization was developed, called in [81] the extended Ritz method. In these papers, approximate solutions were used that were obtained over restrictions of sets of admissible solutions to linear combinations of all n-tuples of functions with varying "free" parameters, instead of linear combinations of first n functions from a basis with fixed ordering as in the classical Ritz method.…”

Neural Networks for Real-Time Traffic Signal Control

“…In the case where all the A i s of the traffic network are designed for continuous traffic signal control, the optimization problem becomes that of an infinite horizon one. A reasonable approximation of such a problem has been presented in [9] based on [10] in the form of the "receding-horizon limited memory" where the requirement for infinite memory storage capacity can be overlooked.…”

“…For the SPSA-NN-based multiagent system, the weight update algorithm follows the form of (9), and the gain sequence (or learning rate) has to satisfy the classical stochastic approximation conditions defined in (4). Choosing an appropriate form for the gain sequence is not a trivial matter as it would affect the performance of the NN in the long run.…”

“…For such a distributed approach, each agent will generate its own control variables based on the local information it receives. Based on the approach in [9], the following are defined: G = (N, L), directed graph with a set N of n nodes and a set L of l links describing the traffic network; A i agent acting at node i N ; T total number of temporal stages at time t;…”