Abstract
In this paper, we adapt the fractional derivative approach to formulate the flow-conservation transportation networks, which consider the propagation dynamics and the users’ behaviors in terms of route choices. We then investigate the controllability of the fractional-order transportation networks by employing the Popov-Belevitch-Hautus rank condition and the QR decomposition algorithm. Furthermore, we provide the exact solutions for the full controllability pricing controller location problem, which includes where to locate the controllers and how many controllers are required at the location positions. Finally, we illustrate two numerical examples to validate the theoretical analysis.
Highlights
In order to assess whether the controlled transportation networks are capable of reaching the desired performance, the concept of “controllability” is introduced, which is originated from the control theoretical concept of complex dynamical networks
There are some existing works that investigate the controllability of transportation problem, such as the traffic signal timing plan and the dynamics equilibrium based on traffic control, which has been discussed in [6, 7]; the relationship between route choice and traffic control has been discussed in [8]; the Gramian-based optimization analysis of the traffic control problem is considered in [9]; the route choice and controller problem in transportation has been clear expressed in [10]
We study the controllability of the flow-conservation transportation networks with fractional-order dynamics and the location and number of controllers based on the controllability of a complex network with flow conservation
Summary
Due to the complexity nature of the fractional-order transportation networks, we should first obtain the minimum number of controller nodes and check whether the PBH rank condition is satisfied. Making a comparison with the existing results, the classical Caputo derivatives because they allow traditional initial and boundary conditions to be included in the formulation of the considered problem were used to model the fractional-order transportation networks in this paper. Flow-conservation networks control theory is based on integer order, but in this paper we apply the Caputo fractional derivative to the flowconservation networks to obtain the dynamic formulation of the fractional-order transportation networks. E exact framework of fractional-order transportation networks control theory is established based on the fact that the PBH condition is not related to the current network congestion; all links in the network are in a stable state in discrete time, regardless of congestion or not.
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