Linear complementarity system approach to macroscopic freeway traffic modelling: uniqueness and convexity

Abstract

The modified cell transmission model (MCTM) is formulated as a linear complementarity system (LCS) in this paper. The LCS formulation presented here consists of a discrete time linear system and a set of complementarity conditions. The discrete time linear system corresponds to the flow conservation equations while the complementarity conditions govern the sending and receiving functions defined by a series of ‘min’ operations in the MCTM. Technical difficulties encountered in application of the CTM and its extensions such as the hard nonlinearity caused by the ‘min’ operator can be avoided by the proposed LCS model. Several basic properties of the proposed LCS formulation, for example, existence and uniqueness of solution, are analysed based on the theory of linear complementarity problem. By this formulation, the theory of LCS developed in control and mathematical programming communities can be applied to the qualitative analysis of the CTM/MCTM. It is shown that the CTM/MCTM is equivalent to a convex programme which can be converted into a constrained linear quadratic control problem. It is found that these results are irrelevant to the cell partition, that is, different cell partitions will not change the uniqueness and convexity of solution. This property is essential for stability analysis and control synthesis. The proposed LCS formulation makes the CTM/MCTM convenient for the design of traffic state estimators, ramp metering controllers.

Keywords:

• the cell transmission model (CTM)
• complementarity condition
• linear complementaritysystem
• convexity
• uniqueness

Acknowledgments

The authors sincerely thank the referees for their insightful and constructive comments which have substantially improved the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

This work was jointly supported by the National Natural Science Foundation of China under Grant No. 51308559, NSFC-RS Exchange Program under Grant No. 513111163, National ‘Twelfth Five-year Plan’ Science & Technology Pillar Program under Grant No. 2014BAG01B05, and Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 20130171120032.

Notes

1. The boundary conditions are not in the exact form as those in Muñoz et al. (2004) and Muñoz (2004) but an equivalent formulation derived to convert the MCTM into a uniform characteristics of the sending and receiving functions as will be shown later. This formulation makes the MCTM more suitable for the LCS modelling paradigm.

2. Most of the time only the on-ramps are metered. Recently, Li, Chang, and Natarajan (2009) proposed a mixed integer model for an integrated control between off-ramp and arterial traffic flows. The off-ramps are controlled to minimise the queue spillback from off-ramps to the freeway mainline that may significantly degrade the performance quality of the entire freeway system.

3. It is worth knowing that the MATLAB codes for complementarity problems adopted in this paper is available at http://www.mathworks.cn/matlabcentral/fileexchange/20952-lcp-mcp-solver-newton-based.