An increasing number of cities are implementing bicycle sharing systems to reduce traffic congestion. Determining the locations of bicycle stations is one of the fundamental challenges in planning of such systems. This paper provides a novel solution on it. The min-plus algebra is introduced to model transport systems for accessibility analysis. A unified model in the sense of the min-plus algebra for an integrated system with both buses and bicycles is presented to dynamically describe the state transitions of passengers in the system. A new accessibility is proposed with regards to a general index ω such as the geographical distance and the travel time. A necessary and sufficient condition on the accessibility is then provided. The minimization of bicycle stations under the accessibility is formulated to be a 0-1 integer programming problem. The case studies on two cities, Ningbo and Hangzhou, were performed, which show that compared with the current layouts of urban transportation networks, the proposed public bicycle sharing systems have remarkable advantages in topological characteristics and robustness against failures.

The problem of leader-following consensus of nonlinear agents in hybrid varying directed topology is considering not only the agent but also that the directed edges can have a time-varying nonlinear dynamics with jump discontinuity, which contains the switching topology as its special case. This paper has the following contributions toward this problem. The leader-following consensus problem in hybrid varying directed topology is first addressed, and an online leader switching method is proposed, which reduces the dependence on some global conditions and the connectivity assumptions on the selection of leaders. Second, we generalize the Lipschitz condition and the combined condition of one-sided Lipschitz and quadratically inner-boundedness conditions to a new generalized linear incremental condition, which gives us a more generalized result in the Lyapunov proof and better performance in simulation. Third, an observer-based consensus protocol is constructed with two sufficient observability and controllability conditions and two optimal control design algorithms. Finally, an example of teleoperating multirobotic manipulator joint network is provided to illustrate the performance improvement by comparing with the existing results.

This paper establishes several bounds on the algebraic connectivity and spectral radius of graphs. Before deriving these bounds, a directed graph with a leader node is first investigated, for which some bounds on the spectral radius and the smallest real part of all the eigenvalues of M = L + D are obtained using the properties of M-matrix and non-negative matrix under a mild assumption, where L is the Laplacian matrix of the graph and D = diag{d1, d2, ..., dN} with di > 0 if node i can access the information of the leader node and 0 otherwise. Subsequently, by virtue of the results on directed graphs, the bounds on the algebraic connectivity and spectral radius of an undirected connected graph are provided. Besides establishing these bounds, another important feature is that all these bounds are distributed in the sense of only knowing the information of edge weights' bounds and the number of nodes in a graph, without using any information of inherent structures of the graph. Therefore, these bounds can be in some sense applied to agent networks for reducing the conservatism where control gains in control protocols depend on the eigenvalues of matrices M or L, which are global information. Also some examples are provided for corroborating the feasibility of the theoretical results.