As an important application of MFD theory, MFD-accumulation-based linear control theory provides a creative perspective on controlling traffic flows at macroscopic network-wide level, such as optimal control, feedback control, robust control and decoupled control. Meanwhile, hysteresis phenomena are important effects of traffic instabilities in networks, which have been experimentally observed and theoretically verified in the past few years. Besides control strategies based on the value of accumulation, a natural question to be asked is: can we learn everything about the dynamical hysteresis behavior for applications in traffic control area. But which worth the whistle is that existing numerical analyses and characterizations for hysteresis are not consistent with mathematical needs of linear control system. The reason is that differential equations of MFD-based linear control system describe the collective traffic flow dynamics based on the form of direct variable signals. To operationalize hysteresis patterns in traffic control applications, the first research question to address is how to interpret hysteresis patterns in a comprehensible and computational way for traffic control system. In this work, we propose a new mathematical formulation of MFD and hysteresis patterns from a vector and matrix perspective. Then we apply this new formulation to create hysteresis-based feed-forward control strategies, which is expected to contribute to traffic flow control in the context of MFD with an unstable congestion state (usually during the rush hours) via feed-forward gating control.

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