線性及非線性跟車模式漸近安定性之模擬分析

A SIMULATION ANALYSIS ON THE ASYMPTOTIC STABILITY OF LINEAR AND NONLINEAR CAR-FOLLOWING MODELS

微觀跟車模式可描述個別車輛駕駛在車流中的行為，惟其數學組成相當複雜，對於模式的安定性，多年來相關研究只完成線性模式的分析，非線性模式的安定性分析一直闕如。本研究以系統模擬的方式分析跟車模式的安定性。對於線性跟車模式，模擬結果除驗證過去理論分析結果外，另發現只要滿足C=1／2的漸近安定性條件，線性跟車模式模擬的最大允許加減速率絕對值會隨車密度（k）增加而減少，但並不受跟車反應時間（T）的影響。對於非線性跟車模式，模擬結果顯示在符合漸近安定性的條件下，模擬的最大允許加減速率絕對值不但隨車密度的增加而降低，亦隨跟車反應時間的增加而降低。更重大的發現是，非線性跟車模式漸近安定性的條件亦為C=1／2，與線性跟車模式同。以Greenshields模式為例，在恰為安定的條件下（C＝1／2），可據以推得不同車流狀態（車密度）下適用的最大允許反應時間Tmax＝kj／（2ufk2），其與高階連續流模式鬆弛時間的定式非常相似，值得後續研究繼續加以探討。

Microscopic car-following models can accurately describe individual driving behaviors within traffic flows. However, due to their complicated model formulation, only linear car-following models have been established for stability analysis, and non-linear models have always been a lackluster. This paper aimed to analyze the stability of car-following models by simulation. Simulation results showed that under the condition of C£1/2 (C=sensitivityÍresponse time), linear models became asymptotically stable and the maximum allowed acceleration for remaining asymptotically stable decreased concurrently with the increased traffic density (k), but is irrelevant to the response time (T) of the vehicle. By contrast, the maximum allowed acceleration of the non-linear models decreased concurrently with increased traffic density and response time. Notably, the researchers found that the condition for the asymptotic stability of non-linear models was consistent to that of linear models (i.e. C£1/2). Using Greenshields model as an example, the maximum allowed response time under various traffic densities can be expressed as Tmax=kj/(2ufk2), which is closely similar to the relaxation time equation of the high order continuum model. This similarity should be further examined.