## 1 Introduction
Specifically, the vehicular platoon with time delays is a third-order system (its states mainly include position, velocity, and acceleration) in essence. This naturally leads to high-dimensional matrix computation. Existing approaches of deriving the kernel and offspring hypersurfaces of the CTCR paradigm, such as the Sylvester resultant [21] and extended Kronecker summation [22], face this challenge. For example, the extended Kronecker summation method is preferable to be combined with the CTCR paradigm, but it needs to construct a high-order auxiliary characteristic equation (ACE). Thus, there is an urgent need for a more efficient and streamlined approach that can alleviate the computational burden while ensuring the accurate analysis of the ETDMs of the vehicular platoon. In addition to the internal stability analysis, studying the string stability of the vehicular platoon with time delays is also an important research topic. The string stability conditions are obtained for a second-order vehicular platoon (its states mainly include position and velocity) with a single delay, and the thresholds of the controller parameters are further provided according to the delay margin [23]. For the vehicular platoon with communication delays, the string stability analysis is achieved by scaling the trigonometric functions with the delays [24]. The delay terms are approximated by using the second-order Taylor series expansion near zero to derive the necessary string stability conditions [25]. The Pade approximation, which offers better precision in approximating the delay terms compared to the Taylor series expansion, has been used to obtain the less conservative string stability conditions [26]. It is worth noting that the aforementioned string stability analysis methods primarily focus on the disturbances acting on the leader vehicle and often neglect any limitations on the frequencies of the spacing errors. However, in practical scenarios, the leader vehicle is typically constrained by the physical and mechanical factors, resulting in less frequent maneuvers. The low-frequency region contains a significant portion of the energy of the spacing errors, making it critical for the string stability analysis. Unfortunately, this aspect has been overlooked in the previous studies. It is demonstrated that the traditional proportional controller, when applied to the predecessor-following (PF) topology with the constant distance (CD) policy, cannot guarantee the string stability of a vehicular platoon [23]. To address this issue, one possible solution is to adopt the constant time headway (CTH) policy, where the distance between the successive vehicles is adjusted based on the velocity of the following vehicles. The CTH policy has shown the improved string stability characteristics compared to the CD policy by increasing the time headway. However, increasing the time headway reduces road capacity and increases fuel consumption. Therefore, it is crucial to determine the minimum allowable time headway with a trade-off. For a second-order platoon with the CTH policy, a sliding-mode controller is designed to guarantee the string stability; meanwhile, the bound of the time headway is obtained by analyzing the sufficient string stability conditions [27]. In [28], the multiple-predecessor following (MPF) topology is used to represent the communication relationship between the followers, and it is proved that the bound of the time headway can be reduced by increasing the number of followers. Then, the communication delay is considered in the controller design of a third-order vehicular platoon, and a lower bound of the time headway, which can compensate for the effects of communications delays, is achieved by introducing the acceleration feedback [29]. The purpose of this paper is to analyze the internal stability and string stability of the vehicular platoon with sensing delay and communication delay under PF topology.