###### Abstract
Furthermore, we [19] revealed that HWI on the RIS has no impact on Doppler shift and spread, but would increase delay spread. Note that [17; 18; 19] mainly studied single-RIS scenarios. Recently, multiple RISs system has been studied in terms of performance enhancement [20; 21; 22; 23]. In [20], the authors formulated and tackled a novel RIS-user/BS association problem that aims to optimally balance the passive beamforming gains from all RISs among different BS-user communication links. In [21], the authors introduced a new RIS-aided communication system, where multiple RISs assist communication between a multi-antenna base station (BS) and a remote single-antenna user by multi-hop signal reflection. In [22], a novel hybrid beamforming scheme for multi-hop RIS-assisted communication is proposed in order to improve the coverage range at THz-band frequencies. Moreover, the authors of [23] studied the statistical characterization and modeling of distributed multi-RIS-assisted communication systems. However, Refs. [20; 21; 22; 23] do not consider multiple RISs in HSC systems, especially in terms of Doppler mitigation and HWI analysis. On the other side, real-time positioning information (PI) is essential to an HSC system to ensure successful operation (e.g., collision prevention) [6; 24]. The trajectory of a train or a car can be sensed or predicated using some techniques such as track circuits [24], RF tags [25], odometers [26], global navigation satellite systems [27; 28; 29; 30], and smartphones [31; 32]. Furthermore, machine learning can also enhance the accuracy of the positioning [32; 33; 34; 35]. It is noteworthy that trajectory positioning is different from conventional channel estimation since the PI is only related to times, speeds, and locations. However, there are few papers designing the phase shift set of RIS using PI. Channel estimation for HSC is always very challenging due to time-varying channels [36]. The estimation process is more costly and inaccurate compared to estimating static/quasi-static channels [37; 38]. Therefore, the research gap can be summarizedas follow. First, although there are some works on Doppler mitigation using RIS in HSC, there is little literature considering the multiple RISs case; Second, it is crucial to study the relationship between Doppler mitigation and HWI on an RIS-assisted system, but there are few papers in this direction; Third, SE analysis is also very necessary when considering HWI on a Doppler-mitigation HSC system with the help of RIS; Fourth, how to reduce the computational complexity and energy consumption of channel estimation for RIS-assisted HSC systems is of great importance. In light of the research gap above, the motivation of this paper is that, instead of using conventional channel estimation techniques, PI (e.g., vehicle positions) could be fully utilized to design the phase shift set in real-time. In particular, the system model proposed in this paper is only related to transmission distance and time; thus, using a well-designed phase shift set, an RIS can reconfigure the reflected phase to align all channels to successfully maximize the received power. The benefit of the proposed scheme is that traditional channel estimations are not needed. Therefore, it can be easily implemented and is more energy-efficient than other conventional RIS-assisted systems. In this paper, we explore the capacity for Doppler mitigation and received power maximization of applying PI-based multiple RISs with HWI (The simulation results can be reproduced using code available at: [[https://github.com/ken0225/Multi-RIS-Doppler-Mitigation-Hardware-Impairments](https://github.com/ken0225/Multi-RIS-Doppler-Mitigation-Hardware-Impairments)]([https://github.com/ken0225/Multi-RIS-Doppler-Mitigation-Hardware-Impairments](https://github.com/ken0225/Multi-RIS-Doppler-Mitigation-Hardware-Impairments)), accessed on 8 July 2022). In order to elaborate, in Table 1, we explicitly contrast our contributions to the state-of-the-art RIS-assisted high-speed communications. In particular, Ref.