###### Abstract
[12] considered RIS with perfect CSI, which is impractical. Moreover, although the multi-RIS scenario is considered, each RIS contains just a few elements in [12], which cannot obtain enough beamforming gains. Ref. [14] does not consider multiple RISs, HWI, or SE analysis, which we already considered in our manuscript. Besides, the channel estimation methods in [37; 38] are all traditionally piloted-based. Although these methods may obtain promising channel parameters, the estimation complexity cannot be ignored. To this end, we utilize PI to replace CSI to obtain the phase shift set in real time, which is less computationally demanding. The main contributions of this paper are as follows. * We model the HWI at multiple RISs as random phase shift errors, and the HWI at the transceiver as additive distortion noises. Based on the modeling of the HWI, we then propose a PI-based time-varying system model for multi-RIS-assisted communications. Furthermore, we also generalize our LoS channel model to the Rician case;
* Using the real-time PI, we design and compare different phase shift sets for multiple RISs, and obtain the phase shift set that can maximize SE, remove Doppler spread, \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Consider** & **Channel** & **Consider** & **Consider** & **Consider** \\  & **Multiple** & **Estimation** & **Method and** & **Hardware** & **Spread Mini-** & **Spectral** \\  & **RISs** & **Complexity.** & **Impairments** & **mization** & **Efficiency** \\ \hline \hline \multirow{2}{*}{Basar [12]} & \multirow{2}{*}{Yes} & Consider & \multirow{2}{*}{No} & \multirow{2}{*}{No} & \multirow{2}{*}{No} & \multirow{2}{*}{No} \\  & & perfect CSI; & & high & \\ \multirow{2}{*}{[PERSON], } & \multirow{2}{*}{No} & Predictive & & & \\  & & information; & & & \\ \multirow{2}{*}{Sun, } & \multirow{2}{*}{No} & Doppler shift & & & \\  & & adjustment; & & & \\ \multirow{2}{*}{[PERSON], } & \multirow{2}{*}{No} & Least squares; & & & \\  & & moderate. & & & \\ \multirow{2}{*}{Ours} & \multirow{2}{*}{Yes} & PI-based; & & & \\  & & low. & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: State-of-the-art RIS-assisted high-speed communications. and keep delay spread at a very low level, even with random trajectory errors. We reveal that the RIS HWI can increase delay spread, but has no impact on Doppler shift and spread;
* We compare the performance of different numbers of RISs in terms of SE and delay spread. We reveal that the delay spread grows with the increasing number of RIS. Therefore, the tradeoff between SE and delay spread should be considered when designing a multi-RIS system. We also mathematically derive the closed-form expression of the SE with respect to the proposed system;
* Detailed simulations are provided to validate the effectiveness and robustness of the proposed system in this paper. In particular, the numerical results reveal that our system is still effective in the Rician channel. Furthermore, the PI-based phase shift set is robust even if the random positioning error exists. The rest of this article is organized as follows. In Section 2, we present a PI-based time-varying multi-RIS-assisted Doppler mitigation communication system model with HWI. Based on the proposed system, in Section 3 we define the expressions for Doppler spread, Doppler shift, and delay spread. In Section 4, we design and compare different phase shift optimization strategies on multiple RISs. In Section 5, we obtain the expression of SE. In Section 6, we provide numerical simulations and discussions. In Section 7, we conclude this paper. In addition, the summary of notations is listed in Table 2.