On optimal fixed time stability: Its application in secure communication using sliding mode control

One of the main contributions of the present article is to provide some sufficient conditions for fixed-time stability (FX_TS) of nonlinear dynamical systems and the superiority of the proposed conditions over several such conditions already available in the literature. In this respect, a theorem on FX_TS is established, and an estimation of the settling time is provided. Furthermore, additional generalized forms are investigated to improve the application of the proposed results. The present results are further applied to a secure communication system to achieve fixed-time synchronization of master-slave system. In secure communication, to prevent data loss or distortion of the transmitted message, it is essential to effectively reject disturbances and uncertainties. In order to address this issue and achieve robust synchronization in the presence of channel noise, a sliding mode control scheme is developed in the present work. The proposed $\mathrm{SMC}$ not only guarantees that the synchronization error converges to zero in fixed time but also provides strong disturbance rejection capability. In this respect, some non-singular terminal sliding surfaces $\left(\mathrm{NSTSS}\right)$ and corresponding controllers are designed. At last, some numerical simulations are provided to verify the theoretical results.