In this article, the Mittag-Leffler synchronization (MLS) problem of a fractional-order permanent magnet synchronous generator (FOPMSG)-based wind turbine system against unknown disturbances, such as external load torque variations and system parameter uncertainties, an adaptive fractional sliding-mode control (AFSMC) method is proposed based on improved convergence rate performance of the FOPMSG to track accuracy, response speed, and robustness. The AFSMC method is based on a fractional-order term incorporated into the new law for reaching the sliding mode, improves the chattering in the control signal, and reduces the time required for the system to reach the sliding-mode surface. Sufficient conditions are derived to ensure the robust MLS for the sliding-mode dynamics by the designed robust controller. In this article, for the first time, an adaptive sliding-mode control (ASMC) with a terminal function that accurately controls the FOPMSG model at a prespecified time is proposed. Moreover, the designed ASMC can effectively attenuate the existence of disturbances and uncertainties by eliminating the reaching phase based on the Lyapunov stability theory. Finally, the simulation results applied to the FOPMSG model show that the proposed control method has better disturbance rejection ability, fast dynamic response, and suppression of the chattering effect.

In recent years, many efficient key exchange protocols have been proposed based on matrices over the tropical semirings. The tropical addition of two elements is the minimum of the elements, while the tropical multiplication is the sum of the two elements. This paper proposes a novel key exchange protocol based on the min-plus semiring ($ \mathbb{Z}\cup\{\infty\}, \oplus, \otimes $) by introducing anti-$ s $-$ p $-circulant matrices, which forms a commutative subset of $ M_{n \times n}(\mathbb{Z}\cup \{\infty\}) $. We have given further analysis of the protocol in detail using upper or lower-$ s $-circulant matrices. Additionally, we prove that the set of all lower-$ s $-circulant matrices is a sub-semiring of the tropical semiring $ M_{n \times n}(\mathbb{Z}\cup \{\infty\}) $. We discuss the detailed security analysis of the protocol with upper or lower-$ s $-circulant matrices and provide cryptographic algorithms for both key exchange protocols with detailed explanations. We compare the protocol based on upper or lower-$ s $-circulant matrices and our proposed protocol in terms of time complexity and memory usage. Finally, we analyse the security and show that our protocol is safe against popular attacks of tropical key exchange protocols. The security of these protocols relies on the difficulty of solving tropical non-linear equations.