In this article, we propose an instantiation, called${\sf Stanislas}$, of a dedicated Self-Synchronizing Stream Cipher (SSSC) involving an automaton with finite input memory using non-triangular state transition functions. Previous existing SSSC are based on automata with shifts or triangular functions ($T$–functions) as state transition functions. Our algorithm${\sf Stanislas}$admits a matrix representation deduced from a general and systematic methodology called Linear Parameter Varying (LPV). This particular representation comes from the automatic theory and from a special property of dynamical systems called flatness. Hardware implementations and comparisons with some state-of-the-art stream ciphers on Xilinx FPGAs are presented. It turns out that${\sf Stanislas}$provides bigger throughput than the considered stream ciphers (synchronous and self-synchronizing) when straightforward implementations are considered. Moreover, its synchronization delay is much smaller than the SSSC Moustique (40 clock cycles instead of 105) and the standard approach CFB1-AES128 (40 clock cycles instead of 128).