## 1 Introduction  
The modern maritime field has developed rapidly under the influence of multiple factors, including technological advancements, globalization demands, environmental protection, and geopolitics. Additionally, with militarization, its importance has been elevated, and countries are competing over maritime rights. Electro-optical tracking systems on shipborne platforms play a crucial role in spatial detection, collaboration, surface ship monitoring, and early-warning processes in maritime domains [1]. Traditional shipborne electro-optical tracking systems typically achieve beam pointing through multi-axis servo gimbals or optical mirrors to capture and track maritime targets. However, such systems have limitations, including large size, limited field of view, and weak anti-jamming performance, making them unsuitable for special application scenarios with restricted space or strict payload requirements [2]. More critically, the traditional PID control method exhibits significant limitations in shipborne dynamic environments: the fixed-parameter design is inadequate for addressing time-varying load characteristics caused by wave disturbances. Experimental results show that it produces an overshoot of 5-15% in step responses, and when the scanning speed increases to 0.4\({}^{\circ}\)/s, the RMS tracking error exceeds 600 \(\upmu\)rad [3, 4]. With the diversification of application demands, a rotating dual-grating electro-optical tracking and pointing device has emerged as a next-generation solution due to its compact structure and low rotational inertia. It achieves beam deflection by rotating two independent, coaxial, parallel gratings. In that way, it can enable efficient laser-beam scanning and beam pointing, as well as target acquisition and tracking tasks, demonstrating excellent scanning and acquisition performance [5]. In 2009, [PERSON]'s team proposed \"dual liquid crystal polarization gratings\". They deflected the beam and completed trajectory scanning within a specific angular range by controlling two independently rotating liquid crystal polarization gratings. This system not only has a high beam-transmission rate but also allows efficient beam pointing [6]. In 2013, [PERSON], [PERSON] et al. derived and verified the forward and reverse solution formulas for dual-prism beam pointing by using a non-axial ray-tracing method, discovering that the reverse calculation formula yields two sets of solutions [7, 8]. In 2021, [PERSON], [PERSON] et al. achieved a back-and-forth spiral-scanning mode by controlling the synchronous rotation of the dual prisms. This mode showed excellent performance in terms of field-of-view coverage, periodicity, and speed stability. They optimized the scanning trajectory through integrated filtering and interpolation methods, which suppressed fluctuations in prism speed and reduced servo-control error from 1100\({}^{\prime\prime}\) to 350\({}^{\prime\prime}\)[9]. In 2024, [PERSON], [PERSON] et al. designed a tracking model based on the decoupling of the optical axis and the reverse solution formulas of the dual liquid crystal polarization gratings. They combined it with Linear Quadratic Regulator (LQR) optimal control theory to test the tracking performance of this new electro-optical tracking system. The root mean square (RMS) value of the tracking accuracy was found to be less than 350 \(\upmu\)rad [10]. However, existing rotating dual-grating control systems often employ predictive-control or adaptive-control model methods. The former relies on an accurate dynamic model (with modeling errors required to be within 5%), while the latter experiences a sharp decline in control accuracy when the parameters of the hydraulic actuator drift by more than \(\pm\)20%. This creates a stark contradiction with the time-varying characteristics of mechanical parameters in shipborne environments (as the scanning speed varies nonlinearly with the scanning range and high-speed scanning may require rapid motor drives in the approach to the target).