## 1 Introduction  
Currently, the forward flight speeds of conventional civilian and military helicopters are limited to approximately 300-350 km/h, which does not satisfy the design goals and mission requirements of high-speed compound helicopters [1]. With innovation and breakthroughs in power plants and configurations, the horizontal flight speed of fifth-generation helicopters has exceeded 400 km/h [2]. Sikorsky displayed an S-97 Raider helicopter demonstrator for 2021, as shown in Figure 1, demonstrating revolutionary speed, maneuverability, and agility, representing a range of technologies required for future helicopters [3, 4]. The novel rotorcraft has two coaxial rotor blades and a single pusher propellor, with a maximum speed exceeding 480 km/h. Its rival, V-280 Valor, features a tilt-rotor design similar to that of V-22 Osprey and is a third-generation tilt-rotor vertical lift helicopter [5]. As shown in Figure 2, it is more efficient and faster in cruise mode. It realized a speed of 555 km/h in a flight test in November 2020 [6] owing to its unique mechanical structure with a variable rotor pull direction. Variable-speed rotor technology was adopted by the S-97 Raider and V-280 Volar [7]. The effectiveness of this technology in addressing low-speed rotor noise, breaking forward flight speeds, increasing payload and range, and reducing fuel burn and operating costs has been demonstrated by NASA [8; 9; 10; 11]. The variable-speed transmission mechanism increases the complexity of the power system, which is a major reason for the high incidence of V-22 Osprey accidents. Boeing's A160T hummingbird unmanned helicopter uses a variable-speed rotor [12], which significantly improves the overall aerodynamic performance and fuel economy of the aircraft [13]. However, during the transition of flight, there is a clear safety concern. This is because matching the power output of the turboshaft engine with the aerodynamic characteristics of the main rotor becomes challenging as the rotor speed changes [14]. In existing studies, the collective pitch of the main rotor is generally used as the cross-linking parameter of the flight control system to reduce the interference of the main rotor load on the turboshaft engine [15; 16]. To ensure the continuous power output of the turboshaft engine, a sequential torque control method was adopted in [17], and a power turbine speed controller with linear matrix inequality control, and torque feed-forward compensation was designed to track the turbine speed and weaken the torque fluctuation in the process of rotor speed change. In a previous study [18], neural networks were used for rotor demand torque prediction to reduce the power turbine speed overshoot by more than 20%. This mitigated the interference of large changes in the rotor demand torque on the power turbine speed with a faster response and higher robustness. The aforementioned studies primarily consider a scenario with a constant rotor speed. In such situations, a minor range of rotor speed variations results in limited changes in engine torque demand [19]. However, when there is a broad variation in rotor speed, the torque required by the main rotor and the engine's output torque must be aligned using variable transmission. The gearshift jerk arising from multi-speed transmission, especially when the motor and clutch reach saturation in a power system, is discussed in ref. [20]. Nevertheless, these studies are predominantly geared toward vehicle systems and are not easily adaptable to variable rotor speeds. Traditional design methods, which  
Figure 1: S-97 Raider. Figure 2: V-280 Valor. ignore the implications of extensive rotor speed variations, fail to align the engine torque output with the dynamic process of variable rotor speed [21]. This makes it challenging to achieve smooth control in systems where the engine, variable speed transmission, and rotor are strongly coupled.