## 1 Introduction  
The free-piston linear power generation system (FPLG) [1, 2, 3] differs from conventional rotary reciprocating internal combustion engines by eliminating the crank-connecting rod mechanism. It uses controlled alternate firing of the opposed-free piston engine [4, 5] to drive the pistons' reciprocating motion, moving the linear motor's mover to cut magnetic flux lines and generate electromotive force, as shown in Figure 1. The FPLG has a more compact structure with high power density. Its modular design suits distributed drive. It also has a flexibly adjustable compression ratio, multi-fuel adaptability [6, 7], and can operate in various combustion modes. However, in actual operation, the lack of a crank-connecting rod structure complicates the piston motion, and this uncertainty becomes a technical bottleneck. It also makes the linear motor's electrical output characteristics [8] differ from those of conventional rotary motors. The composition of the FPLG system and the overall structure of the rectification control strategy adopted in this paper are shown in Figure 1.  
The research on Free-Piston Linear Generators (FPLG) primarily focuses on fuel adaptability, system operating characteristics, and stability control [9; 10; 11]. These studies are often based on piston motion, structural design, and material selection for the linear generator. However, there has been limited research on the power consumption systems at the backend and their impact on system performance and matching. Additionally, thermodynamic or combustion characteristics related to FPLG have been investigated to simplify the design of the backend load. Pure resistive loads are commonly used to dissipate electrical energy, and the output power [12; 13; 14; 15] at the load terminal is considered as the system's output power. To enhance system stability and reduce failures, researchers often choose to control the current to achieve precise piston trajectories or maintain the piston at the dead center within the generator [16]. This strategy typically aims at piston stability or precise trajectory control to manage the generator's output characteristics.  
As research on FPLG operating characteristics progresses, some scholars have found that the system can operate continuously with minimal intervention [17]. This finding accelerates the viability of FPLG in hybrid vehicles and highlights the need to consider the  
Figure 1: FPLG system composition.  
generator system's output characteristics. The adaptability of FPLG to different operating conditions and energy consumption profiles must be carefully matched.  
Uncontrolled rectification can convert the three-phase AC from the linear motor into pulsating DC. Its advantages are a simple structure and no need for complex control strategies. However, the quality of the output DC needs improvement. Three-phase controlled rectification, or Pulse Width Modulation Control (PWMC) [18; 19; 20], regulates the AC voltage and current by controlling the switching of three-phase bridge arms. It uses PI-based dual closed-loop control [21; 22] for voltage and current. This method can stabilize and adjust the DC voltage but has a complex control structure. The system's response speed and anti-interference ability are also weak.  
To address these issues, this paper designs a model predictive control (MPC) controller [23; 24] for the FPLG system. It reduces algorithm optimization steps and improves steady-state performance. An integral sliding mode control algorithm replaces the traditional PI control in the voltage outer loop, forming the Model Predictive Sliding Mode Control (MPSMC) [25; 26; 27; 28; 29]. This compound control simplifies controller parameter tuning, enhances anti-interference and adaptability, and makes the output voltage and frequency highly controllable, better matching the FPLG system's working characteristics.