## 1 Introduction  
Since modern society is mostly based on fossil fuel energy, the depletion of fossil fuels and persistent environmental pollution pose a global problem. There is an increasing demand for the development of new and renewable energy and improving energy utilization efficiency for the use of fossil fuels. Researchers have been actively conducting research to increase energy utilization efficiency through the collection of waste heat energy. In particular, Stirling engines are drawing attention as a device that generates electricity using pure thermal energy [1]. The Stirling engine is a heat engine based on the Stirling cycle, which converts thermal energy into mechanical power by compressing and expanding the enclosed working gas through the reciprocating motions of two pistons at different hot and cold temperatures. The idealized Stirling cycle consists of four thermodynamic processes acting on the working fluid: (1) isothermal expansion, (2) isochoric cooling, (3) isothermal compression, and (4) isochoric heating [2]. Stirling engines are known to have low noise and vibration because they do not involve an explosion stroke. Further, they can utilize, not only existing fossil fuels and waste heats, but also renewable heat energies, such as solar heat, biomass, etc. In particular, the Stirling engine-based micro-CHP system has been successfully utilized as a residential and commercial power generation system and is being evaluated as a powerful alternative energy technology [3, 4, 5]. Structurally, conventional Stirling engines have a mechanical link consisting of a connecting rod, a crankshaft, and a flywheel. In such a kinematic engine, this link mechanically connects the two pistons and precisely adjusts the amplitude and phase angle. Not only are the mechanical links large and heavy, but the side loads on the cylinders cause mechanical friction [6]. To overcome these shortcomings, a free piston Stirling engine (FPSE) has been proposed [7]. In FPSEs, each piston is connected to a spring element instead of a mechanical link structure. Therefore, it has the advantages of having a compact structure, high durability, and high power density [8]. However, since FPSE forms a vibrating system due to spring elements, its dynamic behaviors, such as operating frequency, piston amplitude, and phase angle, change with the operating conditions. In addition, FPSE has the characteristics of operating in resonant conditions, so its operating range is narrow and its sensitivity to operating conditions is high [9]. Therefore, it is important to determine the operating conditions of the engine temperature, charge pressure, and external loads, and to predict the dynamic behavior of the engine under given operating conditions. Recently, FPSEs have been developed with increasing working pressure and operating frequency to achieve higher power density [10]. The working pressure acts as a gas spring and connects the two pistons, which greatly affects the operability and performance of the FPSE. In general, FPSEs have an operable charge pressure range and cannot operate beyond this range. Therefore, the prediction and control of a FPSE's dynamic behavior is an important issue in optimizing operation and performance. [PERSON] and [PERSON] [11] developed a linear dynamics model that predicts the operating frequency, phase angle, and amplitude ratio of an engine using an isothermal model and pressure linearization. They compared the model with the experimental results for RE-1000 [12] and reported prediction errors of approximately 10-40%. This model is considered as a representative linear dynamic model of FPSEs. [PERSON] [13] conducted root locus analysis with a linear dynamics model focusing on the issues of engine stability and operability of FPSEs. [PERSON] proposed that when the real part of the root is greater than zero in the root locus, the engine starts to operate; in addition, the higher values of the real part yield more power output.