## 1 Introduction  
Unsteady thermoacoustic oscillations, also referred to as combustion instability, have emerged as a common problem in modern gas turbine burners and have hindered the development of lean-premixed combustors [1]. Their oscillation amplitude can reach sufficiently high levels to interfere with gas turbine operations and, in some extreme cases, can lead to the failure of a combustion system due to excessive structural vibrations. With the extensive use of lean-premixed combustors in the modern gas turbine industry, the suppression of thermoacoustic oscillations is always crucial for the progression of gas turbine developments. Thus, understanding how to predict and control thermoacoustic oscillations has become important [2]. Mastering and controlling thermoacoustic oscillations has become crucial for future renewable energy power systems [3]. Generally, there are two ways to control thermoacoustic oscillations: active control and passive control [4; 5; 6]. Acoustic dampers are widely applied as a passive control tool to stabilize combustion systems. However, this method cannot respond to the dynamic changes of real operating conditions and tends to be effective only within certain narrow-range frequencies [7]. Compared with passive control approaches, active control, especially in the closed-loop-feedback form, is more flexible and has promotional prospects [8]. Therefore, active control technology is expected to more efficiently suppress thermoacoustic oscillations. This has become a current research hotspot. Active control is generally based on feedback control technologies, either by modulating the acoustic field in the combustion chamber or modulating the unsteady heat-release rate of flames [9]. Currently, reduced-order models of thermoacoustic oscillations are mainly used to investigate the feedback control of combustion instability [10]. [PERSON] et al. [11] studied the feedback control of combustion instabilities with a low-order combustor model and revealed that the open-loop transfer function needed for controller design varies with the oscillation level. [PERSON] et al. [12] explored the feedback control of unstable thermoacoustic modes in an annular Rijke tube; a thermoacoustic network model was constructed and used to derive low-order models for modal control of the system. [PERSON] et al. [13; 14] proved that model-based control is a good method for studying thermoacoustic oscillations in combustion chambers. Different controllers were implemented in simulations using a low-order thermoacoustic model. Although active control, which is based on these models, has significant potential, under different working conditions, the triggering time and climb rate of thermoacoustic oscillations are different [15; 16], which also leads to control system failure or poor control effectiveness. Therefore, it is necessary to further explore the triggering time and climb rate of thermoacoustic oscillations under different conditions. Accordingly, to develop a robust feedback control system based on low-order thermoacoustic oscillation models, exploring the influence of different combustor parameters on the triggering time and climb rate is crucial. The triggering time and climb rate of combustion thermoacoustic oscillations are related to many parameters, and relevant scholars have conducted extensive research on this topic. [PERSON] et al. [17] studied the thermodynamic properties of a thermoacoustic swirl combustor and revealed that oscillation mode switching depends on the equivalence ratio and methane flow rate. [PERSON] et al. [18] investigated the effects of different heat source distributions on a Rijke tube; their experimental results show that optimum spacing exists to minimize the starting power of thermoacoustic oscillations and growth rates and maximize the sound pressure amplitude. Furthermore, it has also been proven that parameters such as the flame equivalence ratio and flow field play a major role in the excitation of thermoacoustic oscillations [19]. [PERSON] et al.