## 1 Introduction  
Nonlinearity is generic in nature, and linear behavior is an exception [1]. Nonlinear vibrating behaviors have been reported in numerous pieces of engineering literature. Some self-excited nonlinear vibrations may lead to a catastrophic disaster, such as the nonlinear flutter in aerospace engineering and the chatter vibrations in manufacture. A few series of literature have discussed the chatter prediction and avoidance as earlier by [PERSON] [2; 3; 4], then followed by [PERSON] [5; 6; 7; 8]. Moreover, the summary could be found in the recent review [9]. In aerospace engineering, as studied by [PERSON] et al. [10], the trapezoidal-plate-like structure may suffer from different types of nonlinear phenomena, such as jump, internal resonance, periodic, quasi-periodic, and chaotic motions. For a real-life structure with non-negligible nonlinearities, it is essential to provide the precise mathematical model to perform accurate and reliable predictions of the structure's dynamic behavior. Therefore, experimental testing and system identification still play a key role because they help the structural dynamicist to reconcile numerical predictions with experimental investigations [1]. The last several decades have witnessed an increasing interest in the system identification of nonlinear mechanical structures for either academic researches or industry applications. A lot of methodologies were well-established through a variety of studies, such as restoring force surface method [11], NARMAX (nonlinear autoregressive moving average with exogeneous inputs) modeling [12; 13], the Hilbert transform [14; 15], and Volterra series [16]. For more details of developments and advances on the topic of nonlinear system identification in structural dynamics, readers can refer to the reviews proposed by [PERSON] et al. [1] and later by [PERSON] [17]. However, these methods were difficult to apply to complicated structures with more than one nonlinearities. On the other hand, it is difficult to apply the identified mathematical model directly in the control field. With the development of system identification and control theory, nonlinear black-box modeling, like the Hammerstein-Wiener model, was proposed and widely used in control of the nonlinear system. A variety of identification methods were developed to estimate the nonlinear and linear model [18; 19; 20; 21; 22]. However, the establishment of the black-box model is based on data alone, lacking physical insight. This is a very ambitious objective knowing that nonlinearity may be caused by many different mechanisms and may result in a plethora of dynamic phenomena [1]. Recently, two types of feedback formulation on nonlinear system identification were developed, namely, reverse path (RP) method and nonlinear identification through feedback of the output (NIFO) method. [PERSON] [23] first introduced the RP method to a single-degree-of-freedom (SDOF) system and by [PERSON] [24] to a multi-degree-of-freedom (MDOF) system. However, the requirement of exciting all the measured DOF was the obstacle to extensive applications for the RP method. Then, [PERSON] and [PERSON] [25] proposed the conditioned reverse path (CRP) method to overcome this limitation, but with more complicated RP formulation and more substantial computational burden. Another type of feedback formulation is the NIFO method put forward by [PERSON] and [PERSON] [26], who introduced a least-squares parameter estimation method to decouple the linear and nonlinear parameters during the nonlinear identification of a multi-input, multi-output (MIMO) system. Then, the subspace-based identification methods were introduced to the identification of nonlinear mechanical systems by taking advantage of this kind of nonlinear feedback interpretation. [PERSON] and [PERSON] [27] proposed the time-domain nonlinear subspace identification (TNSI) method firstly and [PERSON] and [PERSON] [28] developed the counterpart frequency-domain nonlinear subspace identification (FNSI) method.