In computational aeroelasticity, unsteady aerodynamics is computationally expensive compared to the structural calculation. This problem becomes more severe for prediction of flutter or Limit-Cycle oscillation (LCO), which requires multiple runs of simulations at various flow conditions. The former can be captured by linear computational-aerodynamic-fluid (CFD) solver, while the prediction of the latter requires nonlinear CFD solver. Reduced-order modelling (ROM) techniques for unsteady aerodynamics have been investigated extensively. Among these ROMs, Auto-Regression with eXogeneous variables (ARX) has been applied to predict the flutter behaviour successfully for relatively complicated test cases. Motivated by the nice performance of ARX, whether other linear system identification methods, such as Auto-Regression Moving Average with eXogeneous inputs (ARMAX), Output-Error (OE) and Box-Jenkins (BJ), are effective as linear ROMs is investigated. When it comes to the nonlinear ROM, the linear information is expected to use. A polynomial-based state-space model, extended from linear system identification, is defined to combine the linear part with nonlinear functions of the state and input. A direct requirement for the linear method is the representation in state-space form. The nonlinear functions compose of polynomials of degree equal or greater than $2$. The coefficients of this nonlinear model are obtained by solving an optimisation problem with a Levenberg-Marquardt (LM) algorithm. Furthermore, we assume that the linear method is capable of capturing the flutter and keep the linear matrices constant during the nonlinear optimisation to reduce computational cost. As for the test case for LCO, an analytical Van-der-pol (VDP) oscillator is selected. The nonlinear ROM is built to replace the nonlinear term in VDP. Coefficients of the nonlinear ROM are obtained by solving an optimisation problem and the validation is carried out by reproducing the VDP oscillation. Another important factor for a fairly accurate ROM is the training signal. Different training signals are examined to reproduce the VDP oscillation. After checking the theoretical representation and the numerical solution algorithm, ARX and ARMAX are selected as linear ROMs for two reasons: the construction of state-space representation is explicit; ARMAX model is a variation from ARX by adding averaged error terms without changing the stability of the system. The comparison is carried out in two test cases. In the analytical test case, ARX and ARMAX can reproduce the training signal very well after an order selection and capture the flutter boundary. For the second test case, using CFD data, ARX and ARMAX match training signals well, but for test signals, ARMAX shows a lower fitness caused by overestimation of error contribution. For the nonlinear training, the optimiser can follow the nonlinear behaviour of the reference output, which demonstrates significant error reduction compared to the linear model. The validation of the ROM is examined for different training signals: chirps, random phase multi-sine, sequential sinusoids and multi-chirps. Chirps and sequential sinusoids fail to reproduce VDP. For most cases, random phase multi-sine is unable to follow VDP oscillation. Although the bounded oscillation is predicted for certain cases, repetitive tests are not consistent. For the multi-chirps, the optimiser is changed to accustom the cost function. With a couple of tests, this signal can deterministically reproduce the bounded oscillation, but the accuracy is not high. Another assumption that typical frequencies at flutter and LCO are not far apart is considered. Sequential sinusoids with narrow frequency band and wide amplitude range are applied to construct the nonlinear ROM in both random and deterministic cases. The frequencies and amplitudes are randomly chosen with the frequencies predetermined in the random case, while frequencies and amplitudes in the deterministic case are scattered uniformly. The LCO behaviour is predicted quite well in terms of amplitude and frequency in the bounded phase but not in transitional stage for both the cases. Repetitive tests are required for the random case. This study applied ARX and ARMAX as the linear ROM for flutter prediction and built the polynomial-based nonlinear ROM for predicting LCO. The observation that narrowing frequency range and widening amplitude range increase the chance of capturing LCO can be used for nonlinear training signal design.