## 1 Introduction
### Literature Survey  
Rotor unbalance identification is an old problem. The modal method proposed in 1959 [1; 2] and the influence coefficient method proposed in 1964 [3] are the two classical techniques used to solve the problem. Although they have been developed by many other researchers [4; 5; 6; 7; 8; 9; 10; 11; 12; 13], test runs are still demanding. For large field rotors, starting and stopping the machine several times is costly and can reduce its service life [14]. Identifying rotor unbalance without test runs is the current trend [15]. This approach can be divided into two types: a method with external excitation and a method without external excitation. For the external excitation method, [PERSON] and [PERSON] [16] used frequency excitation to estimate rotor unbalance as well as bearing coefficients. [PERSON] [17] applied impulse excitation on a double-disc and single-span rotor-bearing system to identify rotor unbalance as well as the stiffness and damping coefficients of two bearings. [PERSON] [18; 19] estimated rotor unbalance and bearing coefficients using an active magnetic executor to generate external excitation. However, high-power exciters are necessary for large rotors and the excitation may damage the rotors. Consequently, methods that do not require external excitation have been developed. Aiming at a single-span rotor-bearing system, [PERSON] and [PERSON] [20] identified single-disc unbalance parameters from unbalance responses using an unbalance force estimation technique. A rotor unbalance estimation method using the joint-input state estimation technique [21] and least-squares technique was proposed [22] to solve the problem that the required response measurements at different locations on the shaft may not always be accessible. [PERSON] [23] identified single-disc and double-disc unbalance based on modal expansion combined with optimisation algorithms. The errors caused by modal expansion were reduced by an integrated modal expansion inverse problem methodology combined with an optimisation procedure. [PERSON] [24] developed a double-disc unbalance identification method using the finite element model combined with augmented Kalman filter algorithms. While based on the continuous dynamic model and analytical solution methods [25; 26], [PERSON] [27] provided an analytical model-based algorithm to identify single-disc unbalance. Moreover, some scholars have even tried to estimate rotor unbalance and bearing coefficients simultaneously without external excitation. [PERSON] [28; 29] formulated an estimation algorithm using unbalance responses from three different unbalance configurations for both clockwise and anti-clockwise rotor rotations. [PERSON] [30] proposed a simultaneous estimation of the rotor unbalance and bearing coefficients of a continuous single-disc and single-span rotor-bearing system using the Rayleigh beam model. [PERSON] [31] developed a mathematical model of an unbalanced rotor using dimensional analysis and a rigid rotor approach. The factorial regression analysis method is used to solve the model. Based on it, a numerical technique for the detection of unbalance magnitude has been proposed. [PERSON] [32] presented an estimation method for unbalance magnitude and phase from the vibration in frequency domain. [PERSON] [33] proposed an identification method of unbalance for a rotor with residual shaft bent based on the finite element method and correlation analysis. [PERSON] [34] proposed an unbalance identification method for a high-speed rotor without trial weights based on modal analysis and the modal equivalent principle. The shortcomings of the above-mentioned methods are that they cannot incorporate any number of bearings and discs. Therefore, [PERSON] [14] proposed an approach based on the multi-plane influence coefficient balancing method for muti-disc and muti-span rotors.