## 1 Introduction  
Load estimation of severe environmental excitation, such as an earthquake or strong wind, is of great significance in structural design, health monitoring, and condition assessment. Under the practical condition, the external forces of structures cannot be directly measured with force transducers due to the limited number of sensors or an inaccessible force position [1]. Therefore, inverse dynamic analysis is an alternative tool to estimate an external force or excitation with structural response observation, which may be ill-conditioned [2, 3]. Currently, investigations on the dynamic load estimation are being conducted more frequently, but it is still a difficult task for nonlinear structures due to nonlinearity, performance of time-step integration method, and the measurement noise. Numerous indirect force re-construction or force identification methods have been developed for linear structures [4, 5, 6, 7, 8, 9]. Force identification methods in time domain are actively studied in state space with the time history of structural response as the measured information which can be solved with iterative regularization methods, recursive estimation or some other optimization methods [10, 11, 12, 13, 14]. Force identification can also be viewed as an optimization problem the objective of which is to minimize the error between the measured and the estimated structural responses. Compared to the force identification method arranged in one unknown vector solved with regularization method in state space [15], it has been argued that state variables estimated in state space recursively can identify the external force with far fewer unknowns in each time step as introduced in references [16; 17; 18; 19]. The extended Kalman filter (EKF) has been developed as an effective estimation tool to identify the state variable, including the structural response, parameters, and external excitation [17; 18]. This filter can also be applied and combined with least-square estimation to identify the structural parameter and external load recursively with an adaptive factor [20; 21]. To improve accuracy of the nonlinear structural estimation, the unscented Kalman filter (UKF) was proposed with fewer deterministic samplings than the particle filters or other ensemble Kalman filters [19; 22; 23; 24], which will improve computational efficiency. Compared to EKF, it is unnecessary to calculate the Jacobian matrix in the identification process with UKF. Therefore, UKF is a suitable tool to conduct a time-invariant constitutive parameter or nonlinear structural parameter identification for nonlinear civil buildings. In the studies of force identification, the unknown time history of external force or ground motion can be augmented to a state variable [16; 17; 18; 19]. UKF methods have been used for the augmented state variable estimation combined with response or structural parameters. It should be noted that UKF is not suitable for the identification of a time-variant parameter when the dynamic external force or ground motion is a time-variant series. The force identification with a directly augmented state variable may not be as accurate as the time-invariant structural parameter identification. The factor mentioned above could cause large errors in the identification result, especially considering the time-variant statistical characteristics of the external excitation. Chebyshev polynomial decomposition of the time history was proposed [25]. The force time history is assumed to be represented by the Chebyshev polynomials whose coefficients are arranged as a very large vector and identified in state space with regularization methods, as reported in reference [25]. However, it is difficult to determine the order of the polynomial when there is little information about an external excitation. Compared with the generally used Fourier decomposition, the Chebyshev basis would cover a wide range in frequency for each single basis, which could reduce the number of bases and the corresponding coefficients.