Simultaneous State-Input-Stiffness Estimation for Nonlinear Duffing Oscillators Avoiding Jacobian Linearization

This paper presents a simultaneous state-input-stiffness estimation framework for nonlinear systems. The technique combines an unbiased minimum variance estimator (MVE) from the data assimilation context with the Moore-Penrose pseudo-inverse. The investigation employs synthetically generated measurement data with additive Gaussian noise to replicate the field measurements. In the first stage of the algorithm, the MVE estimates a total force component, a function of the unknown system parameters and unknown input excitation. In the second stage, a system of over-determined equations is established by representing the input excitation with a Fourier series (with N coefficients) expansion. A least-squares solution for the unknown stiffness parameters and the Fourier coefficients is then achieved using the Moore-Penrose inverse. A dimensionless transformation handles the scale difference between the Fourier coefficients and unknown parameters. The novelty of the work is that the inputs and the parameters of the nonlinear systems are estimated simultaneously, circumventing any linearization of the system, and the associated computational hassles. The method by construction has a unique solution and an upper bound for stiffness estimation error is derived. The method is demonstrated numerically for Duffing oscillator systems excited by random inputs. The robustness of the technique is assessed by conducting various parametric studies. Numerical results reveal that the developed method accurately estimates the nonlinear cubic stiffness parameter, input force, and state responses.