Among the available approaches to nonlinear system identification are unstructured or black-box models, which rely on neural network models and their function approximation properties. On the other hand, structured or grey-box models are based on the interconnection of linear and nonlinear subsystems.

The most common model structures are the Hammerstein, Wiener, nonlinear feedback, and combined Hammerstein/nonlinear feedback models. These models involve the interconnection of a single linear block and a single nonlinear block. In practice, nonlinear identification with the Wiener model structure is significantly more difficult than identification with the Hammerstein structure since the identification of a nonlinear map is more tractable when a measurement of the input to the map is available.

A general formulation of nonlinear identification in this setting was considered in prior work of the authors using a subspace identification algorithm for the linear dynamics along with a basis expansion for the nonlinear maps. The function expansion was chosen to be linear in the parameters, which allows the nonlinear identification problem to be recast as a linear identification problem with generalized inputs. The multivariable capability of subspace identification algorithms is essential to this approach by allowing an arbitrary number of generalized inputs.

In this paper we employ a two-step procedure for constructing a Hammerstein-Wiener model for magnetospheric data. In the first step a Hammerstein model is identified using the subspace-based technique developed previously. Next, a static nonlinearity is fitted at the output using a least squares fit. The goal of this paper is to apply nonlinear Hammerstein/Wiener identification to data generated by a magnetospheric system.

The magnetic field on the earth is influenced by solar storms arising from coronal mass ejections (CME's) erupting from the sun. Sudden fluctuations in magnetic fields due to large CME's can be detrimental to orbiting spacecraft, power grids, and electronic networks. The ability to predict such an event is valuable to power companies, defense authorities, and space researchers. The CME takes roughly an hour to propagate from the first libration/lagrangian point to the earth, thus providing time to predict the behavior on earth based on readings measured by a spacecraft located at the first lagrangian point.

This goal of this paper is to construct an empirical nonlinear model, where inputs and outputs are provided by the available data; ultimately, these models will be used to augment an existing MHD code. In constructing such models, it is important to have a sufficiently rich model structure that is tractable for optimization-based fitting. Insights from the physics suggest the use of a secondary synthetic input, namely, $B_T V_x \sin^4{(\frac{\theta}{2})}$, which facilitates the model-fitting accuracy. In addition, it is important to avoid overfitting so that the resulting model has good predictive capability.

The inputs to the magnetospheric system are the three components of the magnetic field, the density of the plasma, and the velocity and temperature. The outputs are magnetometer readings on ground-based stations.

An error search algorithm, which seeks inputs to improve both the fit error and prediction error, is used to rank the inputs in order of importance. Based on the results of the algorithm, the linear inputs to the Hammerstein-Wiener model are chosen to be density, B_y, B_z and B_T, while the nonlinear inputs are $B_T V_x \sin^4{(\frac{\theta}{2})}$ and a sinusoidal signal with a period of one day to account for the time-varying nature of the system.

Results are presented for the Thule magnetometer station located in Greenland. Data for fifteen days is used to construct the model. The constructed model is then used to predict the response for the next fifteen days. Plots comparing the magnetometer data and the magnetic field predicted by the model are shown.