Nonlinear Dynamics Identified by Multi-Index Models

We study nonlinear regression models of, for example, NARX-type, where the predicted output is determined as a nonlinear function of known past data. A particular structure of the nonlinear mapping is imposed, which confines the nonlinearities to a subspace of the regression space. Utilizing this structure simplifies the estimation problem and allows more efficient parameterizations as well as visualization of the nonlinearity. We show how the LS fit of polynomials and piecewise affine functions are used as criteria to find the projection that best describes the residual. A study of two particular nonlinear systems illustrates that the regressor can be projected down to 2 dimensions, and still yield a model simulation fit of around 99\%. An electronic circuit can be accurately modeled with far less parameters than conventional, black-box models, of, say, neural network type.