LTI approximation of nonlinear systems via signal distribution theory

L2L2 and L1L1 optimal linear time-invariant (LTI) approximation of discrete-time nonlinear systems, such as nonlinear finite impulse response (NFIR) systems, is studied via a signal distribution theory motivated approach. The use of a signal distribution theoretic framework facilitates the formulation and analysis of many system modelling problems, including system identification problems. Specifically, a very explicit solution to the L2L2 (least squares) LTI approximation problem for NFIR systems is obtained in this manner. Furthermore, the L1L1 (least absolute deviations) LTI approximation problem for NFIR systems is essentially reduced to a linear programming problem. Active LTI modelling emphasizes model quality based on the intended use of the models in linear controller design. Robust stability and LTI approximation concepts are studied here in a nonlinear systems context. Numerical examples are given illustrating the performance of the least squares (LS) method and the least absolute deviations (LAD) method with LTI models against nonlinear unmodelled dynamics.