Iterative-learning procedures for nonlinear-model-order reduction in discrete time

Efficient numerical procedures are developed for model-order reduction of a class of discrete-time nonlinear systems. Based on the solution of a set of linear-matrix inequalities, the Petrov–Galerkin projection concept is utilized to set up the structure of the reduced-order nonlinear model that preserves the input-to-state stability while ensuring an acceptable approximation error. The first numerical algorithm is based on the construction of a constant optimal projection matrix and a constant Lyapunov matrix to form the reduced-order dynamics. The second proposed algorithm aims to incorporate the output of the original system to correct the instantaneous value of the truncation matrix and maintain an acceptable approximation error even with low-order systems. An extension to uncertain systems is provided. The usefulness and the efficacy of the developed procedures are approved by the consideration of two numerical examples treating a nonlinear low-order system and a high-dimensional system, issued from the discretization of the damped heat-transfer partial-differential equation.