Least Squares Model Reduction of Systems with Polynomial Inputs

Abstract

This paper proposes a method for obtaining an optimal reduced order model to approximate a given high-order system with polynomial inputs. The polynomial inputs have many terms with random coefficients and approximate most of inputs operated in practical closed-loop control systems. The index for a reduced model is defined by integrating the square of its output error and averaging the integral over the random coefficients. The optimal reduced order model has the minimum index under the condition that the steady-state portion of its output is equal to the one of the high-orderr system exactly. This paper gives analytical expressions for the evaluation of the index, the gradients with respect to the unknown model parameters and the condition for coincidence of steady-state outputs of high-order system and the reduced order model. These expressions are derived by using a transformation of the state-variables of augmented systems with an input model and reducing the problem with polynomial inputs to a problem with inpulse inputs.