Time domain model reduction of time-delay systems via orthogonal polynomial expansions

Abstract

This paper investigates model reduction of time-delay systems in the time domain based on the expansion of systems over orthogonal polynomials. Time-delay systems are first expanded over generalized Laguerre polynomials. The nice properties of generalized Laguerre polynomials lead to a direct system expansion and a Sylvester equation with special structures which enables an efficient calculation of Laguerre coefficients of systems. Projection methods are then adopted to generate reduced models, and we show that a desired number of Laguerre coefficients are preserved by reduced models. Further, we extend the proposed method to general orthogonal polynomials, where the relationship between Taylor expansion and orthogonal polynomial expansion is examined to achieve the expansion of time-delay systems. Systems with multiple delays and delays appearing in the derivative of the states are also discussed. Two numerical examples are simulated to showcase the efficiency of our approach.