Evaluation of Generalized Quadratic Cost Functionals for Linear Systems with Many Commensurate Time Delays

摘要

In this paper we present a general analytic method for the evaluation of generalized quadratic integrals of the form I_(μ,υ)(σ)= ∫_0~∞ t~μ exp(σt)(d~υe(t)/dt~υ)~2dt, where μ and υ are integers and e(t) is the error variable of a control system having multiple commensurate time delays. More precisely, we derive an analytic solution to the problem of evaluating I_(μ,υ) (σ) from the equivalent frequency-domain complex integral. The method is based on using a mathematically rigorous procedure to perform partial fraction decompositions of the integrand of the complex integral. It involves solving sets of symbolic linear equations and the factorization of 1-D polynomials. The results presented in this paper for the evaluation of complex integrals involving delay operator e~(-TS) are of direct interest to the parametric optimization and balanced realization and reduction of linear time-delay systems.