Evaluation of generalised quadratic cost functionals for linear systems with many commensurate time delays

Abstract

A general analytic method for the evaluation of generalised quadratic integrals of the form Jμ,ν(σ)=∫0∞tμexp (σt)(dνe(t)/dtν)2dt, where μ and ν are integers and e(t) is the error variable of a control system having multiple commensurate time delays is presented. More precisely, the authors derive an analytic solution to the problem of evaluating Jμ,ν(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 factorisation of 1-D polynomials. The results presented for the evaluation of complex integrals involving the delay operator e−τs are of direct interest to the parametric optimisation and balanced realisation and reduction of linear time-delay systems.