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

A general analytic method for the evaluation of generalised quadratic integrals of the form J_μ,ν(σ) = ∫₀^∞t^μ exp (σt)(d^νe(t)/dt^ν)² dt, 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 J_μ,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 I-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.