Evaluation of quadratic cost functionals for a class of distributed-delay systems

The quadratic cost functional or integral square error (ISE) defined as I=int₀ ^infin e²(t) dt has been widely used in the analytical design of optimal control systems. In most control literature, the integral I, by virtue of Parseval´s theorem, is represented by the complex integral (1/i2pi)int_-iinfin ^iinfin E(s)E(-s) ds, i=radic-1, and many efficient parametric expressions derived for the evaluation of I are based on a product-to-sum decomposition E(s)E(-s)=Z(s)+Z(-s). The evaluation of ISE for linear feedback control of systems involving a distributed delay exp(-taus/radic(s²+b²)) is considered. It is shown that because multivalued square root function (s²+b²)^1/2 has a non-removable branch-cut singularity on the imaginary axis, the product-to-sum decomposition approach fails to generate a parametric expression for the evaluation of I. Also shown is that pitfall exists with the use of the Laplace-transform-based representation of Parseval identity when the value of I is computed by a numerical integration of the complex integral in a computer. The findings gained from numerical results indeed clarify the correct use of a useful numerical approach of solving differential equations to compute the quadratic cost functionals.