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

The quadratic cost functional or integral square error (ISE) defined as I = integral(infinity)(0)e(2)(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/i2 pi) integral E-i infinity(-i infinity)(s)E(- s) ds, i = root-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(- tau s/ root(s(2) + b(2))) is considered. It is shown that because multivalued square root function (s(2) + b(2))(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.