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In this note, the authors propose a generalization of the well known Wiener-Hopf design method of optimal controllers and filters, applicable to a certain class of systems described by fractional order differential equations, the so called rational order systems that, in the Laplace domain, are described by transfer functions which are quotients of polynomials in [s.sup.[alpha]], [alpha] = (1/q), q being a positive integer. As can be verified in the literature, such transfer functions arise in the characterization of some industrial processes and physical systems which can be adequately modeled using fractional calculus, or when modeling some distributed parameter systems by finite dimensional models. A brief exposition of the standard Wiener-Hopf method, and some fundamental considerations about rational order systems are given before presenting the proposed procedure. Illustrative examples are discussed. Index Terms--Fractional systems, optimal controllers, rational order, spectral factorization, Wiener-Hopf.

The calculus of stationary points or indirect optimization is noted for rapid speed. It's speed is realized in an analytical representation of the Hessian and a polynomial fit of both the objective function and the constraints. Constraints are treated as active using Lagrangian multipliers. Because the algorithm's speed is so rapid, the problem can be repeated multiple times with random starting guesses to circumvent the possibility of encountering local wells. Both an analytic and an actual field problem are considered by way of providing examples of the technique. Index terms - optimization, functional approximations, gradients