Optimal fractional controllers for commensurate order systems: A special case of the Wiener-Hopf method

In this paper, the authors propose a generalization of the well known Wiener-Hopf design method of optimal controllers and filters, applicable to certain class of systems described by fractional order differential equations, the so called commensurate order systems, i.e., in the Laplace domain, systems described by transfer functions which are not quotients of polynomials in s, but in s^α, α=1/q, being q a positive integer. As can be verified in the literature, such transfer functions arise in the characterization of many industrial processes and physical systems which can be adequately modeled using fractional calculus, or when modelling some distributed parameter systems by finite dimensional models. Taking into account that fractional-order systems and controllers have a limited diffusion, after a brief exposition of the principal results of the traditional Wiener-Hopf method, some fundamental considerations about the dynamical properties of such systems are made. After that, the authors propose a procedure that allows the application of the method to the mentioned class of systems. An illustrative example is given