Use of the Lambert W function for time-domain analysis of feedback fractional delay systems

The Lambert W function is defined as the multivalued inverse of the function w→we^w=z. It has been applied to stability analysis of a class of fractional delay systems whose transcendental characteristic equation (TCE) can be remodelled in the form (as+b)e^cs+d=0. The approach of using the Lambert W function to time-domain analysis of a class of feedback fractional-order time-delay systems is extended. It should be noted that, owing to the multivaluedness of a transfer function of fractional order, the approach has two pitfalls that must be circumvented with care. Because remodelling the TCE of a feedback fractional delay system to allow for the Lambert W function representation of roots introduces superfluous poles to the original TCE, a clarification of the relationship between the roots of the remodelled TCEs and the poles of the system is provided. As a result, the time response function of the system can be approximated by a finite series of eigenmodes written in terms of Lambert W functions. As the singularities of a fractional-order system include both the poles and the branch cut(s) of the transfer function, the neglect of the response portion contributed by the branch cut(s) incurs a significant transient response error. In order to compensate for such a transient response error, three schemes of optimal approximation with specified poles are developed. Simulation results show that the proposed approaches to time-domain analysis of feedback fractional delay systems can indeed enlarge the application scope of the emerging Lambert W function.