Loewner integer-order approximation of MIMO fractional-order systems

A state–space integer–order approximation of commensurate–order systems is obtained using a data–driven interpolation approach based on Loewner matrices. Precisely, given the values of the original fractional–order transfer function at a number of generalised frequencies, a descriptor–form state–space model matching these frequency response values is constructed from a suitable Loewner matrix pencil, as already suggested for the reduction of high–dimensional integer–order systems. Even if the stability of the resulting integer–order system cannot be guaranteed, such an approach is particularly suitable for approximating (infinite–dimensional) fractional–order systems because: (i) the order of the approximation is bounded by half the number of interpolation points, (ii) the procedure is more robust and simple than alternative approximation methods, and (iii) the procedure is fairly flexible and often leads to satisfactory results, as shown by some examples discussed at the end of the article. Clearly, the approximation depends on the location, spacing and number of the generalised interpolation frequencies but there is no particular reason to choose the interpolation frequencies on the imaginary axis, which is a natural choice in integer–order model reduction, since this axis does not correspond to the stability boundary of the original fractional–order system

On robust PID control for time-delay plants

This paper presents an easily implementable method for determining the set of PID controllers that stabilize an LTI system with or without a time delay while satisfying certain robustness requirements. The adopted approach, which does not require approximating the time delay or solving complex non\u2013algebraic equations, draws directly on the graphic approach suggested in [1], [2], [3] for PI controllers and first\u2013order\u2013plus\u2013 dead\u2013time processes. In particular, it is shown that, on suitable cross sections of the parameter space, the boundary of the regions where a given H1 margin is ensured is the envelope of families of ellipses whose centres lie on the stability boundary. The loci of constant crossover frequencies, strictly related to the achievable passbands, are also displayed. The technique is applied to some benchmark examples with the aid of dedicated software