Strictly Positive Real Control for DOSs with a Low Frequency Range

It is well known that each design specification is often given not for the entire frequency range but rather for a low frequency range of relevance, for example a closed-loop shaping control design typically requiring small sensitivity. Weighting method has been proven useful to deal with the finite frequency requirements [150]. However, the additional weights tend to increase the system complexity and the process of selecting appropriate weights is tedious and time-consuming. Another alternative approach has also been used to grid the frequency axis [286], but it suffers from the lack of a rigorous performance guarantee in the design process. Different from weighting function and frequency gridding methods, a generalized KYP lemma has been proposed in [119]. It can avoid computational burden and guarantee gain property performances simultaneously [115]. There are several results in the literature along this line, see for example [114, 116, 117]. Since the notion of positive realness plays a central role in systems theory [3], many researchers have considered positive real control for linear systems such as in [161, 162, 270], and references therein. The finite frequency positive realness approach for dynamical system design from a control perspective has been given in [118].