Robust H -infinity filtering in finite frequency domain for polytopic systems with application to in-band quantisation noise reduction in uncertain cascaded sigma-delta modulators

This study investigates the problem of robust H∞ filtering in finite frequency domain for polytopic systems. Robust fixed-order, infinite-impulse-response (IIR) and finite-impulse-response (FIR) designs via indirect approach and direct method are presented. The aforementioned techniques refer to the conventional loop shaping method and that uses recently introduced generalised Kalman–Yakubovich–Popov (GKYP) lemma, respectively. The authors consider a frequency-domain polytopic model which includes two well-known special cases, that is, the interval plants and the set of transfer functions with affinely dependent uncertain parameters belonging to a convex-bounded polyhedral domain. They derive conditions for the existence of linear-time invariant (LTI) filters. These conditions, irrespective of the uncertain parameters, guarantee both asymptotic stability of the polytopic systems and the filters, and a guaranteed H∞ performance in finite frequency domain. Extension of the results to the multi-band case is straightforward, where the advantages of arbitrary filter order and dropping the use of weighting functions is even more obvious. As an example, the authors apply the proposed methods to in-band quantisation noise reduction for uncertain cascaded sigma-delta modulators. By comprehensive comparisons with existing methods, they show the viability and efficiency of their design.