A dual frequency-selective bounded real lemma and its applications to IIR filter design

Given a transfer function H(s) of order n, the celebrated bounded real lemma characterises the untractable semi-infinite programming (SIP) condition |H(jomega)|² gesgamma ²forallomegaisinR of function bounded realness (BR) by a tractable semi-definite programming (SDP). Some recent results generalise this result for the SIP condition |H(jomega)|² gesgamma ²forall|omega|gesomega of frequency-selective bounded realness (FSBR). The SDP characterisations are given at the expense of an introduced Lyapunov matrix variable of dimension n x n. As a result, the dimension of the resultant SDPs grows so quickly in respect to the function order, making them much less computationally tractable and practicable. Moreover, they do not allow to formulate synthesis problems as SDPs. In this paper, a completely new SDP characterizations for general FSBR for all-pole transfer functions is proposed. Our motivation is the design of infinite-impulse-response (IIR) filters involving a few of simutaneous FS-BRs. Our SDP characterizations are of moderate size and free from Lyapunov variables and thus allow to address problems involving transfer functions of arbitrary order. Examples are also provided to validate the effectiveness of the resulting SDP design formulation. Finally we also raise some issues arising with practicability of SDP for multi-dimensional filter design problems. In particular, any bilinear matrix inequality (BMI) optimization is shown to be solved by a SDP with any prescribed tolerance but the issue is dimensionality of this SDP