Positive Hybrid Real-Trigonometric Polynomials and Applications to Adjustable Filter Design and Absolute Stability Analysis

The present paper points out that a class of positive polynomials deserves special attention due to several interesting applications in signal processing, system analysis and control. We consider positive hybrid polynomials with two variables, one real, the other complex belonging to the unit circle. We present several theoretical results regarding the sum-of-squares representations of such polynomials, treating the cases where positivity occurs globally or on domains. We also give a specific Bounded Real Lemma. All the characterizations of positive hybrid polynomials are expressed in terms of positive semidefinite matrices and can be extended to polynomials with more than two variables. On the applicative side, we show how several problems are numerically tractable via semidefinite programming (SDP) algorithms. The first problem is the minimax design of adjustable FIR filters, using a modified Farrow structure. We discuss linear-phase and approximately linear-phase designs. The second is the absolute stability of time delay feedback systems with unknown delay, for which we treat the cases of bounded and unbounded delay. Finally, we discuss the application of our methods to checking the stability of parameter-dependent systems. The design procedures are illustrated with numerical examples.