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This note considers the problems of robust stability and stabilization for uncertain continuous singular systems with state delay. The parametric uncertainty is assumed to be norm bounded. The purpose of the robust stability problem is to give conditions such that the uncertain singular system is regular, impulse free, and stable for all admissible uncertainties, while the purpose of robust stabilization is to design a state feedback control law such that the resulting closed-loop system is robustly stable. These problems are solved via the notions of generalized quadratic stability and generalized quadratic stabilization, respectively. Necessary and sufficient conditions for generalized quadratic stability and generalized quadratic stabilization are derived. A strict linear matrix inequality (LMI) design approach is developed. An explicit expression for the desired robust state feedback control law is also given. Finally, a numerical example is provided to demonstrate the application of the proposed method.

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The problem under study here is the minimax design of linearphase lowpass FIR filters having variable passband width and implemented through a Farrow structure. We have two main contributions. The first is the design of adjustable FIR filters without discretization, using 2D positive trigonometric polynomials, an approach leading to semidefinite programming (SDP) formulation of the design problem. The second is to modify the design problem by a special choice for the passband and stopband edges of the variable FIR filter. The advantage is a lower implementation complexity. The new problem is solved using positive hybrid real-trigonometric polynomials and their SDP parameterization. Design examples prove the viability of our methods.

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.